Subgroups — #13252660

Order 1792.

606 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×D₈	C₂	1	1	yes	maximal, characteristic
2	#11729368	C₂	1	2	yes	maximal, complemented
3	C₂×D₄⋊(C₇⋊C₈)	C₂	1	2	yes	maximal, complemented
4	C₂×#89253	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₈×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₇×D₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
8	C₄×C₇×D₈	—	2	2	no	—
9	C₄×C₇×D₈	—	2	2	no	—
10	C₂×C₇×#107	C₄	1	1	yes	characteristic
11	C₂×C₇×#102	C₄	1	2	yes	—
12	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊(C₇⋊C₈)	—	2	4	no	—
14	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
15	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
17	D₄⋊(C₇⋊C₈)	—	2	4	no	—
18	#89253	—	2	2	no	—
19	C₈×C₇⋊C₈	—	2	2	no	—
20	#9657	—	7	1	no	maximal, Sylow
21	C₂²×C₇×D₄	C₂×C₄	1	2	yes	—
22	C₂³×C₄×C₇	—	2	2	no	—
23	C₂×C₇×D₈	—	2	2	no	—
24	C₂×C₇×D₈	—	2	2	no	—
25	C₂×C₇×D₈	C₈	1	4	yes	complemented
26	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
28	C₄×C₇×D₄	—	2	2	no	—
29	C₄×C₇×D₄	—	2	2	no	—
30	C₄×C₇×D₄	—	2	2	no	—
31	C₄×C₇×D₄	—	2	2	no	—
32	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
33	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
34	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
35	C₂²×C₇⋊C₈	—	2	1	no	—
36	C₂²×C₇⋊C₈	—	2	2	no	—
37	C₂²×C₇⋊C₈	D₄	1	2	yes	—
38	C₇×#107	—	2	2	no	—
39	C₇×#102	—	2	4	no	—
40	C₄×C₇×C₈	D₄	1	2	yes	—
41	C₄×C₇×C₈	D₄	1	2	yes	—
42	C₂²⋊(C₇⋊C₈)	—	4	8	no	—
43	C₄⋊(C₇⋊C₈)	—	2	2	no	—
44	C₄⋊(C₇⋊C₈)	—	2	2	no	—
45	C₄×C₇⋊C₈	—	2	1	no	—
46	C₄×C₇⋊C₈	—	2	1	no	—
47	C₄×C₇⋊C₈	—	2	1	no	—
48	C₄×C₇⋊C₈	D₄	1	2	yes	—
49	C₂×C₄×D₈	D₇	1	1	yes	characteristic
50	#1954	—	7	2	no	—
51	C₂×#335	—	7	1	no	—
52	C₂×D₄⋊C₈	—	7	2	no	—
53	C₂×C₈²	—	7	1	no	—
54	C₂⁴×C₇	—	2	2	no	—
55	C₂×C₇×D₄	—	2	2	no	—
56	C₂×C₇×D₄	—	2	2	no	—
57	C₂×C₇×D₄	—	2	2	no	—
58	C₂×C₇×D₄	—	2	2	no	—
59	C₂×C₇×D₄	—	2	2	no	—
60	C₂×C₇×D₄	C₂×C₈	1	4	yes	—
61	C₂²×C₄×C₇	—	2	2	no	—
62	C₂²×C₄×C₇	—	4	2	no	—
63	C₂²×C₄×C₇	—	4	2	no	—
64	C₂²×C₄×C₇	—	4	4	no	—
65	C₂²×C₄×C₇	—	4	4	no	—
66	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₇	#41	1	1	yes	characteristic
69	C₇×D₈	—	2	8	no	—
70	C₂×C₇×C₈	—	2	1	no	—
71	C₂×C₇×C₈	—	2	1	no	—
72	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
73	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
74	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
75	C₂×C₇×C₈	M₁₆	1	2	yes	—
76	C₇×C₄⋊C₄	—	2	2	no	—
77	C₇×C₄⋊C₄	—	2	2	no	—
78	C₇×C₂²⋊C₄	—	4	2	no	—
79	C₇×C₂²⋊C₄	—	4	2	no	—
80	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₇	#41	1	1	yes	characteristic
84	C₂×C₇⋊C₈	—	2	4	no	—
85	C₂×C₇⋊C₈	—	2	2	no	—
86	C₂×C₇⋊C₈	—	2	2	no	—
87	C₂×C₇⋊C₈	—	2	2	no	—
88	C₂×C₇⋊C₈	—	4	2	no	—
89	C₂×C₇⋊C₈	—	4	4	no	—
90	C₂×C₇⋊C₈	D₈	1	4	yes	complemented
91	C₂²×D₈	Dic₇	1	1	yes	characteristic
92	C₂×C₄×D₄	C₂×D₇	1	2	yes	—
93	C₄×D₈	—	2	2	no	—
94	C₄×D₈	—	2	2	no	—
95	C₂×#107	Dic₇	1	1	yes	characteristic
96	C₂×C₄⋊C₈	—	7	2	no	—
97	C₂×#102	Dic₇	1	2	yes	—
98	C₂×C₂²⋊C₈	—	14	4	no	—
99	C₂×C₄×C₈	—	7	1	no	—
100	C₂×C₄×C₈	—	7	1	no	—
101	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
