Subgroups — #90349⋊C₃ #13057395

Order 1344.

496 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×#123	C₂	1	1	yes	maximal, characteristic, complemented
2	#6500⋊C₃	C₂	1	1	yes	maximal, characteristic
3	D₄×ΓL(1,𝔽₈)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×(C₂×AGL(1,𝔽₇))⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	#6489⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#6461⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#6461⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#6460⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#6455⋊C₃	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	#90349	C₃	1	1	yes	maximal, characteristic, complemented, Hall
17	C₂³×AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
18	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
19	C₂²×C₄×ΓL(1,𝔽₈)	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	C₂²×ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic
23	C₂²×ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic, complemented
24	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
25	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
26	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	1	yes	characteristic, complemented
27	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented
28	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented
29	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
30	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented
31	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented
32	(C₂²⋊Dic₇)⋊C₃	C₂²	1	1	yes	characteristic
33	(C₂²⋊Dic₇)⋊C₃	C₂²	1	1	yes	characteristic
34	(C₂²⋊Dic₇)⋊C₃	C₂²	1	1	yes	characteristic, complemented
35	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	1	yes	characteristic
36	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	1	yes	characteristic
37	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	1	yes	characteristic
38	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	1	yes	characteristic
39	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	1	yes	characteristic
40	(C₂×AGL(1,𝔽₇))⋊C₄	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	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic, complemented
44	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic, complemented
45	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic, complemented
46	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic, complemented
47	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	1	yes	characteristic, complemented
48	(ΓL(1,𝔽₈)⋊C₄)⋊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	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic, complemented
52	C₇×#123	C₂×C₃	1	1	yes	characteristic, complemented
53	#6500	C₂×C₃	1	1	yes	characteristic
54	D₄×Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
55	C₂×(C₂×D₇)⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
56	#6489	C₂×C₃	1	1	yes	characteristic, complemented
57	#6461	C₂×C₃	1	1	yes	characteristic, complemented
58	#6461	C₂×C₃	1	1	yes	characteristic, complemented
59	#6460	C₂×C₃	1	1	yes	characteristic, complemented
