Subgroups — #90282⋊C₃ #13057326

Order 1344.

668 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×(C₂²⋊D₇)⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#1159⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
3	ΓL(1,𝔽₈)×#120	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×(C₂²⋊Dic₇)⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#6502⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#6502⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#6501⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#6500⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×ΓL(1,𝔽₈)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
10	#6449⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
11	#6448⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#6446⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
13	((C₂²⋊D₇)⋊C₃)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²⋊C₄×AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic, complemented
15	#6441⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
16	#90282	C₃	1	1	yes	maximal, characteristic, complemented, Hall
17	C₂⁴×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
18	C₂³×AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
19	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
20	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented
21	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented
22	C₂×(C₂²⋊D₇)⋊C₃	—	2	4	no	—
23	C₂×(C₂²⋊D₇)⋊C₃	C₂²	1	1	yes	characteristic
24	C₂×(C₂²⋊D₇)⋊C₃	C₂²	1	1	yes	characteristic
25	C₂×(C₂²⋊D₇)⋊C₃	C₂²	1	1	yes	characteristic
26	C₂×(C₂²⋊D₇)⋊C₃	C₂²	1	1	yes	characteristic
27	C₂×(C₂²⋊D₇)⋊C₃	C₂²	1	1	yes	characteristic, complemented
28	C₂×(C₂²⋊D₇)⋊C₃	C₂²	1	1	yes	characteristic, complemented
29	C₂²×ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic
30	C₂²×ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic, complemented
31	C₂²×ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic, complemented
32	C₂²×ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic, complemented
33	#1159⋊C₃	—	2	4	no	—
34	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic, complemented
35	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic, complemented
36	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	1	yes	characteristic, complemented
37	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
38	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented
39	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented
40	(C₂²⋊Dic₇)⋊C₃	C₂²	1	1	yes	characteristic
41	(C₂²⋊Dic₇)⋊C₃	C₂²	1	1	yes	characteristic
42	(C₂²⋊Dic₇)⋊C₃	C₂²	1	1	yes	characteristic, complemented
43	(C₂²⋊Dic₇)⋊C₃	C₂²	1	1	yes	characteristic, complemented
44	(C₂²⋊Dic₇)⋊C₃	C₂²	1	1	yes	characteristic, complemented
45	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	1	yes	characteristic, complemented
