Subgroups — #90313⋊C₃ #13057358

Order 1344.

444 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
2	Q₈×ΓL(1,𝔽₈)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#6507⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#6487⋊C₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×ΓL(1,𝔽₈)⋊C₄	C₂	1	1	yes	maximal, characteristic
6	#6452⋊C₃	C₂	1	2	yes	maximal, complemented
7	#6451⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#6450⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#6450⋊C₃	C₂	1	2	yes	maximal, complemented
10	#6441⋊C₃	C₂	1	2	yes	maximal
11	#6440⋊C₃	C₂	1	2	yes	maximal
12	#90313	C₃	1	1	yes	maximal, characteristic, complemented, Hall
13	C₂×Q₈×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
14	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
15	C₂²×ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic
16	C₂²×ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	2	yes	—
18	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
21	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
22	(C₂²⋊Dic₇)⋊C₃	C₂²	1	2	yes	—
23	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	2	yes	—
24	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic
25	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	—
26	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	—
27	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	—
28	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	—
29	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	—
30	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	2	yes	—
31	C₄×ΓL(1,𝔽₈)⋊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	1	yes	characteristic
36	C₇×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic
37	Q₈×Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
38	#6507	C₂×C₃	1	1	yes	characteristic, complemented
39	#6487	C₂×C₃	1	1	yes	characteristic
40	C₂×C₄×Dic₇	C₂×C₃	1	1	yes	characteristic
41	#6452	C₂×C₃	1	2	yes	complemented
42	#6451	C₂×C₃	1	1	yes	characteristic, complemented
43	#6450	C₂×C₃	1	1	yes	characteristic, complemented
44	#6450	C₂×C₃	1	2	yes	complemented
45	#6441	C₂×C₃	1	2	yes	—
46	#6440	C₂×C₃	1	2	yes	—
47	C₃×#533	—	7	1	no	maximal, Hall
48	C₂³×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
49	Q₈×ΓL(1,𝔽₈)	—	2	2	no	—
50	C₂×C₄×ΓL(1,𝔽₈)	—	2	2	no	—
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×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
55	C₂×ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
56	C₂×ΓL(1,𝔽₈)⋊C₄	Q₈	1	4	yes	complemented
57	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
58	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
59	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
60	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
61	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
62	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
63	ΓL(1,𝔽₈)⋊Q₈	—	2	2	no	—
64	ΓL(1,𝔽₈)⋊Q₈	—	2	4	no	—
65	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
66	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
67	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
68	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
69	C₂×Dic₁₄	C₂²×C₃	1	2	yes	—
70	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
71	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
72	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
73	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
74	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
