Subgroups — #13246738

Order 1792.

378 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#89451	C₂	1	1	yes	maximal, characteristic
2	C₇×#1567	C₂	1	1	yes	maximal, characteristic
3	#11728282	C₂	1	2	yes	maximal
4	#11727484	C₂	1	1	yes	maximal, characteristic
5	#11725517	C₂	1	2	yes	maximal
6	C₂×C₄×Dic₁₄	C₂²	1	1	yes	characteristic
7	C₇×#423	C₂²	1	2	yes	—
8	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₂×#6398	—	2	2	no	—
10	C₂×#6389	C₄	1	2	yes	—
11	C₂×Dic₇⋊C₈	C₄	1	2	yes	—
12	#89696	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
14	#89451	—	2	1	no	—
15	#89451	—	2	1	no	—
16	#89451	—	2	1	no	—
17	#89451	—	2	1	no	—
18	C₇×#335	—	2	1	no	—
19	C₇×#335	—	2	1	no	—
20	#89349	—	2	2	no	—
21	#89252	—	2	1	no	—
22	#89252	—	2	1	no	—
23	#8619	—	7	1	no	maximal, Sylow
24	C₂²×Dic₁₄	D₄	1	1	yes	characteristic
25	C₂×C₇×M₁₆	—	2	2	no	—
26	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊Dic₇	D₄	1	1	yes	characteristic
30	C₂×Dic₇⋊C₄	Q₈	1	2	yes	—
31	C₂×C₄×Dic₇	C₂×C₄	1	2	yes	—
32	C₂×#1136	—	2	2	no	—
33	C₂²×C₇⋊C₈	C₂×C₄	1	2	yes	—
34	C₄×Dic₁₄	—	2	1	no	—
35	C₄×Dic₁₄	—	2	1	no	—
36	C₄×Dic₁₄	—	2	1	no	—
37	C₄×Dic₁₄	—	2	1	no	—
38	C₇×C₄⋊C₈	—	2	2	no	—
39	C₇×C₄⋊C₈	—	2	2	no	—
40	C₄×C₇×C₈	D₄	1	1	yes	characteristic
41	C₄×C₇×C₈	D₄	1	1	yes	characteristic
42	C₄×C₇×C₈	D₄	1	1	yes	characteristic
43	C₄×C₇×C₈	D₄	1	1	yes	characteristic
44	#6398	—	4	4	no	—
45	#6389	—	2	2	no	—
46	#6389	—	2	2	no	—
47	Dic₇⋊C₈	—	2	2	no	—
48	Dic₇⋊C₈	—	2	2	no	—
49	C₄⋊(C₇⋊C₈)	—	2	1	no	—
50	C₄⋊(C₇⋊C₈)	—	2	1	no	—
51	C₄⋊(C₇⋊C₈)	—	2	2	no	—
52	C₄⋊(C₇⋊C₈)	—	2	2	no	—
53	C₂×#446	—	7	1	no	—
54	#1801	—	7	2	no	—
55	#1567	D₇	1	1	yes	characteristic
56	#1566	—	7	1	no	—
57	#1292	—	7	2	no	—
58	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
61	C₂²×Dic₇	C₄⋊C₄	1	2	yes	—
62	C₂²×Dic₇	C₄⋊C₄	1	2	yes	—
63	C₂×Dic₁₄	—	2	1	no	—
64	C₂×Dic₁₄	—	2	1	no	—
65	C₂×Dic₁₄	—	2	2	no	—
66	C₂×Dic₁₄	—	2	2	no	—
67	C₇×M₁₆	—	4	2	no	—
68	C₇×M₁₆	—	4	2	no	—
69	C₂×C₇×C₈	—	2	2	no	—
70	C₂×C₇×C₈	—	4	1	no	—
71	C₂×C₇×C₈	—	4	2	no	—
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	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₇	#41	1	1	yes	characteristic
79	C₄²×C₇	#41	1	1	yes	characteristic
80	C₄⋊Dic₇	—	2	1	no	—
81	C₄⋊Dic₇	—	2	1	no	—
82	Dic₇⋊C₄	—	2	2	no	—
83	Dic₇⋊C₄	—	2	2	no	—
84	C₄×Dic₇	—	2	2	no	—
85	C₄×Dic₇	—	2	2	no	—
86	#1136	—	4	2	no	—
87	#1136	—	4	2	no	—
88	C₂×C₇⋊C₈	—	2	2	no	—
89	C₂×C₇⋊C₈	—	4	2	no	—
90	C₂×C₇⋊C₈	—	4	2	no	—
91	C₂×C₄×Q₈	—	7	1	no	—
92	#446	—	14	1	no	—
93	#446	—	14	1	no	—
94	#446	—	14	1	no	—
95	#446	—	14	1	no	—
96	#423	—	7	2	no	—
97	#423	C₂×D₇	1	2	yes	—
98	C₂×C₄⋊C₈	—	7	2	no	—