46	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	1	yes	characteristic, complemented
47	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	1	yes	characteristic, complemented
48	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	1	yes	characteristic, complemented
49	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic, complemented
50	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic, complemented
51	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	1	yes	characteristic, complemented
52	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	1	yes	characteristic, complemented
53	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic, complemented
54	C₂²×C₂²⋊D₇	C₂×C₃	1	1	yes	characteristic, complemented
55	C₂×#1159	C₂×C₃	1	1	yes	characteristic, complemented
56	C₇×#120	C₂×C₃	1	1	yes	characteristic, complemented
57	C₂×C₂²⋊Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
58	#6502	C₂×C₃	1	1	yes	characteristic, complemented
59	#6502	C₂×C₃	1	1	yes	characteristic, complemented
60	#6501	C₂×C₃	1	1	yes	characteristic, complemented
61	#6500	C₂×C₃	1	1	yes	characteristic, complemented
62	D₄×Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
63	#6449	C₂×C₃	1	1	yes	characteristic, complemented
64	#6448	C₂×C₃	1	1	yes	characteristic, complemented
65	#6446	C₂×C₃	1	1	yes	characteristic, complemented
66	(C₂²⋊D₇)⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
67	D₇×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
68	#6441	C₂×C₃	1	1	yes	characteristic, complemented
69	C₃×#546	—	7	1	no	maximal, Hall
70	C₂³×ΓL(1,𝔽₈)	—	2	2	no	—
71	C₂³×ΓL(1,𝔽₈)	—	2	2	no	—
72	C₂³×ΓL(1,𝔽₈)	—	4	1	no	—
73	C₂³×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
74	C₂³×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
75	C₂³×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
76	C₂³×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
77	C₂²×AGL(1,𝔽₇)	—	2	2	no	—
78	C₂²×AGL(1,𝔽₇)	—	2	2	no	—
79	C₂²×AGL(1,𝔽₇)	—	4	1	no	—
80	C₂²×AGL(1,𝔽₇)	C₂³	1	1	yes	characteristic
81	C₂²×AGL(1,𝔽₇)	C₂³	1	1	yes	characteristic
82	D₄×ΓL(1,𝔽₈)	—	2	2	no	—
83	D₄×ΓL(1,𝔽₈)	—	2	2	no	—
84	D₄×ΓL(1,𝔽₈)	—	4	1	no	—
85	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
86	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
87	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
88	(C₂²⋊D₇)⋊C₃	—	2	2	no	—
89	(C₂²⋊D₇)⋊C₃	—	2	2	no	—
90	(C₂²⋊D₇)⋊C₃	—	2	2	no	—
91	(C₂²⋊D₇)⋊C₃	—	2	2	no	—
92	(C₂²⋊D₇)⋊C₃	—	4	1	no	—
93	(C₂²⋊D₇)⋊C₃	D₄	1	4	yes	complemented
94	C₂×ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
95	C₂×ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
96	C₂×ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
97	C₂×ΓL(1,𝔽₈)⋊C₄	—	4	1	no	—
98	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
99	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
