Subgroups — #90325⋊C₃ #13057370

Order 1344.

484 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)×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#6509⋊C₃	C₂	1	2	yes	maximal, complemented
3	C₂×(C₂×AGL(1,𝔽₇))⋊C₄	C₂	1	1	yes	maximal, characteristic
4	#6487⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#6460⋊C₃	C₂	1	2	yes	maximal, complemented
6	#6459⋊C₃	C₂	1	2	yes	maximal, complemented
7	#6459⋊C₃	C₂	1	2	yes	maximal, complemented
8	C₂²⋊C₄×AGL(1,𝔽₇)	C₂	1	2	yes	maximal
9	#6441⋊C₃	C₂	1	2	yes	maximal, complemented
10	#90325	C₃	1	1	yes	maximal, characteristic, complemented, Hall
11	C₂³×AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
12	C₂×Q₈×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented
13	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
14	C₂²×ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic
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	2	yes	complemented
18	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	1	yes	characteristic, complemented
19	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented
20	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	complemented
21	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
22	(C₂²⋊Dic₇)⋊C₃	C₂²	1	2	yes	—
23	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	2	yes	—
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	2	yes	complemented
28	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic, complemented
29	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	complemented
30	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	complemented
31	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	complemented
32	C₇×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
33	#6509	C₂×C₃	1	2	yes	complemented
34	C₂×(C₂×D₇)⋊C₄	C₂×C₃	1	1	yes	characteristic
35	#6487	C₂×C₃	1	1	yes	characteristic, complemented
36	#6460	C₂×C₃	1	2	yes	complemented
37	#6459	C₂×C₃	1	2	yes	complemented
38	#6459	C₂×C₃	1	2	yes	complemented
39	D₇×C₂²⋊C₄	C₂×C₃	1	2	yes	—
40	#6441	C₂×C₃	1	2	yes	complemented
41	C₃×#543	—	7	1	no	maximal, Hall
42	C₂³×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
43	C₂²×AGL(1,𝔽₇)	—	4	1	no	—
44	C₂²×AGL(1,𝔽₇)	—	4	1	no	—
45	C₂²×AGL(1,𝔽₇)	Q₈	1	4	yes	complemented
46	C₂²×AGL(1,𝔽₇)	C₂³	1	2	yes	—
47	Q₈×ΓL(1,𝔽₈)	—	4	1	no	—
48	C₂×C₄×ΓL(1,𝔽₈)	—	4	1	no	—
49	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
50	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
51	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
52	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
53	C₂×ΓL(1,𝔽₈)⋊C₄	—	4	1	no	—
54	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
55	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
56	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
57	C₄×AGL(1,𝔽₇)	—	4	2	no	—
58	C₄×AGL(1,𝔽₇)	—	4	2	no	—
59	ΓL(1,𝔽₈)⋊Q₈	—	4	1	no	—
60	ΓL(1,𝔽₈)⋊Q₈	—	4	2	no	—
61	C₂³×D₇	C₂²×C₃	1	1	yes	characteristic
62	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
63	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
