Subgroups — #13506456

Order 1792.

716 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3407	C₂	1	1	yes	maximal, characteristic
2	#11741227	C₂	1	1	yes	maximal, characteristic
3	#11741225	C₂	1	1	yes	maximal, characteristic
4	#11729179	C₂	1	6	yes	maximal, complemented
5	C₂×#89307	C₂	1	6	yes	maximal, complemented
6	C₂×C₇×#131	C₂²	1	1	yes	characteristic
7	#41×Dic₇	C₄	1	1	yes	characteristic
8	C₂×#6511	C₄	1	1	yes	characteristic
9	C₇×#504	C₄	1	8	yes	complemented
10	C₂×C₇×M₃₂	C₂²	1	6	yes	—
11	#89912	C₄	1	8	yes	complemented
12	#89894	C₄	1	3	yes	—
13	M₁₆×Dic₇	C₄	1	3	yes	—
14	#89873	C₂²	1	3	yes	—
15	C₂×#6389	C₂²	1	3	yes	—
16	C₂×#6386	C₂²	1	6	yes	—
17	#89307	C₂²	1	4	yes	—
18	#89307	C₂²	1	12	yes	—
19	#33329	—	7	1	no	maximal, Sylow
20	C₂×C₇×#41	C₂×C₄	1	1	yes	characteristic
21	C₇×#131	C₂×C₄	1	4	yes	—
22	C₇×#131	C₂×C₄	1	4	yes	—
23	C₂×C₇×M₁₆	C₂×C₄	1	3	yes	—
24	C₂²×C₇×C₈	C₂³	1	3	yes	—
25	#6511	—	2	4	no	—
26	Q₈×Dic₇	—	2	1	no	—
27	D₄×Dic₇	—	2	3	no	—
28	#6489	C₂×C₄	1	3	yes	—
29	C₂×C₄×Dic₇	C₂×C₄	1	3	yes	—
30	C₂×#1136	C₂×C₄	1	3	yes	—
31	C₂²×C₇⋊C₈	C₂×C₄	1	3	yes	—
32	C₇×M₃₂	C₂×C₄	1	12	yes	complemented
33	C₇×M₃₂	C₂³	1	4	yes	—
34	C₂×C₇×C₁₆	C₂×C₄	1	12	yes	complemented
35	#6389	C₂×C₄	1	3	yes	—
36	#6389	C₂×C₄	1	3	yes	—
37	#6389	C₂³	1	1	yes	characteristic
38	#6389	C₂³	1	3	yes	—
39	C₈×Dic₇	C₂×C₄	1	6	yes	—
40	#6386	C₂×C₄	1	12	yes	complemented
41	#6386	C₂³	1	4	yes	—
42	C₂×C₇⋊C₁₆	C₂×C₄	1	12	yes	complemented
43	#3407	—	7	1	no	—
44	#3407	D₇	1	1	yes	characteristic
45	#2875	—	7	1	no	—
46	#2110	—	7	6	no	—
47	C₂×#347	—	7	6	no	—
48	C₇×#41	C₄²	1	4	yes	—
49	C₇×#41	M₁₆	1	4	yes	—
50	C₂×C₇×Q₈	M₁₆	1	1	yes	characteristic
51	C₂×C₇×D₄	M₁₆	1	3	yes	—
52	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
53	C₂²×Dic₇	—	2	3	no	—
54	C₇×M₁₆	C₄²	1	12	yes	—
55	C₂×C₇×C₈	C₂²×C₄	1	3	yes	—
56	C₂×C₇×C₈	C₂²×C₄	1	6	yes	—
57	C₂×C₇×C₈	C₂²×C₄	1	6	yes	—
58	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
59	C₂²⋊Dic₇	—	2	3	no	—
60	C₄⋊Dic₇	—	2	3	no	—
61	C₄×Dic₇	—	2	3	no	—
62	C₄×Dic₇	C₂²×C₄	1	1	yes	characteristic
63	C₄×Dic₇	C₂²×C₄	1	3	yes	—
