Subgroups — #90371⋊C₃ #13057415

Order 1344.

468 conjugacy classes of subgroups.

The subgroup lattice has 300 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, complemented
2	#6509⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×ΓL(1,𝔽₈)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#6501⋊C₃	C₂	1	1	yes	maximal, characteristic
5	D₄×ΓL(1,𝔽₈)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#6446⋊C₃	C₂	1	2	yes	maximal, complemented
7	((C₂²⋊D₇)⋊C₃)⋊C₄	C₂	1	2	yes	maximal, complemented
8	#6442⋊C₃	C₂	1	2	yes	maximal, complemented
9	#6440⋊C₃	C₂	1	2	yes	maximal, complemented
10	C₄²×AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic, complemented
11	#6433⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#90371	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	C₂×(C₂²⋊D₇)⋊C₃	C₂²	1	2	yes	—
16	C₂²×ΓL(1,𝔽₈)⋊C₄	C₂²	1	2	yes	—
17	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
18	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
19	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
20	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
21	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
22	C₄²×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
23	(C₂²⋊Dic₇)⋊C₃	C₂²	1	2	yes	—
24	(C₂²⋊Dic₇)⋊C₃	C₂²	1	2	yes	—
25	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	2	yes	—
26	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic
27	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic
28	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	2	yes	—
29	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	2	yes	—
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	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	2	yes	—
36	C₇×#124	C₂×C₃	1	1	yes	characteristic, complemented
37	#6509	C₂×C₃	1	1	yes	characteristic, complemented
38	Q₈×Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
39	#6501	C₂×C₃	1	1	yes	characteristic
40	D₄×Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
41	#6446	C₂×C₃	1	2	yes	complemented
42	(C₂²⋊D₇)⋊C₄	C₂×C₃	1	2	yes	complemented
43	#6442	C₂×C₃	1	2	yes	complemented
44	#6440	C₂×C₃	1	2	yes	complemented
45	C₄²×D₇	C₂×C₃	1	1	yes	characteristic, complemented
46	#6433	C₂×C₃	1	1	yes	characteristic, complemented
47	C₃×#529	—	7	1	no	maximal, Hall
48	C₂³×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
49	C₂²×AGL(1,𝔽₇)	C₂³	1	1	yes	characteristic
50	Q₈×ΓL(1,𝔽₈)	—	2	2	no	—
51	D₄×ΓL(1,𝔽₈)	—	2	2	no	—
52	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
53	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
54	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
55	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
56	(C₂²⋊D₇)⋊C₃	—	2	4	no	—
57	C₂×ΓL(1,𝔽₈)⋊C₄	—	2	4	no	—
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	2	yes	—
61	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
62	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
63	C₄×AGL(1,𝔽₇)	—	2	2	no	—
64	C₄×AGL(1,𝔽₇)	—	2	2	no	—
65	C₄×AGL(1,𝔽₇)	—	2	2	no	—
66	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
67	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
69	C₂²×Dic₇	C₂²×C₃	1	2	yes	—
70	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic
71	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic
72	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic
73	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
74	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
75	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
76	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
77	C₂²⋊Dic₇	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	2	yes	—
83	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
