Subgroups — #90224⋊C₃ #13057254

Order 1344.

424 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×#117	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×(C₂²⋊D₇)⋊C₃	C₂	1	1	yes	maximal, characteristic
3	#6487⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#6460⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#6455⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#6451⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#6450⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#6448⋊C₃	C₂	1	2	yes	maximal
9	#6442⋊C₃	C₂	1	2	yes	maximal, complemented
10	#10620712	C₂	1	2	yes	maximal, complemented
11	C₄×ΓL(1,𝔽₈)⋊Q₈	C₂	1	2	yes	maximal, complemented
12	#90224	C₃	1	1	yes	maximal, characteristic, complemented, Hall
13	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
14	C₂×(C₂²⋊D₇)⋊C₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
16	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	2	yes	—
17	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
20	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
21	C₄²×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
22	(C₂²⋊Dic₇)⋊C₃	C₂²	1	1	yes	characteristic
23	(C₂²⋊Dic₇)⋊C₃	C₂²	1	2	yes	—
24	(C₂×AGL(1,𝔽₇))⋊C₄	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	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	2	yes	—
28	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic
29	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	1	yes	characteristic
30	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	—
31	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	—
32	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	—
33	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	2	yes	—
34	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	2	yes	—
35	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic
36	C₇×#117	C₂×C₃	1	1	yes	characteristic, complemented
37	C₄×C₂²⋊D₇	C₂×C₃	1	1	yes	characteristic
38	#6487	C₂×C₃	1	1	yes	characteristic, complemented
39	#6460	C₂×C₃	1	1	yes	characteristic, complemented
40	#6455	C₂×C₃	1	1	yes	characteristic, complemented
41	#6451	C₂×C₃	1	1	yes	characteristic, complemented
42	#6450	C₂×C₃	1	1	yes	characteristic, complemented
43	#6448	C₂×C₃	1	2	yes	—
44	#6442	C₂×C₃	1	2	yes	complemented
45	#6439	C₂×C₃	1	2	yes	complemented
46	C₄×Dic₁₄	C₂×C₃	1	2	yes	complemented
47	C₃×#556	—	7	1	no	maximal, Hall
48	C₂³×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
49	C₂²×AGL(1,𝔽₇)	C₂³	1	1	yes	characteristic
50	C₂×C₄×ΓL(1,𝔽₈)	—	2	2	no	—
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	(C₂²⋊D₇)⋊C₃	—	2	2	no	—
56	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
57	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
58	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
59	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
60	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
61	C₄×AGL(1,𝔽₇)	—	2	2	no	—
62	ΓL(1,𝔽₈)⋊Q₈	—	2	2	no	—
63	ΓL(1,𝔽₈)⋊Q₈	—	2	4	no	—
64	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
65	C₂×C₂²⋊D₇	C₂²×C₃	1	1	yes	characteristic
66	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic
67	C₂×Dic₁₄	C₂²×C₃	1	2	yes	—
68	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
69	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
70	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
71	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
72	C₄²×C₇	C₂²×C₃	1	2	yes	—
73	C₂²⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
74	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
75	(C₂×D₇)⋊C₄	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	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
79	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