60	#6455	C₂×C₃	1	1	yes	characteristic, complemented
61	#6449	C₂×C₃	1	1	yes	characteristic, complemented
62	#6448	C₂×C₃	1	1	yes	characteristic, complemented
63	#6446	C₂×C₃	1	1	yes	characteristic, complemented
64	(C₂²⋊D₇)⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
65	D₇×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
66	#6441	C₂×C₃	1	1	yes	characteristic, complemented
67	C₃×#551	—	7	1	no	maximal, Hall
68	C₂³×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
69	C₂³×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
70	C₂²×AGL(1,𝔽₇)	—	2	2	no	—
71	C₂²×AGL(1,𝔽₇)	—	2	2	no	—
72	C₂²×AGL(1,𝔽₇)	—	4	1	no	—
73	C₂²×AGL(1,𝔽₇)	C₂³	1	1	yes	characteristic
74	C₂²×AGL(1,𝔽₇)	C₂³	1	1	yes	characteristic
75	D₄×ΓL(1,𝔽₈)	—	2	2	no	—
76	C₂×C₄×ΓL(1,𝔽₈)	—	2	2	no	—
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	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
82	(C₂²⋊D₇)⋊C₃	—	2	2	no	—
83	(C₂²⋊D₇)⋊C₃	—	4	1	no	—
84	C₂×ΓL(1,𝔽₈)⋊C₄	—	4	1	no	—
85	C₂×ΓL(1,𝔽₈)⋊C₄	—	4	1	no	—
86	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
87	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
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	C₄×AGL(1,𝔽₇)	—	2	2	no	—
93	C₄×AGL(1,𝔽₇)	—	4	1	no	—
94	ΓL(1,𝔽₈)⋊Q₈	—	4	1	no	—
95	C₂³×D₇	C₂²×C₃	1	1	yes	characteristic
96	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
97	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
98	C₂×C₂²⋊D₇	C₂²×C₃	1	1	yes	characteristic
99	C₂×C₂²⋊D₇	C₂²×C₃	1	1	yes	characteristic
100	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
101	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic, complemented
102	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic
103	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic
104	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic, complemented
105	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
106	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
107	C₇×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
108	C₇×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
109	C₇×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
110	C₂²⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
111	C₂²⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
112	C₂²⋊Dic₇	C₂²×C₃	1	1	yes	characteristic, complemented
113	(C₂×D₇)⋊C₄	C₂²×C₃	1	1	yes	characteristic
114	(C₂×D₇)⋊C₄	C₂²×C₃	1	1	yes	characteristic
115	(C₂×D₇)⋊C₄	C₂²×C₃	1	1	yes	characteristic
116	(C₂×D₇)⋊C₄	C₂²×C₃	1	1	yes	characteristic
117	(C₂×D₇)⋊C₄	C₂²×C₃	1	1	yes	characteristic
118	(C₂×D₇)⋊C₄	C₂²×C₃	1	1	yes	characteristic
119	(C₂×D₇)⋊C₄	C₂²×C₃	1	1	yes	characteristic
120	(C₂×D₇)⋊C₄	C₂²×C₃	1	1	yes	characteristic
121	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic, complemented
122	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic, complemented
123	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic, complemented
124	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic, complemented
125	Dic₇⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
126	Dic₇⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