64	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
65	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
66	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
67	C₂×Dic₁₄	C₂²×C₃	1	2	yes	complemented
68	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic, complemented
69	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
70	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
71	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
72	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
73	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
74	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
75	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
76	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
77	C₄⋊Dic₇	C₂²×C₃	1	2	yes	complemented
78	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic, complemented
79	Dic₇⋊C₄	C₂²×C₃	1	2	yes	complemented
80	Dic₇⋊C₄	C₂²×C₃	1	2	yes	complemented
81	Dic₇⋊C₄	C₂²×C₃	1	2	yes	complemented
82	C₃×C₂²⋊Q₈	—	7	2	no	—
83	C₃×C₂²⋊Q₈	—	7	2	no	—
84	C₃×C₂²⋊Q₈	—	7	2	no	—
85	C₃×C₂²⋊Q₈	—	7	2	no	—
86	C₃×C₂²⋊Q₈	—	7	1	no	—
87	C₃×C₂²⋊Q₈	—	7	2	no	—
88	C₃×C₂²⋊Q₈	—	7	1	no	—
89	C₂×C₃×C₂²⋊C₄	—	7	2	no	—
90	C₂×C₃×C₂²⋊C₄	—	7	1	no	—
91	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
92	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
93	C₂²×ΓL(1,𝔽₈)	C₂×Q₈	1	2	yes	—
94	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
95	C₂×AGL(1,𝔽₇)	—	2	2	no	—
96	C₂×AGL(1,𝔽₇)	—	2	2	no	—
97	C₂×AGL(1,𝔽₇)	—	4	4	no	—
98	C₂×AGL(1,𝔽₇)	C₂×Q₈	1	4	yes	—
99	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
100	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
101	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
102	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
103	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
104	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
105	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
106	#543	—	7	1	no	Sylow
107	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
108	C₂²×D₇	—	4	1	no	—
109	C₂²×D₇	—	4	1	no	—
110	C₂²×D₇	C₃×Q₈	1	4	yes	complemented
111	C₂²×D₇	C₂³×C₃	1	2	yes	—
112	C₇×Q₈	—	4	1	no	—
113	C₂×C₄×C₇	—	4	1	no	—
114	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
115	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
116	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
117	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
118	C₂×Dic₇	—	4	1	no	—
119	C₂×Dic₇	C₂³×C₃	1	2	yes	—
120	C₂×Dic₇	C₂³×C₃	1	2	yes	—
121	C₂×Dic₇	C₂³×C₃	1	2	yes	—
122	C₄×D₇	—	4	2	no	—
123	C₄×D₇	—	4	2	no	—
124	Dic₁₄	—	4	1	no	—
125	Dic₁₄	—	4	2	no	—
126	C₂⁴×C₃	—	7	1	no	—
127	C₂×C₃×Q₈	—	7	2	no	—
128	C₂×C₃×Q₈	—	7	1	no	—
129	C₂×C₃×Q₈	—	7	1	no	—
130	C₂²×C₃×C₄	—	7	2	no	—
131	C₂²×C₃×C₄	—	7	2	no	—
132	C₂²×C₃×C₄	—	7	1	no	—
133	C₂²×C₃×C₄	—	7	1	no	—
134	C₃×C₄⋊C₄	—	7	2	no	—
135	C₃×C₄⋊C₄	—	7	2	no	—
136	C₃×C₄⋊C₄	—	7	2	no	—
137	C₃×C₄⋊C₄	—	7	2	no	—
138	C₃×C₄⋊C₄	—	7	1	no	—
139	C₃×C₄⋊C₄	—	7	1	no	—
140	C₃×C₄⋊C₄	—	7	2	no	—