80	Dic₇⋊C₄	C₂²×C₃	1	1	yes	characteristic
81	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
82	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
83	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
84	C₄×Dic₇	C₂²×C₃	1	2	yes	—
85	C₄×Dic₇	C₂²×C₃	1	2	yes	—
86	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
87	C₃×C₄⋊Q₈	—	7	1	no	—
88	C₃×#126	—	7	2	no	—
89	C₃×#126	—	7	2	no	—
90	C₃×#124	—	7	2	no	—
91	C₃×C₂²⋊Q₈	—	7	1	no	—
92	C₃×C₂²⋊Q₈	—	7	1	no	—
93	C₃×C₄×Q₈	—	7	2	no	—
94	C₃×C₄×Q₈	—	7	1	no	—
95	C₃×C₄×D₄	—	7	1	no	—
96	C₃×#117	—	7	1	no	—
97	C₃×#117	—	7	1	no	—
98	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
99	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
100	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
101	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
102	C₂×AGL(1,𝔽₇)	—	2	1	no	—
103	C₂×AGL(1,𝔽₇)	—	2	1	no	—
104	C₂×AGL(1,𝔽₇)	—	2	1	no	—
105	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
106	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
107	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
108	C₄×ΓL(1,𝔽₈)	#41	1	2	yes	—
109	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
110	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
111	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
112	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
113	ΓL(1,𝔽₈)⋊C₄	#41	1	2	yes	—
114	#556	—	7	1	no	Sylow
115	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
116	C₂²×D₇	C₂³×C₃	1	1	yes	characteristic
117	C₂×C₄×C₇	—	2	2	no	—
118	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
119	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
120	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
121	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
122	C₂²⋊D₇	—	2	2	no	—
123	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
124	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
125	C₂×Dic₇	C₂³×C₃	1	2	yes	—
126	C₂×Dic₇	C₂³×C₃	1	2	yes	—
127	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
128	C₄×D₇	—	2	2	no	—
129	Dic₁₄	—	2	2	no	—
130	Dic₁₄	—	2	4	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	1	no	—
135	C₂²×C₃×C₄	—	7	1	no	—
136	C₃×C₄⋊C₄	—	7	1	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	2	no	—
141	C₃×C₄⋊C₄	—	7	2	no	—
142	C₃×C₄⋊C₄	—	7	1	no	—
143	C₃×C₄⋊C₄	—	7	1	no	—
144	C₃×C₂²⋊C₄	—	7	2	no	—
145	C₃×C₂²⋊C₄	—	7	1	no	—
146	C₃×C₂²⋊C₄	—	7	1	no	—
147	C₃×C₂²⋊C₄	—	7	2	no	—
148	C₃×C₂²⋊C₄	—	7	2	no	—
149	C₃×C₂²⋊C₄	—	7	2	no	—
150	C₃×C₄²	—	7	2	no	—
151	C₃×C₄²	—	7	1	no	—
152	C₃×C₄²	—	7	2	no	—
153	C₃×C₄²	—	7	2	no	—
154	C₂×ΓL(1,𝔽₈)	—	4	1	no	—
155	C₂×ΓL(1,𝔽₈)	C₂×#41	1	1	yes	characteristic
156	C₂×ΓL(1,𝔽₈)	C₂×#41	1	1	yes	characteristic
157	C₂×ΓL(1,𝔽₈)	2₋¹⁺⁴	1	1	yes	characteristic
158	AGL(1,𝔽₇)	—	4	1	no	—
159	C₄⋊Q₈	—	7	1	no	—
160	#126	—	7	2	no	—
161	#126	—	7	2	no	—
162	#124	—	7	2	no	—
163	C₂²⋊Q₈	—	7	1	no	—
164	C₂²⋊Q₈	—	7	1	no	—
165	C₄×Q₈	—	7	1	no	—
166	C₄×Q₈	—	7	2	no	—
167	C₄×D₄	—	7	1	no	—
168	#117	—	7	1	no	—
169	#117	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
170	C₂²×C₇	—	2	1	no	—
171	C₂²×C₇	—	2	1	no	—
172	C₂²×C₇	—	2	1	no	—
173	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
174	C₂×D₇	—	2	1	no	—
175	C₂×D₇	—	2	1	no	—
176	C₂×D₇	—	2	1	no	—
177	C₄×C₇	—	2	2	no	—
178	C₄×C₇	—	2	1	no	—
179	C₄×C₇	—	2	2	no	—
180	C₄×C₇	C₃×#41	1	2	yes	—
181	Dic₇	—	2	2	no	—
182	Dic₇	—	2	2	no	—
183	Dic₇	—	2	1	no	—
184	Dic₇	—	2	1	no	—
185	Dic₇	C₃×#41	1	2	yes	—
186	C₂³×C₃	—	7	1	no	—
187	C₂³×C₃	—	7	1	no	—
188	C₃×Q₈	—	14	4	no	—
189	C₃×Q₈	—	14	2	no	—
190	C₃×D₄	—	14	2	no	—
191	C₂×C₃×C₄	—	7	1	no	—
192	C₂×C₃×C₄	—	7	1	no	—
193	C₂×C₃×C₄	—	7	1	no	—
194	C₂×C₃×C₄	—	7	2	no	—
195	C₂×C₃×C₄	—	7	2	no	—
196	C₂×C₃×C₄	—	7	1	no	—