99	C₂×C₄⋊C₈	—	7	1	no	—
100	C₂×#101	—	14	2	no	—
101	C₂×#97	—	7	2	no	—
102	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
103	#343	—	14	2	no	—
104	#335	—	2	1	no	—
105	#335	—	2	1	no	—
106	#334	—	14	1	no	—
107	#334	—	14	1	no	—
108	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₇	—	2	1	no	—
110	C₂×C₄×C₇	—	2	1	no	—
111	C₂×C₄×C₇	—	4	1	no	—
112	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₇	#117	1	1	yes	characteristic
114	C₂×C₄×C₇	#121	1	1	yes	characteristic
115	C₂×C₄×C₇	#121	1	1	yes	characteristic
116	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
119	C₂×C₄×C₇	#137	1	2	yes	—
120	C₂×Dic₇	—	2	2	no	—
121	C₂×Dic₇	—	2	2	no	—
122	C₂×Dic₇	—	2	4	no	—
123	C₂×Dic₇	—	4	2	no	—
124	Dic₁₄	—	4	2	no	—
125	Dic₁₄	—	4	1	no	—
126	Dic₁₄	—	4	1	no	—
127	C₇×C₈	—	2	1	no	—
128	C₇×C₈	—	2	1	no	—
129	C₇×C₈	—	2	1	no	—
130	C₇×C₈	—	8	2	no	—
131	C₇×C₈	#101	1	2	yes	complemented
132	C₇⋊C₈	—	4	2	no	—
133	C₇⋊C₈	—	8	2	no	—
134	C₂²×Q₈	—	7	1	no	—
135	C₂×M₁₆	—	2	2	no	—
136	C₂×M₁₆	—	14	2	no	—
137	C₂²×C₈	—	7	2	no	—
138	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
139	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
140	C₄×Q₈	—	14	1	no	—
141	C₄×Q₈	—	14	1	no	—
142	C₄×Q₈	—	14	1	no	—
143	C₄×Q₈	—	14	1	no	—
144	C₂×C₄⋊C₄	—	7	1	no	—
145	C₂×C₄⋊C₄	—	7	2	no	—
146	C₂×C₄²	—	7	2	no	—
147	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
148	C₄⋊C₈	—	2	2	no	—
149	C₄⋊C₈	—	2	2	no	—
150	C₄⋊C₈	—	14	2	no	—
151	C₄⋊C₈	—	14	2	no	—
152	C₄⋊C₈	—	14	1	no	—
153	C₄⋊C₈	—	14	2	no	—
154	C₄⋊C₈	—	14	2	no	—
155	C₄⋊C₈	—	14	1	no	—
156	#101	—	28	4	no	—
157	#97	—	14	2	no	—
158	#97	—	14	2	no	—
159	C₄×C₈	D₂₈	1	1	yes	characteristic
160	C₄×C₈	D₂₈	1	1	yes	characteristic
161	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
162	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
163	C₂²×C₇	—	2	1	no	—
164	C₂²×C₇	—	2	1	no	—
165	C₂²×C₇	#380	1	1	yes	characteristic
166	C₂²×C₇	#428	1	2	yes	—
167	C₄×C₇	—	2	1	no	—
168	C₄×C₇	—	2	1	no	—
169	C₄×C₇	—	2	1	no	—
170	C₄×C₇	C₂×#101	1	1	yes	characteristic
171	C₄×C₇	#413	1	1	yes	characteristic
172	C₄×C₇	#468	1	2	yes	—
173	C₄×C₇	#469	1	1	yes	characteristic
174	C₄×C₇	#470	1	1	yes	characteristic
175	Dic₇	—	4	2	no	—
176	Dic₇	—	4	2	no	—
177	C₂×Q₈	—	14	1	no	—
178	C₂×Q₈	—	14	2	no	—
179	C₂×Q₈	—	14	2	no	—
180	C₂×Q₈	—	14	1	no	—
181	C₂²×C₄	—	7	2	no	—
182	C₂²×C₄	—	7	2	no	—
183	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
184	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
185	C₂²×C₄	Q₈×D₇	1	1	yes	characteristic
186	M₁₆	—	4	2	no	—
187	M₁₆	—	4	2	no	—
188	M₁₆	—	28	2	no	—
189	M₁₆	—	28	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	4	1	no	—