64	#1136	—	2	6	no	—
65	C₂×C₇⋊C₈	—	2	6	no	—
66	C₂×C₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₇⋊C₈	C₂²×C₄	1	3	yes	—
68	C₇×C₁₆	C₂²×C₄	1	8	yes	—
69	C₇⋊C₁₆	C₂²×C₄	1	8	yes	—
70	C₂×#131	—	7	1	no	—
71	C₂×#131	C₂×D₇	1	1	yes	characteristic
72	C₄×#41	—	7	1	no	—
73	#504	—	7	8	no	—
74	#504	Dic₇	1	8	yes	complemented
75	C₂×M₃₂	—	7	6	no	—
76	C₂×M₃₂	C₂×D₇	1	6	yes	—
77	#405	—	7	3	no	—
78	#405	—	7	3	no	—
79	C₄×M₁₆	—	7	3	no	—
80	C₂×#97	—	7	3	no	—
81	#347	—	7	4	no	—
82	#347	—	7	12	no	—
83	C₂³×C₇	C₂×M₁₆	1	3	yes	—
84	C₇×Q₈	#97	1	4	yes	—
85	C₇×D₄	#97	1	12	yes	—
86	C₂×C₄×C₇	C₂×C₄²	1	6	yes	—
87	C₂×C₄×C₇	C₂×M₁₆	1	6	yes	—
88	C₂×C₄×C₇	C₂×M₁₆	1	3	yes	—
89	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
90	C₂×Dic₇	—	2	1	no	—
91	C₂×Dic₇	—	2	3	no	—
92	C₂×Dic₇	—	8	3	no	—
93	C₇×C₈	C₂×C₄²	1	6	yes	—
94	C₇×C₈	C₂³×C₄	1	1	yes	characteristic
95	C₇×C₈	C₂³×C₄	1	1	yes	characteristic
96	C₇⋊C₈	—	2	1	no	—
97	C₇⋊C₈	—	2	3	no	—
98	C₂×#41	C₄×D₇	1	1	yes	characteristic
99	#131	—	14	4	no	—
100	#131	C₄×D₇	1	4	yes	—
101	#131	C₂×Dic₇	1	4	yes	—
102	C₂×M₁₆	—	7	3	no	—
103	C₂×M₁₆	C₄×D₇	1	3	yes	—
104	C₂²×C₈	—	7	3	no	—
105	C₂²×C₈	C₂²×D₇	1	3	yes	—
106	C₄×Q₈	—	14	1	no	—
107	C₄×D₄	—	14	3	no	—
108	#117	—	7	3	no	—
109	C₂×C₄²	—	7	3	no	—
110	M₃₂	—	7	12	no	—
111	M₃₂	—	7	4	no	—
112	M₃₂	C₂×Dic₇	1	12	yes	complemented
113	M₃₂	C₂²×D₇	1	4	yes	—
114	C₂×C₁₆	—	7	12	no	—
115	C₂×C₁₆	C₂×Dic₇	1	12	yes	complemented
116	#97	—	7	3	no	—
117	#97	—	7	3	no	—
118	#97	—	7	1	no	—
119	#97	—	7	3	no	—
120	C₄×C₈	—	7	6	no	—
121	C₂²×C₇	—	4	3	no	—
122	C₂²×C₇	C₂×#97	1	6	yes	—
123	C₂²×C₇	C₂²×M₁₆	1	1	yes	characteristic
124	C₄×C₇	C₂×#97	1	6	yes	—
125	C₄×C₇	C₂²×C₄²	1	1	yes	characteristic
126	C₄×C₇	C₂²×M₁₆	1	1	yes	characteristic
127	Dic₇	—	4	3	no	—
128	Dic₇	—	4	1	no	—
129	#41	#1131	1	4	yes	—
130	#41	C₄×Dic₇	1	4	yes	—
131	C₂×Q₈	#1131	1	1	yes	characteristic
132	C₂×D₄	#1131	1	3	yes	—
133	C₂²×C₄	—	14	3	no	—
134	C₂²×C₄	C₂×C₄×D₇	1	3	yes	—
135	M₁₆	—	14	6	no	—
136	M₁₆	C₄×Dic₇	1	12	yes	—
137	C₂×C₈	—	7	3	no	—
138	C₂×C₈	—	7	1	no	—
139	C₂×C₈	—	14	6	no	—
140	C₂×C₈	C₂×C₄×D₇	1	3	yes	—
