Subgroups — #90381⋊C₃ #13057425

Order 1344.

456 conjugacy classes of subgroups.

The subgroup lattice has 284 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	#6460⋊C₃	C₂	1	2	yes	maximal, complemented
3	#6459⋊C₃	C₂	1	2	yes	maximal, complemented
4	#6459⋊C₃	C₂	1	2	yes	maximal, complemented
5	#6456⋊C₃	C₂	1	1	yes	maximal, characteristic
6	#6456⋊C₃	C₂	1	1	yes	maximal, characteristic
7	C₄⋊C₄×AGL(1,𝔽₇)	C₂	1	2	yes	maximal, complemented
8	#6453⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#6451⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₄⋊AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic
11	C₄×ΓL(1,𝔽₈)⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
12	#90381	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	2	yes	—
16	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	2	yes	—
17	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	1	yes	characteristic
18	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	2	yes	—
19	C₄⋊C₄×ΓL(1,𝔽₈)	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	2	yes	—
23	C₄²×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
24	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	2	yes	—
25	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	2	yes	—
26	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	2	yes	—
27	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic
28	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic
29	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	2	yes	—
30	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	1	yes	characteristic
31	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	1	yes	characteristic
32	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	—
33	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	—
34	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic
35	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic
36	C₇×#125	C₂×C₃	1	1	yes	characteristic, complemented
37	#6460	C₂×C₃	1	2	yes	complemented
38	#6459	C₂×C₃	1	2	yes	complemented
39	#6459	C₂×C₃	1	2	yes	complemented
40	#6456	C₂×C₃	1	1	yes	characteristic
41	#6456	C₂×C₃	1	1	yes	characteristic
42	D₇×C₄⋊C₄	C₂×C₃	1	2	yes	complemented
43	#6453	C₂×C₃	1	1	yes	characteristic, complemented
44	#6451	C₂×C₃	1	1	yes	characteristic, complemented
45	C₄×D₂₈	C₂×C₃	1	1	yes	characteristic
46	C₄×Dic₁₄	C₂×C₃	1	1	yes	characteristic, complemented
47	C₃×#554	—	7	1	no	maximal, Hall
48	C₂²×AGL(1,𝔽₇)	C₂³	1	2	yes	—
49	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
50	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
51	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
52	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
53	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
54	C₂×ΓL(1,𝔽₈)⋊C₄	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)	Q₈	1	4	yes	complemented
60	C₄×AGL(1,𝔽₇)	—	2	4	no	—
61	C₄×AGL(1,𝔽₇)	—	4	2	no	—
62	C₄×AGL(1,𝔽₇)	—	4	2	no	—
63	ΓL(1,𝔽₈)⋊Q₈	—	2	2	no	—
64	ΓL(1,𝔽₈)⋊Q₈	—	4	2	no	—
65	C₂×D₂₈	C₂²×C₃	1	1	yes	characteristic
66	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
67	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
68	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
69	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
70	C₂×Dic₁₄	C₂²×C₃	1	2	yes	—
71	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
72	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
73	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
74	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
75	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
76	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
77	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
