Subgroups — #90468⋊C₃ #13057511

Order 1344.

524 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#66659	C₂	1	1	yes	maximal, characteristic, complemented
3	ΓL(1,𝔽₈)×#137	C₂	1	1	yes	maximal, characteristic, complemented
4	#6512⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#6505⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#6482⋊C₃	C₂	1	2	yes	maximal, complemented
7	#6481⋊C₃	C₂	1	2	yes	maximal, complemented
8	#6479⋊C₃	C₂	1	2	yes	maximal, complemented
9	#6477⋊C₃	C₂	1	2	yes	maximal, complemented
10	#6471⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
11	M₁₆×AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic, complemented
12	#90468	C₃	1	1	yes	maximal, characteristic, complemented, Hall
13	#41×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
14	C₂×Q₈×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented
15	#66659	—	2	1	no	—
16	#66659	—	2	1	no	—
17	#66659	C₂²	1	2	yes	—
18	#66659	C₂²	1	2	yes	complemented
19	#66659	C₂²	1	1	yes	characteristic, complemented
20	Q₈×AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic, complemented
21	#1159⋊C₃	—	2	1	no	—
22	#1159⋊C₃	C₂²	1	1	yes	characteristic, complemented
23	D₄×AGL(1,𝔽₇)	—	2	1	no	—
24	D₄×AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic, complemented
25	#1157⋊C₃	—	2	1	no	—
26	#1157⋊C₃	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊AGL(1,𝔽₇)	—	2	1	no	—
28	C₂×C₄⋊AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic, complemented
29	C₂×C₄×AGL(1,𝔽₇)	—	2	1	no	—
30	C₂×C₄×AGL(1,𝔽₇)	—	2	1	no	—
31	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
32	Q₁₆×ΓL(1,𝔽₈)	C₂²	1	2	yes	complemented
33	SD₁₆×ΓL(1,𝔽₈)	C₂²	1	2	yes	complemented
34	M₁₆×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented
35	#1144⋊C₃	C₂²	1	2	yes	complemented
36	#1143⋊C₃	C₂²	1	2	yes	complemented
37	#1143⋊C₃	C₂²	1	2	yes	complemented
38	#1141⋊C₃	C₂²	1	2	yes	complemented
39	#1136⋊C₃	C₂²	1	1	yes	characteristic, complemented
40	D₅₆⋊C₃	C₂²	1	2	yes	complemented
41	#1132⋊C₃	C₂²	1	2	yes	complemented
42	#1131⋊C₃	C₂²	1	2	yes	complemented
43	C₈×AGL(1,𝔽₇)	C₂²	1	2	yes	complemented
44	D₇×#41	C₂×C₃	1	1	yes	characteristic, complemented
45	C₂×#1161	C₂×C₃	1	1	yes	characteristic, complemented