197	C₂×C₃×C₄	—	7	2	no	—
198	C₂×C₃×C₄	—	7	1	no	—
199	C₂×C₃×C₄	—	7	2	no	—
200	C₂×C₃×C₄	—	14	2	no	—
201	C₂×C₃×C₄	—	14	2	no	—
202	ΓL(1,𝔽₈)	#556	1	1	yes	characteristic, complemented, Hall
203	C₂×Q₈	—	7	1	no	—
204	C₂×Q₈	—	7	2	no	—
205	C₂×D₄	—	7	1	no	—
206	C₂²×C₄	—	7	1	no	—
207	C₂²×C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
208	C₄⋊C₄	—	7	1	no	—
209	C₄⋊C₄	—	7	2	no	—
210	C₄⋊C₄	—	7	2	no	—
211	C₄⋊C₄	—	7	2	no	—
212	C₄⋊C₄	—	7	1	no	—
213	C₄⋊C₄	—	7	2	no	—
214	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
215	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
216	C₂²⋊C₄	—	7	1	no	—
217	C₂²⋊C₄	—	7	2	no	—
218	C₂²⋊C₄	—	7	2	no	—
219	C₂²⋊C₄	—	7	1	no	—
220	C₂²⋊C₄	—	7	2	no	—
221	C₂²⋊C₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
222	C₄²	—	7	2	no	—
223	C₄²	—	7	1	no	—
224	C₄²	—	7	2	no	—
225	C₄²	C₂×AGL(1,𝔽₇)	1	2	yes	—
226	C₂×C₇	—	4	1	no	—
227	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
228	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
229	C₂×C₇	C₃×2₋¹⁺⁴	1	1	yes	characteristic
230	D₇	—	4	1	no	—
231	C₂²×C₃	—	7	1	no	—
232	C₂²×C₃	—	14	1	no	—
233	C₂²×C₃	—	14	1	no	—
234	C₂²×C₃	—	14	1	no	—
235	C₂²×C₃	—	14	1	no	—
236	C₂²×C₃	—	14	1	no	—
237	C₂²×C₃	—	14	1	no	—
238	C₃×C₄	—	7	2	no	—
239	C₃×C₄	—	7	2	no	—
240	C₃×C₄	—	14	1	no	—
241	C₃×C₄	—	14	2	no	—
242	C₃×C₄	—	14	2	no	—
243	C₃×C₄	—	14	1	no	—
244	C₃×C₄	—	14	2	no	—
245	C₃×C₄	—	14	2	no	—
246	C₃×C₄	—	14	1	no	—
247	C₂³	—	7	1	no	—
248	C₂³	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
249	Q₈	—	14	4	no	—
250	Q₈	—	14	2	no	—
251	D₄	—	14	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	7	1	no	—
254	C₂×C₄	—	7	1	no	—
255	C₂×C₄	—	7	1	no	—
256	C₂×C₄	—	7	2	no	—
257	C₂×C₄	—	7	2	no	—
258	C₂×C₄	—	14	2	no	—
259	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
260	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
261	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
262	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
263	C₇	C₃×#556	1	1	yes	characteristic, complemented, Sylow
264	C₂×C₃	—	7	1	no	—
265	C₂×C₃	—	7	1	no	—
266	C₂×C₃	—	7	1	no	—
267	C₂×C₃	—	28	1	no	—
268	C₂×C₃	—	28	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	14	1	no	—
273	C₂²	—	14	1	no	—
274	C₂²	—	14	1	no	—
275	C₂²	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	—	7	2	no	—
280	C₄	—	14	1	no	—
281	C₄	—	14	2	no	—
282	C₄	—	14	2	no	—
283	C₄	—	14	1	no	—
284	C₄	#1157⋊C₃	1	2	yes	—
285	C₃	—	7	1	no	Sylow
286	C₂	—	4	1	no	—
287	C₂	—	28	1	no	—
288	C₂	C₂×#1157⋊C₃	1	1	yes	characteristic
289	C₂	#41×AGL(1,𝔽₇)	1	1	yes	characteristic
290	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×#117	C₂×C₃
prime radical	C₂×(C₂²⋊D₇)⋊C₃	C₂²
hypercenter	#117	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#556
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
1. ΓL(1,𝔽₈)×#117	C₂	semidirect
3. #6487⋊C₃	C₂	semidirect
4. #6460⋊C₃	C₂	semidirect
5. #6455⋊C₃	C₂	semidirect
6. #6451⋊C₃	C₂	semidirect
7. #6450⋊C₃	C₂	semidirect
9. #6442⋊C₃	C₂	semidirect
10. #10620712	C₂	semidirect
11. C₄×ΓL(1,𝔽₈)⋊Q₈	C₂	semidirect
12. #90224	C₃	semidirect
36. C₇×#117	C₂×C₃	semidirect
38. #6487	C₂×C₃	semidirect
39. #6460	C₂×C₃	semidirect
40. #6455	C₂×C₃	semidirect
41. #6451	C₂×C₃	semidirect
42. #6450	C₂×C₃	semidirect
44. #6442	C₂×C₃	semidirect
45. #6439	C₂×C₃	semidirect
46. C₄×Dic₁₄	C₂×C₃	semidirect
169. #117	AGL(1,𝔽₇)	semidirect
202. ΓL(1,𝔽₈)	#556	semidirect
263. C₇	C₃×#556	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)×#117	2
C₄×(C₂²⋊D₇)⋊C₃	2
#6487⋊C₃	2
#6460⋊C₃	2
#6455⋊C₃	2
#6451⋊C₃	2
#6450⋊C₃	2
#6448⋊C₃	2
#6442⋊C₃	2
#10620712	2
C₄×ΓL(1,𝔽₈)⋊Q₈	2
#90224	3
C₃×#556	7

Subgroups — #90224⋊C3 #13057254

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #90224⋊C₃ #13057254