Subgroups — #90382⋊C₃ #13057426

Order 1344.

468 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	#6461⋊C₃	C₂	1	2	yes	maximal, complemented
3	#6460⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#6458⋊C₃	C₂	1	1	yes	maximal, characteristic
5	#6457⋊C₃	C₂	1	2	yes	maximal, complemented
6	#6456⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#6456⋊C₃	C₂	1	2	yes	maximal, complemented
8	#6455⋊C₃	C₂	1	2	yes	maximal, complemented
9	#6450⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#6438⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄²×AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic, complemented
12	#90382	C₃	1	1	yes	maximal, characteristic, complemented, Hall
13	C₂×C₄⋊AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊AGL(1,𝔽₇)	C₂²	1	2	yes	—
15	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
16	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
17	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
18	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	2	yes	—
19	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
22	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
23	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
24	C₄²×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
25	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	2	yes	—
26	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	2	yes	—
27	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	2	yes	—
28	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	2	yes	—
29	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	2	yes	—
30	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	2	yes	—
31	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	—
32	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic
33	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic
34	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic
35	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	2	yes	—
36	C₇×#125	C₂×C₃	1	1	yes	characteristic, complemented
37	#6461	C₂×C₃	1	2	yes	complemented
38	#6460	C₂×C₃	1	1	yes	characteristic, complemented
39	#6458	C₂×C₃	1	1	yes	characteristic
40	#6457	C₂×C₃	1	2	yes	complemented
41	#6456	C₂×C₃	1	1	yes	characteristic, complemented
42	#6456	C₂×C₃	1	2	yes	complemented
43	#6455	C₂×C₃	1	2	yes	complemented
44	#6450	C₂×C₃	1	1	yes	characteristic, complemented
45	#6438	C₂×C₃	1	1	yes	characteristic, complemented
46	C₄²×D₇	C₂×C₃	1	1	yes	characteristic, complemented
47	C₃×#529	—	7	1	no	maximal, Hall
48	C₂²×AGL(1,𝔽₇)	C₂³	1	2	yes	—
49	C₂²×AGL(1,𝔽₇)	C₂³	1	1	yes	characteristic
50	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
51	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
52	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
53	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
54	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
55	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
56	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
57	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
58	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
59	C₄⋊AGL(1,𝔽₇)	—	2	4	no	—
60	C₄⋊AGL(1,𝔽₇)	—	2	2	no	—
61	C₄×AGL(1,𝔽₇)	—	2	2	no	—
62	C₄×AGL(1,𝔽₇)	—	2	2	no	—
63	C₄×AGL(1,𝔽₇)	—	2	2	no	—
64	C₄×AGL(1,𝔽₇)	—	2	4	no	—
65	ΓL(1,𝔽₈)⋊Q₈	—	2	2	no	—
66	C₂×D₂₈	C₂²×C₃	1	1	yes	characteristic
67	C₂×D₂₈	C₂²×C₃	1	2	yes	—
68	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic
69	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic
70	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic
71	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
72	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
73	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
74	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
75	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
76	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
77	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