78	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
79	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
80	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
81	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
82	Dic₇⋊C₄	C₂²×C₃	1	1	yes	characteristic
83	Dic₇⋊C₄	C₂²×C₃	1	1	yes	characteristic
84	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
85	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
86	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
87	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
88	C₃×C₄⋊Q₈	—	7	1	no	—
89	C₃×#125	—	7	1	no	—
90	C₃×#125	—	7	1	no	—
91	C₃×C₂²⋊Q₈	—	7	2	no	—
92	C₃×C₂²⋊Q₈	—	7	2	no	—
93	C₃×C₂²⋊Q₈	—	7	2	no	—
94	C₃×C₄×Q₈	—	7	1	no	—
95	C₃×C₄×D₄	—	7	1	no	—
96	C₃×C₄×D₄	—	7	1	no	—
97	C₃×C₄×D₄	—	7	1	no	—
98	C₂×C₃×C₄⋊C₄	—	7	2	no	—
99	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
100	C₂×AGL(1,𝔽₇)	—	2	2	no	—
101	C₂×AGL(1,𝔽₇)	—	2	2	no	—
102	C₂×AGL(1,𝔽₇)	C₂×Q₈	1	4	yes	—
103	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
104	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
105	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
106	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
107	C₄×ΓL(1,𝔽₈)	C₂×Q₈	1	2	yes	—
108	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
109	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
110	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
111	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
112	ΓL(1,𝔽₈)⋊C₄	#41	1	2	yes	—
113	#554	—	7	1	no	Sylow
114	C₂²×D₇	C₂³×C₃	1	2	yes	—
115	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
116	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
117	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
118	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
119	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
120	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
121	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
122	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
123	C₂×Dic₇	C₂³×C₃	1	2	yes	—
124	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
125	D₂₈	C₃×Q₈	1	4	yes	complemented
126	C₄×D₇	—	2	4	no	—
127	C₄×D₇	—	4	2	no	—
128	C₄×D₇	—	4	2	no	—
129	Dic₁₄	—	2	2	no	—
130	Dic₁₄	—	4	2	no	—
131	C₂×C₃×Q₈	—	7	1	no	—
132	C₂×C₃×Q₈	—	7	2	no	—
133	C₂×C₃×D₄	—	7	1	no	—
134	C₂²×C₃×C₄	—	7	2	no	—
135	C₂²×C₃×C₄	—	7	2	no	—
136	C₂²×C₃×C₄	—	7	2	no	—
137	C₃×C₄⋊C₄	—	7	1	no	—
138	C₃×C₄⋊C₄	—	7	2	no	—
139	C₃×C₄⋊C₄	—	7	2	no	—
140	C₃×C₄⋊C₄	—	7	1	no	—
141	C₃×C₄⋊C₄	—	7	1	no	—
142	C₃×C₄⋊C₄	—	7	1	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	1	no	—
147	C₃×C₄⋊C₄	—	7	2	no	—
148	C₃×C₂²⋊C₄	—	7	2	no	—
149	C₃×C₂²⋊C₄	—	7	2	no	—
150	C₃×C₂²⋊C₄	—	7	2	no	—
151	C₃×C₄²	—	7	1	no	—
152	C₃×C₄²	—	7	1	no	—
153	C₃×C₄²	—	7	1	no	—
154	C₂×ΓL(1,𝔽₈)	C₂²×Q₈	1	1	yes	characteristic
155	C₂×ΓL(1,𝔽₈)	C₂×#41	1	1	yes	characteristic
156	C₂×ΓL(1,𝔽₈)	H(5,𝔽₂)	1	1	yes	characteristic
157	AGL(1,𝔽₇)	—	2	4	no	—
158	C₄⋊Q₈	—	7	1	no	—
159	#125	—	7	1	no	—
160	#125	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
161	C₂²⋊Q₈	—	7	2	no	—
162	C₂²⋊Q₈	—	7	2	no	—
163	C₂²⋊Q₈	—	7	2	no	—
164	C₄×Q₈	—	7	1	no	—
165	C₄×D₄	—	7	1	no	—
166	C₄×D₄	—	7	1	no	—
167	C₄×D₄	—	7	1	no	—
168	C₂×C₄⋊C₄	—	7	2	no	—
169	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
170	C₂×D₇	—	2	2	no	—
171	C₂×D₇	—	2	2	no	—
172	C₂×D₇	C₂×C₃×Q₈	1	4	yes	—
173	C₄×C₇	—	2	1	no	—
174	C₄×C₇	—	2	2	no	—
175	C₄×C₇	—	2	2	no	—
176	C₄×C₇	—	2	1	no	—
177	C₄×C₇	C₂×C₃×Q₈	1	2	yes	—
178	Dic₇	—	2	1	no	—
179	Dic₇	—	2	1	no	—
180	Dic₇	—	2	2	no	—
181	Dic₇	—	2	1	no	—
182	Dic₇	C₃×#41	1	2	yes	—
183	C₂³×C₃	—	7	2	no	—
184	C₃×Q₈	—	14	2	no	—
185	C₃×Q₈	—	28	2	no	—
186	C₃×D₄	—	7	4	no	—
187	C₂×C₃×C₄	—	7	1	no	—
188	C₂×C₃×C₄	—	7	1	no	—
