Subgroups — #13268652

Order 1792.

586 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	C₂×C₇×#405	C₂	1	1	yes	maximal, characteristic
2	C₂²×#6386	C₂	1	2	yes	maximal
3	#11725464	C₂	1	4	yes	maximal, complemented
4	C₂²×C₇×M₁₆	C₄	1	1	yes	characteristic
5	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
6	C₂×C₇×#117	C₄	1	1	yes	characteristic
7	C₇×#405	—	2	4	no	—
8	C₂×C₇×#97	C₄	1	2	yes	—
9	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
10	C₂×#6386	—	2	2	no	—
11	C₂×#6386	—	2	4	no	—
12	C₂×#6386	C₄	1	8	yes	complemented
13	C₂×#6386	C₂²	1	4	yes	—
14	C₂²×C₇⋊C₁₆	—	2	2	no	—
15	#12549	—	7	1	no	maximal, Sylow
16	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₇×M₁₆	—	2	2	no	—
18	C₂×C₇×M₁₆	—	2	2	no	—
19	C₂×C₇×M₁₆	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	C₇×#117	—	2	2	no	—
28	C₇×#117	C₈	1	4	yes	—
29	C₂×C₇×C₄⋊C₄	—	2	1	no	—
30	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
31	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
32	C₇×#97	—	2	4	no	—
33	C₄×C₇×C₈	—	2	4	no	—
34	#6386	—	2	8	no	—
35	#6386	—	2	4	no	—
36	#6386	C₂×C₄	1	8	yes	—
37	C₂×C₇⋊C₁₆	—	2	8	no	—
38	C₂×C₇⋊C₁₆	—	2	4	no	—
39	C₂²×M₃₂	—	7	2	no	—
40	C₂×#405	D₇	1	1	yes	characteristic
41	#1376	—	7	4	no	—
42	C₂⁴×C₇	M₁₆	1	1	yes	characteristic
43	C₂²×C₄×C₇	—	2	2	no	—
44	C₂²×C₄×C₇	C₄²	1	2	yes	—
45	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
50	C₂²×C₄×C₇	M₁₆	1	2	yes	—
51	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
52	C₇×M₁₆	—	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	4	yes	—
58	C₂×C₇×C₈	C₂²⋊C₄	1	4	yes	—
59	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
60	C₂×C₇×C₈	C₄⋊C₄	1	8	yes	—
61	C₂×C₇×C₈	C₂×C₈	1	4	yes	—
62	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
63	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
64	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
65	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
66	C₇×C₄⋊C₄	—	2	4	no	—
67	C₇×C₂²⋊C₄	—	2	4	no	—
68	C₄²×C₇	—	2	2	no	—
69	C₄²×C₇	C₂×C₈	1	4	yes	—
70	C₇⋊C₁₆	—	2	8	no	—
71	C₂²×M₁₆	Dic₇	1	1	yes	characteristic
72	C₂³×C₈	C₂×D₇	1	1	yes	characteristic
73	C₂×#117	Dic₇	1	1	yes	characteristic
74	C₂×M₃₂	—	7	8	no	—
75	C₂×M₃₂	—	7	4	no	—
76	C₂×M₃₂	—	14	2	no	—
77	C₂×M₃₂	—	14	4	no	—
78	C₂²×C₁₆	—	14	2	no	—
79	#405	—	2	4	no	—
80	C₂×#97	Dic₇	1	2	yes	—
81	C₂×C₄×C₈	C₂×D₇	1	2	yes	—
82	C₂³×C₇	—	4	2	no	—
83	C₂³×C₇	#97	1	2	yes	—
84	C₂³×C₇	C₂²⋊C₈	1	2	yes	—
85	C₂³×C₇	C₄⋊C₈	1	2	yes	—
86	C₂³×C₇	C₂×M₁₆	1	1	yes	characteristic
87	C₂×C₄×C₇	—	2	4	no	—
88	C₂×C₄×C₇	—	4	4	no	—
89	C₂×C₄×C₇	#95	1	4	yes	—
90	C₂×C₄×C₇	C₄×C₈	1	4	yes	—
91	C₂×C₄×C₇	#97	1	2	yes	—
92	C₂×C₄×C₇	C₂²⋊C₈	1	2	yes	—
93	C₂×C₄×C₇	C₂²⋊C₈	1	4	yes	—
94	C₂×C₄×C₇	C₄⋊C₈	1	2	yes	—
95	C₂×C₄×C₇	C₄⋊C₈	1	4	yes	—
96	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
97	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
101	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
102	C₇×C₈	—	2	4	no	—
103	C₇×C₈	#95	1	4	yes	—
104	C₇×C₈	C₂×C₂²⋊C₄	1	2	yes	—
105	C₇×C₈	C₂×C₄⋊C₄	1	2	yes	—
106	C₂³×C₄	C₂×Dic₇	1	1	yes	characteristic
107	C₂×M₁₆	—	2	2	no	—
108	C₂×M₁₆	—	2	2	no	—