78	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
79	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
80	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
81	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
82	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
83	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
84	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
85	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
86	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
87	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
88	C₄×Dic₇	C₂²×C₃	1	2	yes	—
89	C₃×#126	—	7	2	no	—
90	C₃×#125	—	7	1	no	—
91	C₃×#124	—	7	1	no	—
92	C₃×#123	—	7	2	no	—
93	C₃×C₂²⋊Q₈	—	7	1	no	—
94	C₃×#121	—	7	1	no	—
95	C₃×C₄×Q₈	—	7	1	no	—
96	C₃×C₄×D₄	—	7	1	no	—
97	C₃×C₄×D₄	—	7	2	no	—
98	C₃×#117	—	7	2	no	—
99	C₂×C₃×C₄²	—	7	1	no	—
100	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
101	C₂×AGL(1,𝔽₇)	—	2	1	no	—
102	C₂×AGL(1,𝔽₇)	—	2	1	no	—
103	C₂×AGL(1,𝔽₇)	—	2	2	no	—
104	C₂×AGL(1,𝔽₇)	—	2	2	no	—
105	C₂×AGL(1,𝔽₇)	—	2	2	no	—
106	C₂×AGL(1,𝔽₇)	#41	1	2	yes	—
107	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
108	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
109	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
110	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
111	C₄×ΓL(1,𝔽₈)	#41	1	2	yes	—
112	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
113	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
114	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
115	ΓL(1,𝔽₈)⋊C₄	#41	1	2	yes	—
116	#529	—	7	1	no	Sylow
117	C₂²×D₇	C₂³×C₃	1	2	yes	—
118	C₂²×D₇	C₂³×C₃	1	1	yes	characteristic
119	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
120	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
121	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
122	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
123	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
124	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
125	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
126	C₂×Dic₇	C₂³×C₃	1	2	yes	—
127	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
128	D₂₈	—	2	4	no	—
129	D₂₈	—	2	2	no	—
130	C₄×D₇	—	2	2	no	—
131	C₄×D₇	—	2	2	no	—
132	C₄×D₇	—	2	2	no	—
133	C₄×D₇	—	2	4	no	—
134	Dic₁₄	—	2	2	no	—
135	C₂×C₃×Q₈	—	7	1	no	—
136	C₂×C₃×D₄	—	7	2	no	—
137	C₂×C₃×D₄	—	7	1	no	—
138	C₂²×C₃×C₄	—	7	1	no	—
139	C₂²×C₃×C₄	—	7	1	no	—
140	C₂²×C₃×C₄	—	7	1	no	—
141	C₂²×C₃×C₄	—	7	2	no	—
142	C₃×C₄⋊C₄	—	7	2	no	—
143	C₃×C₄⋊C₄	—	7	2	no	—
144	C₃×C₄⋊C₄	—	7	2	no	—
145	C₃×C₄⋊C₄	—	7	1	no	—
146	C₃×C₄⋊C₄	—	7	2	no	—
147	C₃×C₄⋊C₄	—	7	1	no	—
148	C₃×C₂²⋊C₄	—	7	2	no	—
149	C₃×C₂²⋊C₄	—	7	2	no	—
150	C₃×C₂²⋊C₄	—	7	2	no	—
151	C₃×C₂²⋊C₄	—	7	2	no	—
152	C₃×C₂²⋊C₄	—	7	2	no	—
153	C₃×C₄²	—	7	1	no	—
154	C₃×C₄²	—	7	1	no	—
155	C₃×C₄²	—	7	1	no	—
156	C₃×C₄²	—	7	2	no	—
157	C₃×C₄²	—	7	1	no	—
158	C₂×ΓL(1,𝔽₈)	C₂×#41	1	1	yes	characteristic
159	C₂×ΓL(1,𝔽₈)	C₂×#41	1	1	yes	characteristic
160	C₂×ΓL(1,𝔽₈)	C₂×#41	1	1	yes	characteristic
161	AGL(1,𝔽₇)	—	2	2	no	—
162	AGL(1,𝔽₇)	—	4	2	no	—
163	#126	—	7	2	no	—
164	#125	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
165	#124	—	7	1	no	—
166	#123	—	7	2	no	—
167	C₂²⋊Q₈	—	7	1	no	—
168	#121	—	7	1	no	—
169	C₄×Q₈	—	7	1	no	—
170	C₄×D₄	—	7	1	no	—
171	C₄×D₄	—	7	2	no	—
172	#117	—	7	2	no	—
173	C₂×C₄²	—	7	1	no	—
174	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
175	C₂×D₇	—	2	1	no	—
176	C₂×D₇	—	2	1	no	—
177	C₂×D₇	—	2	2	no	—
178	C₂×D₇	—	2	2	no	—
179	C₂×D₇	—	2	2	no	—
180	C₂×D₇	C₃×#41	1	2	yes	—
181	C₄×C₇	—	2	1	no	—
182	C₄×C₇	—	2	1	no	—
183	C₄×C₇	—	2	2	no	—
184	C₄×C₇	—	2	2	no	—
185	C₄×C₇	C₃×#41	1	2	yes	—
186	Dic₇	—	2	2	no	—
187	Dic₇	—	2	1	no	—
188	Dic₇	—	2	1	no	—
189	Dic₇	C₃×#41	1	2	yes	—
190	C₂³×C₃	—	7	1	no	—
191	C₂³×C₃	—	7	2	no	—
192	C₃×Q₈	—	14	2	no	—
193	C₃×D₄	—	14	2	no	—
194	C₃×D₄	—	14	4	no	—
195	C₂×C₃×C₄	—	7	1	no	—
196	C₂×C₃×C₄	—	7	1	no	—
197	C₂×C₃×C₄	—	7	1	no	—
198	C₂×C₃×C₄	—	7	2	no	—
199	C₂×C₃×C₄	—	7	1	no	—
