Subgroups — #90346⋊C₃ #13057393

Order 1344.

584 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1159⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#1157⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
3	ΓL(1,𝔽₈)×#123	C₂	1	1	yes	maximal, characteristic, complemented
4	#6502⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#6459⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#6457⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄⋊C₄×AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic, complemented
9	#6451⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#6449⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
11	#6447⋊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⋊D₇)⋊C₃)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
15	#6441⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
16	#90346	C₃	1	1	yes	maximal, characteristic, complemented, Hall
17	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
18	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
19	C₂×(C₂²⋊D₇)⋊C₃	C₂²	1	1	yes	characteristic
20	C₂×(C₂²⋊D₇)⋊C₃	C₂²	1	1	yes	characteristic
21	C₂×(C₂²⋊D₇)⋊C₃	C₂²	1	1	yes	characteristic
22	C₂×(C₂²⋊D₇)⋊C₃	C₂²	1	1	yes	characteristic
23	C₂²×ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic
24	C₂²×ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic
25	C₂²×ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic
26	#1159⋊C₃	—	2	4	no	—
27	#1157⋊C₃	—	2	4	no	—
28	C₂×C₄⋊AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
29	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
30	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
31	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
32	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
33	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	1	yes	characteristic
34	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
37	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
38	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
39	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
40	(C₂²⋊Dic₇)⋊C₃	C₂²	1	1	yes	characteristic
41	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	1	yes	characteristic
42	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	1	yes	characteristic
43	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	1	yes	characteristic
44	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	1	yes	characteristic
45	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic
46	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic
47	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	1	yes	characteristic
48	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	1	yes	characteristic
49	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	1	yes	characteristic
50	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	1	yes	characteristic
51	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	1	yes	characteristic
52	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	1	yes	characteristic
53	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic
54	C₂×#1159	C₂×C₃	1	1	yes	characteristic, complemented
55	C₂×#1157	C₂×C₃	1	1	yes	characteristic, complemented
56	C₇×#123	C₂×C₃	1	1	yes	characteristic, complemented
57	#6502	C₂×C₃	1	1	yes	characteristic, complemented
58	C₂×Dic₇⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
59	#6459	C₂×C₃	1	1	yes	characteristic, complemented
60	#6457	C₂×C₃	1	1	yes	characteristic, complemented
61	D₇×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
62	#6451	C₂×C₃	1	1	yes	characteristic, complemented
63	#6449	C₂×C₃	1	1	yes	characteristic, complemented
64	#6447	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	(C₂²⋊D₇)⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
68	#6441	C₂×C₃	1	1	yes	characteristic, complemented
69	C₃×#547	—	7	1	no	maximal, Hall
70	C₂³×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
71	C₂³×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
72	C₂²×AGL(1,𝔽₇)	C₂³	1	1	yes	characteristic
73	C₂²×AGL(1,𝔽₇)	C₂³	1	1	yes	characteristic
74	D₄×ΓL(1,𝔽₈)	—	2	2	no	—
75	C₂×C₄×ΓL(1,𝔽₈)	—	2	2	no	—
76	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
77	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
78	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
79	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
80	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
81	(C₂²⋊D₇)⋊C₃	—	2	2	no	—
82	(C₂²⋊D₇)⋊C₃	—	2	2	no	—
83	(C₂²⋊D₇)⋊C₃	—	2	2	no	—
84	(C₂²⋊D₇)⋊C₃	D₄	1	4	yes	complemented
85	C₂×ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
86	C₂×ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
87	C₂×ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
88	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
89	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
90	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
91	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
92	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
93	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
94	C₄⋊AGL(1,𝔽₇)	—	2	2	no	—
95	C₄×AGL(1,𝔽₇)	—	2	2	no	—
96	C₄×AGL(1,𝔽₇)	—	2	2	no	—
97	C₄×AGL(1,𝔽₇)	—	2	2	no	—
98	C₄×AGL(1,𝔽₇)	—	2	2	no	—
99	ΓL(1,𝔽₈)⋊Q₈	—	2	2	no	—
100	ΓL(1,𝔽₈)⋊Q₈	—	2	2	no	—
101	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
102	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
103	C₂×C₂²⋊D₇	C₂²×C₃	1	1	yes	characteristic
104	C₂×C₂²⋊D₇	C₂²×C₃	1	1	yes	characteristic
105	C₂×C₂²⋊D₇	C₂²×C₃	1	1	yes	characteristic
106	C₂×C₂²⋊D₇	C₂²×C₃	1	1	yes	characteristic
107	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
108	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
109	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
110	#1159	—	2	4	no	—
111	#1157	—	2	4	no	—
112	C₂×D₂₈	C₂²×C₃	1	1	yes	characteristic
113	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic
114	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic
115	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic
116	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic
117	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
118	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