100	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
101	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
102	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
103	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
104	C₄×AGL(1,𝔽₇)	—	2	2	no	—
105	C₄×AGL(1,𝔽₇)	—	4	1	no	—
106	ΓL(1,𝔽₈)⋊Q₈	—	2	2	no	—
107	C₂⁴×C₇	C₂²×C₃	1	1	yes	characteristic
108	C₂³×D₇	C₂²×C₃	1	1	yes	characteristic
109	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
110	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic, complemented
111	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic, complemented
112	C₂×C₂²⋊D₇	—	2	4	no	—
113	C₂×C₂²⋊D₇	C₂²×C₃	1	1	yes	characteristic
114	C₂×C₂²⋊D₇	C₂²×C₃	1	1	yes	characteristic
115	C₂×C₂²⋊D₇	C₂²×C₃	1	1	yes	characteristic
116	C₂×C₂²⋊D₇	C₂²×C₃	1	1	yes	characteristic
117	C₂×C₂²⋊D₇	C₂²×C₃	1	1	yes	characteristic, complemented
118	C₂×C₂²⋊D₇	C₂²×C₃	1	1	yes	characteristic, complemented
119	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
120	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic, complemented
121	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic, complemented
122	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic, complemented
123	#1159	—	2	4	no	—
124	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic, complemented
125	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic, complemented
126	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic, complemented
127	C₇×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
128	C₇×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
129	C₇×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
130	C₂²⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
131	C₂²⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
132	C₂²⋊Dic₇	C₂²×C₃	1	1	yes	characteristic, complemented
133	C₂²⋊Dic₇	C₂²×C₃	1	1	yes	characteristic, complemented
134	C₂²⋊Dic₇	C₂²×C₃	1	1	yes	characteristic, complemented
135	(C₂×D₇)⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
136	(C₂×D₇)⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
137	(C₂×D₇)⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
138	(C₂×D₇)⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
139	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic, complemented
140	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic, complemented
141	Dic₇⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
142	Dic₇⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
143	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic, complemented
144	C₂×C₃×#41	—	7	1	no	—
145	C₂²×C₃×D₄	—	7	1	no	—
146	C₃×#124	—	7	1	no	—
147	C₃×#123	—	7	1	no	—
148	C₃×#123	—	7	1	no	—
149	C₃×C₂²⋊Q₈	—	7	1	no	—
150	C₃×#121	—	7	1	no	—
151	C₃×#121	—	7	1	no	—
152	C₃×#121	—	7	1	no	—
153	C₃×#120	—	7	1	no	—
154	C₃×#120	—	7	1	no	—
155	C₃×C₄×D₄	—	7	1	no	—
156	C₃×C₄×D₄	—	7	1	no	—
