Subgroups — #13246925

Order 1792.

394 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	#11729376	C₂	1	2	yes	maximal
3	#11727546	C₂	1	1	yes	maximal, characteristic
4	#11727483	C₂	1	1	yes	maximal, characteristic
5	#11725513	C₂	1	2	yes	maximal
6	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
7	C₇×C₈×Q₈	—	2	1	no	—
8	C₇×C₈×Q₈	—	2	1	no	—
9	C₇×C₈×Q₈	C₄	1	2	yes	—
10	C₇×C₈×Q₈	C₄	1	2	yes	—
11	C₂×C₇×C₄⋊C₈	—	2	1	no	—
12	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₇×C₈	—	2	1	no	—
14	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
15	C₂×#6393	—	2	1	no	—
16	C₂×#6393	—	2	1	no	—
17	#89697	C₂²	1	2	yes	—
18	#89696	C₂²	1	2	yes	—
19	Q₈⋊(C₇⋊C₈)	—	2	4	no	—
20	#89253	—	2	1	no	—
21	#89252	—	2	1	no	—
22	#8659	—	7	1	no	maximal, Sylow
23	C₂²×C₇×Q₈	Q₈	1	1	yes	characteristic
24	C₂²×C₇×C₈	—	2	1	no	—
25	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
26	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
27	C₄×C₇×Q₈	—	2	1	no	—
28	C₄×C₇×Q₈	—	2	1	no	—
29	C₄×C₇×Q₈	C₂×C₄	1	2	yes	—
30	C₄×C₇×Q₈	D₄	1	2	yes	—
31	C₂×C₇×C₄⋊C₄	—	2	1	no	—
32	C₂×C₇×C₄⋊C₄	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₇	—	2	1	no	—
34	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
35	C₂×#1136	—	2	2	no	—
36	C₇×C₄⋊C₈	—	2	2	no	—
37	C₇×C₄⋊C₈	—	2	2	no	—
38	C₇×C₄⋊C₈	—	2	1	no	—
39	C₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
40	C₄×C₇×C₈	—	2	2	no	—
41	C₄×C₇×C₈	—	2	2	no	—
42	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
43	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
44	C₄×C₇×C₈	D₄	1	1	yes	characteristic
45	C₄×C₇×C₈	D₄	1	1	yes	characteristic
46	C₂²⋊(C₇⋊C₈)	—	2	2	no	—
47	#6393	—	4	2	no	—
48	#6393	—	4	2	no	—
49	C₄⋊(C₇⋊C₈)	—	2	2	no	—
50	C₄⋊(C₇⋊C₈)	—	2	2	no	—
51	C₄⋊(C₇⋊C₈)	—	2	2	no	—
52	C₄×C₇⋊C₈	—	2	2	no	—
53	C₂×C₈×Q₈	D₇	1	1	yes	characteristic
54	#1848	—	7	1	no	—
55	#1564	—	7	1	no	—
56	#1487	—	7	2	no	—
57	#1301	—	7	2	no	—
58	C₂×C₇×Q₈	—	2	4	no	—
59	C₂×C₇×Q₈	—	2	1	no	—
60	C₂×C₇×Q₈	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₇	—	2	1	no	—
62	C₂²×C₄×C₇	—	2	1	no	—
63	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₇	#41	1	1	yes	characteristic
66	C₂×C₇×C₈	—	2	1	no	—
67	C₂×C₇×C₈	—	2	1	no	—
68	C₂×C₇×C₈	—	2	2	no	—
69	C₂×C₇×C₈	—	4	1	no	—
70	C₂×C₇×C₈	—	4	1	no	—
71	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₇×C₈	SD₁₆	1	2	yes	—
76	C₂×C₇×C₈	Q₁₆	1	2	yes	—
77	C₇×C₄⋊C₄	—	2	2	no	—
78	C₇×C₄⋊C₄	—	2	2	no	—
79	C₇×C₄⋊C₄	—	2	1	no	—
80	C₇×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
81	C₄²×C₇	—	2	2	no	—
82	C₄²×C₇	—	2	2	no	—
83	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₇	#41	1	1	yes	characteristic
87	#1136	—	4	2	no	—
88	#1136	—	4	2	no	—
89	C₂×C₇⋊C₈	—	4	2	no	—
90	C₂×C₇⋊C₈	—	4	2	no	—
91	C₂×C₄×Q₈	C₂×D₇	1	1	yes	characteristic
92	C₈×Q₈	—	2	1	no	—
93	C₈×Q₈	—	2	1	no	—
94	C₈×Q₈	Dic₇	1	2	yes	—
95	C₈×Q₈	Dic₇	1	2	yes	—