119	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
120	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
121	C₇×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
122	C₇×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
123	C₇×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
124	C₂²⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
125	(C₂×D₇)⋊C₄	C₂²×C₃	1	1	yes	characteristic
126	(C₂×D₇)⋊C₄	C₂²×C₃	1	1	yes	characteristic
127	(C₂×D₇)⋊C₄	C₂²×C₃	1	1	yes	characteristic
128	(C₂×D₇)⋊C₄	C₂²×C₃	1	1	yes	characteristic
129	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
130	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
131	Dic₇⋊C₄	C₂²×C₃	1	1	yes	characteristic
132	Dic₇⋊C₄	C₂²×C₃	1	1	yes	characteristic
133	Dic₇⋊C₄	C₂²×C₃	1	1	yes	characteristic
134	Dic₇⋊C₄	C₂²×C₃	1	1	yes	characteristic
135	Dic₇⋊C₄	C₂²×C₃	1	1	yes	characteristic
136	Dic₇⋊C₄	C₂²×C₃	1	1	yes	characteristic
137	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
138	C₂×C₃×#41	—	7	1	no	—
139	C₂×C₃×#41	—	7	1	no	—
140	C₃×C₄⋊Q₈	—	7	1	no	—
141	C₃×#123	—	7	1	no	—
142	C₃×#123	—	7	1	no	—
143	C₃×#123	—	7	1	no	—
144	C₃×#123	—	7	1	no	—
145	C₃×C₂²⋊Q₈	—	7	1	no	—
146	C₃×C₂²⋊Q₈	—	7	1	no	—
147	C₃×#121	—	7	1	no	—
148	C₃×#121	—	7	1	no	—
149	C₃×C₄×D₄	—	7	1	no	—
150	C₃×C₄×D₄	—	7	1	no	—
151	C₂×C₃×C₄⋊C₄	—	7	1	no	—
152	C₂×C₃×C₄⋊C₄	—	7	1	no	—
153	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
154	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
155	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
156	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
157	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
158	C₂²×ΓL(1,𝔽₈)	C₂×D₄	1	2	yes	—
159	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
160	C₂×AGL(1,𝔽₇)	—	2	1	no	—
161	C₂×AGL(1,𝔽₇)	—	2	1	no	—
162	C₂×AGL(1,𝔽₇)	—	2	1	no	—
163	C₂×AGL(1,𝔽₇)	—	2	1	no	—
164	C₂×AGL(1,𝔽₇)	—	2	1	no	—
165	C₂×AGL(1,𝔽₇)	C₂×D₄	1	2	yes	—
166	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
167	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
168	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
169	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
170	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
171	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
172	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
173	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
174	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
175	ΓL(1,𝔽₈)⋊C₄	C₂×D₄	1	2	yes	—
176	ΓL(1,𝔽₈)⋊C₄	#41	1	2	yes	complemented
177	#547	—	7	1	no	Sylow
178	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
179	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
180	C₂²×D₇	C₂³×C₃	1	1	yes	characteristic
181	C₂²×D₇	C₂³×C₃	1	1	yes	characteristic
182	C₇×D₄	—	2	2	no	—
183	C₂×C₄×C₇	—	2	2	no	—
184	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
185	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
186	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
187	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
188	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