75	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
76	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
77	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
78	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
79	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
80	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
81	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
82	C₄×Dic₇	C₂²×C₃	1	2	yes	—
83	C₄×Dic₇	C₂²×C₃	1	2	yes	—
84	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
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	C₃×C₄⋊Q₈	—	7	1	no	—
89	C₃×C₄⋊Q₈	—	7	1	no	—
90	C₃×#125	—	7	2	no	—
91	C₃×C₂²⋊Q₈	—	7	2	no	—
92	C₃×C₂²⋊Q₈	—	7	1	no	—
93	C₃×C₂²⋊Q₈	—	7	1	no	—
94	C₃×C₄×Q₈	—	7	1	no	—
95	C₃×C₄×Q₈	—	7	2	no	—
96	C₃×C₄×Q₈	—	7	1	no	—
97	C₃×#117	—	7	2	no	—
98	C₂×C₃×C₄²	—	7	1	no	—
99	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
100	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
101	C₂²×ΓL(1,𝔽₈)	C₂×Q₈	1	2	yes	—
102	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
103	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
104	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
105	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
106	C₄×ΓL(1,𝔽₈)	#41	1	2	yes	—
107	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
108	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
109	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
110	ΓL(1,𝔽₈)⋊C₄	C₂×Q₈	1	2	yes	—
111	ΓL(1,𝔽₈)⋊C₄	C₂×Q₈	1	2	yes	—
112	ΓL(1,𝔽₈)⋊C₄	#41	1	2	yes	—
113	#533	—	7	1	no	Sylow
114	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
115	C₇×Q₈	—	2	2	no	—
116	C₂×C₄×C₇	—	2	2	no	—
117	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
118	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
119	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
120	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
121	C₂×Dic₇	—	2	2	no	—
122	C₂×Dic₇	C₃×Q₈	1	4	yes	complemented
123	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
124	C₂×Dic₇	C₂³×C₃	1	2	yes	—
125	C₂×Dic₇	C₂³×C₃	1	2	yes	—
126	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
127	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
128	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
129	Dic₁₄	—	2	2	no	—
130	Dic₁₄	—	2	4	no	—
131	C₂×C₃×Q₈	—	7	2	no	—
132	C₂×C₃×Q₈	—	7	1	no	—
133	C₂×C₃×Q₈	—	7	1	no	—
134	C₂²×C₃×C₄	—	7	1	no	—
135	C₂²×C₃×C₄	—	7	1	no	—
136	C₂²×C₃×C₄	—	7	1	no	—
137	C₃×C₄⋊C₄	—	7	2	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	2	no	—
143	C₃×C₄⋊C₄	—	7	1	no	—
144	C₃×C₄⋊C₄	—	7	2	no	—
145	C₃×C₄⋊C₄	—	7	1	no	—
146	C₃×C₂²⋊C₄	—	7	2	no	—
147	C₃×C₂²⋊C₄	—	7	2	no	—
148	C₃×C₄²	—	7	1	no	—
149	C₃×C₄²	—	7	1	no	—
150	C₃×C₄²	—	7	1	no	—
151	C₃×C₄²	—	7	1	no	—
152	C₃×C₄²	—	7	2	no	—
153	C₃×C₄²	—	7	2	no	—
154	C₂×ΓL(1,𝔽₈)	—	2	2	no	—
155	C₂×ΓL(1,𝔽₈)	C₂²×Q₈	1	1	yes	characteristic
156	C₂×ΓL(1,𝔽₈)	C₂×#41	1	1	yes	characteristic
157	C₂×ΓL(1,𝔽₈)	C₂×#41	1	1	yes	characteristic
158	C₄⋊Q₈	—	7	1	no	—
159	C₄⋊Q₈	—	7	1	no	—
160	#125	—	7	2	no	—
161	C₂²⋊Q₈	—	7	1	no	—
162	C₂²⋊Q₈	—	7	2	no	—
163	C₂²⋊Q₈	AGL(1,𝔽₇)	1	1	yes	characteristic
164	C₄×Q₈	—	7	1	no	—
165	C₄×Q₈	—	7	1	no	—
166	C₄×Q₈	—	7	2	no	—
167	#117	—	7	2	no	—
168	C₂×C₄²	—	7	1	no	—
169	C₂²×C₇	—	2	1	no	—
170	C₂²×C₇	—	2	1	no	—
171	C₂²×C₇	C₂×C₃×Q₈	1	2	yes	—
172	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
173	C₄×C₇	—	2	2	no	—
174	C₄×C₇	—	2	1	no	—
175	C₄×C₇	—	2	2	no	—
176	C₄×C₇	C₃×#41	1	2	yes	—
177	Dic₇	—	2	2	no	—
178	Dic₇	—	2	2	no	—