200	C₂×C₃×C₄	—	7	1	no	—
201	C₂×C₃×C₄	—	7	1	no	—
202	C₂×C₃×C₄	—	7	2	no	—
203	C₂×C₃×C₄	—	7	2	no	—
204	C₂×C₃×C₄	—	14	2	no	—
205	C₂×C₃×C₄	—	14	4	no	—
206	C₂×C₃×C₄	—	14	2	no	—
207	C₂×C₃×C₄	—	14	2	no	—
208	ΓL(1,𝔽₈)	#529	1	1	yes	characteristic, complemented, Hall
209	C₂×Q₈	—	7	1	no	—
210	C₂×D₄	—	7	2	no	—
211	C₂×D₄	—	7	1	no	—
212	C₂²×C₄	—	7	1	no	—
213	C₂²×C₄	—	7	1	no	—
214	C₂²×C₄	—	7	1	no	—
215	C₂²×C₄	—	7	2	no	—
216	C₄⋊C₄	—	7	2	no	—
217	C₄⋊C₄	—	7	2	no	—
218	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
219	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
220	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
221	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
222	C₂²⋊C₄	—	7	2	no	—
223	C₂²⋊C₄	—	7	2	no	—
224	C₂²⋊C₄	—	7	2	no	—
225	C₂²⋊C₄	—	7	2	no	—
226	C₂²⋊C₄	—	7	2	no	—
227	C₄²	—	7	1	no	—
228	C₄²	—	7	1	no	—
229	C₄²	—	7	1	no	—
230	C₄²	—	7	2	no	—
231	C₄²	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
232	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
233	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
234	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
235	D₇	—	2	2	no	—
236	D₇	—	4	2	no	—
237	C₂²×C₃	—	7	2	no	—
238	C₂²×C₃	—	7	1	no	—
239	C₂²×C₃	—	14	1	no	—
240	C₂²×C₃	—	14	1	no	—
241	C₂²×C₃	—	14	2	no	—
242	C₂²×C₃	—	14	2	no	—
243	C₂²×C₃	—	14	2	no	—
244	C₃×C₄	—	7	2	no	—
245	C₃×C₄	—	7	2	no	—
246	C₃×C₄	—	14	1	no	—
247	C₃×C₄	—	14	2	no	—
248	C₃×C₄	—	14	1	no	—
249	C₃×C₄	—	14	2	no	—
250	C₃×C₄	—	14	1	no	—
251	C₃×C₄	—	14	1	no	—
252	C₃×C₄	—	14	2	no	—
253	C₂³	—	7	1	no	—
254	C₂³	—	7	2	no	—
255	Q₈	—	14	2	no	—
256	D₄	—	14	4	no	—
257	D₄	—	14	2	no	—
258	C₂×C₄	—	7	1	no	—
259	C₂×C₄	—	7	1	no	—
260	C₂×C₄	—	7	1	no	—
261	C₂×C₄	—	7	2	no	—
262	C₂×C₄	—	14	2	no	—
263	C₂×C₄	—	14	4	no	—
264	C₂×C₄	—	14	2	no	—
265	C₂×C₄	—	14	2	no	—
266	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
267	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
268	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
269	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
270	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
271	C₇	C₃×#529	1	1	yes	characteristic, complemented, Sylow
272	C₂×C₃	—	7	1	no	—
273	C₂×C₃	—	7	1	no	—
274	C₂×C₃	—	7	1	no	—
275	C₂×C₃	—	14	2	no	—
276	C₂×C₃	—	28	2	no	—
277	C₂²	—	7	2	no	—
278	C₂²	—	14	1	no	—
279	C₂²	—	14	1	no	—
280	C₂²	—	14	2	no	—
281	C₂²	—	14	2	no	—
282	C₂²	—	14	2	no	—
283	C₂²	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	2	no	—
288	C₄	—	7	2	no	—
289	C₄	—	14	1	no	—
290	C₄	—	14	2	no	—
291	C₄	—	14	1	no	—
292	C₄	#66659	1	2	yes	—
293	C₃	—	7	1	no	Sylow
294	C₂	—	14	2	no	—
295	C₂	—	28	2	no	—
296	C₂	C₂×#66659	1	1	yes	characteristic
297	C₂	#41×AGL(1,𝔽₇)	1	1	yes	characteristic
298	C₂	#41×AGL(1,𝔽₇)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×AGL(1,𝔽₇)
commutator, socle, cosocle	C₂²×C₇	C₂⁴×C₃
Fitting	C₇×#125	C₂×C₃
prime radical	#6458⋊C₃	C₂
hypercenter	#125	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#529
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
1. ΓL(1,𝔽₈)×#125	C₂	semidirect
2. #6461⋊C₃	C₂	semidirect
3. #6460⋊C₃	C₂	semidirect
5. #6457⋊C₃	C₂	semidirect
6. #6456⋊C₃	C₂	semidirect
7. #6456⋊C₃	C₂	semidirect
8. #6455⋊C₃	C₂	semidirect
9. #6450⋊C₃	C₂	semidirect
10. #6438⋊C₃	C₂	semidirect
11. C₄²×AGL(1,𝔽₇)	C₂	semidirect
12. #90382	C₃	semidirect
36. C₇×#125	C₂×C₃	semidirect
37. #6461	C₂×C₃	semidirect
38. #6460	C₂×C₃	semidirect
40. #6457	C₂×C₃	semidirect
41. #6456	C₂×C₃	semidirect
42. #6456	C₂×C₃	semidirect
43. #6455	C₂×C₃	semidirect
44. #6450	C₂×C₃	semidirect
45. #6438	C₂×C₃	semidirect
46. C₄²×D₇	C₂×C₃	semidirect
164. #125	AGL(1,𝔽₇)	semidirect
208. ΓL(1,𝔽₈)	#529	semidirect
271. C₇	C₃×#529	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)×#125	2
#6461⋊C₃	2
#6460⋊C₃	2
#6458⋊C₃	2
#6457⋊C₃	2
#6456⋊C₃	2
#6455⋊C₃	2
#6450⋊C₃	2
#6438⋊C₃	2
C₄²×AGL(1,𝔽₇)	2
#90382	3
C₃×#529	7

Subgroups — #90382⋊C3 #13057426

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #90382⋊C₃ #13057426