Subgroups — #90372⋊C₃ #13057406

Order 1344.

472 conjugacy classes of subgroups.

The subgroup lattice has 304 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	#6510⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×ΓL(1,𝔽₈)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#6503⋊C₃	C₂	1	1	yes	maximal, characteristic
5	D₄×ΓL(1,𝔽₈)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#6449⋊C₃	C₂	1	2	yes	maximal, complemented
7	#6448⋊C₃	C₂	1	2	yes	maximal, complemented
8	#6446⋊C₃	C₂	1	2	yes	maximal, complemented
9	((C₂²⋊D₇)⋊C₃)⋊C₄	C₂	1	2	yes	maximal, complemented
10	C₄×C₄⋊AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×ΓL(1,𝔽₈)⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
12	#90372	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	C₂×(C₂²⋊D₇)⋊C₃	C₂²	1	2	yes	—
17	C₂²×ΓL(1,𝔽₈)⋊C₄	C₂²	1	2	yes	—
18	C₂×C₄⋊AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
19	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	2	yes	—
20	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	1	yes	characteristic
21	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
22	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
23	C₄²×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
24	(C₂²⋊Dic₇)⋊C₃	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	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	2	yes	—
28	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic
29	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	1	yes	characteristic
30	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	2	yes	—
31	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	1	yes	characteristic
32	(ΓL(1,𝔽₈)⋊C₄)⋊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	#6510	C₂×C₃	1	1	yes	characteristic, complemented
38	Q₈×Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
39	#6503	C₂×C₃	1	1	yes	characteristic
40	D₄×Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
41	#6449	C₂×C₃	1	2	yes	complemented
42	#6448	C₂×C₃	1	2	yes	complemented
43	#6446	C₂×C₃	1	2	yes	complemented
44	(C₂²⋊D₇)⋊C₄	C₂×C₃	1	2	yes	complemented
45	C₄×D₂₈	C₂×C₃	1	1	yes	characteristic, complemented
46	C₄×Dic₁₄	C₂×C₃	1	1	yes	characteristic, complemented
47	C₃×#559	—	7	1	no	maximal, Hall
48	C₂³×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
49	C₂²×AGL(1,𝔽₇)	C₂³	1	2	yes	—
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	(C₂²⋊D₇)⋊C₃	—	4	2	no	—
58	C₂×ΓL(1,𝔽₈)⋊C₄	—	4	2	no	—
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	1	yes	characteristic
62	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
63	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
64	C₄⋊AGL(1,𝔽₇)	—	2	2	no	—
65	C₄×AGL(1,𝔽₇)	—	4	2	no	—
66	ΓL(1,𝔽₈)⋊Q₈	—	2	2	no	—
67	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
69	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
70	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
71	C₂²×Dic₇	C₂²×C₃	1	2	yes	—
72	C₂×D₂₈	C₂²×C₃	1	1	yes	characteristic
73	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
74	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
75	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
76	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
77	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
78	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
79	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
80	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
81	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
82	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
83	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
84	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
85	Dic₇⋊C₄	C₂²×C₃	1	1	yes	characteristic
86	Dic₇⋊C₄	C₂²×C₃	1	1	yes	characteristic
