Subgroups — #13265163

Order 1792.

292 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11735887	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#364	C₂	1	1	yes	maximal, characteristic
3	C₂×#89258	C₂	1	1	yes	maximal, characteristic
4	#11725379	C₂	1	4	yes	maximal, complemented
5	#90161	C₄	1	2	yes	—
6	C₂×C₇×M₃₂	—	2	1	no	—
7	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
8	#89878	—	2	1	no	—
9	#89878	—	2	1	no	—
10	C₂×#6386	—	2	1	no	—
11	#89455	C₄	1	2	yes	—
12	#89455	C₂²	1	2	yes	—
13	#89454	—	2	1	no	—
14	#89452	—	2	1	no	—
15	C₇×#364	—	2	1	no	—
16	C₇×#364	C₂²	1	2	yes	—
17	#89258	—	2	1	no	—
18	#89258	C₂²	1	2	yes	—
19	#11956	—	7	1	no	maximal, Sylow
20	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
21	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
22	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
23	#6489	—	2	2	no	—
24	#6487	—	4	2	no	—
25	#6433	C₂×C₄	1	4	yes	—
26	#6432	—	2	2	no	—
27	C₄×Dic₁₄	—	4	2	no	—
28	C₇×M₃₂	—	2	2	no	—
29	C₇×M₃₂	—	4	1	no	—
30	C₂×C₇×C₁₆	—	4	1	no	—
31	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
32	C₄×C₇×C₈	D₄	1	1	yes	characteristic
33	C₄×C₇×C₈	D₄	1	1	yes	characteristic
34	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
35	#6392	—	2	2	no	—
36	#6392	—	2	2	no	—
37	#6391	—	2	2	no	—
38	#6391	—	2	2	no	—
39	#6386	—	2	2	no	—
40	#6386	—	4	1	no	—
41	C₂×C₇⋊C₁₆	—	4	1	no	—
42	#3160	—	7	1	no	—
43	C₂×#364	—	7	1	no	—
44	C₂×#364	D₇	1	1	yes	characteristic
45	#1366	—	7	4	no	—
46	C₂²×C₄×C₇	—	2	1	no	—
47	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×Dic₁₄	—	4	2	no	—
49	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₇×C₈	SD₁₆	1	1	yes	characteristic
52	C₂×C₇×C₈	SD₁₆	1	1	yes	characteristic
53	C₂×C₇×C₈	SD₁₆	1	1	yes	characteristic
54	C₂×C₇×C₈	SD₁₆	1	1	yes	characteristic
55	C₂×C₇×C₈	Q₁₆	1	1	yes	characteristic
56	C₂×C₇×C₈	Q₁₆	1	1	yes	characteristic
57	C₂×C₇×C₈	Q₁₆	1	1	yes	characteristic
58	C₂×C₇×C₈	Q₁₆	1	1	yes	characteristic
59	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
60	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₇	—	2	1	no	—
62	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²⋊Dic₇	—	4	2	no	—
65	C₄⋊Dic₇	—	2	4	no	—
66	C₄⋊Dic₇	—	4	2	no	—
67	Dic₇⋊C₄	—	4	4	no	—
68	C₄×Dic₇	—	4	2	no	—
69	C₇×C₁₆	—	4	2	no	—
70	C₇⋊C₁₆	—	4	2	no	—
71	#533	—	7	2	no	—
72	C₂×M₃₂	—	2	1	no	—
73	C₂×M₃₂	—	14	1	no	—
74	#500	—	14	1	no	—
75	#499	—	7	2	no	—
76	#499	—	7	2	no	—
77	#498	—	14	1	no	—
78	#427	—	14	1	no	—
79	#427	—	14	1	no	—
80	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
81	#364	—	2	1	no	—
82	#364	—	7	2	no	—
83	#364	—	14	1	no	—
84	#364	C₂×D₇	1	2	yes	—
85	C₂³×C₇	#100	1	1	yes	characteristic
86	C₂×C₄×C₇	—	2	1	no	—
87	C₂×C₄×C₇	—	2	1	no	—
88	C₂×C₄×C₇	—	4	2	no	—
89	C₂×C₄×C₇	#100	1	1	yes	characteristic
90	C₂×C₄×C₇	#103	1	1	yes	characteristic
91	C₂×C₄×C₇	#103	1	1	yes	characteristic
92	C₂×C₄×C₇	#103	1	1	yes	characteristic
93	C₂×C₄×C₇	#103	1	1	yes	characteristic
94	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×Dic₇	—	4	4	no	—
96	Dic₁₄	—	8	4	no	—
