Subgroups — #90362⋊C₃ #13057405

Order 1344.

424 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×#124	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	#6499⋊C₃	C₂	1	1	yes	maximal, characteristic
5	D₄×ΓL(1,𝔽₈)⋊C₄	C₂	1	1	yes	maximal, characteristic
6	#6442⋊C₃	C₂	1	2	yes	maximal, complemented
7	#6442⋊C₃	C₂	1	2	yes	maximal, complemented
8	#6441⋊C₃	C₂	1	2	yes	maximal, complemented
9	#6440⋊C₃	C₂	1	2	yes	maximal, complemented
10	C₄×ΓL(1,𝔽₈)⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×ΓL(1,𝔽₈)⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
12	#90362	C₃	1	1	yes	maximal, characteristic, complemented, Hall
13	C₂×Q₈×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
14	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
15	C₂²×ΓL(1,𝔽₈)⋊C₄	C₂²	1	2	yes	—
16	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	1	yes	characteristic
18	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
19	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
20	C₄²×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
21	(C₂²⋊Dic₇)⋊C₃	C₂²	1	2	yes	—
22	(C₂²⋊Dic₇)⋊C₃	C₂²	1	2	yes	—
23	(C₂²⋊Dic₇)⋊C₃	C₂²	1	2	yes	—
24	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic
25	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic
26	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	2	yes	—
27	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	1	yes	characteristic
28	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	—
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	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	2	yes	—
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×#124	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	#6499	C₂×C₃	1	1	yes	characteristic
40	D₄×Dic₇	C₂×C₃	1	1	yes	characteristic
41	#6442	C₂×C₃	1	2	yes	complemented
42	#6442	C₂×C₃	1	2	yes	complemented
43	#6441	C₂×C₃	1	2	yes	complemented
44	#6440	C₂×C₃	1	2	yes	complemented
45	C₄×Dic₁₄	C₂×C₃	1	1	yes	characteristic, complemented
46	C₄×Dic₁₄	C₂×C₃	1	1	yes	characteristic, complemented
47	C₃×#556	—	7	1	no	maximal, Hall
48	C₂³×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
49	Q₈×ΓL(1,𝔽₈)	—	2	2	no	—
50	D₄×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	1	yes	characteristic
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	—	2	4	no	—
56	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
57	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
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	ΓL(1,𝔽₈)⋊Q₈	—	2	2	no	—
63	ΓL(1,𝔽₈)⋊Q₈	—	2	2	no	—
64	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
65	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
66	C₂²×Dic₇	C₂²×C₃	1	2	yes	—
67	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
68	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
69	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
70	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
71	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
72	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
73	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
74	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
75	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
76	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
77	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
78	Dic₇⋊C₄	C₂²×C₃	1	1	yes	characteristic
79	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
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	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	2	yes	—
87	C₃×C₄⋊Q₈	—	7	1	no	—
88	C₃×#126	—	7	2	no	—
89	C₃×#126	—	7	2	no	—
90	C₃×#124	—	7	1	no	—
91	C₃×#124	—	7	1	no	—
92	C₃×C₂²⋊Q₈	—	7	2	no	—
93	C₃×C₄×Q₈	—	7	1	no	—
94	C₃×C₄×Q₈	—	7	1	no	—
95	C₃×C₄×Q₈	—	7	1	no	—
96	C₃×C₄×D₄	—	7	1	no	—
97	C₃×#117	—	7	2	no	—
98	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
99	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
100	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
101	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
102	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
103	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
104	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
105	C₄×ΓL(1,𝔽₈)	#41	1	2	yes	—
106	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
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	—	2	1	no	—
111	ΓL(1,𝔽₈)⋊C₄	#41	1	2	yes	—
112	#556	—	7	1	no	Sylow
113	C₂³×C₇	C₂³×C₃	1	2	yes	—
114	C₇×Q₈	—	2	2	no	—