109	C₂×M₁₆	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	#117	—	2	2	no	—
118	#117	C₇⋊C₈	1	4	yes	—
119	C₂×C₄⋊C₄	—	2	1	no	—
120	C₂×C₂²⋊C₄	—	2	1	no	—
121	C₂×C₄²	C₂×Dic₇	1	2	yes	—
122	M₃₂	—	7	8	no	—
123	M₃₂	—	14	8	no	—
124	M₃₂	—	14	4	no	—
125	C₂×C₁₆	—	14	8	no	—
126	C₂×C₁₆	—	14	4	no	—
127	#97	—	2	4	no	—
128	C₄×C₈	—	2	4	no	—
129	C₂²×C₇	—	2	2	no	—
130	C₂²×C₇	—	2	4	no	—
131	C₂²×C₇	—	4	2	no	—
132	C₂²×C₇	—	4	2	no	—
133	C₂²×C₇	#336	1	4	yes	—
134	C₂²×C₇	C₂×#97	1	1	yes	characteristic
135	C₂²×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
136	C₂²×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
137	C₄×C₇	—	4	4	no	—
138	C₄×C₇	#336	1	4	yes	—
139	C₄×C₇	C₂×#95	1	1	yes	characteristic
140	C₄×C₇	C₂×C₄×C₈	1	1	yes	characteristic
141	C₄×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
142	C₄×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
143	C₂⁴	#1136	1	1	yes	characteristic
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	C₂×C₇⋊C₈	1	2	yes	—
146	C₂²×C₄	C₂×C₇⋊C₈	1	2	yes	—
147	C₂²×C₄	#1136	1	2	yes	—
148	C₂²×C₄	#1136	1	1	yes	characteristic
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	M₁₆	—	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	4	yes	—
161	C₂×C₈	(C₂×D₇)⋊C₄	1	4	yes	—
162	C₂×C₈	C₂²⋊Dic₇	1	2	yes	—
163	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
164	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
165	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
166	C₂×C₈	C₂×C₂²⋊D₇	1	2	yes	—
167	C₄⋊C₄	—	2	4	no	—
168	C₂²⋊C₄	—	2	4	no	—
169	C₄²	—	2	2	no	—
170	C₄²	C₂×C₇⋊C₈	1	4	yes	—
171	C₁₆	—	14	8	no	—
172	C₂×C₇	—	2	1	no	—
173	C₂×C₇	—	2	4	no	—
174	C₂×C₇	—	2	1	no	—
175	C₂×C₇	C₂×#336	1	1	yes	characteristic
176	C₂×C₇	#2153	1	2	yes	—
177	C₂³	—	4	2	no	—
178	C₂³	#6377	1	2	yes	—
179	C₂³	C₄⋊(C₇⋊C₈)	1	2	yes	—
180	C₂³	C₂²⋊(C₇⋊C₈)	1	2	yes	—
181	C₂³	C₂×#1136	1	1	yes	characteristic
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	C₄×C₇⋊C₈	1	4	yes	—
185	C₂×C₄	#6377	1	2	yes	—
186	C₂×C₄	C₄⋊(C₇⋊C₈)	1	4	yes	—
187	C₂×C₄	C₄⋊(C₇⋊C₈)	1	2	yes	—
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	C₂²×C₇⋊C₈	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	#12549	1	1	yes	characteristic, complemented, Sylow
202	C₂²	—	2	2	no	—
203	C₂²	—	2	4	no	—
204	C₂²	—	4	2	no	—
205	C₂²	—	4	2	no	—
206	C₂²	#89319	1	4	yes	—
207	C₂²	C₂×#6377	1	1	yes	characteristic
208	C₂²	C₂×C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
209	C₂²	C₂×C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
210	C₄	—	4	4	no	—
211	C₄	#89319	1	4	yes	—
212	C₄	C₂×C₄×C₇⋊C₈	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	1	no	—
217	C₂	—	2	1	no	—
218	C₂	—	2	4	no	—
219	C₂	#11727639	1	2	yes	—
220	C₂	C₂×#89319	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₈
commutator	C₄×C₇	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₂×C₇×#405	C₂
socle	C₂²×C₇	C₂×C₄⋊C₈
prime radical	C₂⁴×C₇	M₁₆
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₂×#405	D₇
hypocenter	C₇	#12549
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
3. #11725464	C₂	semidirect
12. C₂×#6386	C₄	semidirect
201. C₇	#12549	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
#11725464	2
#12549	7