189	C₂²⋊D₇	—	2	2	no	—
190	C₂²⋊D₇	—	2	2	no	—
191	C₂²⋊D₇	—	2	2	no	—
192	C₂²⋊D₇	C₃×D₄	1	4	yes	complemented
193	C₂×Dic₇	—	2	2	no	—
194	C₂×Dic₇	—	2	2	no	—
195	C₂×Dic₇	—	2	2	no	—
196	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
197	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
198	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
199	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
200	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
201	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
202	D₂₈	—	2	2	no	—
203	C₄×D₇	—	2	2	no	—
204	C₄×D₇	—	2	2	no	—
205	C₄×D₇	—	2	2	no	—
206	C₄×D₇	—	2	2	no	—
207	Dic₁₄	—	2	2	no	—
208	Dic₁₄	—	2	2	no	—
209	C₃×#41	—	14	4	no	—
210	C₃×#41	—	14	4	no	—
211	C₂×C₃×Q₈	—	7	1	no	—
212	C₂×C₃×Q₈	—	7	1	no	—
213	C₂×C₃×D₄	—	7	1	no	—
214	C₂×C₃×D₄	—	7	1	no	—
215	C₂×C₃×D₄	—	7	1	no	—
216	C₂×C₃×D₄	—	7	1	no	—
217	C₂×C₃×D₄	—	7	1	no	—
218	C₂×C₃×D₄	—	7	1	no	—
219	C₂²×C₃×C₄	—	7	1	no	—
220	C₂²×C₃×C₄	—	7	1	no	—
221	C₂²×C₃×C₄	—	7	1	no	—
222	C₂²×C₃×C₄	—	7	1	no	—
223	C₂²×C₃×C₄	—	7	1	no	—
224	C₂²×C₃×C₄	—	7	1	no	—
225	C₂²×C₃×C₄	—	7	1	no	—
226	C₂²×C₃×C₄	—	7	1	no	—
227	C₃×C₄⋊C₄	—	7	1	no	—
228	C₃×C₄⋊C₄	—	7	1	no	—
229	C₃×C₄⋊C₄	—	7	1	no	—
230	C₃×C₄⋊C₄	—	7	1	no	—
231	C₃×C₄⋊C₄	—	7	1	no	—
232	C₃×C₄⋊C₄	—	7	1	no	—
233	C₃×C₄⋊C₄	—	7	1	no	—
234	C₃×C₄⋊C₄	—	7	1	no	—
235	C₃×C₄⋊C₄	—	7	1	no	—
236	C₃×C₄⋊C₄	—	7	1	no	—
237	C₃×C₂²⋊C₄	—	7	1	no	—
238	C₃×C₂²⋊C₄	—	7	1	no	—
239	C₃×C₂²⋊C₄	—	7	1	no	—
240	C₃×C₂²⋊C₄	—	7	1	no	—
241	C₃×C₂²⋊C₄	—	7	1	no	—
242	C₃×C₂²⋊C₄	—	7	1	no	—
243	C₃×C₂²⋊C₄	—	7	1	no	—
244	C₃×C₂²⋊C₄	—	7	1	no	—
245	C₃×C₄²	—	7	1	no	—
246	C₂×ΓL(1,𝔽₈)	—	2	2	no	—
247	C₂×ΓL(1,𝔽₈)	—	4	1	no	—
248	C₂×ΓL(1,𝔽₈)	C₂²×D₄	1	1	yes	characteristic
249	C₂×ΓL(1,𝔽₈)	C₂×#41	1	1	yes	characteristic
250	C₂×ΓL(1,𝔽₈)	2₋¹⁺⁴	1	1	yes	characteristic
251	AGL(1,𝔽₇)	—	2	2	no	—
252	AGL(1,𝔽₇)	—	4	1	no	—
253	C₂×#41	—	7	1	no	—
254	C₂×#41	—	7	1	no	—
255	C₄⋊Q₈	—	7	1	no	—
256	#123	—	7	1	no	—
257	#123	—	7	1	no	—
258	#123	—	7	1	no	—
259	#123	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
260	C₂²⋊Q₈	—	7	1	no	—
261	C₂²⋊Q₈	—	7	1	no	—
262	#121	—	7	1	no	—
263	#121	—	7	1	no	—
264	C₄×D₄	—	7	1	no	—
265	C₄×D₄	—	7	1	no	—
266	C₂×C₄⋊C₄	—	7	1	no	—
267	C₂×C₄⋊C₄	—	7	1	no	—
268	C₂²×C₇	—	2	1	no	—
269	C₂²×C₇	—	2	1	no	—
270	C₂²×C₇	—	2	1	no	—
271	C₂²×C₇	—	2	1	no	—
272	C₂²×C₇	—	2	1	no	—
273	C₂²×C₇	C₂×C₃×D₄	1	2	yes	—
274	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
275	C₂×D₇	—	2	1	no	—
276	C₂×D₇	—	2	1	no	—
277	C₂×D₇	—	2	1	no	—
278	C₂×D₇	—	2	1	no	—
279	C₂×D₇	—	2	1	no	—
280	C₂×D₇	C₂×C₃×D₄	1	2	yes	—
281	C₄×C₇	—	2	1	no	—
282	C₄×C₇	—	2	1	no	—
283	C₄×C₇	—	2	1	no	—
284	C₄×C₇	—	2	1	no	—
285	C₄×C₇	—	2	1	no	—
286	Dic₇	—	2	1	no	—
287	Dic₇	—	2	1	no	—
288	Dic₇	—	2	1	no	—
289	Dic₇	—	2	1	no	—
290	Dic₇	C₂×C₃×D₄	1	2	yes	—
291	Dic₇	C₃×#41	1	2	yes	complemented
292	C₂³×C₃	—	7	1	no	—
293	C₂³×C₃	—	7	1	no	—
294	C₂³×C₃	—	7	1	no	—
295	C₂³×C₃	—	7	1	no	—
296	C₃×Q₈	—	14	2	no	—
297	C₃×Q₈	—	14	2	no	—
298	C₃×D₄	—	7	4	no	—
299	C₃×D₄	—	14	2	no	—
300	C₃×D₄	—	14	2	no	—
301	C₃×D₄	—	14	2	no	—
302	C₃×D₄	—	14	2	no	—
303	C₃×D₄	—	14	2	no	—
304	C₂×C₃×C₄	—	7	1	no	—