127	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic, complemented
128	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic, complemented
129	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic, complemented
130	C₃×#126	—	7	1	no	—
131	C₃×#126	—	7	1	no	—
132	C₃×#124	—	7	1	no	—
133	C₃×#123	—	7	1	no	—
134	C₃×#123	—	7	1	no	—
135	C₃×#123	—	7	1	no	—
136	C₃×C₂²⋊Q₈	—	7	1	no	—
137	C₃×C₂²⋊Q₈	—	7	1	no	—
138	C₃×#120	—	7	1	no	—
139	C₃×C₄×D₄	—	7	1	no	—
140	C₃×C₄×D₄	—	7	1	no	—
141	C₃×#117	—	7	1	no	—
142	C₃×#117	—	7	1	no	—
143	C₂×C₃×C₂²⋊C₄	—	7	1	no	—
144	C₂×C₃×C₂²⋊C₄	—	7	1	no	—
145	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
146	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
147	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
148	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
149	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
150	C₂²×ΓL(1,𝔽₈)	#41	1	2	yes	—
151	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
152	C₂×AGL(1,𝔽₇)	—	2	1	no	—
153	C₂×AGL(1,𝔽₇)	—	2	1	no	—
154	C₂×AGL(1,𝔽₇)	—	2	1	no	—
155	C₂×AGL(1,𝔽₇)	—	2	1	no	—
156	C₂×AGL(1,𝔽₇)	—	2	1	no	—
157	C₂×AGL(1,𝔽₇)	—	4	4	no	—
158	C₂×AGL(1,𝔽₇)	#41	1	2	yes	—
159	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
160	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
161	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
162	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
163	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
164	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
165	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
166	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
167	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
168	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
169	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
170	#551	—	7	1	no	Sylow
171	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
172	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
173	C₂²×D₇	—	2	2	no	—
174	C₂²×D₇	—	2	2	no	—
175	C₂²×D₇	—	4	1	no	—
176	C₂²×D₇	C₂³×C₃	1	1	yes	characteristic
177	C₂²×D₇	C₂³×C₃	1	1	yes	characteristic
178	C₇×D₄	—	2	2	no	—
179	C₂×C₄×C₇	—	2	2	no	—
180	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
181	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
182	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
183	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
184	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
185	C₂²⋊D₇	—	2	2	no	—
186	C₂²⋊D₇	—	4	1	no	—
187	C₂×Dic₇	—	4	1	no	—
188	C₂×Dic₇	—	4	1	no	—
189	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
190	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
191	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
192	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
193	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
194	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
195	C₄×D₇	—	2	2	no	—
196	C₄×D₇	—	4	1	no	—
197	Dic₁₄	—	4	1	no	—
198	C₂⁴×C₃	—	7	1	no	—
199	C₂×C₃×Q₈	—	7	1	no	—
200	C₂×C₃×D₄	—	7	1	no	—
201	C₂×C₃×D₄	—	7	1	no	—
202	C₂×C₃×D₄	—	7	1	no	—
203	C₂²×C₃×C₄	—	7	1	no	—
204	C₂²×C₃×C₄	—	7	1	no	—
