Subgroups — #13268636

Order 1792.

678 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×#6386	C₂	1	2	yes	maximal
3	#11725463	C₂	1	4	yes	maximal, complemented
4	C₂³×C₇×C₈	C₄	1	1	yes	characteristic
5	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₇	C₄	1	1	yes	characteristic
7	C₂×C₄×C₇×C₈	—	2	4	no	—
8	C₂×C₄×C₇×C₈	C₄	1	2	yes	—
9	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
10	C₂×#6386	—	2	4	no	—
11	C₂×#6386	—	2	4	no	—
12	C₂×#6386	C₄	1	8	yes	complemented
13	C₂²×C₇⋊C₁₆	C₂²	1	4	yes	—
14	#12544	—	7	1	no	maximal, Sylow
15	C₂³×C₄×C₇	C₈	1	2	yes	—
16	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₇×C₈	—	2	2	no	—
18	C₂²×C₇×C₈	—	2	2	no	—
19	C₂²×C₇×C₈	C₈	1	4	yes	—
20	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
23	C₂²×C₇×C₈	D₄	1	2	yes	—
24	C₂²×C₇×C₈	D₄	1	4	yes	—
25	C₂²×C₇×C₈	Q₈	1	2	yes	—
26	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₇	—	2	2	no	—
28	C₂×C₄²×C₇	—	2	2	no	—
29	C₂×C₄²×C₇	C₈	1	2	yes	—
30	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
31	C₄×C₇×C₈	—	2	4	no	—
32	C₄×C₇×C₈	—	2	4	no	—
33	#6386	—	2	16	no	—
34	C₂×C₇⋊C₁₆	—	2	4	no	—
35	C₂×C₇⋊C₁₆	—	2	4	no	—
36	C₂×C₇⋊C₁₆	C₂×C₄	1	8	yes	complemented
37	C₂²×M₃₂	—	7	2	no	—
38	C₂²×C₄×C₈	D₇	1	1	yes	characteristic
39	#1374	—	7	4	no	—
40	C₂⁴×C₇	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₇	—	2	4	no	—
42	C₂²×C₄×C₇	—	2	8	no	—
43	C₂²×C₄×C₇	C₄²	1	2	yes	—
44	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
46	C₂²×C₄×C₇	C₂×C₈	1	4	yes	—
47	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₇	M₁₆	1	2	yes	—
50	C₂²×C₄×C₇	M₁₆	1	2	yes	—
51	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₇×C₈	—	2	8	no	—
53	C₂×C₇×C₈	—	2	2	no	—
54	C₂×C₇×C₈	—	2	2	no	—
55	C₂×C₇×C₈	C₄²	1	2	yes	—
56	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
57	C₂×C₇×C₈	C₂²⋊C₄	1	8	yes	—
58	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
59	C₂×C₇×C₈	C₄⋊C₄	1	8	yes	—
60	C₂×C₇×C₈	C₂×C₈	1	4	yes	—
61	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
62	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
63	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
64	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
65	C₄²×C₇	—	2	4	no	—
66	C₄²×C₇	—	2	2	no	—
67	C₄²×C₇	—	2	2	no	—
68	C₇⋊C₁₆	—	2	8	no	—
69	C₂³×C₈	Dic₇	1	1	yes	characteristic
70	C₂³×C₈	C₂×D₇	1	1	yes	characteristic
71	C₂²×C₄²	Dic₇	1	1	yes	characteristic
72	C₂×M₃₂	—	7	8	no	—
73	C₂×M₃₂	—	14	4	no	—
74	C₂×M₃₂	—	14	4	no	—
75	C₂²×C₁₆	—	7	4	no	—
76	C₂×C₄×C₈	—	2	4	no	—
77	C₂×C₄×C₈	Dic₇	1	2	yes	—
78	C₂×C₄×C₈	C₂×D₇	1	2	yes	—
79	C₂³×C₇	—	2	4	no	—
80	C₂³×C₇	C₄×C₈	1	2	yes	—