189	C₂×C₃×C₄	—	7	1	no	—
190	C₂×C₃×C₄	—	7	1	no	—
191	C₂×C₃×C₄	—	7	2	no	—
192	C₂×C₃×C₄	—	7	1	no	—
193	C₂×C₃×C₄	—	7	1	no	—
194	C₂×C₃×C₄	—	7	1	no	—
195	C₂×C₃×C₄	—	7	2	no	—
196	C₂×C₃×C₄	—	7	2	no	—
197	C₂×C₃×C₄	—	14	4	no	—
198	C₂×C₃×C₄	—	28	2	no	—
199	C₂×C₃×C₄	—	28	2	no	—
200	ΓL(1,𝔽₈)	#554	1	1	yes	characteristic, complemented, Hall
201	C₂×Q₈	—	7	1	no	—
202	C₂×Q₈	—	7	2	no	—
203	C₂×D₄	—	7	1	no	—
204	C₂²×C₄	—	7	2	no	—
205	C₂²×C₄	—	7	2	no	—
206	C₂²×C₄	—	7	2	no	—
207	C₄⋊C₄	—	7	1	no	—
208	C₄⋊C₄	—	7	2	no	—
209	C₄⋊C₄	—	7	1	no	—
210	C₄⋊C₄	—	7	2	no	—
211	C₄⋊C₄	—	7	2	no	—
212	C₄⋊C₄	—	7	1	no	—
213	C₄⋊C₄	—	7	1	no	—
214	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
215	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
216	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
217	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
218	C₂²⋊C₄	—	7	2	no	—
219	C₂²⋊C₄	—	7	2	no	—
220	C₂²⋊C₄	—	7	2	no	—
221	C₄²	—	7	1	no	—
222	C₄²	—	7	1	no	—
223	C₄²	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
224	C₂×C₇	C₂²×C₃×Q₈	1	1	yes	characteristic
225	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
226	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
227	D₇	—	2	4	no	—
228	C₂²×C₃	—	7	4	no	—
229	C₂²×C₃	—	7	1	no	—
230	C₂²×C₃	—	14	2	no	—
231	C₂²×C₃	—	14	2	no	—
232	C₃×C₄	—	7	2	no	—
233	C₃×C₄	—	7	2	no	—
234	C₃×C₄	—	14	2	no	—
235	C₃×C₄	—	14	1	no	—
236	C₃×C₄	—	14	2	no	—
237	C₃×C₄	—	14	1	no	—
238	C₃×C₄	—	14	1	no	—
239	C₃×C₄	—	14	2	no	—
240	C₃×C₄	—	14	1	no	—
241	C₃×C₄	—	14	1	no	—
242	C₂³	—	7	2	no	—
243	Q₈	—	14	2	no	—
244	Q₈	—	28	2	no	—
245	D₄	—	7	4	no	—
246	C₂×C₄	—	7	1	no	—
247	C₂×C₄	—	7	1	no	—
248	C₂×C₄	—	7	1	no	—
249	C₂×C₄	—	7	1	no	—
250	C₂×C₄	—	7	2	no	—
251	C₂×C₄	—	14	4	no	—
252	C₂×C₄	—	28	2	no	—
253	C₂×C₄	—	28	2	no	—
254	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
255	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
256	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
257	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
258	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
259	C₇	C₃×#554	1	1	yes	characteristic, complemented, Sylow
260	C₂×C₃	—	7	1	no	—
261	C₂×C₃	—	7	1	no	—
262	C₂×C₃	—	7	1	no	—
263	C₂×C₃	—	14	4	no	—
264	C₂²	—	7	4	no	—
265	C₂²	—	14	2	no	—
266	C₂²	—	14	2	no	—
267	C₂²	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	2	2	no	—
272	C₄	—	7	2	no	—
273	C₄	—	14	1	no	—
274	C₄	—	14	1	no	—
275	C₄	—	14	2	no	—
276	C₄	—	14	1	no	—
277	C₄	Q₈×AGL(1,𝔽₇)	1	2	yes	—
278	C₃	—	7	1	no	Sylow
279	C₂	—	14	4	no	—
280	C₂	C₂×Q₈×AGL(1,𝔽₇)	1	1	yes	characteristic
281	C₂	#41×AGL(1,𝔽₇)	1	1	yes	characteristic
282	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×#125	C₂×C₃
prime radical	C₂×C₄⋊AGL(1,𝔽₇)	C₂²
hypercenter	#125	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#554
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. #6460⋊C₃	C₂	semidirect
3. #6459⋊C₃	C₂	semidirect
4. #6459⋊C₃	C₂	semidirect
7. C₄⋊C₄×AGL(1,𝔽₇)	C₂	semidirect
8. #6453⋊C₃	C₂	semidirect
9. #6451⋊C₃	C₂	semidirect
11. C₄×ΓL(1,𝔽₈)⋊Q₈	C₂	semidirect
12. #90381	C₃	semidirect
36. C₇×#125	C₂×C₃	semidirect
37. #6460	C₂×C₃	semidirect
38. #6459	C₂×C₃	semidirect
39. #6459	C₂×C₃	semidirect
42. D₇×C₄⋊C₄	C₂×C₃	semidirect
43. #6453	C₂×C₃	semidirect
44. #6451	C₂×C₃	semidirect
46. C₄×Dic₁₄	C₂×C₃	semidirect
59. C₄⋊AGL(1,𝔽₇)	Q₈	semidirect
125. D₂₈	C₃×Q₈	semidirect
160. #125	AGL(1,𝔽₇)	semidirect
200. ΓL(1,𝔽₈)	#554	semidirect
259. C₇	C₃×#554	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
#6460⋊C₃	2
#6459⋊C₃	2
#6456⋊C₃	2
C₄⋊C₄×AGL(1,𝔽₇)	2
#6453⋊C₃	2
#6451⋊C₃	2
C₄×C₄⋊AGL(1,𝔽₇)	2
C₄×ΓL(1,𝔽₈)⋊Q₈	2
#90381	3
C₃×#554	7

Subgroups — #90381⋊C3 #13057425

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #90381⋊C₃ #13057425