Subgroups — #13265383

Order 1792.

242 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#335	C₂	1	1	yes	maximal, characteristic
2	C₄×#6386	C₂	1	1	yes	maximal, characteristic
3	#11727483	C₂	1	1	yes	maximal, characteristic
4	#11725276	C₂	1	2	yes	maximal, complemented
5	#11725275	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×C₄⋊C₈	—	2	1	no	—
7	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₂×#6386	C₄	1	1	yes	characteristic
9	C₂×#6386	C₄	1	1	yes	characteristic
10	#89697	—	2	1	no	—
11	C₇×#335	—	2	1	no	—
12	C₇×#335	C₂²	1	2	yes	—
13	#89256	C₄	1	2	yes	—
14	C₄×C₇⋊C₁₆	C₂²	1	2	yes	—
15	#89253	C₂²	1	1	yes	characteristic
16	#89253	C₂²	1	1	yes	characteristic
17	#89252	—	2	1	no	—
18	#12009	—	7	1	no	maximal, Sylow
19	C₂²×C₇×C₈	—	4	1	no	—
20	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₇×C₈	Q₈	1	1	yes	characteristic
22	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
23	C₇×C₄⋊C₈	—	2	2	no	—
24	C₇×C₄⋊C₈	—	4	1	no	—
25	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₇×C₈	D₄	1	1	yes	characteristic
27	C₄×C₇×C₈	Q₈	1	1	yes	characteristic
28	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
29	C₂²⋊(C₇⋊C₈)	—	4	1	no	—
30	#6386	—	2	2	no	—
31	#6386	C₈	1	4	yes	complemented
32	C₂×C₇⋊C₁₆	C₂×C₄	1	2	yes	—
33	C₂×C₇⋊C₁₆	C₂×C₄	1	2	yes	—
34	C₄⋊(C₇⋊C₈)	—	2	1	no	—
35	C₄⋊(C₇⋊C₈)	—	2	1	no	—
36	C₄×C₇⋊C₈	—	4	1	no	—
37	C₄×M₃₂	—	7	1	no	—
38	#1564	—	7	1	no	—
39	C₂×#335	D₇	1	1	yes	characteristic
40	#1372	—	7	2	no	—
41	#1371	—	7	2	no	—
42	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
44	C₂²×C₄×C₇	Q₁₆	1	1	yes	characteristic
45	C₂×C₇×C₈	—	2	2	no	—
46	C₂×C₇×C₈	—	4	2	no	—
47	C₂×C₇×C₈	—	8	2	no	—
48	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
51	C₂×C₇×C₈	M₁₆	1	2	yes	—
52	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
54	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
55	C₄²×C₇	D₈	1	1	yes	characteristic
56	C₄²×C₇	Q₁₆	1	1	yes	characteristic
57	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₇⋊C₈	—	4	1	no	—
59	C₂×C₇⋊C₈	—	4	1	no	—
60	C₇⋊C₁₆	—	2	2	no	—
61	C₇⋊C₁₆	C₂×C₈	1	4	yes	complemented
62	C₂×M₃₂	—	7	1	no	—
63	C₂×M₃₂	—	7	1	no	—
64	#431	—	14	1	no	—
65	C₂×C₄⋊C₈	—	2	1	no	—
66	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
67	#346	—	7	2	no	—
68	C₄×C₁₆	—	7	2	no	—
69	#335	—	2	1	no	—
70	#335	—	7	1	no	—
71	#335	—	7	1	no	—
72	#335	C₂×D₇	1	2	yes	—
73	#334	—	14	1	no	—
74	C₂³×C₇	#107	1	1	yes	characteristic
75	C₂×C₄×C₇	—	2	1	no	—
76	C₂×C₄×C₇	—	2	1	no	—
77	C₂×C₄×C₇	—	4	1	no	—
78	C₂×C₄×C₇	C₄⋊C₈	1	2	yes	—
79	C₂×C₄×C₇	#107	1	1	yes	characteristic
80	C₂×C₄×C₇	#107	1	1	yes	characteristic
81	C₂×C₄×C₇	#107	1	1	yes	characteristic
82	C₂×C₄×C₇	#108	1	2	yes	—
83	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
86	C₇×C₈	—	2	1	no	—
87	C₇×C₈	—	2	1	no	—
88	C₇×C₈	—	8	2	no	—
89	C₇×C₈	C₄⋊C₈	1	2	yes	—
90	C₇×C₈	C₂²×C₈	1	1	yes	characteristic
91	C₇×C₈	C₂×M₁₆	1	1	yes	characteristic
92	C₇⋊C₈	—	8	1	no	—
93	C₇⋊C₈	—	8	1	no	—
94	C₂²×C₈	—	4	1	no	—
95	C₂²×C₈	Dic₁₄	1	1	yes	characteristic
96	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
97	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
98	M₃₂	—	7	4	no	—
99	M₃₂	—	14	2	no	—
100	C₂×C₁₆	—	7	2	no	—
101	C₂×C₁₆	—	7	2	no	—
102	C₄⋊C₈	—	2	2	no	—