102	#335	—	14	2	no	—
103	D₄⋊C₈	—	14	4	no	—
104	C₈²	—	14	2	no	—
105	C₂³×C₇	—	2	4	no	—
106	C₂³×C₇	—	4	2	no	—
107	C₂³×C₇	—	4	2	no	—
108	C₂³×C₇	—	4	4	no	—
109	C₂³×C₇	—	4	4	no	—
110	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
111	C₇×D₄	—	2	4	no	—
112	C₇×D₄	—	2	4	no	—
113	C₇×D₄	—	4	2	no	—
114	C₇×D₄	—	4	2	no	—
115	C₂×C₄×C₇	—	2	2	no	—
116	C₂×C₄×C₇	—	4	2	no	—
117	C₂×C₄×C₇	—	4	4	no	—
118	C₂×C₄×C₇	—	8	4	no	—
119	C₂×C₄×C₇	—	8	2	no	—
120	C₂×C₄×C₇	—	8	2	no	—
121	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₇	#121	1	1	yes	characteristic
124	C₂×C₄×C₇	#121	1	1	yes	characteristic
125	C₂×C₄×C₇	#124	1	1	yes	characteristic
126	C₂×C₄×C₇	#127	1	1	yes	characteristic
127	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
129	C₂×C₄×C₇	#131	1	1	yes	characteristic
130	C₂×C₄×C₇	#131	1	1	yes	characteristic
131	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
132	C₂×C₄×C₇	#135	1	2	yes	—
133	C₇×C₈	—	2	2	no	—
134	C₇×C₈	C₂²⋊C₈	1	4	yes	complemented
135	C₇⋊C₈	—	2	4	no	—
136	C₇⋊C₈	—	2	4	no	—
137	C₇⋊C₈	—	8	2	no	—
138	C₂²×D₄	C₂×Dic₇	1	2	yes	—
139	C₂³×C₄	—	2	2	no	—
140	C₂×D₈	—	2	2	no	—
141	C₂×D₈	—	2	2	no	—
142	C₂×D₈	C₇⋊C₈	1	4	yes	complemented
143	C₂²×C₈	—	7	2	no	—
144	C₂²×C₈	—	14	1	no	—
145	C₂²×C₈	—	14	2	no	—
146	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
147	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
148	C₄×D₄	—	2	2	no	—
149	C₄×D₄	—	2	2	no	—
150	C₄×D₄	—	2	2	no	—
151	C₄×D₄	—	2	2	no	—
152	C₂×C₄⋊C₄	C₂×Dic₇	1	2	yes	—
153	C₂×C₂²⋊C₄	—	2	2	no	—
154	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
155	#107	—	2	2	no	—
156	C₄⋊C₈	—	14	2	no	—
157	C₄⋊C₈	—	14	2	no	—
158	#102	—	2	4	no	—
159	C₂²⋊C₈	—	28	8	no	—
160	C₄×C₈	—	7	2	no	—
161	C₄×C₈	—	14	1	no	—
162	C₄×C₈	—	14	1	no	—
163	C₄×C₈	—	14	1	no	—
164	C₄×C₈	C₂²⋊D₇	1	2	yes	—
165	C₄×C₈	C₂²⋊D₇	1	2	yes	—
166	C₂²×C₇	—	4	8	no	—
167	C₂²×C₇	—	4	4	no	—
168	C₂²×C₇	—	8	4	no	—
169	C₂²×C₇	—	8	2	no	—
170	C₂²×C₇	—	8	2	no	—
171	C₂²×C₇	#390	1	1	yes	characteristic
172	C₂²×C₇	C₈×D₄	1	1	yes	characteristic
173	C₂²×C₇	#436	1	1	yes	characteristic
174	C₂²×C₇	C₄×D₈	1	2	yes	—
175	C₂²×C₇	#443	1	2	yes	—
176	C₄×C₇	—	2	1	no	—
177	C₄×C₇	—	2	1	no	—
178	C₄×C₇	—	8	2	no	—
179	C₄×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
180	C₄×C₇	#408	1	1	yes	characteristic
181	C₄×C₇	C₈×D₄	1	1	yes	characteristic
182	C₄×C₇	#436	1	1	yes	characteristic
183	C₄×C₇	#466	1	2	yes	—
184	C₄×C₇	#493	1	1	yes	characteristic
185	C₄×C₇	#495	1	1	yes	characteristic
186	C₂⁴	—	2	2	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	C₂×C₇⋊C₈	1	4	yes	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	4	4	no	—
195	C₂²×C₄	—	4	4	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	—	4	2	no	—
198	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
199	C₂²×C₄	#1159	1	1	yes	characteristic
200	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
201	D₈	—	2	8	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	7	4	no	—
205	C₂×C₈	—	14	4	no	—
206	C₂×C₈	—	14	2	no	—
207	C₂×C₈	—	14	2	no	—
208	C₂×C₈	—	14	2	no	—
209	C₂×C₈	—	28	4	no	—
210	C₂×C₈	—	28	2	no	—
211	C₂×C₈	C₂×C₇⋊C₈	1	2	yes	—