84	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
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	C₄×Dic₇	C₂²×C₃	1	2	yes	—
89	C₃×#126	—	7	2	no	—
90	C₃×#125	—	7	1	no	—
91	C₃×#124	—	7	1	no	—
92	C₃×#123	—	7	2	no	—
93	C₃×C₂²⋊Q₈	—	7	1	no	—
94	C₃×#121	—	7	1	no	—
95	C₃×C₄×Q₈	—	7	1	no	—
96	C₃×C₄×D₄	—	7	2	no	—
97	C₃×C₄×D₄	—	7	1	no	—
98	C₃×#117	—	7	2	no	—
99	C₂×C₃×C₄²	—	7	1	no	—
100	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
101	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
102	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
103	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
104	C₂×AGL(1,𝔽₇)	—	2	1	no	—
105	C₂×AGL(1,𝔽₇)	—	2	1	no	—
106	C₂×AGL(1,𝔽₇)	#41	1	2	yes	—
107	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
108	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
109	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
110	C₄×ΓL(1,𝔽₈)	#41	1	2	yes	—
111	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
112	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
113	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
114	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
115	ΓL(1,𝔽₈)⋊C₄	#41	1	2	yes	—
116	#529	—	7	1	no	Sylow
117	C₂³×C₇	C₂³×C₃	1	2	yes	—
118	C₂²×D₇	C₂³×C₃	1	1	yes	characteristic
119	C₇×Q₈	—	2	2	no	—
120	C₇×D₄	—	2	2	no	—
121	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
122	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
123	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
124	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
125	C₂²⋊D₇	—	2	4	no	—
126	C₂×Dic₇	—	2	4	no	—
127	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
128	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
129	C₂×Dic₇	C₂³×C₃	1	2	yes	—
130	C₂×Dic₇	C₂³×C₃	1	2	yes	—
131	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
132	C₄×D₇	—	2	2	no	—
133	C₄×D₇	—	2	2	no	—
134	C₄×D₇	—	2	2	no	—
135	C₂×C₃×Q₈	—	7	1	no	—
136	C₂×C₃×D₄	—	7	2	no	—
137	C₂×C₃×D₄	—	7	1	no	—
138	C₂²×C₃×C₄	—	7	1	no	—
139	C₂²×C₃×C₄	—	7	1	no	—
140	C₂²×C₃×C₄	—	7	1	no	—
141	C₂²×C₃×C₄	—	7	2	no	—
142	C₃×C₄⋊C₄	—	7	2	no	—
143	C₃×C₄⋊C₄	—	7	2	no	—
144	C₃×C₄⋊C₄	—	7	1	no	—
145	C₃×C₄⋊C₄	—	7	2	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×C₂²⋊C₄	—	7	2	no	—
152	C₃×C₂²⋊C₄	—	7	2	no	—
153	C₃×C₄²	—	7	1	no	—
154	C₃×C₄²	—	7	1	no	—
155	C₃×C₄²	—	7	1	no	—
156	C₃×C₄²	—	7	2	no	—
157	C₃×C₄²	—	7	1	no	—
158	C₂×ΓL(1,𝔽₈)	—	4	2	no	—
159	C₂×ΓL(1,𝔽₈)	C₂×#41	1	1	yes	characteristic
160	C₂×ΓL(1,𝔽₈)	C₂×#41	1	1	yes	characteristic
161	C₂×ΓL(1,𝔽₈)	C₂×#41	1	1	yes	characteristic
162	AGL(1,𝔽₇)	—	2	2	no	—
163	#126	—	7	2	no	—
164	#125	—	7	1	no	—
165	#124	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
166	#123	—	7	2	no	—
167	C₂²⋊Q₈	—	7	1	no	—
168	#121	—	7	1	no	—
169	C₄×Q₈	—	7	1	no	—
170	C₄×D₄	—	7	1	no	—
171	C₄×D₄	—	7	2	no	—
172	#117	—	7	2	no	—
173	C₂×C₄²	—	7	1	no	—
174	C₂²×C₇	—	2	2	no	—
175	C₂²×C₇	—	2	2	no	—
176	C₂²×C₇	—	2	2	no	—
177	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
178	C₂×D₇	—	2	1	no	—
179	C₂×D₇	—	2	1	no	—
180	C₂×D₇	C₃×#41	1	2	yes	—
181	C₄×C₇	—	2	1	no	—
182	C₄×C₇	—	2	1	no	—
183	C₄×C₇	—	2	2	no	—
184	C₄×C₇	C₃×#41	1	2	yes	—
185	Dic₇	—	2	1	no	—
186	Dic₇	—	2	1	no	—
187	Dic₇	—	2	2	no	—
188	Dic₇	—	2	2	no	—
189	Dic₇	C₃×#41	1	2	yes	—
190	C₂³×C₃	—	7	1	no	—
191	C₂³×C₃	—	7	2	no	—
192	C₃×Q₈	—	14	2	no	—
193	C₃×D₄	—	14	2	no	—
194	C₃×D₄	—	14	4	no	—
195	C₂×C₃×C₄	—	7	1	no	—
196	C₂×C₃×C₄	—	7	1	no	—
197	C₂×C₃×C₄	—	7	1	no	—
198	C₂×C₃×C₄	—	7	2	no	—
199	C₂×C₃×C₄	—	7	2	no	—
200	C₂×C₃×C₄	—	7	1	no	—
201	C₂×C₃×C₄	—	7	1	no	—
202	C₂×C₃×C₄	—	7	1	no	—
203	C₂×C₃×C₄	—	7	2	no	—
204	C₂×C₃×C₄	—	14	2	no	—
205	C₂×C₃×C₄	—	14	2	no	—
206	C₂×C₃×C₄	—	14	4	no	—
207	C₂×C₃×C₄	—	14	2	no	—