141	C₃×C₂²⋊C₄	—	7	2	no	—
142	C₃×C₂²⋊C₄	—	7	2	no	—
143	C₃×C₂²⋊C₄	—	7	2	no	—
144	C₃×C₂²⋊C₄	—	7	2	no	—
145	C₃×C₂²⋊C₄	—	7	2	no	—
146	C₃×C₂²⋊C₄	—	7	2	no	—
147	C₂×ΓL(1,𝔽₈)	—	2	2	no	—
148	C₂×ΓL(1,𝔽₈)	C₂²×Q₈	1	1	yes	characteristic
149	C₂×ΓL(1,𝔽₈)	H(5,𝔽₂)	1	1	yes	characteristic
150	C₂×ΓL(1,𝔽₈)	H(5,𝔽₂)	1	1	yes	characteristic
151	AGL(1,𝔽₇)	—	2	4	no	—
152	C₂²⋊Q₈	—	7	2	no	—
153	C₂²⋊Q₈	—	7	1	no	—
154	C₂²⋊Q₈	—	7	2	no	—
155	C₂²⋊Q₈	—	7	2	no	—
156	C₂²⋊Q₈	—	7	2	no	—
157	C₂²⋊Q₈	—	7	2	no	—
158	C₂²⋊Q₈	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
159	C₂×C₂²⋊C₄	—	7	1	no	—
160	C₂×C₂²⋊C₄	—	7	2	no	—
161	C₂²×C₇	—	2	1	no	—
162	C₂²×C₇	—	2	1	no	—
163	C₂²×C₇	C₂×C₃×Q₈	1	2	yes	—
164	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
165	C₂×D₇	—	2	2	no	—
166	C₂×D₇	—	2	2	no	—
167	C₂×D₇	—	4	4	no	—
168	C₂×D₇	C₂×C₃×Q₈	1	4	yes	—
169	C₄×C₇	—	2	2	no	—
170	C₄×C₇	—	2	1	no	—
171	C₄×C₇	—	2	2	no	—
172	C₄×C₇	—	2	1	no	—
173	Dic₇	—	2	2	no	—
174	Dic₇	—	2	2	no	—
175	Dic₇	—	2	2	no	—
176	C₂³×C₃	—	7	2	no	—
177	C₂³×C₃	—	7	4	no	—
178	C₂³×C₃	—	7	1	no	—
179	C₂³×C₃	—	28	1	no	—
180	C₂³×C₃	—	28	1	no	—
181	C₃×Q₈	—	28	1	no	—
182	C₃×Q₈	—	28	1	no	—
183	C₃×Q₈	—	28	2	no	—
184	C₂×C₃×C₄	—	7	2	no	—
185	C₂×C₃×C₄	—	7	2	no	—
186	C₂×C₃×C₄	—	7	2	no	—
187	C₂×C₃×C₄	—	7	1	no	—
188	C₂×C₃×C₄	—	7	1	no	—
189	C₂×C₃×C₄	—	7	2	no	—
190	C₂×C₃×C₄	—	7	2	no	—
191	C₂×C₃×C₄	—	28	1	no	—
192	C₂×C₃×C₄	—	28	2	no	—
193	C₂×C₃×C₄	—	28	2	no	—
194	C₂×C₃×C₄	—	28	1	no	—
195	ΓL(1,𝔽₈)	#543	1	1	yes	characteristic, complemented, Hall
196	C₂⁴	—	7	1	no	—
197	C₂×Q₈	—	7	1	no	—
198	C₂×Q₈	—	7	2	no	—
199	C₂×Q₈	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
200	C₂²×C₄	—	7	1	no	—
201	C₂²×C₄	—	7	2	no	—
202	C₂²×C₄	—	7	2	no	—
203	C₂²×C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
204	C₄⋊C₄	—	7	2	no	—
205	C₄⋊C₄	—	7	2	no	—
206	C₄⋊C₄	—	7	2	no	—
207	C₄⋊C₄	—	7	1	no	—
208	C₄⋊C₄	—	7	2	no	—
209	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
210	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	2	yes	complemented
211	C₂²⋊C₄	—	7	2	no	—
212	C₂²⋊C₄	—	7	2	no	—
213	C₂²⋊C₄	—	7	2	no	—
214	C₂²⋊C₄	—	7	2	no	—
215	C₂²⋊C₄	—	7	2	no	—
216	C₂²⋊C₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
217	C₂×C₇	—	2	2	no	—
218	C₂×C₇	C₂²×C₃×Q₈	1	1	yes	characteristic
219	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
220	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
221	D₇	—	2	4	no	—
222	C₂²×C₃	—	7	4	no	—
223	C₂²×C₃	—	7	2	no	—
224	C₂²×C₃	—	7	1	no	—
225	C₂²×C₃	—	14	1	no	—
226	C₂²×C₃	—	14	1	no	—
227	C₂²×C₃	—	14	2	no	—
228	C₂²×C₃	—	14	2	no	—
229	C₂²×C₃	—	28	4	no	—
230	C₃×C₄	—	14	1	no	—
231	C₃×C₄	—	14	1	no	—
232	C₃×C₄	—	14	2	no	—
233	C₃×C₄	—	14	2	no	—
234	C₃×C₄	—	14	2	no	—
235	C₃×C₄	—	14	2	no	—
236	C₃×C₄	—	14	2	no	—
237	C₂³	—	7	2	no	—
238	C₂³	—	7	4	no	—
239	C₂³	—	28	1	no	—
240	C₂³	—	28	1	no	—
241	C₂³	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
242	Q₈	—	4	1	no	—
243	Q₈	—	28	1	no	—
244	Q₈	—	28	2	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	7	2	no	—
247	C₂×C₄	—	7	2	no	—
248	C₂×C₄	—	7	2	no	—