46	C₇×#137	C₂×C₃	1	1	yes	characteristic, complemented
47	#6512	C₂×C₃	1	1	yes	characteristic, complemented
48	#6505	C₂×C₃	1	1	yes	characteristic, complemented
49	#6482	C₂×C₃	1	2	yes	complemented
50	#6481	C₂×C₃	1	2	yes	complemented
51	#6479	C₂×C₃	1	2	yes	complemented
52	#6477	C₂×C₃	1	2	yes	complemented
53	#6471	C₂×C₃	1	1	yes	characteristic, complemented
54	D₇×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
55	C₃×#575	—	7	1	no	maximal, Hall
56	C₂²×AGL(1,𝔽₇)	—	2	1	no	—
57	C₂²×AGL(1,𝔽₇)	—	2	1	no	—
58	C₂²×AGL(1,𝔽₇)	D₄	1	1	yes	characteristic
59	Q₈×ΓL(1,𝔽₈)	—	2	1	no	—
60	Q₈×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
61	Q₈×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
62	D₄×ΓL(1,𝔽₈)	—	2	1	no	—
63	D₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
64	C₂×C₄×ΓL(1,𝔽₈)	—	2	1	no	—
65	C₂×C₄×ΓL(1,𝔽₈)	—	2	1	no	—
66	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
67	(C₂²⋊D₇)⋊C₃	—	2	1	no	—
68	(C₂²⋊D₇)⋊C₃	—	2	1	no	—
69	(C₂²⋊D₇)⋊C₃	—	2	1	no	—
70	C₂×ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
71	C₂×ΓL(1,𝔽₈)⋊C₄	D₄	1	1	yes	characteristic
72	C₄⋊AGL(1,𝔽₇)	—	2	2	no	—
73	C₄⋊AGL(1,𝔽₇)	—	2	2	no	—
74	C₄⋊AGL(1,𝔽₇)	—	2	1	no	—
75	C₄⋊AGL(1,𝔽₇)	—	2	1	no	—
76	C₄⋊AGL(1,𝔽₇)	C₂³	1	2	yes	—
77	C₄⋊AGL(1,𝔽₇)	C₂³	1	1	yes	characteristic
78	C₄×AGL(1,𝔽₇)	—	2	1	no	—
79	C₄×AGL(1,𝔽₇)	—	2	1	no	—
80	C₄×AGL(1,𝔽₇)	—	2	1	no	—
81	C₄×AGL(1,𝔽₇)	—	2	2	no	—
82	C₄×AGL(1,𝔽₇)	—	2	2	no	—
83	C₄×AGL(1,𝔽₇)	D₄	1	1	yes	characteristic
84	C₄×AGL(1,𝔽₇)	D₄	1	1	yes	characteristic
85	C₄×AGL(1,𝔽₇)	C₂³	1	1	yes	characteristic
86	C₄×AGL(1,𝔽₇)	C₂³	1	1	yes	characteristic
87	ΓL(1,𝔽₈)⋊Q₈	—	2	1	no	—
88	ΓL(1,𝔽₈)⋊Q₈	C₂³	1	1	yes	characteristic
89	C₈×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
90	ΓL(1,𝔽₈)⋊C₈	C₂³	1	2	yes	—
91	C₇×#41	C₂²×C₃	1	1	yes	characteristic
92	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
93	#1161	—	2	1	no	—
94	#1161	—	2	1	no	—
95	#1161	C₂²×C₃	1	2	yes	—
96	#1161	C₂²×C₃	1	2	yes	complemented
97	#1161	C₂²×C₃	1	1	yes	characteristic, complemented
98	Q₈×D₇	C₂²×C₃	1	1	yes	characteristic, complemented
99	#1159	—	2	1	no	—
100	#1159	C₂²×C₃	1	1	yes	characteristic, complemented
101	D₄×D₇	—	2	1	no	—
102	D₄×D₇	C₂²×C₃	1	1	yes	characteristic, complemented