212	C₂×C₈	#1136	1	2	yes	—
213	C₂×C₈	C₂²⋊Dic₇	1	2	yes	—
214	C₂×C₈	C₂²⋊Dic₇	1	2	yes	—
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	—	2	2	no	—
217	C₂²⋊C₄	—	4	2	no	—
218	C₂²⋊C₄	—	4	2	no	—
219	C₄²	D₄×D₇	1	1	yes	characteristic
220	C₄²	#1159	1	1	yes	characteristic
221	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
222	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
223	C₂×C₇	—	8	4	no	—
224	C₂×C₇	C₈×D₈	1	2	yes	—
225	C₂×C₇	#1589	1	2	yes	—
226	C₂×C₇	#1954	1	1	yes	characteristic
227	C₂×C₇	#1970	1	1	yes	characteristic
228	C₂×C₇	#1974	1	1	yes	characteristic
229	C₂³	—	2	4	no	—
230	C₂³	—	4	4	no	—
231	C₂³	—	4	4	no	—
232	C₂³	—	4	2	no	—
233	C₂³	—	4	2	no	—
234	C₂³	D₄×Dic₇	1	1	yes	characteristic
235	D₄	—	2	4	no	—
236	D₄	—	2	4	no	—
237	D₄	—	4	2	no	—
238	D₄	—	4	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	8	2	no	—
243	C₂×C₄	—	8	2	no	—
244	C₂×C₄	—	8	4	no	—
245	C₂×C₄	D₇×D₈	1	2	yes	—
246	C₂×C₄	#6475	1	2	yes	—
247	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
248	C₂×C₄	C₂×#1136	1	1	yes	characteristic
249	C₂×C₄	D₄×Dic₇	1	1	yes	characteristic
250	C₂×C₄	#6499	1	1	yes	characteristic
251	C₂×C₄	#6501	1	1	yes	characteristic
252	C₂×C₄	#6501	1	1	yes	characteristic
253	C₂×C₄	#6503	1	1	yes	characteristic
254	C₂×C₄	#6511	1	1	yes	characteristic
255	C₂×C₄	#6511	1	1	yes	characteristic
256	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
257	C₈	—	2	2	no	—
258	C₈	—	14	4	no	—
259	C₈	—	14	4	no	—
260	C₈	—	56	2	no	—
261	C₈	C₂²⋊(C₇⋊C₈)	1	4	yes	complemented
262	C₇	#9657	1	1	yes	characteristic, complemented, Sylow
263	C₂²	—	4	4	no	—
264	C₂²	—	4	8	no	—
265	C₂²	—	8	2	no	—
266	C₂²	—	8	2	no	—
267	C₂²	—	8	4	no	—
268	C₂²	D₄×C₇⋊C₈	1	1	yes	characteristic
269	C₂²	#89784	1	1	yes	characteristic
270	C₂²	D₈×Dic₇	1	2	yes	—
271	C₂²	#89968	1	2	yes	—
272	C₂²	#89993	1	1	yes	characteristic
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	8	2	no	—
276	C₄	D₄×C₇⋊C₈	1	1	yes	characteristic
277	C₄	#89784	1	1	yes	characteristic
278	C₄	#89923	1	1	yes	characteristic
279	C₄	#89927	1	1	yes	characteristic
280	C₄	#89929	1	2	yes	—
281	C₄	C₂×C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
282	C₄	#90006	1	1	yes	characteristic
283	C₂	—	8	4	no	—
284	C₂	D₈×C₇⋊C₈	1	2	yes	—
285	C₂	#11728635	1	2	yes	—
286	C₂	#11729368	1	1	yes	characteristic
287	C₂	#11729792	1	1	yes	characteristic
288	C₂	#11729838	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₇
commutator	C₂×C₄×C₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₂×C₄×C₇×D₈	C₂
socle	C₂³×C₇	C₄×D₄
prime radical	C₂²×C₇×D₈	C₄
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	C₂×C₄×D₈	D₇
hypocenter	C₇	#9657
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
2. #11729368	C₂	semidirect
3. C₂×D₄⋊(C₇⋊C₈)	C₂	semidirect
4. C₂×#89253	C₂	semidirect
5. C₂×C₈×C₇⋊C₈	C₂	semidirect
25. C₂×C₇×D₈	C₈	semidirect
90. C₂×C₇⋊C₈	D₈	semidirect
134. C₇×C₈	C₂²⋊C₈	semidirect
142. C₂×D₈	C₇⋊C₈	semidirect
261. C₈	C₂²⋊(C₇⋊C₈)	semidirect
262. C₇	#9657	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄×C₇×D₈	2
#11729368	2
C₂×D₄⋊(C₇⋊C₈)	2
C₂×#89253	2
C₂×C₈×C₇⋊C₈	2
#9657	7