Subgroups — #13268639

Order 1792.

614 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#405	C₂	1	1	yes	maximal, characteristic
2	C₂²×#6386	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	#11725463	C₂	1	4	yes	maximal, complemented
5	C₂²×C₇×M₁₆	C₄	1	1	yes	characteristic
6	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₇×#117	C₄	1	1	yes	characteristic
8	C₇×#405	—	2	2	no	—
9	C₇×#405	—	2	2	no	—
10	C₂×C₇×#97	C₄	1	2	yes	—
11	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
12	C₂×#6386	—	2	2	no	—
13	C₂×#6386	—	2	2	no	—
14	C₂×#6386	C₄	1	4	yes	complemented
15	C₂²×C₇⋊C₁₆	—	2	2	no	—
16	C₂²×C₇⋊C₁₆	—	2	2	no	—
17	C₂²×C₇⋊C₁₆	C₄	1	4	yes	complemented
18	C₂²×C₇⋊C₁₆	C₂²	1	2	yes	—
19	C₂²×C₇⋊C₁₆	C₂²	1	2	yes	—
20	#12545	—	7	1	no	maximal, Sylow
21	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₇×M₁₆	—	2	2	no	—
23	C₂×C₇×M₁₆	—	2	2	no	—
24	C₂×C₇×M₁₆	C₈	1	4	yes	—
25	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
26	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
28	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
29	C₂²×C₇×C₈	D₄	1	2	yes	—
30	C₂²×C₇×C₈	D₄	1	2	yes	—
31	C₂²×C₇×C₈	D₄	1	2	yes	—
32	C₂²×C₇×C₈	Q₈	1	2	yes	—
33	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
34	C₇×#117	—	2	2	no	—
35	C₇×#117	—	2	2	no	—
36	C₂×C₇×C₄⋊C₄	C₈	1	2	yes	—
37	C₂×C₇×C₂²⋊C₄	C₈	1	2	yes	—
38	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
39	C₇×#97	—	2	4	no	—
40	C₄×C₇×C₈	—	2	2	no	—
41	C₄×C₇×C₈	—	2	2	no	—
42	#6386	—	2	8	no	—
43	C₂×C₇⋊C₁₆	—	2	8	no	—
44	C₂×C₇⋊C₁₆	—	2	2	no	—
45	C₂×C₇⋊C₁₆	—	2	2	no	—
46	C₂×C₇⋊C₁₆	—	2	2	no	—
47	C₂×C₇⋊C₁₆	—	2	2	no	—
48	C₂×C₇⋊C₁₆	C₂×C₄	1	4	yes	—
49	C₂×C₇⋊C₁₆	C₂×C₄	1	4	yes	—
50	C₂²×M₃₂	—	7	1	no	—
51	C₂³×C₁₆	—	7	1	no	—
52	C₂×#405	D₇	1	1	yes	characteristic
53	#1374	—	7	4	no	—
54	C₂⁴×C₇	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₇	C₄²	1	2	yes	—
56	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₇	C₂×C₈	1	4	yes	—
59	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₇	M₁₆	1	2	yes	—
62	C₂²×C₄×C₇	M₁₆	1	2	yes	—
63	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
64	C₇×M₁₆	—	2	8	no	—
65	C₂×C₇×C₈	—	2	2	no	—
66	C₂×C₇×C₈	—	2	2	no	—
67	C₂×C₇×C₈	C₄²	1	2	yes	—
68	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
69	C₂×C₇×C₈	C₂²⋊C₄	1	4	yes	—
70	C₂×C₇×C₈	C₂²⋊C₄	1	4	yes	—
71	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
72	C₂×C₇×C₈	C₄⋊C₄	1	4	yes	—
73	C₂×C₇×C₈	C₄⋊C₄	1	4	yes	—
74	C₂×C₇×C₈	C₂×C₈	1	4	yes	—
75	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
78	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
79	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
80	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
81	C₇×C₄⋊C₄	—	2	4	no	—
82	C₇×C₂²⋊C₄	—	2	4	no	—
83	C₄²×C₇	—	2	2	no	—
84	C₄²×C₇	—	2	2	no	—
85	C₇⋊C₁₆	—	2	4	no	—
86	C₇⋊C₁₆	—	2	4	no	—
87	C₂²×M₁₆	Dic₇	1	1	yes	characteristic
88	C₂³×C₈	C₂×D₇	1	1	yes	characteristic
89	C₂×#117	Dic₇	1	1	yes	characteristic
90	C₂×M₃₂	—	7	4	no	—
91	C₂×M₃₂	—	14	2	no	—