103	#1157	—	2	1	no	—
104	#1157	C₂²×C₃	1	1	yes	characteristic
105	C₂×D₂₈	—	2	1	no	—
106	C₂×D₂₈	C₂²×C₃	1	1	yes	characteristic, complemented
107	C₂×C₄×D₇	—	2	1	no	—
108	C₂×C₄×D₇	—	2	1	no	—
109	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic
110	C₇×Q₁₆	C₂²×C₃	1	2	yes	complemented
111	C₇×SD₁₆	C₂²×C₃	1	2	yes	complemented
112	C₇×M₁₆	C₂²×C₃	1	1	yes	characteristic, complemented
113	#1144	C₂²×C₃	1	2	yes	complemented
114	#1143	C₂²×C₃	1	2	yes	complemented
115	#1143	C₂²×C₃	1	2	yes	complemented
116	#1141	C₂²×C₃	1	2	yes	complemented
117	#1136	C₂²×C₃	1	1	yes	characteristic, complemented
118	D₅₆	C₂²×C₃	1	2	yes	complemented
119	#1132	C₂²×C₃	1	2	yes	complemented
120	#1131	C₂²×C₃	1	2	yes	complemented
121	C₈×D₇	C₂²×C₃	1	2	yes	complemented
122	C₂×C₃×#41	—	7	1	no	—
123	C₂×C₃×#41	—	7	1	no	—
124	C₃×#137	—	7	2	no	—
125	C₃×#137	—	7	1	no	—
126	C₃×#137	—	7	1	no	—
127	C₃×AGL(1,ℤ/8)	—	7	1	no	—
128	C₃×AGL(1,ℤ/8)	—	7	2	no	—
129	C₃×AGL(1,ℤ/8)	—	7	1	no	—
130	C₃×#135	—	7	2	no	—
131	C₃×#135	—	7	2	no	—
132	C₂×C₃×M₁₆	—	7	1	no	—
133	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
134	C₂²×ΓL(1,𝔽₈)	C₂×D₄	1	1	yes	characteristic
135	C₂×AGL(1,𝔽₇)	—	2	2	no	—
136	C₂×AGL(1,𝔽₇)	—	2	1	no	—
137	C₂×AGL(1,𝔽₇)	—	4	1	no	—
138	C₂×AGL(1,𝔽₇)	—	4	2	no	—
139	C₂×AGL(1,𝔽₇)	—	4	2	no	—
140	C₂×AGL(1,𝔽₇)	C₂×D₄	1	1	yes	characteristic
141	C₂×AGL(1,𝔽₇)	C₂×D₄	1	1	yes	characteristic
142	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
143	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
144	C₄×ΓL(1,𝔽₈)	C₂×D₄	1	1	yes	characteristic
145	C₄×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
146	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
147	ΓL(1,𝔽₈)⋊C₄	C₂×D₄	1	1	yes	characteristic
148	ΓL(1,𝔽₈)⋊C₄	C₂×D₄	1	1	yes	characteristic
149	#575	—	7	1	no	Sylow
150	C₂²×D₇	—	2	1	no	—
151	C₂²×D₇	—	2	1	no	—
152	C₂²×D₇	C₃×D₄	1	1	yes	characteristic
153	C₇×Q₈	—	2	1	no	—
154	C₇×Q₈	C₂³×C₃	1	1	yes	characteristic
155	C₇×Q₈	C₂³×C₃	1	2	yes	—
156	C₇×D₄	—	2	1	no	—
157	C₇×D₄	C₂³×C₃	1	1	yes	characteristic
158	C₂×C₄×C₇	—	2	1	no	—
159	C₂×C₄×C₇	—	2	1	no	—
160	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
161	C₂²⋊D₇	—	2	1	no	—
162	C₂²⋊D₇	—	2	1	no	—
163	C₂²⋊D₇	—	2	1	no	—
164	C₂×Dic₇	—	2	1	no	—