87	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
88	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
89	C₄×Dic₇	C₂²×C₃	1	2	yes	—
90	C₃×#127	—	7	1	no	—
91	C₃×#124	—	7	2	no	—
92	C₃×#124	—	7	1	no	—
93	C₃×#124	—	7	1	no	—
94	C₃×#123	—	7	2	no	—
95	C₃×#123	—	7	2	no	—
96	C₃×C₄×Q₈	—	7	1	no	—
97	C₃×C₄×Q₈	—	7	1	no	—
98	C₃×C₄×D₄	—	7	1	no	—
99	C₃×C₄×D₄	—	7	2	no	—
100	C₃×C₄×D₄	—	7	1	no	—
101	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
102	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
103	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
104	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
105	C₂×AGL(1,𝔽₇)	—	2	2	no	—
106	C₂×AGL(1,𝔽₇)	—	2	2	no	—
107	C₂×AGL(1,𝔽₇)	—	2	2	no	—
108	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
109	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
110	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
111	C₄×ΓL(1,𝔽₈)	#41	1	2	yes	—
112	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
113	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
114	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
115	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
116	ΓL(1,𝔽₈)⋊C₄	#41	1	2	yes	—
117	#559	—	7	1	no	Sylow
118	C₂³×C₇	C₂³×C₃	1	2	yes	—
119	C₂²×D₇	C₂³×C₃	1	2	yes	—
120	C₇×Q₈	—	2	2	no	—
121	C₇×D₄	—	2	2	no	—
122	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
123	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
124	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
125	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
126	C₂²⋊D₇	—	2	4	no	—
127	C₂²⋊D₇	—	4	2	no	—
128	C₂×Dic₇	—	4	2	no	—
129	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
130	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
131	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
132	C₂×Dic₇	C₂³×C₃	1	2	yes	—
133	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
134	D₂₈	—	2	2	no	—
135	C₄×D₇	—	4	2	no	—
136	Dic₁₄	—	2	2	no	—
137	C₂×C₃×Q₈	—	7	1	no	—
138	C₂×C₃×Q₈	—	7	1	no	—
139	C₂×C₃×D₄	—	7	1	no	—
140	C₂×C₃×D₄	—	7	2	no	—
141	C₂×C₃×D₄	—	7	2	no	—
142	C₂×C₃×D₄	—	7	1	no	—
143	C₂²×C₃×C₄	—	7	2	no	—
144	C₂²×C₃×C₄	—	7	2	no	—
145	C₃×C₄⋊C₄	—	7	1	no	—
146	C₃×C₄⋊C₄	—	7	1	no	—
147	C₃×C₄⋊C₄	—	7	1	no	—
148	C₃×C₄⋊C₄	—	7	1	no	—
149	C₃×C₄⋊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×C₂²⋊C₄	—	7	2	no	—
154	C₃×C₂²⋊C₄	—	7	2	no	—
155	C₃×C₂²⋊C₄	—	7	2	no	—
156	C₃×C₄²	—	7	2	no	—
157	C₃×C₄²	—	7	1	no	—
158	C₃×C₄²	—	7	1	no	—
159	C₃×C₄²	—	7	1	no	—
160	C₂×ΓL(1,𝔽₈)	—	4	2	no	—
161	C₂×ΓL(1,𝔽₈)	C₂×#41	1	1	yes	characteristic
162	C₂×ΓL(1,𝔽₈)	C₂×#41	1	1	yes	characteristic
163	C₂×ΓL(1,𝔽₈)	H(5,𝔽₂)	1	1	yes	characteristic
164	AGL(1,𝔽₇)	—	4	2	no	—
165	#127	—	7	1	no	—
166	#124	—	7	1	no	—
167	#124	—	7	2	no	—
168	#124	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
169	#123	—	7	2	no	—
170	#123	—	7	2	no	—
171	C₄×Q₈	—	7	1	no	—
172	C₄×Q₈	—	7	1	no	—
173	C₄×D₄	—	7	1	no	—
174	C₄×D₄	—	7	2	no	—
175	C₄×D₄	—	7	1	no	—
176	C₂²×C₇	—	2	2	no	—
177	C₂²×C₇	—	2	2	no	—
178	C₂²×C₇	—	2	2	no	—
179	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
180	C₂×D₇	—	2	2	no	—
181	C₂×D₇	—	2	2	no	—
182	C₂×D₇	—	2	2	no	—
183	C₄×C₇	—	2	1	no	—
184	C₄×C₇	—	2	2	no	—
185	C₄×C₇	—	2	1	no	—
186	C₄×C₇	C₃×#41	1	2	yes	—
187	Dic₇	—	2	1	no	—
188	Dic₇	—	2	2	no	—
189	Dic₇	—	2	1	no	—
190	Dic₇	—	2	1	no	—
191	Dic₇	C₃×#41	1	2	yes	—
192	C₂³×C₃	—	7	2	no	—
193	C₂³×C₃	—	7	2	no	—
194	C₃×Q₈	—	14	2	no	—
195	C₃×Q₈	—	14	2	no	—
196	C₃×D₄	—	14	2	no	—
197	C₃×D₄	—	14	2	no	—
198	C₃×D₄	—	14	4	no	—
199	C₃×D₄	—	28	2	no	—
200	C₂×C₃×C₄	—	7	1	no	—
201	C₂×C₃×C₄	—	7	1	no	—
202	C₂×C₃×C₄	—	7	2	no	—
203	C₂×C₃×C₄	—	7	1	no	—
204	C₂×C₃×C₄	—	7	1	no	—
205	C₂×C₃×C₄	—	7	1	no	—
206	C₂×C₃×C₄	—	7	2	no	—
207	C₂×C₃×C₄	—	7	1	no	—
208	C₂×C₃×C₄	—	7	1	no	—
209	C₂×C₃×C₄	—	28	2	no	—
210	C₂×C₃×C₄	—	28	2	no	—