81	C₂³×C₇	C₂²⋊C₈	1	2	yes	—
82	C₂³×C₇	C₄⋊C₈	1	2	yes	—
83	C₂³×C₇	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₇	—	2	16	no	—
85	C₂×C₄×C₇	—	2	4	no	—
86	C₂×C₄×C₇	—	2	4	no	—
87	C₂×C₄×C₇	#95	1	4	yes	—
88	C₂×C₄×C₇	C₄×C₈	1	2	yes	—
89	C₂×C₄×C₇	#97	1	4	yes	—
90	C₂×C₄×C₇	C₂²⋊C₈	1	2	yes	—
91	C₂×C₄×C₇	C₂²⋊C₈	1	4	yes	—
92	C₂×C₄×C₇	C₄⋊C₈	1	2	yes	—
93	C₂×C₄×C₇	C₄⋊C₈	1	4	yes	—
94	C₂×C₄×C₇	M₃₂	1	8	yes	complemented
95	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
96	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
100	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
101	C₇×C₈	—	2	4	no	—
102	C₇×C₈	#95	1	4	yes	—
103	C₇×C₈	C₂×C₂²⋊C₄	1	2	yes	—
104	C₇×C₈	C₂×C₄⋊C₄	1	2	yes	—
105	C₂³×C₄	C₇⋊C₈	1	2	yes	—
106	C₂³×C₄	C₂×Dic₇	1	1	yes	characteristic
107	C₂²×C₈	—	2	2	no	—
108	C₂²×C₈	—	2	2	no	—
109	C₂²×C₈	C₇⋊C₈	1	4	yes	—
110	C₂²×C₈	Dic₁₄	1	2	yes	—
111	C₂²×C₈	C₄×D₇	1	4	yes	—
112	C₂²×C₈	D₂₈	1	2	yes	—
113	C₂²×C₈	C₂×Dic₇	1	2	yes	—
114	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
115	C₂²×C₈	C₂²⋊D₇	1	4	yes	—
116	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
117	C₂×C₄²	—	2	2	no	—
118	C₂×C₄²	—	2	2	no	—
119	C₂×C₄²	C₇⋊C₈	1	2	yes	—
120	C₂×C₄²	C₂×Dic₇	1	2	yes	—
121	M₃₂	—	14	16	no	—
122	C₂×C₁₆	—	7	8	no	—
123	C₂×C₁₆	—	14	4	no	—
124	C₂×C₁₆	—	14	4	no	—
125	C₄×C₈	—	2	4	no	—
126	C₄×C₈	—	2	4	no	—
127	C₂²×C₇	—	2	4	no	—
128	C₂²×C₇	—	2	4	no	—
129	C₂²×C₇	—	2	4	no	—
130	C₂²×C₇	#336	1	4	yes	—
131	C₂²×C₇	C₂×C₄×C₈	1	1	yes	characteristic
132	C₂²×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
133	C₂²×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
134	C₂²×C₇	C₂×M₃₂	1	4	yes	—
135	C₄×C₇	—	2	8	no	—
136	C₄×C₇	#336	1	4	yes	—
137	C₄×C₇	C₂×#95	1	1	yes	characteristic
138	C₄×C₇	C₂×#97	1	1	yes	characteristic
139	C₄×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
140	C₄×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
141	C₂⁴	C₂×C₇⋊C₈	1	1	yes	characteristic
142	C₂²×C₄	—	2	4	no	—
143	C₂²×C₄	—	2	8	no	—
144	C₂²×C₄	C₂×C₇⋊C₈	1	4	yes	—
145	C₂²×C₄	C₂×C₇⋊C₈	1	2	yes	—
146	C₂²×C₄	C₂×C₇⋊C₈	1	1	yes	characteristic
147	C₂²×C₄	#1136	1	2	yes	—
148	C₂²×C₄	#1136	1	2	yes	—
149	C₂²×C₄	C₄×Dic₇	1	2	yes	—
150	C₂²×C₄	C₄⋊Dic₇	1	2	yes	—
151	C₂²×C₄	C₂²⋊Dic₇	1	2	yes	—
152	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
153	C₂×C₈	—	2	8	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	C₂×C₇⋊C₈	1	4	yes	—
157	C₂×C₈	C₄×Dic₇	1	2	yes	—
158	C₂×C₈	Dic₇⋊C₄	1	8	yes	—
159	C₂×C₈	C₄⋊Dic₇	1	2	yes	—
160	C₂×C₈	(C₂×D₇)⋊C₄	1	8	yes	—
161	C₂×C₈	C₂²⋊Dic₇	1	2	yes	—
162	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