157	C₂×C₃×C₂²⋊C₄	—	7	1	no	—
158	C₂×C₃×C₂²⋊C₄	—	7	1	no	—
159	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
160	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
161	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
162	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
163	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
164	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
165	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
166	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
167	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
168	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
169	C₂²×ΓL(1,𝔽₈)	—	4	4	no	—
170	C₂²×ΓL(1,𝔽₈)	C₂×D₄	1	2	yes	—
171	C₂²×ΓL(1,𝔽₈)	#41	1	2	yes	complemented
172	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
173	C₂×AGL(1,𝔽₇)	—	2	1	no	—
174	C₂×AGL(1,𝔽₇)	—	2	1	no	—
175	C₂×AGL(1,𝔽₇)	—	2	1	no	—
176	C₂×AGL(1,𝔽₇)	—	2	1	no	—
177	C₂×AGL(1,𝔽₇)	—	2	1	no	—
178	C₂×AGL(1,𝔽₇)	—	4	4	no	—
179	C₂×AGL(1,𝔽₇)	C₂×D₄	1	2	yes	—
180	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
181	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
182	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
183	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
184	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
185	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
186	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
187	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
188	ΓL(1,𝔽₈)⋊C₄	C₂×D₄	1	2	yes	—
189	#546	—	7	1	no	Sylow
190	C₂³×C₇	—	2	2	no	—
191	C₂³×C₇	—	2	2	no	—
192	C₂³×C₇	—	4	1	no	—
193	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
194	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
195	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
196	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
197	C₂²×D₇	—	2	2	no	—
198	C₂²×D₇	—	2	2	no	—
199	C₂²×D₇	—	4	1	no	—
200	C₂²×D₇	C₂³×C₃	1	1	yes	characteristic
201	C₂²×D₇	C₂³×C₃	1	1	yes	characteristic
202	C₇×D₄	—	2	2	no	—
203	C₇×D₄	—	2	2	no	—
204	C₇×D₄	—	4	1	no	—
205	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
206	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
207	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
208	C₂²⋊D₇	—	2	2	no	—
209	C₂²⋊D₇	—	2	2	no	—
210	C₂²⋊D₇	—	2	2	no	—
211	C₂²⋊D₇	—	2	2	no	—
212	C₂²⋊D₇	—	4	1	no	—
213	C₂²⋊D₇	C₃×D₄	1	4	yes	complemented
214	C₂×Dic₇	—	2	2	no	—
215	C₂×Dic₇	—	2	2	no	—
216	C₂×Dic₇	—	2	2	no	—
217	C₂×Dic₇	—	4	1	no	—
218	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
219	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
220	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
221	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
222	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
223	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
224	C₄×D₇	—	2	2	no	—