115	C₇×D₄	—	2	2	no	—
116	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
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	—	2	4	no	—
121	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
122	C₂×Dic₇	C₂³×C₃	1	2	yes	—
123	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
124	C₂×Dic₇	C₂³×C₃	1	2	yes	—
125	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
126	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
127	Dic₁₄	—	2	2	no	—
128	Dic₁₄	—	2	2	no	—
129	C₂×C₃×Q₈	—	7	1	no	—
130	C₂×C₃×Q₈	—	7	1	no	—
131	C₂×C₃×Q₈	—	7	1	no	—
132	C₂×C₃×D₄	—	7	1	no	—
133	C₂²×C₃×C₄	—	7	2	no	—
134	C₃×C₄⋊C₄	—	7	1	no	—
135	C₃×C₄⋊C₄	—	7	2	no	—
136	C₃×C₄⋊C₄	—	7	1	no	—
137	C₃×C₄⋊C₄	—	7	2	no	—
138	C₃×C₄⋊C₄	—	7	2	no	—
139	C₃×C₄⋊C₄	—	7	1	no	—
140	C₃×C₄⋊C₄	—	7	1	no	—
141	C₃×C₄⋊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	C₃×C₄²	—	7	2	no	—
148	C₃×C₄²	—	7	2	no	—
149	C₃×C₄²	—	7	1	no	—
150	C₃×C₄²	—	7	1	no	—
151	C₃×C₄²	—	7	1	no	—
152	C₂×ΓL(1,𝔽₈)	—	4	2	no	—
153	C₂×ΓL(1,𝔽₈)	C₂×#41	1	1	yes	characteristic
154	C₂×ΓL(1,𝔽₈)	C₂×#41	1	1	yes	characteristic
155	C₂×ΓL(1,𝔽₈)	2₋¹⁺⁴	1	1	yes	characteristic
156	C₄⋊Q₈	—	7	1	no	—
157	#126	—	7	2	no	—
158	#126	—	7	2	no	—
159	#124	—	7	1	no	—
160	#124	AGL(1,𝔽₇)	1	1	yes	characteristic
161	C₂²⋊Q₈	—	7	2	no	—
162	C₄×Q₈	—	7	1	no	—
163	C₄×Q₈	—	7	1	no	—
164	C₄×Q₈	—	7	1	no	—
165	C₄×D₄	—	7	1	no	—
166	#117	—	7	2	no	—
167	C₂²×C₇	—	2	2	no	—
168	C₂²×C₇	—	2	2	no	—
169	C₂²×C₇	—	2	2	no	—
170	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
171	C₄×C₇	—	2	1	no	—
172	C₄×C₇	—	2	2	no	—
173	C₄×C₇	—	2	1	no	—
174	C₄×C₇	C₃×#41	1	2	yes	—
175	Dic₇	—	2	2	no	—
176	Dic₇	—	2	1	no	—
177	Dic₇	—	2	2	no	—
178	Dic₇	—	2	1	no	—
179	Dic₇	—	2	1	no	—
180	Dic₇	C₃×#41	1	2	yes	—
181	C₂³×C₃	—	7	2	no	—
182	C₃×Q₈	—	14	2	no	—
183	C₃×Q₈	—	14	2	no	—
184	C₃×Q₈	—	14	2	no	—
185	C₃×D₄	—	14	2	no	—
186	C₂×C₃×C₄	—	7	1	no	—
187	C₂×C₃×C₄	—	7	1	no	—
188	C₂×C₃×C₄	—	7	2	no	—
189	C₂×C₃×C₄	—	7	1	no	—
190	C₂×C₃×C₄	—	7	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	1	no	—
196	C₂×C₃×C₄	—	14	4	no	—
197	ΓL(1,𝔽₈)	#556	1	1	yes	characteristic, complemented, Hall
198	C₂×Q₈	—	7	1	no	—
199	C₂×Q₈	—	7	1	no	—
200	C₂×Q₈	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
201	C₂×D₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
202	C₂²×C₄	—	7	2	no	—
203	C₄⋊C₄	—	7	1	no	—
204	C₄⋊C₄	—	7	2	no	—
205	C₄⋊C₄	—	7	2	no	—
206	C₄⋊C₄	—	7	1	no	—
207	C₄⋊C₄	—	7	2	no	—
208	C₄⋊C₄	—	7	1	no	—
209	C₄⋊C₄	—	7	2	no	—
210	C₄⋊C₄	—	7	1	no	—
211	C₂²⋊C₄	—	7	2	no	—
212	C₂²⋊C₄	—	7	2	no	—
213	C₂²⋊C₄	—	7	2	no	—
214	C₂²⋊C₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
215	C₂²⋊C₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
216	C₄²	—	7	2	no	—
217	C₄²	—	7	2	no	—
218	C₄²	—	7	1	no	—
219	C₄²	—	7	1	no	—
220	C₄²	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
221	C₂×C₇	—	4	2	no	—
222	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
223	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
224	C₂×C₇	C₃×2₋¹⁺⁴	1	1	yes	characteristic
225	C₂²×C₃	—	7	1	no	—
226	C₂²×C₃	—	14	2	no	—
227	C₂²×C₃	—	14	2	no	—
228	C₂²×C₃	—	14	2	no	—
229	C₃×C₄	—	7	2	no	—
230	C₃×C₄	—	7	2	no	—
231	C₃×C₄	—	14	2	no	—
232	C₃×C₄	—	14	1	no	—
233	C₃×C₄	—	14	1	no	—
234	C₃×C₄	—	14	2	no	—
235	C₃×C₄	—	14	2	no	—
236	C₃×C₄	—	14	1	no	—
237	C₃×C₄	—	14	1	no	—
238	C₃×C₄	—	14	1	no	—
239	C₂³	C₂²×AGL(1,𝔽₇)	1	2	yes	—
240	Q₈	—	2	2	no	—
241	Q₈	—	14	2	no	—
242	Q₈	—	14	2	no	—
243	D₄	—	2	2	no	—
244	C₂×C₄	—	7	1	no	—
245	C₂×C₄	—	7	1	no	—
246	C₂×C₄	—	7	2	no	—
247	C₂×C₄	—	7	1	no	—
248	C₂×C₄	—	7	2	no	—
249	C₂×C₄	—	7	1	no	—
250	C₂×C₄	—	14	4	no	—
251	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
252	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
253	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
254	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
255	C₇	C₃×#556	1	1	yes	characteristic, complemented, Sylow
256	C₂×C₃	—	7	1	no	—
257	C₂×C₃	—	7	1	no	—
258	C₂×C₃	—	7	1	no	—
259	C₂×C₃	—	28	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	7	2	no	—
268	C₄	—	14	2	no	—
269	C₄	—	14	2	no	—
270	C₄	—	14	1	no	—
271	C₄	—	14	1	no	—
272	C₄	—	14	1	no	—
273	C₄	#1159⋊C₃	1	2	yes	—
274	C₃	—	7	1	no	Sylow
275	C₂	—	4	2	no	—
276	C₂	C₂×#1159⋊C₃	1	1	yes	characteristic
277	C₂	#41×AGL(1,𝔽₇)	1	1	yes	characteristic
278	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×#124	C₂×C₃
prime radical	C₂×D₄×ΓL(1,𝔽₈)	C₂²
hypercenter	#124	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#556
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #124