192	C₂×C₈	—	4	2	no	—
193	C₂×C₈	—	14	2	no	—
194	C₂×C₈	—	28	2	no	—
195	C₂×C₈	—	28	2	no	—
196	C₂×C₈	(C₂×D₇)⋊C₄	1	1	yes	characteristic
197	C₂×C₈	(C₂×D₇)⋊C₄	1	1	yes	characteristic
198	C₂×C₈	(C₂×D₇)⋊C₄	1	1	yes	characteristic
199	C₂×C₈	(C₂×D₇)⋊C₄	1	1	yes	characteristic
200	C₄⋊C₄	—	14	1	no	—
201	C₄⋊C₄	—	14	2	no	—
202	C₄⋊C₄	—	14	1	no	—
203	C₄⋊C₄	—	14	2	no	—
204	C₄²	—	14	2	no	—
205	C₄²	—	14	2	no	—
206	C₄²	C₂×D₂₈	1	1	yes	characteristic
207	C₄²	#1159	1	1	yes	characteristic
208	C₄²	#1161	1	1	yes	characteristic
209	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
210	C₂×C₇	—	2	2	no	—
211	C₂×C₇	#1801	1	1	yes	characteristic
212	C₂×C₇	#1809	1	1	yes	characteristic
213	C₂×C₇	#1813	1	1	yes	characteristic
214	C₂³	D₇×C₄⋊C₄	1	1	yes	characteristic
215	Q₈	—	28	2	no	—
216	Q₈	—	28	1	no	—
217	Q₈	—	28	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	4	1	no	—
221	C₂×C₄	—	14	4	no	—
222	C₂×C₄	—	14	2	no	—
223	C₂×C₄	—	14	2	no	—
224	C₂×C₄	—	28	2	no	—
225	C₂×C₄	#6455	1	1	yes	characteristic
226	C₂×C₄	#6458	1	1	yes	characteristic
227	C₂×C₄	#6460	1	1	yes	characteristic
228	C₂×C₄	#6474	1	2	yes	—
229	C₂×C₄	#6481	1	2	yes	—
230	C₂×C₄	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
231	C₂×C₄	#6501	1	1	yes	characteristic
232	C₂×C₄	#6509	1	1	yes	characteristic
233	C₈	—	2	1	no	—
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	8	2	no	—
237	C₈	—	28	2	no	—
238	C₈	—	56	2	no	—
239	C₈	#6398	1	2	yes	complemented
240	C₇	#8619	1	1	yes	characteristic, complemented, Sylow
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	#89653	1	2	yes	—
244	C₂²	#89759	1	1	yes	characteristic
245	C₄	—	2	1	no	—
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	28	2	no	—
249	C₄	—	28	2	no	—
250	C₄	#89658	1	2	yes	—
251	C₄	#89904	1	1	yes	characteristic
252	C₄	C₂×#6398	1	1	yes	characteristic
253	C₄	#89931	1	1	yes	characteristic
254	C₄	#89961	1	1	yes	characteristic
255	C₂	—	2	2	no	—
256	C₂	#11728282	1	1	yes	characteristic
257	C₂	#11728385	1	1	yes	characteristic
258	C₂	#11728472	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#89759
commutator	C₂²×C₄×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1567	C₂
socle	C₂²×C₇	#380
prime radical	C₂³×C₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#1567	D₇
hypocenter	C₇	#8619
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
131. C₇×C₈	#101	semidirect
239. C₈	#6398	semidirect
240. C₇	#8619	semidirect

Maximal subgroups

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

subgroup	index
C₂×#89451	2
C₇×#1567	2
#11728282	2
#11727484	2
#11725517	2
#8619	7