225	C₄×D₇	—	4	1	no	—
226	Dic₁₄	—	2	2	no	—
227	C₂⁴×C₃	—	7	1	no	—
228	C₂⁴×C₃	—	7	1	no	—
229	C₃×#41	—	14	4	no	—
230	C₂×C₃×Q₈	—	7	1	no	—
231	C₂×C₃×D₄	—	7	1	no	—
232	C₂×C₃×D₄	—	7	1	no	—
233	C₂×C₃×D₄	—	7	1	no	—
234	C₂×C₃×D₄	—	7	1	no	—
235	C₂×C₃×D₄	—	7	1	no	—
236	C₂×C₃×D₄	—	7	1	no	—
237	C₂×C₃×D₄	—	7	1	no	—
238	C₂×C₃×D₄	—	7	1	no	—
239	C₂×C₃×D₄	—	7	1	no	—
240	C₂×C₃×D₄	—	14	4	no	—
241	C₂²×C₃×C₄	—	7	1	no	—
242	C₂²×C₃×C₄	—	7	1	no	—
243	C₂²×C₃×C₄	—	7	1	no	—
244	C₂²×C₃×C₄	—	7	1	no	—
245	C₂²×C₃×C₄	—	7	1	no	—
246	C₂²×C₃×C₄	—	7	1	no	—
247	C₃×C₄⋊C₄	—	7	1	no	—
248	C₃×C₄⋊C₄	—	7	1	no	—
249	C₃×C₄⋊C₄	—	7	1	no	—
250	C₃×C₄⋊C₄	—	7	1	no	—
251	C₃×C₂²⋊C₄	—	7	1	no	—
252	C₃×C₂²⋊C₄	—	7	1	no	—
253	C₃×C₂²⋊C₄	—	7	1	no	—
254	C₃×C₂²⋊C₄	—	7	1	no	—
255	C₃×C₂²⋊C₄	—	7	1	no	—
256	C₃×C₂²⋊C₄	—	7	1	no	—
257	C₃×C₂²⋊C₄	—	7	1	no	—
258	C₃×C₂²⋊C₄	—	7	1	no	—
259	C₃×C₂²⋊C₄	—	7	1	no	—
260	C₃×C₂²⋊C₄	—	7	1	no	—
261	C₃×C₂²⋊C₄	—	7	1	no	—
262	C₃×C₂²⋊C₄	—	7	1	no	—
263	C₃×C₄²	—	7	1	no	—
264	C₂×ΓL(1,𝔽₈)	—	2	2	no	—
265	C₂×ΓL(1,𝔽₈)	—	2	2	no	—
266	C₂×ΓL(1,𝔽₈)	—	4	1	no	—
267	C₂×ΓL(1,𝔽₈)	—	4	1	no	—
268	C₂×ΓL(1,𝔽₈)	C₂²×D₄	1	1	yes	characteristic
269	C₂×ΓL(1,𝔽₈)	C₂×#41	1	1	yes	characteristic
270	C₂×ΓL(1,𝔽₈)	H(5,𝔽₂)	1	1	yes	characteristic
271	AGL(1,𝔽₇)	—	2	2	no	—
272	AGL(1,𝔽₇)	—	4	1	no	—
273	C₂×#41	—	7	1	no	—
274	C₂²×D₄	—	7	1	no	—
275	#124	—	7	1	no	—
276	#123	—	7	1	no	—
277	#123	—	7	1	no	—
278	C₂²⋊Q₈	—	7	1	no	—
279	#121	—	7	1	no	—
280	#121	—	7	1	no	—
281	#121	—	7	1	no	—
282	#120	—	7	1	no	—
283	#120	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
284	C₄×D₄	—	7	1	no	—
285	C₄×D₄	—	7	1	no	—
286	C₂×C₂²⋊C₄	—	7	1	no	—
287	C₂×C₂²⋊C₄	—	7	1	no	—
288	C₂²×C₇	—	2	1	no	—
289	C₂²×C₇	—	2	1	no	—
290	C₂²×C₇	—	2	1	no	—
291	C₂²×C₇	—	2	1	no	—
292	C₂²×C₇	—	2	1	no	—
293	C₂²×C₇	—	2	1	no	—
294	C₂²×C₇	—	2	1	no	—
295	C₂²×C₇	—	2	1	no	—
296	C₂²×C₇	—	2	1	no	—
297	C₂²×C₇	—	2	1	no	—
298	C₂²×C₇	—	4	4	no	—
299	C₂²×C₇	C₂×C₃×D₄	1	2	yes	—
300	C₂²×C₇	C₃×#41	1	2	yes	complemented
301	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
302	C₂×D₇	—	2	1	no	—
303	C₂×D₇	—	2	1	no	—
304	C₂×D₇	—	2	1	no	—
305	C₂×D₇	—	2	1	no	—
306	C₂×D₇	—	2	1	no	—
307	C₂×D₇	—	4	4	no	—
308	C₂×D₇	C₂×C₃×D₄	1	2	yes	—
309	C₄×C₇	—	2	1	no	—
310	C₄×C₇	—	2	1	no	—
311	C₄×C₇	—	2	1	no	—
312	Dic₇	—	2	1	no	—
313	Dic₇	—	2	1	no	—
314	Dic₇	—	2	1	no	—
315	Dic₇	—	2	1	no	—
316	Dic₇	—	2	1	no	—
317	Dic₇	C₂×C₃×D₄	1	2	yes	—
318	C₂³×C₃	—	7	1	no	—
319	C₂³×C₃	—	7	1	no	—
320	C₂³×C₃	—	7	1	no	—
321	C₂³×C₃	—	7	1	no	—
322	C₂³×C₃	—	7	1	no	—
323	C₂³×C₃	—	7	1	no	—
324	C₂³×C₃	—	14	2	no	—
325	C₂³×C₃	—	14	2	no	—
326	C₂³×C₃	—	14	2	no	—
327	C₂³×C₃	—	14	2	no	—
328	C₂³×C₃	—	28	1	no	—
329	C₂³×C₃	—	28	1	no	—
330	C₃×Q₈	—	14	2	no	—
331	C₃×D₄	—	7	4	no	—
332	C₃×D₄	—	14	2	no	—
333	C₃×D₄	—	14	2	no	—
334	C₃×D₄	—	14	2	no	—
335	C₃×D₄	—	14	2	no	—
336	C₃×D₄	—	14	2	no	—
337	C₃×D₄	—	14	2	no	—
338	C₃×D₄	—	28	1	no	—
339	C₃×D₄	—	28	1	no	—
340	C₂×C₃×C₄	—	7	1	no	—