165	C₂×Dic₇	C₃×D₄	1	1	yes	characteristic
166	D₂₈	—	2	2	no	—
167	D₂₈	—	2	2	no	—
168	D₂₈	—	2	1	no	—
169	D₂₈	—	2	1	no	—
170	D₂₈	C₂³×C₃	1	2	yes	—
171	D₂₈	C₂³×C₃	1	1	yes	characteristic
172	C₄×D₇	—	2	1	no	—
173	C₄×D₇	—	2	1	no	—
174	C₄×D₇	—	2	1	no	—
175	C₄×D₇	—	2	2	no	—
176	C₄×D₇	—	2	2	no	—
177	C₄×D₇	C₃×D₄	1	1	yes	characteristic
178	C₄×D₇	C₃×D₄	1	1	yes	characteristic
179	C₄×D₇	C₂³×C₃	1	1	yes	characteristic
180	C₄×D₇	C₂³×C₃	1	1	yes	characteristic
181	Dic₁₄	—	2	1	no	—
182	Dic₁₄	C₂³×C₃	1	1	yes	characteristic
183	C₇×C₈	C₂³×C₃	1	2	yes	—
184	C₇⋊C₈	C₂³×C₃	1	2	yes	—
185	C₃×#41	—	7	1	no	—
186	C₃×#41	—	7	1	no	—
187	C₃×#41	—	7	2	no	—
188	C₃×#41	—	7	2	no	—
189	C₃×#41	—	7	1	no	—
190	C₃×#41	—	7	1	no	—
191	C₃×#41	—	14	1	no	—
192	C₃×#41	—	14	1	no	—
193	C₃×#41	—	14	1	no	—
194	C₃×#41	—	14	1	no	—
195	C₂×C₃×Q₈	—	7	1	no	—
196	C₂×C₃×Q₈	—	7	1	no	—
197	C₂×C₃×D₄	—	7	1	no	—
198	C₂×C₃×D₄	—	7	1	no	—
199	C₂×C₃×D₄	—	14	1	no	—
200	C₂×C₃×D₄	—	14	1	no	—
201	C₂²×C₃×C₄	—	7	1	no	—
202	C₂²×C₃×C₄	—	14	1	no	—
203	C₂²×C₃×C₄	—	14	1	no	—
204	C₃×Q₁₆	—	7	2	no	—
205	C₃×Q₁₆	—	7	2	no	—
206	C₃×SD₁₆	—	7	2	no	—
207	C₃×SD₁₆	—	7	2	no	—
208	C₃×SD₁₆	—	7	2	no	—
209	C₃×SD₁₆	—	7	2	no	—
210	C₃×D₈	—	7	2	no	—
211	C₃×D₈	—	7	2	no	—
212	C₃×M₁₆	—	7	2	no	—
213	C₃×M₁₆	—	7	1	no	—
214	C₃×M₁₆	—	7	1	no	—
215	C₂×C₃×C₈	—	7	2	no	—
216	C₂×ΓL(1,𝔽₈)	—	2	1	no	—
217	C₂×ΓL(1,𝔽₈)	—	4	1	no	—
218	C₂×ΓL(1,𝔽₈)	C₂²×D₄	1	1	yes	characteristic
219	AGL(1,𝔽₇)	—	2	1	no	—
220	AGL(1,𝔽₇)	—	2	1	no	—
221	AGL(1,𝔽₇)	—	4	2	no	—
222	AGL(1,𝔽₇)	—	4	1	no	—
223	C₂×#41	—	7	1	no	—
224	C₂×#41	—	7	1	no	—
225	#137	—	7	1	no	—
226	#137	—	7	2	no	—
227	#137	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
228	AGL(1,ℤ/8)	—	7	1	no	—
229	AGL(1,ℤ/8)	—	7	2	no	—
230	AGL(1,ℤ/8)	—	7	1	no	—
231	#135	—	7	2	no	—
232	#135	—	7	2	no	—
233	C₂×M₁₆	—	7	1	no	—
234	C₂²×C₇	—	2	1	no	—
235	C₂²×C₇	C₂×C₃×D₄	1	1	yes	characteristic
236	C₂×D₇	—	2	2	no	—
237	C₂×D₇	—	2	1	no	—
238	C₂×D₇	—	4	1	no	—
239	C₂×D₇	—	4	2	no	—
240	C₂×D₇	—	4	2	no	—
241	C₂×D₇	C₂×C₃×D₄	1	1	yes	characteristic
242	C₂×D₇	C₂×C₃×D₄	1	1	yes	characteristic
243	C₄×C₇	—	2	1	no	—
244	C₄×C₇	—	2	2	no	—
245	C₄×C₇	C₂×C₃×D₄	1	1	yes	characteristic
246	C₄×C₇	C₂⁴×C₃	1	1	yes	characteristic
247	Dic₇	—	2	1	no	—