163	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
164	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
165	C₂×C₈	C₂×C₂²⋊D₇	1	2	yes	—
166	C₄²	—	2	4	no	—
167	C₄²	—	2	2	no	—
168	C₄²	—	2	2	no	—
169	C₁₆	—	14	8	no	—
170	C₂×C₇	—	2	4	no	—
171	C₂×C₇	C₂×#336	1	1	yes	characteristic
172	C₂×C₇	C₄×M₃₂	1	2	yes	—
173	C₂×C₇	#2112	1	2	yes	—
174	C₂×C₇	#2150	1	2	yes	—
175	C₂³	—	2	4	no	—
176	C₂³	C₄×C₇⋊C₈	1	2	yes	—
177	C₂³	C₄⋊(C₇⋊C₈)	1	2	yes	—
178	C₂³	C₂²⋊(C₇⋊C₈)	1	2	yes	—
179	C₂³	C₂²×C₇⋊C₈	1	1	yes	characteristic
180	C₂×C₄	—	2	16	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	C₄×C₇⋊C₈	1	2	yes	—
184	C₂×C₄	#6377	1	4	yes	—
185	C₂×C₄	C₄⋊(C₇⋊C₈)	1	4	yes	—
186	C₂×C₄	C₄⋊(C₇⋊C₈)	1	2	yes	—
187	C₂×C₄	#6386	1	8	yes	complemented
188	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	4	yes	—
189	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	2	yes	—
190	C₂×C₄	C₇⋊#95	1	4	yes	—
191	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
192	C₂×C₄	C₂×#1136	1	1	yes	characteristic
193	C₂×C₄	C₂×#1136	1	1	yes	characteristic
194	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
195	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
196	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
197	C₈	—	2	4	no	—
198	C₈	C₇⋊#95	1	4	yes	—
199	C₈	C₂×Dic₇⋊C₄	1	2	yes	—
200	C₈	C₂×(C₂×D₇)⋊C₄	1	2	yes	—
201	C₇	#12544	1	1	yes	characteristic, complemented, Sylow
202	C₂²	—	2	4	no	—
203	C₂²	—	2	4	no	—
204	C₂²	—	2	4	no	—
205	C₂²	#89319	1	4	yes	—
206	C₂²	C₂×C₄×C₇⋊C₈	1	1	yes	characteristic
207	C₂²	C₂×C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
208	C₂²	C₂×#6386	1	4	yes	—
209	C₂²	C₂×C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
210	C₄	—	2	8	no	—
211	C₄	#89319	1	4	yes	—
212	C₄	C₂×#6377	1	1	yes	characteristic
213	C₄	C₂×C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
214	C₄	C₂×C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
215	C₄	C₂×C₇⋊#95	1	1	yes	characteristic
216	C₂	—	2	4	no	—
217	C₂	C₄×#6386	1	2	yes	—
218	C₂	#11727497	1	2	yes	—
219	C₂	C₂×#89319	1	1	yes	characteristic
220	C₂	#11729717	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇
commutator	C₂²×C₇	C₂×C₄×C₈
Fitting	C₂²×C₄×C₇×C₈	C₂
socle	C₂³×C₇	C₂²×C₈
prime radical	C₂⁴×C₇	C₂×C₈
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₂²×C₄×C₈	D₇
hypocenter	C₇	#12544
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #11725463	C₂	semidirect
12. C₂×#6386	C₄	semidirect
36. C₂×C₇⋊C₁₆	C₂×C₄	semidirect
94. C₂×C₄×C₇	M₃₂	semidirect
187. C₂×C₄	#6386	semidirect
201. C₇	#12544	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₇×C₈	2
C₂²×#6386	2
#11725463	2
#12544	7