211	ΓL(1,𝔽₈)	#559	1	1	yes	characteristic, complemented, Hall
212	C₂×Q₈	—	7	1	no	—
213	C₂×Q₈	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
214	C₂×D₄	—	7	2	no	—
215	C₂×D₄	—	7	2	no	—
216	C₂×D₄	—	7	1	no	—
217	C₂×D₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
218	C₂²×C₄	—	7	2	no	—
219	C₂²×C₄	—	7	2	no	—
220	C₄⋊C₄	—	7	1	no	—
221	C₄⋊C₄	—	7	1	no	—
222	C₄⋊C₄	—	7	1	no	—
223	C₄⋊C₄	—	7	2	no	—
224	C₄⋊C₄	—	7	1	no	—
225	C₂²⋊C₄	—	7	2	no	—
226	C₂²⋊C₄	—	7	2	no	—
227	C₂²⋊C₄	—	7	2	no	—
228	C₂²⋊C₄	—	7	2	no	—
229	C₂²⋊C₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
230	C₂²⋊C₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
231	C₄²	—	7	2	no	—
232	C₄²	—	7	1	no	—
233	C₄²	—	7	1	no	—
234	C₄²	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
235	C₂×C₇	—	4	2	no	—
236	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
237	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
238	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
239	D₇	—	4	2	no	—
240	C₂²×C₃	—	7	1	no	—
241	C₂²×C₃	—	14	2	no	—
242	C₂²×C₃	—	14	2	no	—
243	C₂²×C₃	—	14	2	no	—
244	C₂²×C₃	—	14	2	no	—
245	C₂²×C₃	—	14	2	no	—
246	C₂²×C₃	—	14	2	no	—
247	C₃×C₄	—	7	2	no	—
248	C₃×C₄	—	7	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	C₃×C₄	—	14	1	no	—
254	C₃×C₄	—	14	1	no	—
255	C₃×C₄	—	14	1	no	—
256	C₂³	—	7	2	no	—
257	C₂³	C₂²×AGL(1,𝔽₇)	1	2	yes	—
258	Q₈	—	2	2	no	—
259	Q₈	—	14	2	no	—
260	D₄	—	2	2	no	—
261	D₄	—	14	4	no	—
262	D₄	—	14	2	no	—
263	D₄	—	28	2	no	—
264	C₂×C₄	—	7	1	no	—
265	C₂×C₄	—	7	1	no	—
266	C₂×C₄	—	7	2	no	—
267	C₂×C₄	—	7	1	no	—
268	C₂×C₄	—	7	1	no	—
269	C₂×C₄	—	28	2	no	—
270	C₂×C₄	—	28	2	no	—
271	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
272	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
273	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
274	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
275	C₇	C₃×#559	1	1	yes	characteristic, complemented, Sylow
276	C₂×C₃	—	7	1	no	—
277	C₂×C₃	—	7	1	no	—
278	C₂×C₃	—	7	1	no	—
279	C₂×C₃	—	28	2	no	—
280	C₂×C₃	—	28	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	14	2	no	—
285	C₂²	—	14	2	no	—
286	C₂²	—	14	2	no	—
287	C₂²	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic
288	C₄	—	2	2	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	7	2	no	—
292	C₄	—	14	1	no	—
293	C₄	—	14	2	no	—
294	C₄	—	14	1	no	—
295	C₄	—	14	1	no	—
296	C₄	#1159⋊C₃	1	2	yes	—
297	C₃	—	7	1	no	Sylow
298	C₂	—	4	2	no	—
299	C₂	—	28	2	no	—
300	C₂	C₂×#1159⋊C₃	1	1	yes	characteristic
301	C₂	#41×AGL(1,𝔽₇)	1	1	yes	characteristic
302	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×#124	C₂×C₃
prime radical	#6503⋊C₃	C₂
hypercenter	#124	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#559
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. #6510⋊C₃	C₂	semidirect
3. Q₈×ΓL(1,𝔽₈)⋊C₄	C₂	semidirect
5. D₄×ΓL(1,𝔽₈)⋊C₄	C₂	semidirect
6. #6449⋊C₃	C₂	semidirect
7. #6448⋊C₃	C₂	semidirect
8. #6446⋊C₃	C₂	semidirect
9. ((C₂²⋊D₇)⋊C₃)⋊C₄	C₂	semidirect
10. C₄×C₄⋊AGL(1,𝔽₇)	C₂	semidirect
11. C₄×ΓL(1,𝔽₈)⋊Q₈	C₂	semidirect
12. #90372	C₃	semidirect
36. C₇×#124	C₂×C₃	semidirect
37. #6510	C₂×C₃	semidirect
38. Q₈×Dic₇	C₂×C₃	semidirect
40. D₄×Dic₇	C₂×C₃	semidirect
41. #6449	C₂×C₃	semidirect
42. #6448	C₂×C₃	semidirect
43. #6446	C₂×C₃	semidirect
44. (C₂²⋊D₇)⋊C₄	C₂×C₃	semidirect
45. C₄×D₂₈	C₂×C₃	semidirect
46. C₄×Dic₁₄	C₂×C₃	semidirect
168. #124	AGL(1,𝔽₇)	semidirect
211. ΓL(1,𝔽₈)	#559	semidirect
275. C₇	C₃×#559	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
#6510⋊C₃	2
Q₈×ΓL(1,𝔽₈)⋊C₄	2
#6503⋊C₃	2
D₄×ΓL(1,𝔽₈)⋊C₄	2
#6449⋊C₃	2
#6448⋊C₃	2
#6446⋊C₃	2
((C₂²⋊D₇)⋊C₃)⋊C₄	2
C₄×C₄⋊AGL(1,𝔽₇)	2
C₄×ΓL(1,𝔽₈)⋊Q₈	2
#90372	3
C₃×#559	7

Subgroups — #90372⋊C3 #13057406

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #90372⋊C₃ #13057406