103	C₄⋊C₈	—	4	1	no	—
104	C₄⋊C₈	—	14	1	no	—
105	C₄⋊C₈	—	14	1	no	—
106	C₂²⋊C₈	—	28	1	no	—
107	C₄×C₈	—	28	1	no	—
108	C₄×C₈	Dic₁₄	1	1	yes	characteristic
109	C₄×C₈	C₄×D₇	1	1	yes	characteristic
110	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
111	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
112	C₂²×C₇	—	2	1	no	—
113	C₂²×C₇	—	2	1	no	—
114	C₂²×C₇	#335	1	2	yes	—
115	C₂²×C₇	C₂×#107	1	1	yes	characteristic
116	C₄×C₇	—	2	1	no	—
117	C₄×C₇	—	2	1	no	—
118	C₄×C₇	—	2	1	no	—
119	C₄×C₇	#335	1	2	yes	—
120	C₄×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
121	C₄×C₇	C₂×#108	1	1	yes	characteristic
122	C₄×C₇	#509	1	1	yes	characteristic
123	C₄×C₇	#510	1	1	yes	characteristic
124	C₂²×C₄	Dic₇⋊C₄	1	1	yes	characteristic
125	C₂²×C₄	#1141	1	1	yes	characteristic
126	C₂²×C₄	#1144	1	1	yes	characteristic
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	4	2	no	—
129	C₂×C₈	—	8	2	no	—
130	C₂×C₈	—	28	1	no	—
131	C₂×C₈	—	28	1	no	—
132	C₂×C₈	C₈×D₇	1	2	yes	—
133	C₂×C₈	#1131	1	2	yes	—
134	C₂×C₈	Dic₇⋊C₄	1	1	yes	characteristic
135	C₂×C₈	Dic₇⋊C₄	1	1	yes	characteristic
136	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
137	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
138	C₄²	Dic₇⋊C₄	1	1	yes	characteristic
139	C₄²	#1141	1	1	yes	characteristic
140	C₄²	#1144	1	1	yes	characteristic
141	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
142	C₁₆	—	7	4	no	—
143	C₁₆	—	14	2	no	—
144	C₂×C₇	—	2	2	no	—
145	C₂×C₇	C₂×#335	1	1	yes	characteristic
146	C₂×C₇	#2159	1	1	yes	characteristic
147	C₂×C₇	#2161	1	1	yes	characteristic
148	C₂³	#6381	1	1	yes	characteristic
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	#6381	1	1	yes	characteristic
153	C₂×C₄	#6381	1	1	yes	characteristic
154	C₂×C₄	#6381	1	1	yes	characteristic
155	C₂×C₄	Dic₇⋊C₈	1	2	yes	—
156	C₂×C₄	#6396	1	2	yes	—
157	C₂×C₄	C₂×Dic₇⋊C₄	1	1	yes	characteristic
158	C₂×C₄	C₂×#1141	1	1	yes	characteristic
159	C₂×C₄	C₂×#1144	1	1	yes	characteristic
160	C₈	—	2	1	no	—
161	C₈	—	2	1	no	—
162	C₈	—	8	2	no	—
163	C₈	—	56	1	no	—
164	C₈	—	56	1	no	—
165	C₈	Dic₇⋊C₈	1	2	yes	—
166	C₈	C₂×C₈×D₇	1	1	yes	characteristic
167	C₈	C₂×#1131	1	1	yes	characteristic
168	C₇	#12009	1	1	yes	characteristic, complemented, Sylow
169	C₂²	—	2	1	no	—
170	C₂²	—	2	1	no	—
171	C₂²	#89274	1	2	yes	—
172	C₂²	C₂×#6381	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	#89274	1	2	yes	—
177	C₄	C₂×Dic₇⋊C₈	1	1	yes	characteristic
178	C₄	C₂×#6396	1	1	yes	characteristic
179	C₄	#89919	1	1	yes	characteristic
180	C₄	#89951	1	1	yes	characteristic
181	C₂	—	2	2	no	—
182	C₂	C₂×#89274	1	1	yes	characteristic
183	C₂	#11728350	1	1	yes	characteristic
184	C₂	#11728437	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#1141
commutator	C₇×C₈	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₂×C₇×#335	C₂
socle	C₂²×C₇	C₂×#107
prime radical	C₂³×C₇	#107
cosocle	C₄×C₇×C₈	C₂³
hypercenter	C₂×#335	D₇
hypocenter	C₇	#12009
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
4. #11725276	C₂	semidirect
5. #11725275	C₂	semidirect
31. #6386	C₈	semidirect
61. C₇⋊C₁₆	C₂×C₈	semidirect
168. C₇	#12009	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#335	2
C₄×#6386	2
#11727483	2
#11725276	2
#11725275	2
#12009	7