248	Dic₇	C₂×C₃×D₄	1	1	yes	characteristic
249	Dic₇	C₂×C₃×D₄	1	1	yes	characteristic
250	C₂³×C₃	—	7	1	no	—
251	C₂³×C₃	—	14	1	no	—
252	C₂³×C₃	—	14	1	no	—
253	C₃×Q₈	—	7	1	no	—
254	C₃×Q₈	—	7	1	no	—
255	C₃×Q₈	—	7	2	no	—
256	C₃×Q₈	—	14	1	no	—
257	C₃×Q₈	—	14	1	no	—
258	C₃×D₄	—	7	2	no	—
259	C₃×D₄	—	7	1	no	—
260	C₃×D₄	—	7	1	no	—
261	C₃×D₄	—	14	1	no	—
262	C₃×D₄	—	14	1	no	—
263	C₃×D₄	—	14	1	no	—
264	C₃×D₄	—	14	2	no	—
265	C₃×D₄	—	14	2	no	—
266	C₃×D₄	—	14	1	no	—
267	C₃×D₄	—	14	1	no	—
268	C₃×D₄	—	14	1	no	—
269	C₂×C₃×C₄	—	7	1	no	—
270	C₂×C₃×C₄	—	7	1	no	—
271	C₂×C₃×C₄	—	7	1	no	—
272	C₂×C₃×C₄	—	7	1	no	—
273	C₂×C₃×C₄	—	7	1	no	—
274	C₂×C₃×C₄	—	7	1	no	—
275	C₂×C₃×C₄	—	14	1	no	—
276	C₂×C₃×C₄	—	14	1	no	—
277	C₂×C₃×C₄	—	14	1	no	—
278	C₂×C₃×C₄	—	14	2	no	—
279	C₂×C₃×C₄	—	14	2	no	—
280	C₂×C₃×C₄	—	14	1	no	—
281	C₂×C₃×C₄	—	14	1	no	—
282	C₂×C₃×C₄	—	14	1	no	—
283	C₃×C₈	—	7	2	no	—
284	C₃×C₈	—	7	2	no	—
285	ΓL(1,𝔽₈)	#575	1	1	yes	characteristic, complemented, Hall
286	#41	—	7	1	no	—
287	#41	—	7	2	no	—
288	#41	—	7	2	no	—
289	#41	—	7	1	no	—
290	#41	—	7	1	no	—
291	#41	—	14	1	no	—
292	#41	—	14	1	no	—
293	#41	—	14	1	no	—
294	#41	—	14	1	no	—
295	#41	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
296	C₂×Q₈	—	7	1	no	—
297	C₂×Q₈	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
298	C₂×D₄	—	7	1	no	—
299	C₂×D₄	—	7	1	no	—
300	C₂×D₄	—	14	1	no	—
301	C₂×D₄	—	14	1	no	—
302	C₂²×C₄	—	7	1	no	—
303	C₂²×C₄	—	14	1	no	—
304	C₂²×C₄	—	14	1	no	—
305	Q₁₆	—	7	2	no	—
306	Q₁₆	C₂×AGL(1,𝔽₇)	1	2	yes	complemented
307	SD₁₆	—	7	2	no	—
308	SD₁₆	—	7	2	no	—
309	SD₁₆	—	7	2	no	—
310	SD₁₆	C₂×AGL(1,𝔽₇)	1	2	yes	complemented
311	D₈	—	7	2	no	—
312	D₈	—	7	2	no	—
313	M₁₆	—	7	1	no	—
314	M₁₆	—	7	2	no	—
315	M₁₆	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
316	C₂×C₈	—	7	2	no	—
317	C₂×C₇	—	2	1	no	—
318	C₂×C₇	—	4	1	no	—
319	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
320	D₇	—	2	1	no	—
321	D₇	—	2	1	no	—
322	D₇	—	4	2	no	—
323	D₇	—	4	1	no	—
324	C₂²×C₃	—	7	1	no	—
325	C₂²×C₃	—	7	1	no	—
326	C₂²×C₃	—	7	1	no	—
327	C₂²×C₃	—	14	1	no	—
328	C₂²×C₃	—	14	2	no	—
329	C₂²×C₃	—	14	1	no	—
330	C₂²×C₃	—	28	1	no	—
331	C₂²×C₃	—	28	2	no	—
332	C₂²×C₃	—	28	2	no	—
333	C₃×C₄	—	7	1	no	—
334	C₃×C₄	—	7	1	no	—
335	C₃×C₄	—	7	1	no	—