341	C₂×C₃×C₄	—	7	1	no	—
342	C₂×C₃×C₄	—	7	1	no	—
343	C₂×C₃×C₄	—	7	1	no	—
344	C₂×C₃×C₄	—	7	1	no	—
345	C₂×C₃×C₄	—	7	1	no	—
346	C₂×C₃×C₄	—	7	1	no	—
347	C₂×C₃×C₄	—	7	1	no	—
348	C₂×C₃×C₄	—	7	1	no	—
349	C₂×C₃×C₄	—	14	2	no	—
350	C₂×C₃×C₄	—	14	2	no	—
351	C₂×C₃×C₄	—	14	2	no	—
352	C₂×C₃×C₄	—	14	2	no	—
353	C₂×C₃×C₄	—	28	1	no	—
354	C₂×C₃×C₄	—	28	1	no	—
355	ΓL(1,𝔽₈)	#546	1	1	yes	characteristic, complemented, Hall
356	C₂⁴	—	7	1	no	—
357	C₂⁴	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
358	#41	—	14	4	no	—
359	C₂×Q₈	—	7	1	no	—
360	C₂×D₄	—	7	1	no	—
361	C₂×D₄	—	7	1	no	—
362	C₂×D₄	—	7	1	no	—
363	C₂×D₄	—	7	1	no	—
364	C₂×D₄	—	7	1	no	—
365	C₂×D₄	—	7	1	no	—
366	C₂×D₄	—	14	4	no	—
367	C₂×D₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
368	C₂×D₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
369	C₂×D₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
370	C₂²×C₄	—	7	1	no	—
371	C₂²×C₄	—	7	1	no	—
372	C₂²×C₄	—	7	1	no	—
373	C₂²×C₄	—	7	1	no	—
374	C₂²×C₄	—	7	1	no	—
375	C₂²×C₄	—	7	1	no	—
376	C₄⋊C₄	—	7	1	no	—
377	C₄⋊C₄	—	7	1	no	—
378	C₄⋊C₄	—	7	1	no	—
379	C₄⋊C₄	—	7	1	no	—
380	C₂²⋊C₄	—	7	1	no	—
381	C₂²⋊C₄	—	7	1	no	—
382	C₂²⋊C₄	—	7	1	no	—
383	C₂²⋊C₄	—	7	1	no	—
384	C₂²⋊C₄	—	7	1	no	—
385	C₂²⋊C₄	—	7	1	no	—
386	C₂²⋊C₄	—	7	1	no	—
387	C₂²⋊C₄	—	7	1	no	—
388	C₂²⋊C₄	—	7	1	no	—
389	C₂²⋊C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
390	C₂²⋊C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
391	C₂²⋊C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
392	C₄²	—	7	1	no	—
393	C₂×C₇	—	2	2	no	—
394	C₂×C₇	—	2	2	no	—
395	C₂×C₇	—	4	1	no	—
396	C₂×C₇	—	4	1	no	—
397	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
398	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
399	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
400	D₇	—	2	2	no	—
401	D₇	—	4	1	no	—
402	C₂²×C₃	—	7	2	no	—
403	C₂²×C₃	—	7	2	no	—
404	C₂²×C₃	—	7	2	no	—
405	C₂²×C₃	—	7	1	no	—
406	C₂²×C₃	—	14	1	no	—
407	C₂²×C₃	—	14	1	no	—
408	C₂²×C₃	—	14	1	no	—
409	C₂²×C₃	—	14	1	no	—
410	C₂²×C₃	—	14	1	no	—
411	C₂²×C₃	—	14	1	no	—
412	C₂²×C₃	—	14	1	no	—
413	C₂²×C₃	—	14	1	no	—
414	C₂²×C₃	—	14	1	no	—
415	C₂²×C₃	—	14	1	no	—
416	C₂²×C₃	—	14	1	no	—
417	C₂²×C₃	—	14	1	no	—
418	C₂²×C₃	—	14	1	no	—
419	C₂²×C₃	—	14	1	no	—
420	C₂²×C₃	—	14	1	no	—
421	C₂²×C₃	—	28	4	no	—
422	C₂²×C₃	—	28	4	no	—
423	C₃×C₄	—	7	2	no	—
424	C₃×C₄	—	14	1	no	—
425	C₃×C₄	—	14	1	no	—
426	C₃×C₄	—	14	1	no	—
427	C₃×C₄	—	14	1	no	—
428	C₃×C₄	—	14	1	no	—
429	C₃×C₄	—	14	1	no	—
430	C₃×C₄	—	14	1	no	—
431	C₃×C₄	—	14	1	no	—
432	C₂³	—	2	2	no	—
433	C₂³	—	2	2	no	—
434	C₂³	—	4	1	no	—
435	C₂³	—	7	1	no	—
436	C₂³	—	7	1	no	—
437	C₂³	—	14	2	no	—
438	C₂³	—	14	2	no	—
439	C₂³	—	28	1	no	—
440	C₂³	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
441	C₂³	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
442	C₂³	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