179	Dic₇	—	2	1	no	—
180	Dic₇	C₂×C₃×Q₈	1	2	yes	—
181	Dic₇	C₂×C₃×Q₈	1	2	yes	—
182	Dic₇	C₃×#41	1	2	yes	—
183	C₂³×C₃	—	7	1	no	—
184	C₃×Q₈	—	14	2	no	—
185	C₃×Q₈	—	14	4	no	—
186	C₃×Q₈	—	14	2	no	—
187	C₂×C₃×C₄	—	7	4	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	1	no	—
192	C₂×C₃×C₄	—	7	2	no	—
193	C₂×C₃×C₄	—	7	2	no	—
194	C₂×C₃×C₄	—	7	1	no	—
195	C₂×C₃×C₄	—	7	1	no	—
196	C₂×C₃×C₄	—	7	2	no	—
197	C₂×C₃×C₄	—	7	2	no	—
198	C₂×C₃×C₄	—	14	2	no	—
199	C₂×C₃×C₄	—	14	2	no	—
200	ΓL(1,𝔽₈)	#533	1	1	yes	characteristic, complemented, Hall
201	C₂×Q₈	—	7	2	no	—
202	C₂×Q₈	—	7	1	no	—
203	C₂×Q₈	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
204	C₂²×C₄	—	7	1	no	—
205	C₂²×C₄	—	7	1	no	—
206	C₂²×C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
207	C₄⋊C₄	—	7	2	no	—
208	C₄⋊C₄	—	7	2	no	—
209	C₄⋊C₄	—	7	2	no	—
210	C₄⋊C₄	—	7	2	no	—
211	C₄⋊C₄	—	7	2	no	—
212	C₄⋊C₄	—	7	2	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	C₂²⋊C₄	—	7	2	no	—
217	C₂²⋊C₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
218	C₄²	—	7	1	no	—
219	C₄²	—	7	1	no	—
220	C₄²	—	7	1	no	—
221	C₄²	—	7	1	no	—
222	C₄²	—	7	2	no	—
223	C₄²	—	7	2	no	—
224	C₂×C₇	—	2	2	no	—
225	C₂×C₇	C₂²×C₃×Q₈	1	1	yes	characteristic
226	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
227	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
228	C₂²×C₃	—	7	2	no	—
229	C₂²×C₃	—	7	1	no	—
230	C₂²×C₃	—	14	1	no	—
231	C₂²×C₃	—	14	1	no	—
232	C₃×C₄	—	7	2	no	—
233	C₃×C₄	—	7	2	no	—
234	C₃×C₄	—	7	2	no	—
235	C₃×C₄	—	7	2	no	—
236	C₃×C₄	—	14	1	no	—
237	C₃×C₄	—	14	2	no	—
238	C₃×C₄	—	14	2	no	—
239	C₃×C₄	—	14	1	no	—
240	C₃×C₄	—	14	2	no	—
241	C₃×C₄	—	14	2	no	—
242	C₂³	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
243	Q₈	—	2	2	no	—
244	Q₈	—	14	4	no	—
245	Q₈	—	14	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	7	4	no	—
248	C₂×C₄	—	7	1	no	—
249	C₂×C₄	—	7	1	no	—
250	C₂×C₄	—	7	1	no	—
251	C₂×C₄	—	7	1	no	—
252	C₂×C₄	—	7	2	no	—
253	C₂×C₄	—	7	2	no	—
254	C₂×C₄	—	14	2	no	—
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×#533	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	2	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	Q₈×AGL(1,𝔽₇)	1	2	yes	—
267	C₂²	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	7	2	no	—
272	C₄	—	7	2	no	—
273	C₄	—	7	2	no	—
274	C₄	—	14	2	no	—
275	C₄	—	14	1	no	—
276	C₄	—	14	2	no	—
277	C₄	#1159⋊C₃	1	2	yes	—
278	C₃	—	7	1	no	Sylow
279	C₂	—	2	2	no	—
280	C₂	C₂×#1159⋊C₃	1	1	yes	characteristic
281	C₂	C₂×Q₈×AGL(1,𝔽₇)	1	1	yes	characteristic
282	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×C₂²⋊Q₈	C₂×C₃
prime radical	C₂³×ΓL(1,𝔽₈)	C₂³
hypercenter	C₂²⋊Q₈	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#533
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
2. Q₈×ΓL(1,𝔽₈)⋊C₄	C₂	semidirect
3. #6507⋊C₃	C₂	semidirect
6. #6452⋊C₃	C₂	semidirect
7. #6451⋊C₃	C₂	semidirect
8. #6450⋊C₃	C₂	semidirect
9. #6450⋊C₃	C₂	semidirect
12. #90313	C₃	semidirect
37. Q₈×Dic₇	C₂×C₃	semidirect
38. #6507	C₂×C₃	semidirect
41. #6452	C₂×C₃	semidirect
42. #6451	C₂×C₃	semidirect
43. #6450	C₂×C₃	semidirect
44. #6450	C₂×C₃	semidirect
56. C₂×ΓL(1,𝔽₈)⋊C₄	Q₈	semidirect
122. C₂×Dic₇	C₃×Q₈	semidirect
200. ΓL(1,𝔽₈)	#533	semidirect
259. C₇	C₃×#533	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
Q₈×ΓL(1,𝔽₈)⋊C₄	2
#6507⋊C₃	2
#6487⋊C₃	2
C₂×C₄×ΓL(1,𝔽₈)⋊C₄	2
#6452⋊C₃	2
#6451⋊C₃	2
#6450⋊C₃	2
#6441⋊C₃	2
#6440⋊C₃	2
#90313	3
C₃×#533	7

Subgroups — #90313⋊C3 #13057358

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #90313⋊C₃ #13057358