336	C₃×C₄	—	7	1	no	—
337	C₃×C₄	—	14	1	no	—
338	C₃×C₄	—	14	2	no	—
339	C₃×C₄	—	14	1	no	—
340	C₂³	—	7	1	no	—
341	C₂³	—	14	1	no	—
342	C₂³	—	14	1	no	—
343	Q₈	—	2	1	no	—
344	Q₈	—	7	1	no	—
345	Q₈	—	14	1	no	—
346	Q₈	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
347	Q₈	C₂²×AGL(1,𝔽₇)	1	2	yes	—
348	D₄	—	2	1	no	—
349	D₄	—	7	2	no	—
350	D₄	—	7	1	no	—
351	D₄	—	14	1	no	—
352	D₄	—	14	2	no	—
353	D₄	—	14	2	no	—
354	D₄	—	14	1	no	—
355	D₄	—	14	1	no	—
356	D₄	—	14	1	no	—
357	D₄	—	14	1	no	—
358	D₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	7	1	no	—
362	C₂×C₄	—	7	1	no	—
363	C₂×C₄	—	7	1	no	—
364	C₂×C₄	—	7	1	no	—
365	C₂×C₄	—	7	1	no	—
366	C₂×C₄	—	14	1	no	—
367	C₂×C₄	—	14	2	no	—
368	C₂×C₄	—	14	2	no	—
369	C₂×C₄	—	14	1	no	—
370	C₂×C₄	—	14	1	no	—
371	C₂×C₄	—	14	1	no	—
372	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
373	C₈	—	7	2	no	—
374	C₈	C₂²×AGL(1,𝔽₇)	1	2	yes	—
375	C₇	C₃×#575	1	1	yes	characteristic, complemented, Sylow
376	C₂×C₃	—	7	1	no	—
377	C₂×C₃	—	14	1	no	—
378	C₂×C₃	—	14	1	no	—
379	C₂×C₃	—	14	1	no	—
380	C₂×C₃	—	28	1	no	—
381	C₂×C₃	—	28	2	no	—
382	C₂×C₃	—	28	1	no	—
383	C₂²	—	2	1	no	—
384	C₂²	—	7	1	no	—
385	C₂²	—	7	1	no	—
386	C₂²	—	14	2	no	—
387	C₂²	—	14	1	no	—
388	C₂²	—	28	1	no	—
389	C₂²	—	28	2	no	—
390	C₂²	—	28	2	no	—
391	C₂²	D₄×AGL(1,𝔽₇)	1	1	yes	characteristic
392	C₄	—	2	1	no	—
393	C₄	—	2	2	no	—
394	C₄	—	7	1	no	—
395	C₄	—	7	1	no	—
396	C₄	—	14	1	no	—
397	C₄	D₄×AGL(1,𝔽₇)	1	1	yes	characteristic
398	C₄	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic
399	C₃	—	7	1	no	Sylow
400	C₂	—	2	1	no	—
401	C₂	—	4	1	no	—
402	C₂	—	14	1	no	—
403	C₂	—	14	1	no	—
404	C₂	—	28	2	no	—
405	C₂	—	28	1	no	—
406	C₂	C₂×D₄×AGL(1,𝔽₇)	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×AGL(1,𝔽₇)
commutator, cosocle	C₄×C₇	C₂⁴×C₃
Frattini	C₄	C₂³×AGL(1,𝔽₇)
Fitting	C₇×#137	C₂×C₃
socle	C₂×C₇	C₂²×C₃×D₄
hypercenter	#137	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#575
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

Complements

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

N	H ≅ G/N	type
1. #41×AGL(1,𝔽₇)	C₂	semidirect
2. C₂×#66659	C₂	semidirect
3. ΓL(1,𝔽₈)×#137	C₂	semidirect
4. #6512⋊C₃	C₂	semidirect
5. #6505⋊C₃	C₂	semidirect