96	#431	—	7	2	no	—
97	C₂×#108	—	14	1	no	—
98	C₂×#108	—	14	1	no	—
99	#423	—	7	2	no	—
100	C₂×C₄⋊C₈	—	2	1	no	—
101	C₂×C₄⋊C₈	C₂×D₇	1	1	yes	characteristic
102	C₂×C₄×C₈	—	2	1	no	—
103	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
104	#335	—	14	1	no	—
105	#334	—	14	1	no	—
106	Q₈⋊C₈	—	14	4	no	—
107	C₂³×C₇	C₂²⋊Q₈	1	1	yes	characteristic
108	C₇×Q₈	—	2	4	no	—
109	C₇×Q₈	—	2	2	no	—
110	C₇×Q₈	#108	1	4	yes	complemented
111	C₂×C₄×C₇	—	2	4	no	—
112	C₂×C₄×C₇	—	2	2	no	—
113	C₂×C₄×C₇	—	2	2	no	—
114	C₂×C₄×C₇	—	2	2	no	—
115	C₂×C₄×C₇	—	4	1	no	—
116	C₂×C₄×C₇	—	4	2	no	—
117	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₇	#117	1	1	yes	characteristic
120	C₂×C₄×C₇	#121	1	1	yes	characteristic
121	C₂×C₄×C₇	#121	1	1	yes	characteristic
122	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
123	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
125	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₂×C₄×C₇	#137	1	1	yes	characteristic
127	C₇×C₈	—	2	1	no	—
128	C₇×C₈	—	2	1	no	—
129	C₇×C₈	—	4	2	no	—
130	C₇×C₈	#103	1	2	yes	—
131	C₇×C₈	#103	1	2	yes	—
132	C₇⋊C₈	—	8	2	no	—
133	C₇⋊C₈	—	8	2	no	—
134	C₂²×Q₈	Dic₁₄	1	1	yes	characteristic
135	C₂×M₁₆	—	14	2	no	—
136	C₂²×C₈	—	2	1	no	—
137	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
138	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
139	C₄×Q₈	—	2	1	no	—
140	C₄×Q₈	—	2	1	no	—
141	C₄×Q₈	D₂₈	1	2	yes	—
142	C₄×Q₈	C₂×Dic₇	1	2	yes	—
143	C₂×C₄⋊C₄	—	2	1	no	—
144	C₂×C₄⋊C₄	Dic₁₄	1	1	yes	characteristic
145	C₂×C₄²	—	2	1	no	—
146	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
147	#108	—	28	2	no	—
148	#108	—	28	2	no	—
149	C₄⋊C₈	—	2	2	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	2	1	no	—
152	C₄⋊C₈	—	14	2	no	—
153	C₄⋊C₈	—	14	2	no	—
154	C₄⋊C₈	—	14	2	no	—
155	C₄⋊C₈	C₂×Dic₇	1	2	yes	—
156	C₂²⋊C₈	—	14	2	no	—
157	C₄×C₈	—	2	2	no	—
158	C₄×C₈	—	2	2	no	—
159	C₄×C₈	—	14	2	no	—
160	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
161	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
162	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
163	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
164	C₂²×C₇	—	2	1	no	—
165	C₂²×C₇	—	2	1	no	—
166	C₂²×C₇	#380	1	1	yes	characteristic
167	C₂²×C₇	#475	1	1	yes	characteristic
168	C₂²×C₇	#477	1	1	yes	characteristic
169	C₄×C₇	—	2	4	no	—
170	C₄×C₇	—	2	1	no	—
171	C₄×C₇	—	2	1	no	—
172	C₄×C₇	—	2	1	no	—
173	C₄×C₇	—	4	2	no	—
174	C₄×C₇	C₂×#103	1	1	yes	characteristic
175	C₄×C₇	#417	1	1	yes	characteristic
176	C₄×C₇	C₂×#108	1	2	yes	—
177	C₄×C₇	#460	1	1	yes	characteristic
178	C₄×C₇	#462	1	1	yes	characteristic
179	C₂×Q₈	—	2	4	no	—
180	C₂×Q₈	—	2	1	no	—
181	C₂×Q₈	C₄⋊Dic₇	1	2	yes	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	C₂×Dic₁₄	1	1	yes	characteristic
185	C₂²×C₄	#1157	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
187	M₁₆	—	28	2	no	—