205	C₂²×C₃×C₄	—	7	1	no	—
206	C₂²×C₃×C₄	—	7	1	no	—
207	C₂²×C₃×C₄	—	7	1	no	—
208	C₃×C₄⋊C₄	—	7	1	no	—
209	C₃×C₄⋊C₄	—	7	1	no	—
210	C₃×C₄⋊C₄	—	7	1	no	—
211	C₃×C₄⋊C₄	—	7	1	no	—
212	C₃×C₄⋊C₄	—	7	1	no	—
213	C₃×C₄⋊C₄	—	7	1	no	—
214	C₃×C₄⋊C₄	—	7	1	no	—
215	C₃×C₄⋊C₄	—	7	1	no	—
216	C₃×C₂²⋊C₄	—	7	1	no	—
217	C₃×C₂²⋊C₄	—	7	1	no	—
218	C₃×C₂²⋊C₄	—	7	1	no	—
219	C₃×C₂²⋊C₄	—	7	1	no	—
220	C₃×C₂²⋊C₄	—	7	1	no	—
221	C₃×C₂²⋊C₄	—	7	1	no	—
222	C₃×C₂²⋊C₄	—	7	1	no	—
223	C₃×C₂²⋊C₄	—	7	1	no	—
224	C₃×C₂²⋊C₄	—	7	1	no	—
225	C₃×C₂²⋊C₄	—	7	1	no	—
226	C₃×C₂²⋊C₄	—	7	1	no	—
227	C₃×C₂²⋊C₄	—	7	1	no	—
228	C₃×C₂²⋊C₄	—	7	1	no	—
229	C₃×C₂²⋊C₄	—	7	1	no	—
230	C₃×C₄²	—	7	1	no	—
231	C₃×C₄²	—	7	1	no	—
232	C₃×C₄²	—	7	1	no	—
233	C₂×ΓL(1,𝔽₈)	—	2	2	no	—
234	C₂×ΓL(1,𝔽₈)	—	4	1	no	—
235	C₂×ΓL(1,𝔽₈)	C₂×#41	1	1	yes	characteristic
236	C₂×ΓL(1,𝔽₈)	C₂×#41	1	1	yes	characteristic
237	C₂×ΓL(1,𝔽₈)	H(5,𝔽₂)	1	1	yes	characteristic
238	AGL(1,𝔽₇)	—	2	2	no	—
239	AGL(1,𝔽₇)	—	4	1	no	—
240	#126	—	7	1	no	—
241	#126	—	7	1	no	—
242	#124	—	7	1	no	—
243	#123	—	7	1	no	—
244	#123	—	7	1	no	—
245	#123	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
246	C₂²⋊Q₈	—	7	1	no	—
247	C₂²⋊Q₈	—	7	1	no	—
248	#120	—	7	1	no	—
249	C₄×D₄	—	7	1	no	—
250	C₄×D₄	—	7	1	no	—
251	#117	—	7	1	no	—
252	#117	—	7	1	no	—
253	C₂×C₂²⋊C₄	—	7	1	no	—
254	C₂×C₂²⋊C₄	—	7	1	no	—
255	C₂²×C₇	—	2	1	no	—
256	C₂²×C₇	—	2	1	no	—
257	C₂²×C₇	—	2	1	no	—
258	C₂²×C₇	—	2	1	no	—
259	C₂²×C₇	—	2	1	no	—
260	C₂²×C₇	C₃×#41	1	2	yes	—
261	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
262	C₂×D₇	—	2	1	no	—
263	C₂×D₇	—	2	1	no	—
264	C₂×D₇	—	2	1	no	—
265	C₂×D₇	—	2	1	no	—
266	C₂×D₇	—	2	1	no	—
267	C₂×D₇	—	4	4	no	—
268	C₂×D₇	C₃×#41	1	2	yes	—
269	C₄×C₇	—	2	1	no	—
270	C₄×C₇	—	2	1	no	—
271	C₄×C₇	—	2	1	no	—
272	C₄×C₇	—	2	1	no	—
273	C₄×C₇	—	2	1	no	—
274	Dic₇	—	2	1	no	—
275	Dic₇	—	2	1	no	—
276	Dic₇	—	2	1	no	—
277	Dic₇	—	2	1	no	—
278	Dic₇	—	2	1	no	—
279	Dic₇	—	2	1	no	—
280	C₂³×C₃	—	7	1	no	—
281	C₂³×C₃	—	7	1	no	—
282	C₂³×C₃	—	7	1	no	—
283	C₂³×C₃	—	7	1	no	—
284	C₂³×C₃	—	14	2	no	—
285	C₂³×C₃	—	14	2	no	—
286	C₂³×C₃	—	28	1	no	—
287	C₃×Q₈	—	28	1	no	—
288	C₃×D₄	—	14	2	no	—
289	C₃×D₄	—	14	2	no	—
290	C₃×D₄	—	28	1	no	—
291	C₂×C₃×C₄	—	7	1	no	—
292	C₂×C₃×C₄	—	7	1	no	—
293	C₂×C₃×C₄	—	7	1	no	—
294	C₂×C₃×C₄	—	7	1	no	—
295	C₂×C₃×C₄	—	7	1	no	—
296	C₂×C₃×C₄	—	7	1	no	—
297	C₂×C₃×C₄	—	7	1	no	—
298	C₂×C₃×C₄	—	7	1	no	—
299	C₂×C₃×C₄	—	7	1	no	—
300	C₂×C₃×C₄	—	7	1	no	—
301	C₂×C₃×C₄	—	7	1	no	—
302	C₂×C₃×C₄	—	14	2	no	—
303	C₂×C₃×C₄	—	14	2	no	—
304	C₂×C₃×C₄	—	28	1	no	—
305	C₂×C₃×C₄	—	28	1	no	—
306	C₂×C₃×C₄	—	28	1	no	—
307	ΓL(1,𝔽₈)	#551	1	1	yes	characteristic, complemented, Hall
308	C₂⁴	—	7	1	no	—
309	C₂×Q₈	—	7	1	no	—
310	C₂×D₄	—	7	1	no	—
311	C₂×D₄	—	7	1	no	—
312	C₂×D₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
313	C₂²×C₄	—	7	1	no	—
314	C₂²×C₄	—	7	1	no	—
315	C₂²×C₄	—	7	1	no	—
316	C₂²×C₄	—	7	1	no	—
317	C₂²×C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
318	C₄⋊C₄	—	7	1	no	—
319	C₄⋊C₄	—	7	1	no	—
320	C₄⋊C₄	—	7	1	no	—
321	C₄⋊C₄	—	7	1	no	—
322	C₄⋊C₄	—	7	1	no	—
323	C₄⋊C₄	—	7	1	no	—
324	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
325	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