6. #6482⋊C₃	C₂	semidirect
7. #6481⋊C₃	C₂	semidirect
8. #6479⋊C₃	C₂	semidirect
9. #6477⋊C₃	C₂	semidirect
10. #6471⋊C₃	C₂	semidirect
11. M₁₆×AGL(1,𝔽₇)	C₂	semidirect
12. #90468	C₃	semidirect
14. C₂×Q₈×ΓL(1,𝔽₈)	C₂²	semidirect
18. #66659	C₂²	semidirect
19. #66659	C₂²	semidirect
20. Q₈×AGL(1,𝔽₇)	C₂²	semidirect
22. #1159⋊C₃	C₂²	semidirect
24. D₄×AGL(1,𝔽₇)	C₂²	semidirect
28. C₂×C₄⋊AGL(1,𝔽₇)	C₂²	semidirect
32. Q₁₆×ΓL(1,𝔽₈)	C₂²	semidirect
33. SD₁₆×ΓL(1,𝔽₈)	C₂²	semidirect
34. M₁₆×ΓL(1,𝔽₈)	C₂²	semidirect
35. #1144⋊C₃	C₂²	semidirect
36. #1143⋊C₃	C₂²	semidirect
37. #1143⋊C₃	C₂²	semidirect
38. #1141⋊C₃	C₂²	semidirect
39. #1136⋊C₃	C₂²	semidirect
40. D₅₆⋊C₃	C₂²	semidirect
41. #1132⋊C₃	C₂²	semidirect
42. #1131⋊C₃	C₂²	semidirect
43. C₈×AGL(1,𝔽₇)	C₂²	semidirect
44. D₇×#41	C₂×C₃	semidirect
45. C₂×#1161	C₂×C₃	semidirect
46. C₇×#137	C₂×C₃	semidirect
47. #6512	C₂×C₃	semidirect
48. #6505	C₂×C₃	semidirect
49. #6482	C₂×C₃	semidirect
50. #6481	C₂×C₃	semidirect
51. #6479	C₂×C₃	semidirect
52. #6477	C₂×C₃	semidirect
53. #6471	C₂×C₃	semidirect
54. D₇×M₁₆	C₂×C₃	semidirect
92. C₂×C₇×Q₈	C₂²×C₃	semidirect
96. #1161	C₂²×C₃	semidirect
97. #1161	C₂²×C₃	semidirect
98. Q₈×D₇	C₂²×C₃	semidirect
100. #1159	C₂²×C₃	semidirect
102. D₄×D₇	C₂²×C₃	semidirect
106. C₂×D₂₈	C₂²×C₃	semidirect
110. C₇×Q₁₆	C₂²×C₃	semidirect
111. C₇×SD₁₆	C₂²×C₃	semidirect
112. C₇×M₁₆	C₂²×C₃	semidirect
113. #1144	C₂²×C₃	semidirect
114. #1143	C₂²×C₃	semidirect
115. #1143	C₂²×C₃	semidirect
116. #1141	C₂²×C₃	semidirect
117. #1136	C₂²×C₃	semidirect
118. D₅₆	C₂²×C₃	semidirect
119. #1132	C₂²×C₃	semidirect
120. #1131	C₂²×C₃	semidirect
121. C₈×D₇	C₂²×C₃	semidirect
227. #137	AGL(1,𝔽₇)	semidirect
285. ΓL(1,𝔽₈)	#575	semidirect
297. C₂×Q₈	C₂×AGL(1,𝔽₇)	semidirect
306. Q₁₆	C₂×AGL(1,𝔽₇)	semidirect
310. SD₁₆	C₂×AGL(1,𝔽₇)	semidirect
315. M₁₆	C₂×AGL(1,𝔽₇)	semidirect
375. C₇	C₃×#575	semidirect

Maximal subgroups

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

subgroup	index
#41×AGL(1,𝔽₇)	2
C₂×#66659	2
ΓL(1,𝔽₈)×#137	2
#6512⋊C₃	2
#6505⋊C₃	2
#6482⋊C₃	2
#6481⋊C₃	2
#6479⋊C₃	2
#6477⋊C₃	2
#6471⋊C₃	2
M₁₆×AGL(1,𝔽₇)	2
#90468	3
C₃×#575	7

Subgroups — #90468⋊C3 #13057511

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #90468⋊C₃ #13057511