443	C₂³	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
444	Q₈	—	14	2	no	—
445	D₄	—	2	2	no	—
446	D₄	—	2	2	no	—
447	D₄	—	4	1	no	—
448	D₄	—	7	4	no	—
449	D₄	—	14	2	no	—
450	D₄	—	14	2	no	—
451	D₄	—	14	2	no	—
452	D₄	—	14	2	no	—
453	D₄	—	28	1	no	—
454	C₂×C₄	—	7	1	no	—
455	C₂×C₄	—	7	1	no	—
456	C₂×C₄	—	7	1	no	—
457	C₂×C₄	—	7	1	no	—
458	C₂×C₄	—	7	1	no	—
459	C₂×C₄	—	7	1	no	—
460	C₂×C₄	—	14	2	no	—
461	C₂×C₄	—	14	2	no	—
462	C₂×C₄	—	14	2	no	—
463	C₂×C₄	—	14	2	no	—
464	C₂×C₄	—	28	1	no	—
465	C₂×C₄	—	28	1	no	—
466	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
467	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
468	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
469	C₇	C₃×#546	1	1	yes	characteristic, complemented, Sylow
470	C₂×C₃	—	7	1	no	—
471	C₂×C₃	—	7	1	no	—
472	C₂×C₃	—	7	1	no	—
473	C₂×C₃	—	14	2	no	—
474	C₂×C₃	—	14	2	no	—
475	C₂×C₃	—	14	2	no	—
476	C₂×C₃	—	28	1	no	—
477	C₂×C₃	—	28	1	no	—
478	C₂×C₃	—	28	1	no	—
479	C₂²	—	2	1	no	—
480	C₂²	—	2	1	no	—
481	C₂²	—	2	1	no	—
482	C₂²	—	2	1	no	—
483	C₂²	—	2	1	no	—
484	C₂²	—	2	1	no	—
485	C₂²	—	2	1	no	—
486	C₂²	—	2	1	no	—
487	C₂²	—	2	1	no	—
488	C₂²	—	2	1	no	—
489	C₂²	—	4	4	no	—
490	C₂²	—	7	2	no	—
491	C₂²	—	14	1	no	—
492	C₂²	—	14	1	no	—
493	C₂²	—	14	1	no	—
494	C₂²	—	14	1	no	—
495	C₂²	—	14	1	no	—
496	C₂²	—	28	4	no	—
497	C₂²	D₄×AGL(1,𝔽₇)	1	2	yes	—
498	C₂²	#1159⋊C₃	1	2	yes	complemented
499	C₂²	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic
500	C₄	—	2	1	no	—
501	C₄	—	2	1	no	—
502	C₄	—	2	1	no	—
503	C₄	—	7	2	no	—
504	C₄	—	14	1	no	—
505	C₄	—	14	1	no	—
506	C₄	—	14	1	no	—
507	C₄	—	14	1	no	—
508	C₄	—	14	1	no	—
509	C₃	—	7	1	no	Sylow
510	C₂	—	2	2	no	—
511	C₂	—	2	2	no	—
512	C₂	—	4	1	no	—
513	C₂	—	4	1	no	—
514	C₂	—	14	2	no	—
515	C₂	—	28	1	no	—
516	C₂	C₂×D₄×AGL(1,𝔽₇)	1	1	yes	characteristic
517	C₂	C₂×#1159⋊C₃	1	1	yes	characteristic
518	C₂	#6548⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×AGL(1,𝔽₇)
commutator, socle, cosocle	C₂²×C₇	C₂⁴×C₃
Fitting	C₇×#120	C₂×C₃
hypercenter	#120	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#546
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #120

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂²×(C₂²⋊D₇)⋊C₃	C₂	semidirect
2. C₂×#1159⋊C₃	C₂	semidirect
3. ΓL(1,𝔽₈)×#120	C₂	semidirect
4. C₂×(C₂²⋊Dic₇)⋊C₃	C₂	semidirect
5. #6502⋊C₃	C₂	semidirect
6. #6502⋊C₃	C₂	semidirect
7. #6501⋊C₃	C₂	semidirect
8. #6500⋊C₃	C₂	semidirect
9. D₄×ΓL(1,𝔽₈)⋊C₄	C₂	semidirect
10. #6449⋊C₃	C₂	semidirect
11. #6448⋊C₃	C₂	semidirect
12. #6446⋊C₃	C₂	semidirect
13. ((C₂²⋊D₇)⋊C₃)⋊C₄	C₂	semidirect
14. C₂²⋊C₄×AGL(1,𝔽₇)	C₂	semidirect
15. #6441⋊C₃	C₂	semidirect
16. #90282	C₃	semidirect
20. C₂×D₄×ΓL(1,𝔽₈)	C₂²	semidirect
21. C₂×D₄×ΓL(1,𝔽₈)	C₂²	semidirect
27. C₂×(C₂²⋊D₇)⋊C₃	C₂²	semidirect
28. C₂×(C₂²⋊D₇)⋊C₃	C₂²	semidirect
30. C₂²×ΓL(1,𝔽₈)⋊C₄	C₂²	semidirect
31. C₂²×ΓL(1,𝔽₈)⋊C₄	C₂²	semidirect
32. C₂²×ΓL(1,𝔽₈)⋊C₄	C₂²	semidirect