326	C₂²⋊C₄	—	7	1	no	—
327	C₂²⋊C₄	—	7	1	no	—
328	C₂²⋊C₄	—	7	1	no	—
329	C₂²⋊C₄	—	7	1	no	—
330	C₂²⋊C₄	—	7	1	no	—
331	C₂²⋊C₄	—	7	1	no	—
332	C₂²⋊C₄	—	7	1	no	—
333	C₂²⋊C₄	—	7	1	no	—
334	C₂²⋊C₄	—	7	1	no	—
335	C₂²⋊C₄	—	7	1	no	—
336	C₂²⋊C₄	—	7	1	no	—
337	C₂²⋊C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
338	C₂²⋊C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
339	C₂²⋊C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
340	C₄²	—	7	1	no	—
341	C₄²	—	7	1	no	—
342	C₄²	—	7	1	no	—
343	C₂×C₇	—	2	2	no	—
344	C₂×C₇	—	4	1	no	—
345	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
346	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
347	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
348	D₇	—	2	2	no	—
349	D₇	—	4	1	no	—
350	C₂²×C₃	—	7	2	no	—
351	C₂²×C₃	—	7	2	no	—
352	C₂²×C₃	—	7	1	no	—
353	C₂²×C₃	—	14	1	no	—
354	C₂²×C₃	—	14	1	no	—
355	C₂²×C₃	—	14	1	no	—
356	C₂²×C₃	—	14	1	no	—
357	C₂²×C₃	—	14	1	no	—
358	C₂²×C₃	—	14	1	no	—
359	C₂²×C₃	—	14	1	no	—
360	C₂²×C₃	—	14	1	no	—
361	C₂²×C₃	—	14	1	no	—
362	C₂²×C₃	—	14	1	no	—
363	C₂²×C₃	—	28	4	no	—
364	C₃×C₄	—	14	1	no	—
365	C₃×C₄	—	14	1	no	—
366	C₃×C₄	—	14	1	no	—
367	C₃×C₄	—	14	1	no	—
368	C₃×C₄	—	14	1	no	—
369	C₃×C₄	—	14	1	no	—
370	C₃×C₄	—	14	1	no	—
371	C₃×C₄	—	14	1	no	—
372	C₃×C₄	—	14	1	no	—
373	C₃×C₄	—	14	1	no	—
374	C₃×C₄	—	14	1	no	—
375	C₂³	—	7	1	no	—
376	C₂³	—	7	1	no	—
377	C₂³	—	14	2	no	—
378	C₂³	—	14	2	no	—
379	C₂³	—	28	1	no	—
380	C₂³	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
381	C₂³	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
382	Q₈	—	28	1	no	—
383	D₄	—	2	2	no	—
384	D₄	—	14	2	no	—
385	D₄	—	28	1	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	7	1	no	—
388	C₂×C₄	—	7	1	no	—
389	C₂×C₄	—	7	1	no	—
390	C₂×C₄	—	7	1	no	—
391	C₂×C₄	—	7	1	no	—
392	C₂×C₄	—	7	1	no	—
393	C₂×C₄	—	14	2	no	—
394	C₂×C₄	—	28	1	no	—
395	C₂×C₄	—	28	1	no	—
396	C₂×C₄	—	28	1	no	—
397	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
398	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
399	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
400	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
401	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
402	C₇	C₃×#551	1	1	yes	characteristic, complemented, Sylow
403	C₂×C₃	—	7	1	no	—
404	C₂×C₃	—	7	1	no	—
405	C₂×C₃	—	7	1	no	—
406	C₂×C₃	—	14	2	no	—
407	C₂×C₃	—	14	2	no	—
408	C₂×C₃	—	28	1	no	—
409	C₂×C₃	—	28	1	no	—
410	C₂²	—	2	1	no	—
411	C₂²	—	2	1	no	—
412	C₂²	—	2	1	no	—
413	C₂²	—	2	1	no	—
414	C₂²	—	2	1	no	—
415	C₂²	—	7	2	no	—
416	C₂²	—	14	1	no	—
417	C₂²	—	14	1	no	—
418	C₂²	—	14	1	no	—
419	C₂²	—	14	1	no	—
420	C₂²	—	14	1	no	—
421	C₂²	—	28	4	no	—
422	C₂²	#1159⋊C₃	1	2	yes	—
423	C₂²	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic
424	C₄	—	2	1	no	—
425	C₄	—	2	1	no	—
426	C₄	—	2	1	no	—
427	C₄	—	2	1	no	—
428	C₄	—	2	1	no	—
429	C₄	—	14	1	no	—
430	C₄	—	14	1	no	—
431	C₄	—	14	1	no	—
432	C₄	—	14	1	no	—
433	C₄	—	14	1	no	—
434	C₄	—	14	1	no	—
435	C₃	—	7	1	no	Sylow
436	C₂	—	2	2	no	—
437	C₂	—	4	1	no	—
438	C₂	—	14	2	no	—
439	C₂	—	28	1	no	—
440	C₂	C₂×#1159⋊C₃	1	1	yes	characteristic
441	C₂	#41×AGL(1,𝔽₇)	1	1	yes	characteristic