249	C₂×C₄	—	28	2	no	—
250	C₂×C₄	—	28	2	no	—
251	C₂×C₄	—	28	1	no	—
252	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
253	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
254	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
255	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
256	C₇	C₃×#543	1	1	yes	characteristic, complemented, Sylow
257	C₂×C₃	—	7	1	no	—
258	C₂×C₃	—	7	1	no	—
259	C₂×C₃	—	7	1	no	—
260	C₂×C₃	—	14	2	no	—
261	C₂×C₃	—	14	4	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	7	4	no	—
265	C₂²	—	14	2	no	—
266	C₂²	—	14	2	no	—
267	C₂²	—	28	4	no	—
268	C₂²	Q₈×AGL(1,𝔽₇)	1	2	yes	—
269	C₂²	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	14	2	no	—
275	C₄	—	14	2	no	—
276	C₄	—	14	2	no	—
277	C₃	—	7	1	no	Sylow
278	C₂	—	2	2	no	—
279	C₂	—	14	4	no	—
280	C₂	#6548⋊C₃	1	1	yes	characteristic
281	C₂	C₂×Q₈×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×C₂²⋊Q₈	C₂×C₃
prime radical	C₂³×AGL(1,𝔽₇)	C₂²
hypercenter	C₂²⋊Q₈	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#543
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. ΓL(1,𝔽₈)×C₂²⋊Q₈	C₂	semidirect
2. #6509⋊C₃	C₂	semidirect
4. #6487⋊C₃	C₂	semidirect
5. #6460⋊C₃	C₂	semidirect
6. #6459⋊C₃	C₂	semidirect
7. #6459⋊C₃	C₂	semidirect
9. #6441⋊C₃	C₂	semidirect
10. #90325	C₃	semidirect
12. C₂×Q₈×ΓL(1,𝔽₈)	C₂²	semidirect
17. C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	semidirect
18. C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	semidirect
19. C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	semidirect
20. C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	semidirect
27. C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	semidirect
28. C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	semidirect
29. (ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	semidirect
30. (ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	semidirect
31. (ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	semidirect
32. C₇×C₂²⋊Q₈	C₂×C₃	semidirect
33. #6509	C₂×C₃	semidirect
35. #6487	C₂×C₃	semidirect
36. #6460	C₂×C₃	semidirect
37. #6459	C₂×C₃	semidirect
38. #6459	C₂×C₃	semidirect
40. #6441	C₂×C₃	semidirect
45. C₂²×AGL(1,𝔽₇)	Q₈	semidirect
62. C₂×C₇×Q₈	C₂²×C₃	semidirect
67. C₂×Dic₁₄	C₂²×C₃	semidirect
68. C₂×Dic₁₄	C₂²×C₃	semidirect
69. C₇×C₄⋊C₄	C₂²×C₃	semidirect
70. C₇×C₄⋊C₄	C₂²×C₃	semidirect
77. C₄⋊Dic₇	C₂²×C₃	semidirect
78. C₄⋊Dic₇	C₂²×C₃	semidirect
79. Dic₇⋊C₄	C₂²×C₃	semidirect
80. Dic₇⋊C₄	C₂²×C₃	semidirect
81. Dic₇⋊C₄	C₂²×C₃	semidirect
110. C₂²×D₇	C₃×Q₈	semidirect
158. C₂²⋊Q₈	AGL(1,𝔽₇)	semidirect
195. ΓL(1,𝔽₈)	#543	semidirect
199. C₂×Q₈	C₂×AGL(1,𝔽₇)	semidirect
209. C₄⋊C₄	C₂×AGL(1,𝔽₇)	semidirect
210. C₄⋊C₄	C₂×AGL(1,𝔽₇)	semidirect
256. C₇	C₃×#543	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)×C₂²⋊Q₈	2
#6509⋊C₃	2
C₂×(C₂×AGL(1,𝔽₇))⋊C₄	2
#6487⋊C₃	2
#6460⋊C₃	2
#6459⋊C₃	2
C₂²⋊C₄×AGL(1,𝔽₇)	2
#6441⋊C₃	2
#90325	3
C₃×#543	7

Subgroups — #90325⋊C3 #13057370

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #90325⋊C₃ #13057370