305	C₂×C₃×C₄	—	7	1	no	—
306	C₂×C₃×C₄	—	7	1	no	—
307	C₂×C₃×C₄	—	7	1	no	—
308	C₂×C₃×C₄	—	7	1	no	—
309	C₂×C₃×C₄	—	7	1	no	—
310	C₂×C₃×C₄	—	7	1	no	—
311	C₂×C₃×C₄	—	7	1	no	—
312	C₂×C₃×C₄	—	7	1	no	—
313	C₂×C₃×C₄	—	7	1	no	—
314	C₂×C₃×C₄	—	7	1	no	—
315	C₂×C₃×C₄	—	14	2	no	—
316	C₂×C₃×C₄	—	14	2	no	—
317	C₂×C₃×C₄	—	14	2	no	—
318	C₂×C₃×C₄	—	14	2	no	—
319	C₂×C₃×C₄	—	14	2	no	—
320	C₂×C₃×C₄	—	14	2	no	—
321	C₂×C₃×C₄	—	14	2	no	—
322	C₂×C₃×C₄	—	14	2	no	—
323	ΓL(1,𝔽₈)	#547	1	1	yes	characteristic, complemented, Hall
324	#41	—	14	4	no	—
325	#41	—	14	4	no	—
326	C₂×Q₈	—	7	1	no	—
327	C₂×Q₈	—	7	1	no	—
328	C₂×D₄	—	7	1	no	—
329	C₂×D₄	—	7	1	no	—
330	C₂×D₄	—	7	1	no	—
331	C₂×D₄	—	7	1	no	—
332	C₂×D₄	—	7	1	no	—
333	C₂×D₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
334	C₂²×C₄	—	7	1	no	—
335	C₂²×C₄	—	7	1	no	—
336	C₂²×C₄	—	7	1	no	—
337	C₂²×C₄	—	7	1	no	—
338	C₂²×C₄	—	7	1	no	—
339	C₂²×C₄	—	7	1	no	—
340	C₂²×C₄	—	7	1	no	—
341	C₂²×C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
342	C₄⋊C₄	—	7	1	no	—
343	C₄⋊C₄	—	7	1	no	—
344	C₄⋊C₄	—	7	1	no	—
345	C₄⋊C₄	—	7	1	no	—
346	C₄⋊C₄	—	7	1	no	—
347	C₄⋊C₄	—	7	1	no	—
348	C₄⋊C₄	—	7	1	no	—
349	C₄⋊C₄	—	7	1	no	—
350	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
351	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
352	C₂²⋊C₄	—	7	1	no	—
353	C₂²⋊C₄	—	7	1	no	—
354	C₂²⋊C₄	—	7	1	no	—
355	C₂²⋊C₄	—	7	1	no	—
356	C₂²⋊C₄	—	7	1	no	—
357	C₂²⋊C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
358	C₂²⋊C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
359	C₂²⋊C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
360	C₄²	—	7	1	no	—
361	C₂×C₇	—	2	2	no	—
362	C₂×C₇	—	4	1	no	—
363	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
364	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
365	C₂×C₇	C₃×2₋¹⁺⁴	1	1	yes	characteristic
366	D₇	—	2	2	no	—
367	D₇	—	4	1	no	—
368	C₂²×C₃	—	7	2	no	—
369	C₂²×C₃	—	7	2	no	—
370	C₂²×C₃	—	7	1	no	—
371	C₂²×C₃	—	14	1	no	—
372	C₂²×C₃	—	14	1	no	—
373	C₂²×C₃	—	14	1	no	—
374	C₂²×C₃	—	14	1	no	—
375	C₂²×C₃	—	14	1	no	—
376	C₂²×C₃	—	14	1	no	—
377	C₂²×C₃	—	14	1	no	—
378	C₂²×C₃	—	14	1	no	—
379	C₂²×C₃	—	14	1	no	—
380	C₂²×C₃	—	14	1	no	—
381	C₃×C₄	—	7	2	no	—
382	C₃×C₄	—	7	2	no	—
383	C₃×C₄	—	14	1	no	—
384	C₃×C₄	—	14	1	no	—
385	C₃×C₄	—	14	1	no	—
386	C₃×C₄	—	14	1	no	—
387	C₃×C₄	—	14	1	no	—
388	C₃×C₄	—	14	1	no	—
389	C₃×C₄	—	14	1	no	—
390	C₃×C₄	—	14	1	no	—
391	C₃×C₄	—	14	1	no	—
392	C₂³	—	7	1	no	—
393	C₂³	—	7	1	no	—
394	C₂³	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
395	C₂³	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
396	Q₈	—	14	2	no	—
397	Q₈	—	14	2	no	—
398	D₄	—	2	2	no	—
399	D₄	—	7	4	no	—
400	D₄	—	14	2	no	—
401	D₄	—	14	2	no	—
402	D₄	—	14	2	no	—
403	D₄	—	14	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	7	1	no	—
406	C₂×C₄	—	7	1	no	—
407	C₂×C₄	—	7	1	no	—
408	C₂×C₄	—	7	1	no	—
409	C₂×C₄	—	7	1	no	—
410	C₂×C₄	—	7	1	no	—
411	C₂×C₄	—	14	2	no	—
412	C₂×C₄	—	14	2	no	—
413	C₂×C₄	—	14	2	no	—
414	C₂×C₄	—	14	2	no	—
415	C₂×C₄	—	14	2	no	—
416	C₂×C₄	—	14	2	no	—
417	C₂×C₄	—	14	2	no	—
418	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