208	ΓL(1,𝔽₈)	#529	1	1	yes	characteristic, complemented, Hall
209	C₂×Q₈	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
210	C₂×D₄	—	7	2	no	—
211	C₂×D₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
212	C₂²×C₄	—	7	2	no	—
213	C₂²×C₄	—	7	1	no	—
214	C₂²×C₄	—	7	1	no	—
215	C₂²×C₄	—	7	1	no	—
216	C₄⋊C₄	—	7	2	no	—
217	C₄⋊C₄	—	7	2	no	—
218	C₄⋊C₄	—	7	2	no	—
219	C₄⋊C₄	—	7	1	no	—
220	C₄⋊C₄	—	7	1	no	—
221	C₄⋊C₄	—	7	2	no	—
222	C₂²⋊C₄	—	7	2	no	—
223	C₂²⋊C₄	—	7	2	no	—
224	C₂²⋊C₄	—	7	2	no	—
225	C₂²⋊C₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
226	C₂²⋊C₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
227	C₄²	—	7	1	no	—
228	C₄²	—	7	1	no	—
229	C₄²	—	7	1	no	—
230	C₄²	—	7	2	no	—
231	C₄²	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
232	C₂×C₇	—	4	2	no	—
233	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
234	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
235	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
236	D₇	—	2	2	no	—
237	C₂²×C₃	—	7	2	no	—
238	C₂²×C₃	—	7	1	no	—
239	C₂²×C₃	—	14	2	no	—
240	C₂²×C₃	—	14	2	no	—
241	C₂²×C₃	—	14	2	no	—
242	C₂²×C₃	—	14	1	no	—
243	C₂²×C₃	—	14	1	no	—
244	C₃×C₄	—	7	2	no	—
245	C₃×C₄	—	7	2	no	—
246	C₃×C₄	—	14	1	no	—
247	C₃×C₄	—	14	1	no	—
248	C₃×C₄	—	14	2	no	—
249	C₃×C₄	—	14	1	no	—
250	C₃×C₄	—	14	2	no	—
251	C₃×C₄	—	14	1	no	—
252	C₃×C₄	—	14	2	no	—
253	C₂³	—	7	1	no	—
254	C₂³	C₂²×AGL(1,𝔽₇)	1	2	yes	—
255	Q₈	—	2	2	no	—
256	D₄	—	2	2	no	—
257	D₄	—	14	4	no	—
258	C₂×C₄	—	7	1	no	—
259	C₂×C₄	—	7	1	no	—
260	C₂×C₄	—	7	1	no	—
261	C₂×C₄	—	7	2	no	—
262	C₂×C₄	—	7	2	no	—
263	C₂×C₄	—	14	2	no	—
264	C₂×C₄	—	14	2	no	—
265	C₂×C₄	—	14	4	no	—
266	C₂×C₄	—	14	2	no	—
267	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
268	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
269	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
270	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
271	C₇	C₃×#529	1	1	yes	characteristic, complemented, Sylow
272	C₂×C₃	—	7	1	no	—
273	C₂×C₃	—	7	1	no	—
274	C₂×C₃	—	7	1	no	—
275	C₂×C₃	—	14	2	no	—
276	C₂×C₃	—	28	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	7	2	no	—
281	C₂²	—	14	1	no	—
282	C₂²	—	14	1	no	—
283	C₂²	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	7	2	no	—
288	C₄	—	14	1	no	—
289	C₄	—	14	2	no	—
290	C₄	—	14	1	no	—
291	C₄	—	14	2	no	—
292	C₄	#1159⋊C₃	1	2	yes	—
293	C₃	—	7	1	no	Sylow
294	C₂	—	4	2	no	—
295	C₂	—	14	2	no	—
296	C₂	C₂×#1159⋊C₃	1	1	yes	characteristic
297	C₂	#41×AGL(1,𝔽₇)	1	1	yes	characteristic
298	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×#124	C₂×C₃
prime radical	#6501⋊C₃	C₂
hypercenter	#124	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#529
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
1. ΓL(1,𝔽₈)×#124	C₂	semidirect
2. #6509⋊C₃	C₂	semidirect
3. Q₈×ΓL(1,𝔽₈)⋊C₄	C₂	semidirect
5. D₄×ΓL(1,𝔽₈)⋊C₄	C₂	semidirect
6. #6446⋊C₃	C₂	semidirect
7. ((C₂²⋊D₇)⋊C₃)⋊C₄	C₂	semidirect
8. #6442⋊C₃	C₂	semidirect
9. #6440⋊C₃	C₂	semidirect
10. C₄²×AGL(1,𝔽₇)	C₂	semidirect
11. #6433⋊C₃	C₂	semidirect
12. #90371	C₃	semidirect
36. C₇×#124	C₂×C₃	semidirect
37. #6509	C₂×C₃	semidirect
38. Q₈×Dic₇	C₂×C₃	semidirect
40. D₄×Dic₇	C₂×C₃	semidirect
41. #6446	C₂×C₃	semidirect
42. (C₂²⋊D₇)⋊C₄	C₂×C₃	semidirect
43. #6442	C₂×C₃	semidirect
44. #6440	C₂×C₃	semidirect
45. C₄²×D₇	C₂×C₃	semidirect
46. #6433	C₂×C₃	semidirect
165. #124	AGL(1,𝔽₇)	semidirect
208. ΓL(1,𝔽₈)	#529	semidirect
271. C₇	C₃×#529	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
#6509⋊C₃	2
Q₈×ΓL(1,𝔽₈)⋊C₄	2
#6501⋊C₃	2
D₄×ΓL(1,𝔽₈)⋊C₄	2
#6446⋊C₃	2
((C₂²⋊D₇)⋊C₃)⋊C₄	2
#6442⋊C₃	2
#6440⋊C₃	2
C₄²×AGL(1,𝔽₇)	2
#6433⋊C₃	2
#90371	3
C₃×#529	7

Subgroups — #90371⋊C3 #13057415

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #90371⋊C₃ #13057415