92	C₂×M₃₂	—	14	2	no	—
93	C₂²×C₁₆	—	7	4	no	—
94	C₂²×C₁₆	—	7	2	no	—
95	C₂²×C₁₆	—	7	2	no	—
96	C₂²×C₁₆	—	14	2	no	—
97	C₂²×C₁₆	—	14	2	no	—
98	#405	—	2	2	no	—
99	#405	—	2	2	no	—
100	C₂×#97	Dic₇	1	2	yes	—
101	C₂×C₄×C₈	C₂×D₇	1	2	yes	—
102	C₂³×C₇	—	2	4	no	—
103	C₂³×C₇	C₄×C₈	1	2	yes	—
104	C₂³×C₇	C₂²⋊C₈	1	2	yes	—
105	C₂³×C₇	C₄⋊C₈	1	2	yes	—
106	C₂³×C₇	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₇	—	2	4	no	—
108	C₂×C₄×C₇	—	2	8	no	—
109	C₂×C₄×C₇	#95	1	4	yes	—
110	C₂×C₄×C₇	C₄×C₈	1	2	yes	—
111	C₂×C₄×C₇	#97	1	4	yes	—
112	C₂×C₄×C₇	C₂²⋊C₈	1	2	yes	—
113	C₂×C₄×C₇	C₂²⋊C₈	1	4	yes	—
114	C₂×C₄×C₇	C₄⋊C₈	1	2	yes	—
115	C₂×C₄×C₇	C₄⋊C₈	1	4	yes	—
116	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
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	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
121	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
122	C₇×C₈	—	2	4	no	—
123	C₇×C₈	#95	1	4	yes	—
124	C₇×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₇×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂³×C₄	C₂×Dic₇	1	1	yes	characteristic
129	C₂×M₁₆	—	2	2	no	—
130	C₂×M₁₆	—	2	2	no	—
131	C₂×M₁₆	C₇⋊C₈	1	4	yes	—
132	C₂²×C₈	Dic₁₄	1	2	yes	—
133	C₂²×C₈	C₄×D₇	1	2	yes	—
134	C₂²×C₈	C₄×D₇	1	2	yes	—
135	C₂²×C₈	D₂₈	1	2	yes	—
136	C₂²×C₈	C₂×Dic₇	1	2	yes	—
137	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
138	C₂²×C₈	C₂²⋊D₇	1	2	yes	—
139	C₂²×C₈	C₂²⋊D₇	1	2	yes	—
140	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
141	#117	—	2	2	no	—
142	#117	—	2	2	no	—
143	C₂×C₄⋊C₄	C₇⋊C₈	1	2	yes	—
144	C₂×C₂²⋊C₄	C₇⋊C₈	1	2	yes	—
145	C₂×C₄²	C₂×Dic₇	1	2	yes	—
146	M₃₂	—	14	8	no	—
147	C₂×C₁₆	—	7	4	no	—
148	C₂×C₁₆	—	7	4	no	—
149	C₂×C₁₆	—	14	8	no	—
150	C₂×C₁₆	—	14	2	no	—
151	C₂×C₁₆	—	14	2	no	—
152	C₂×C₁₆	—	14	2	no	—
153	C₂×C₁₆	—	14	2	no	—
154	#97	—	2	4	no	—
155	C₄×C₈	—	2	2	no	—
156	C₄×C₈	—	2	2	no	—
157	C₂²×C₇	—	2	4	no	—
158	C₂²×C₇	—	2	4	no	—
159	C₂²×C₇	—	2	4	no	—
160	C₂²×C₇	#336	1	4	yes	—
161	C₂²×C₇	C₂×C₄×C₈	1	1	yes	characteristic
162	C₂²×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
163	C₂²×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
164	C₂²×C₇	#504	1	4	yes	—
165	C₄×C₇	—	4	4	no	—
166	C₄×C₇	#336	1	4	yes	—
167	C₄×C₇	C₂×#95	1	1	yes	characteristic
168	C₄×C₇	C₂×#97	1	1	yes	characteristic
169	C₄×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
170	C₄×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
171	C₂⁴	C₂×C₇⋊C₈	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₇⋊C₈	1	4	yes	—
173	C₂²×C₄	C₂×C₇⋊C₈	1	2	yes	—
174	C₂²×C₄	C₂×C₇⋊C₈	1	1	yes	characteristic
175	C₂²×C₄	#1136	1	2	yes	—
176	C₂²×C₄	#1136	1	2	yes	—
177	C₂²×C₄	C₄×Dic₇	1	2	yes	—
178	C₂²×C₄	C₄⋊Dic₇	1	2	yes	—
179	C₂²×C₄	C₂²⋊Dic₇	1	2	yes	—
180	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
181	M₁₆	—	2	8	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	C₂×C₇⋊C₈	1	4	yes	—
185	C₂×C₈	C₄×Dic₇	1	2	yes	—
186	C₂×C₈	Dic₇⋊C₄	1	4	yes	—
187	C₂×C₈	Dic₇⋊C₄	1	4	yes	—
188	C₂×C₈	C₄⋊Dic₇	1	2	yes	—
189	C₂×C₈	(C₂×D₇)⋊C₄	1	4	yes	—