188	M₁₆	—	28	2	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	4	1	no	—
193	C₂×C₈	—	4	1	no	—
194	C₂×C₈	—	28	2	no	—
195	C₂×C₈	—	28	2	no	—
196	C₂×C₈	#1143	1	2	yes	—
197	C₂×C₈	#1144	1	2	yes	—
198	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
199	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
200	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
201	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
202	C₄⋊C₄	—	2	1	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	C₄⋊Dic₇	1	2	yes	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	2	no	—
208	C₄²	C₂×D₂₈	1	1	yes	characteristic
209	C₄²	#1157	1	1	yes	characteristic
210	C₄²	C₂²×Dic₇	1	1	yes	characteristic
211	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
212	C₂×C₇	—	2	1	no	—
213	C₂×C₇	—	2	1	no	—
214	C₂×C₇	#1803	1	1	yes	characteristic
215	C₂×C₇	#1810	1	1	yes	characteristic
216	C₂×C₇	#1814	1	1	yes	characteristic
217	C₂³	#6487	1	1	yes	characteristic
218	Q₈	—	2	4	no	—
219	Q₈	—	2	2	no	—
220	Q₈	#6393	1	4	yes	complemented
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	#6487	1	1	yes	characteristic
228	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
229	C₂×C₄	#6489	1	1	yes	characteristic
230	C₂×C₄	#6493	1	1	yes	characteristic
231	C₂×C₄	#6493	1	1	yes	characteristic
232	C₂×C₄	C₂×#1143	1	1	yes	characteristic
233	C₂×C₄	C₂×#1144	1	1	yes	characteristic
234	C₂×C₄	#6512	1	1	yes	characteristic
235	C₂×C₄	#6514	1	1	yes	characteristic
236	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
237	C₈	—	2	1	no	—
238	C₈	—	2	1	no	—
239	C₈	—	4	2	no	—
240	C₈	—	56	2	no	—
241	C₈	—	56	2	no	—
242	C₈	#6409	1	2	yes	—
243	C₈	#6409	1	2	yes	—
244	C₇	#8659	1	1	yes	characteristic, complemented, Sylow
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	#89792	1	1	yes	characteristic
248	C₂²	#89793	1	1	yes	characteristic
249	C₂²	#89984	1	1	yes	characteristic
250	C₄	—	2	4	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	4	2	no	—
255	C₄	#89799	1	1	yes	characteristic
256	C₄	#89803	1	1	yes	characteristic
257	C₄	C₂×#6393	1	2	yes	—
258	C₄	C₂×#6409	1	1	yes	characteristic
259	C₄	#90004	1	1	yes	characteristic
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	#11729380	1	1	yes	characteristic
263	C₂	#11729731	1	1	yes	characteristic
264	C₂	#11729767	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6493
commutator	C₄²×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₂×C₇×C₈×Q₈	C₂
socle	C₂²×C₇	#380
prime radical	C₂³×C₇	C₂²⋊Q₈
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	C₂×C₈×Q₈	D₇
hypocenter	C₇	#8659
LCS	G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
110. C₇×Q₈	#108	semidirect
220. Q₈	#6393	semidirect
244. C₇	#8659	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×C₈×Q₈	2
#11729376	2
#11727546	2
#11727483	2
#11725513	2
#8659	7