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. #6442⋊C₃	C₂	semidirect
7. #6442⋊C₃	C₂	semidirect
8. #6441⋊C₃	C₂	semidirect
9. #6440⋊C₃	C₂	semidirect
10. C₄×ΓL(1,𝔽₈)⋊Q₈	C₂	semidirect
11. C₄×ΓL(1,𝔽₈)⋊Q₈	C₂	semidirect
12. #90362	C₃	semidirect
37. Q₈×Dic₇	C₂×C₃	semidirect
38. #6507	C₂×C₃	semidirect
41. #6442	C₂×C₃	semidirect
42. #6442	C₂×C₃	semidirect
43. #6441	C₂×C₃	semidirect
44. #6440	C₂×C₃	semidirect
45. C₄×Dic₁₄	C₂×C₃	semidirect
46. C₄×Dic₁₄	C₂×C₃	semidirect
197. ΓL(1,𝔽₈)	#556	semidirect
255. 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)×#124	2
Q₈×ΓL(1,𝔽₈)⋊C₄	2
#6507⋊C₃	2
#6499⋊C₃	2
D₄×ΓL(1,𝔽₈)⋊C₄	2
#6442⋊C₃	2
#6441⋊C₃	2
#6440⋊C₃	2
C₄×ΓL(1,𝔽₈)⋊Q₈	2
#90362	3
C₃×#556	7

Subgroups — #90362⋊C3 #13057405

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #90362⋊C₃ #13057405