141	C₂×C₈	C₂×C₄×D₇	1	6	yes	—
142	C₂×C₈	C₂²×Dic₇	1	6	yes	—
143	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
144	C₄⋊C₄	—	14	3	no	—
145	C₂²⋊C₄	—	14	3	no	—
146	C₄²	—	7	3	no	—
147	C₄²	—	7	1	no	—
148	C₄²	—	14	3	no	—
149	C₁₆	—	7	8	no	—
150	C₁₆	C₂²×Dic₇	1	8	yes	—
151	C₂×C₇	—	2	1	no	—
152	C₂×C₇	—	2	6	no	—
153	C₂×C₇	C₂²×#97	1	1	yes	characteristic
154	C₂³	C₂×#1131	1	3	yes	—
155	Q₈	#6389	1	4	yes	—
156	D₄	#6389	1	12	yes	—
157	C₂×C₄	—	14	1	no	—
158	C₂×C₄	—	14	3	no	—
159	C₂×C₄	—	56	3	no	—
160	C₂×C₄	C₂×#1131	1	3	yes	—
161	C₂×C₄	C₂×#1131	1	6	yes	—
162	C₂×C₄	C₂×C₄×Dic₇	1	6	yes	—
163	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
164	C₈	—	14	1	no	—
165	C₈	—	14	3	no	—
166	C₈	C₂×C₄×Dic₇	1	6	yes	—
167	C₈	C₂²×C₄×D₇	1	1	yes	characteristic
168	C₈	C₂³×Dic₇	1	1	yes	characteristic
169	C₇	#33329	1	1	yes	characteristic, complemented, Sylow
170	C₂²	—	4	3	no	—
171	C₂²	C₂×#6389	1	6	yes	—
172	C₂²	C₂²×#1131	1	1	yes	characteristic
173	C₄	—	28	1	no	—
174	C₄	—	28	3	no	—
175	C₄	C₂×#6389	1	6	yes	—
176	C₄	C₂²×#1131	1	1	yes	characteristic
177	C₄	C₂²×C₄×Dic₇	1	1	yes	characteristic
178	C₂	—	2	1	no	—
179	C₂	—	2	6	no	—
180	C₂	C₂²×#6389	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₄×Dic₇
commutator	C₄×C₇	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₇×#3407	C₂
socle	C₂×C₇	C₂²×#97
prime radical	C₂×C₇×#41	C₂×C₄
cosocle	C₂×C₇×C₈	C₂⁴
hypercenter	#3407	D₇
hypocenter	C₇	#33329
LCS	G ▷ C₄×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ #3407

Complements

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

N	H ≅ G/N	type
4. #11729179	C₂	semidirect
5. C₂×#89307	C₂	semidirect
9. C₇×#504	C₄	semidirect
11. #89912	C₄	semidirect
32. C₇×M₃₂	C₂×C₄	semidirect
34. C₂×C₇×C₁₆	C₂×C₄	semidirect
40. #6386	C₂×C₄	semidirect
42. C₂×C₇⋊C₁₆	C₂×C₄	semidirect
74. #504	Dic₇	semidirect
112. M₃₂	C₂×Dic₇	semidirect
115. C₂×C₁₆	C₂×Dic₇	semidirect
169. C₇	#33329	semidirect

Maximal subgroups

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

subgroup	index
C₇×#3407	2
#11741227	2
#11741225	2
#11729179	2
C₂×#89307	2
#33329	7