442	C₂	#10621518	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
prime radical	#6500⋊C₃	C₂
hypercenter	#123	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#551
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. ΓL(1,𝔽₈)×#123	C₂	semidirect
3. D₄×ΓL(1,𝔽₈)⋊C₄	C₂	semidirect
4. C₂×(C₂×AGL(1,𝔽₇))⋊C₄	C₂	semidirect
5. #6489⋊C₃	C₂	semidirect
6. #6461⋊C₃	C₂	semidirect
7. #6461⋊C₃	C₂	semidirect
8. #6460⋊C₃	C₂	semidirect
9. #6455⋊C₃	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. #90349	C₃	semidirect
23. C₂²×ΓL(1,𝔽₈)⋊C₄	C₂²	semidirect
26. C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	semidirect
27. C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	semidirect
28. C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	semidirect
30. C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	semidirect
31. C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	semidirect
34. (C₂²⋊Dic₇)⋊C₃	C₂²	semidirect
43. C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	semidirect
44. C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	semidirect
45. C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	semidirect
46. C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	semidirect
47. (ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	semidirect
48. (ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	semidirect
49. C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	semidirect
50. C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	semidirect
51. C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	semidirect
52. C₇×#123	C₂×C₃	semidirect
54. D₄×Dic₇	C₂×C₃	semidirect
55. C₂×(C₂×D₇)⋊C₄	C₂×C₃	semidirect
56. #6489	C₂×C₃	semidirect
57. #6461	C₂×C₃	semidirect
58. #6461	C₂×C₃	semidirect
59. #6460	C₂×C₃	semidirect
60. #6455	C₂×C₃	semidirect
61. #6449	C₂×C₃	semidirect
62. #6448	C₂×C₃	semidirect
63. #6446	C₂×C₃	semidirect
64. (C₂²⋊D₇)⋊C₄	C₂×C₃	semidirect
65. D₇×C₂²⋊C₄	C₂×C₃	semidirect
66. #6441	C₂×C₃	semidirect
101. C₂²×Dic₇	C₂²×C₃	semidirect
104. C₂×Dic₁₄	C₂²×C₃	semidirect
105. C₇×C₄⋊C₄	C₂²×C₃	semidirect
106. C₇×C₄⋊C₄	C₂²×C₃	semidirect
108. C₇×C₂²⋊C₄	C₂²×C₃	semidirect
109. C₇×C₂²⋊C₄	C₂²×C₃	semidirect
112. C₂²⋊Dic₇	C₂²×C₃	semidirect
121. C₄⋊Dic₇	C₂²×C₃	semidirect
122. C₄⋊Dic₇	C₂²×C₃	semidirect
123. C₄⋊Dic₇	C₂²×C₃	semidirect
124. C₄⋊Dic₇	C₂²×C₃	semidirect
125. Dic₇⋊C₄	C₂²×C₃	semidirect
126. Dic₇⋊C₄	C₂²×C₃	semidirect
127. C₄×Dic₇	C₂²×C₃	semidirect
128. C₄×Dic₇	C₂²×C₃	semidirect
129. C₄×Dic₇	C₂²×C₃	semidirect
245. #123	AGL(1,𝔽₇)	semidirect
307. ΓL(1,𝔽₈)	#551	semidirect
324. C₄⋊C₄	C₂×AGL(1,𝔽₇)	semidirect
325. C₄⋊C₄	C₂×AGL(1,𝔽₇)	semidirect
338. C₂²⋊C₄	C₂×AGL(1,𝔽₇)	semidirect
339. C₂²⋊C₄	C₂×AGL(1,𝔽₇)	semidirect
402. C₇	C₃×#551	semidirect

Maximal subgroups

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

subgroup	index
ΓL(1,𝔽₈)×#123	2
#6500⋊C₃	2
D₄×ΓL(1,𝔽₈)⋊C₄	2
C₂×(C₂×AGL(1,𝔽₇))⋊C₄	2
#6489⋊C₃	2
#6461⋊C₃	2
#6460⋊C₃	2
#6455⋊C₃	2
#6449⋊C₃	2
#6448⋊C₃	2
#6446⋊C₃	2
((C₂²⋊D₇)⋊C₃)⋊C₄	2
C₂²⋊C₄×AGL(1,𝔽₇)	2
#6441⋊C₃	2
#90349	3
C₃×#551	7

Subgroups — #90349⋊C3 #13057395

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #90349⋊C₃ #13057395