419	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
420	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
421	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
422	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
423	C₇	C₃×#547	1	1	yes	characteristic, complemented, Sylow
424	C₂×C₃	—	7	1	no	—
425	C₂×C₃	—	7	1	no	—
426	C₂×C₃	—	7	1	no	—
427	C₂×C₃	—	14	2	no	—
428	C₂×C₃	—	14	2	no	—
429	C₂×C₃	—	28	1	no	—
430	C₂×C₃	—	28	1	no	—
431	C₂²	—	2	1	no	—
432	C₂²	—	2	1	no	—
433	C₂²	—	2	1	no	—
434	C₂²	—	2	1	no	—
435	C₂²	—	2	1	no	—
436	C₂²	—	7	2	no	—
437	C₂²	—	14	1	no	—
438	C₂²	—	14	1	no	—
439	C₂²	—	14	1	no	—
440	C₂²	—	14	1	no	—
441	C₂²	—	14	1	no	—
442	C₂²	D₄×AGL(1,𝔽₇)	1	2	yes	—
443	C₂²	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic
444	C₄	—	2	1	no	—
445	C₄	—	2	1	no	—
446	C₄	—	2	1	no	—
447	C₄	—	2	1	no	—
448	C₄	—	2	1	no	—
449	C₄	—	7	2	no	—
450	C₄	—	7	2	no	—
451	C₄	—	14	1	no	—
452	C₄	—	14	1	no	—
453	C₄	—	14	1	no	—
454	C₄	—	14	1	no	—
455	C₃	—	7	1	no	Sylow
456	C₂	—	2	2	no	—
457	C₂	—	4	1	no	—
458	C₂	—	14	2	no	—
459	C₂	—	28	1	no	—
460	C₂	C₂×D₄×AGL(1,𝔽₇)	1	1	yes	characteristic
461	C₂	#41×AGL(1,𝔽₇)	1	1	yes	characteristic
462	C₂	#10621519	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×AGL(1,𝔽₇)
commutator, socle, cosocle	C₂²×C₇	C₂⁴×C₃
Fitting	C₇×#123	C₂×C₃
hypercenter	#123	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#547
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #123

Complements

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

N	H ≅ G/N	type
1. C₂×#1159⋊C₃	C₂	semidirect
2. C₂×#1157⋊C₃	C₂	semidirect
3. ΓL(1,𝔽₈)×#123	C₂	semidirect
4. #6502⋊C₃	C₂	semidirect
5. C₂×(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂	semidirect
6. #6459⋊C₃	C₂	semidirect
7. #6457⋊C₃	C₂	semidirect
8. C₄⋊C₄×AGL(1,𝔽₇)	C₂	semidirect
9. #6451⋊C₃	C₂	semidirect
10. #6449⋊C₃	C₂	semidirect
11. #6447⋊C₃	C₂	semidirect
12. #6446⋊C₃	C₂	semidirect
13. ((C₂²⋊D₇)⋊C₃)⋊C₄	C₂	semidirect
14. ((C₂²⋊D₇)⋊C₃)⋊C₄	C₂	semidirect
15. #6441⋊C₃	C₂	semidirect
16. #90346	C₃	semidirect
54. C₂×#1159	C₂×C₃	semidirect
55. C₂×#1157	C₂×C₃	semidirect
56. C₇×#123	C₂×C₃	semidirect
57. #6502	C₂×C₃	semidirect
58. C₂×Dic₇⋊C₄	C₂×C₃	semidirect
59. #6459	C₂×C₃	semidirect
60. #6457	C₂×C₃	semidirect
61. D₇×C₄⋊C₄	C₂×C₃	semidirect
62. #6451	C₂×C₃	semidirect
63. #6449	C₂×C₃	semidirect
64. #6447	C₂×C₃	semidirect
65. #6446	C₂×C₃	semidirect
66. (C₂²⋊D₇)⋊C₄	C₂×C₃	semidirect
67. (C₂²⋊D₇)⋊C₄	C₂×C₃	semidirect
68. #6441	C₂×C₃	semidirect
84. (C₂²⋊D₇)⋊C₃	D₄	semidirect
176. ΓL(1,𝔽₈)⋊C₄	#41	semidirect
192. C₂²⋊D₇	C₃×D₄	semidirect
259. #123	AGL(1,𝔽₇)	semidirect
291. Dic₇	C₃×#41	semidirect
323. ΓL(1,𝔽₈)	#547	semidirect
423. C₇	C₃×#547	semidirect

Maximal subgroups

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

subgroup	index
C₂×#1159⋊C₃	2
C₂×#1157⋊C₃	2
ΓL(1,𝔽₈)×#123	2
#6502⋊C₃	2
C₂×(ΓL(1,𝔽₈)⋊C₄)⋊C₄	2
#6459⋊C₃	2
#6457⋊C₃	2
C₄⋊C₄×AGL(1,𝔽₇)	2
#6451⋊C₃	2
#6449⋊C₃	2
#6447⋊C₃	2
#6446⋊C₃	2
((C₂²⋊D₇)⋊C₃)⋊C₄	2
#6441⋊C₃	2
#90346	3
C₃×#547	7

Subgroups — #90346⋊C3 #13057393

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #90346⋊C₃ #13057393