34. C₂×C₄×AGL(1,𝔽₇)	C₂²	semidirect
35. C₂×C₄×AGL(1,𝔽₇)	C₂²	semidirect
36. C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	semidirect
38. C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	semidirect
39. C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	semidirect
42. (C₂²⋊Dic₇)⋊C₃	C₂²	semidirect
43. (C₂²⋊Dic₇)⋊C₃	C₂²	semidirect
44. (C₂²⋊Dic₇)⋊C₃	C₂²	semidirect
45. (C₂×AGL(1,𝔽₇))⋊C₄	C₂²	semidirect
46. (C₂×AGL(1,𝔽₇))⋊C₄	C₂²	semidirect
47. (C₂×AGL(1,𝔽₇))⋊C₄	C₂²	semidirect
48. (C₂×AGL(1,𝔽₇))⋊C₄	C₂²	semidirect
49. C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	semidirect
50. C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	semidirect
51. (ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	semidirect
52. (ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	semidirect
53. C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	semidirect
54. C₂²×C₂²⋊D₇	C₂×C₃	semidirect
55. C₂×#1159	C₂×C₃	semidirect
56. C₇×#120	C₂×C₃	semidirect
57. C₂×C₂²⋊Dic₇	C₂×C₃	semidirect
58. #6502	C₂×C₃	semidirect
59. #6502	C₂×C₃	semidirect
60. #6501	C₂×C₃	semidirect
61. #6500	C₂×C₃	semidirect
62. D₄×Dic₇	C₂×C₃	semidirect
63. #6449	C₂×C₃	semidirect
64. #6448	C₂×C₃	semidirect
65. #6446	C₂×C₃	semidirect
66. (C₂²⋊D₇)⋊C₄	C₂×C₃	semidirect
67. D₇×C₂²⋊C₄	C₂×C₃	semidirect
68. #6441	C₂×C₃	semidirect
93. (C₂²⋊D₇)⋊C₃	D₄	semidirect
110. C₂×C₇×D₄	C₂²×C₃	semidirect
111. C₂×C₇×D₄	C₂²×C₃	semidirect
117. C₂×C₂²⋊D₇	C₂²×C₃	semidirect
118. C₂×C₂²⋊D₇	C₂²×C₃	semidirect
120. C₂²×Dic₇	C₂²×C₃	semidirect
121. C₂²×Dic₇	C₂²×C₃	semidirect
122. C₂²×Dic₇	C₂²×C₃	semidirect
124. C₂×C₄×D₇	C₂²×C₃	semidirect
125. C₂×C₄×D₇	C₂²×C₃	semidirect
126. C₂×Dic₁₄	C₂²×C₃	semidirect
128. C₇×C₂²⋊C₄	C₂²×C₃	semidirect
129. C₇×C₂²⋊C₄	C₂²×C₃	semidirect
132. C₂²⋊Dic₇	C₂²×C₃	semidirect
133. C₂²⋊Dic₇	C₂²×C₃	semidirect
134. C₂²⋊Dic₇	C₂²×C₃	semidirect
135. (C₂×D₇)⋊C₄	C₂²×C₃	semidirect
136. (C₂×D₇)⋊C₄	C₂²×C₃	semidirect
137. (C₂×D₇)⋊C₄	C₂²×C₃	semidirect
138. (C₂×D₇)⋊C₄	C₂²×C₃	semidirect
139. C₄⋊Dic₇	C₂²×C₃	semidirect
140. C₄⋊Dic₇	C₂²×C₃	semidirect
141. Dic₇⋊C₄	C₂²×C₃	semidirect
142. Dic₇⋊C₄	C₂²×C₃	semidirect
143. C₄×Dic₇	C₂²×C₃	semidirect
171. C₂²×ΓL(1,𝔽₈)	#41	semidirect
213. C₂²⋊D₇	C₃×D₄	semidirect
283. #120	AGL(1,𝔽₇)	semidirect
300. C₂²×C₇	C₃×#41	semidirect
355. ΓL(1,𝔽₈)	#546	semidirect
368. C₂×D₄	C₂×AGL(1,𝔽₇)	semidirect
369. C₂×D₄	C₂×AGL(1,𝔽₇)	semidirect
390. C₂²⋊C₄	C₂×AGL(1,𝔽₇)	semidirect
391. C₂²⋊C₄	C₂×AGL(1,𝔽₇)	semidirect
469. C₇	C₃×#546	semidirect
498. C₂²	#1159⋊C₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×(C₂²⋊D₇)⋊C₃	2
C₂×#1159⋊C₃	2
ΓL(1,𝔽₈)×#120	2
C₂×(C₂²⋊Dic₇)⋊C₃	2
#6502⋊C₃	2
#6501⋊C₃	2
#6500⋊C₃	2
D₄×ΓL(1,𝔽₈)⋊C₄	2
#6449⋊C₃	2
#6448⋊C₃	2
#6446⋊C₃	2
((C₂²⋊D₇)⋊C₃)⋊C₄	2
C₂²⋊C₄×AGL(1,𝔽₇)	2
#6441⋊C₃	2
#90282	3
C₃×#546	7

Subgroups — #90282⋊C3 #13057326

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #90282⋊C₃ #13057326