190	C₂×C₈	(C₂×D₇)⋊C₄	1	4	yes	—
191	C₂×C₈	C₂²⋊Dic₇	1	2	yes	—
192	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
193	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
194	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
195	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
196	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
197	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
198	C₄⋊C₄	—	2	4	no	—
199	C₂²⋊C₄	—	2	4	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	2	no	—
202	C₁₆	—	14	4	no	—
203	C₁₆	—	14	4	no	—
204	C₂×C₇	—	2	4	no	—
205	C₂×C₇	C₂×#336	1	1	yes	characteristic
206	C₂×C₇	#2108	1	2	yes	—
207	C₂×C₇	#2113	1	2	yes	—
208	C₂×C₇	#2151	1	2	yes	—
209	C₂³	—	2	4	no	—
210	C₂³	C₄×C₇⋊C₈	1	2	yes	—
211	C₂³	C₄⋊(C₇⋊C₈)	1	2	yes	—
212	C₂³	C₂²⋊(C₇⋊C₈)	1	2	yes	—
213	C₂³	C₂²×C₇⋊C₈	1	1	yes	characteristic
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	C₄×C₇⋊C₈	1	2	yes	—
217	C₂×C₄	#6377	1	4	yes	—
218	C₂×C₄	C₄⋊(C₇⋊C₈)	1	4	yes	—
219	C₂×C₄	C₄⋊(C₇⋊C₈)	1	2	yes	—
220	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	4	yes	—
221	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	2	yes	—
222	C₂×C₄	C₇⋊#95	1	4	yes	—
223	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
224	C₂×C₄	C₂×#1136	1	1	yes	characteristic
225	C₂×C₄	C₂×#1136	1	1	yes	characteristic
226	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
227	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
228	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
229	C₈	—	2	4	no	—
230	C₈	C₇⋊#95	1	4	yes	—
231	C₈	C₂×Dic₇⋊C₄	1	1	yes	characteristic
232	C₈	C₂×Dic₇⋊C₄	1	1	yes	characteristic
233	C₈	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
234	C₈	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
235	C₇	#12545	1	1	yes	characteristic, complemented, Sylow
236	C₂²	—	2	4	no	—
237	C₂²	—	2	4	no	—
238	C₂²	—	2	4	no	—
239	C₂²	#89319	1	4	yes	—
240	C₂²	C₂×C₄×C₇⋊C₈	1	1	yes	characteristic
241	C₂²	C₂×C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
242	C₂²	#89912	1	4	yes	—
243	C₂²	C₂×C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
244	C₄	—	4	4	no	—
245	C₄	#89319	1	4	yes	—
246	C₄	C₂×#6377	1	1	yes	characteristic
247	C₄	C₂×C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
248	C₄	C₂×C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
249	C₄	C₂×C₇⋊#95	1	1	yes	characteristic
250	C₂	—	2	4	no	—
251	C₂	#11727635	1	2	yes	—
252	C₂	#11727636	1	2	yes	—
253	C₂	C₂×#89319	1	1	yes	characteristic
254	C₂	#11729758	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇
commutator	C₂²×C₇	C₂×C₄×C₈
Fitting	C₂×C₇×#405	C₂
socle	C₂³×C₇	C₂²×C₈
prime radical	C₂⁴×C₇	C₂×C₈
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₂×#405	D₇
hypocenter	C₇	#12545
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
4. #11725463	C₂	semidirect
14. C₂×#6386	C₄	semidirect
17. C₂²×C₇⋊C₁₆	C₄	semidirect
235. C₇	#12545	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#405	2
C₂²×#6386	2
C₂³×C₇⋊C₁₆	2
#11725463	2
#12545	7