97	C₇×C₈	#103	1	1	yes	characteristic
98	C₇×C₈	#103	1	1	yes	characteristic
99	C₇×C₈	#103	1	1	yes	characteristic
100	C₇×C₈	#103	1	1	yes	characteristic
101	C₇×C₈	C₂×SD₁₆	1	1	yes	characteristic
102	C₇×C₈	C₂×SD₁₆	1	1	yes	characteristic
103	C₇×C₈	C₂×Q₁₆	1	1	yes	characteristic
104	C₇×C₈	C₂×Q₁₆	1	1	yes	characteristic
105	C₂²×C₈	D₂₈	1	1	yes	characteristic
106	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
107	C₄⋊Q₈	—	14	2	no	—
108	#125	—	7	4	no	—
109	C₂²⋊Q₈	—	28	2	no	—
110	C₄×Q₈	—	28	2	no	—
111	#117	—	14	2	no	—
112	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
113	M₃₂	—	2	2	no	—
114	M₃₂	—	4	1	no	—
115	M₃₂	—	14	2	no	—
116	M₃₂	—	28	1	no	—
117	C₂×C₁₆	—	4	1	no	—
118	C₂×C₁₆	—	28	1	no	—
119	#107	—	14	2	no	—
120	#107	—	14	2	no	—
121	#106	—	14	2	no	—
122	#106	—	14	2	no	—
123	C₄×C₈	C₄×D₇	1	1	yes	characteristic
124	C₄×C₈	D₂₈	1	1	yes	characteristic
125	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
126	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
127	C₂²×C₇	—	4	1	no	—
128	C₂²×C₇	#333	1	1	yes	characteristic
129	C₂²×C₇	#333	1	1	yes	characteristic
130	C₂²×C₇	C₂×#100	1	1	yes	characteristic
131	C₄×C₇	—	2	2	no	—
132	C₄×C₇	—	4	1	no	—
133	C₄×C₇	#333	1	1	yes	characteristic
134	C₄×C₇	#333	1	1	yes	characteristic
135	C₄×C₇	C₂×#103	1	1	yes	characteristic
136	C₄×C₇	C₂×#103	1	1	yes	characteristic
137	Dic₇	—	8	4	no	—
138	C₂×Q₈	—	28	2	no	—
139	C₂²×C₄	—	2	1	no	—
140	C₂²×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
141	C₂×C₈	#1132	1	1	yes	characteristic
142	C₂×C₈	#1132	1	1	yes	characteristic
143	C₂×C₈	Dic₂₈	1	1	yes	characteristic
144	C₂×C₈	Dic₂₈	1	1	yes	characteristic
145	C₂×C₈	(C₂×D₇)⋊C₄	1	1	yes	characteristic
146	C₂×C₈	(C₂×D₇)⋊C₄	1	1	yes	characteristic
147	C₂×C₈	#1142	1	1	yes	characteristic
148	C₂×C₈	#1142	1	1	yes	characteristic
149	C₂×C₈	#1144	1	1	yes	characteristic
150	C₂×C₈	#1144	1	1	yes	characteristic
151	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
152	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
153	C₄⋊C₄	—	14	4	no	—
154	C₄⋊C₄	—	28	4	no	—
155	C₄⋊C₄	—	28	2	no	—
156	C₂²⋊C₄	—	28	2	no	—
157	C₄²	—	2	1	no	—
158	C₄²	—	28	2	no	—
159	C₄²	(C₂×D₇)⋊C₄	1	1	yes	characteristic
160	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
161	C₁₆	—	4	2	no	—
162	C₁₆	—	28	2	no	—
163	C₂×C₇	—	2	1	no	—
164	C₂×C₇	—	2	1	no	—
165	C₂×C₇	—	2	1	no	—
166	C₂×C₇	C₂×#333	1	1	yes	characteristic
167	C₂³	#6397	1	1	yes	characteristic
168	Q₈	—	56	4	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	28	4	no	—
173	C₂×C₄	#6384	1	1	yes	characteristic
174	C₂×C₄	#6384	1	1	yes	characteristic
175	C₂×C₄	#6390	1	1	yes	characteristic
176	C₂×C₄	#6390	1	1	yes	characteristic
177	C₂×C₄	#6397	1	1	yes	characteristic
178	C₂×C₄	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
179	C₈	#6384	1	1	yes	characteristic
180	C₈	#6384	1	1	yes	characteristic
181	C₈	#6390	1	1	yes	characteristic
182	C₈	#6390	1	1	yes	characteristic
183	C₈	C₂×#1132	1	1	yes	characteristic
184	C₈	C₂×Dic₂₈	1	1	yes	characteristic
185	C₈	C₂×#1142	1	1	yes	characteristic
186	C₈	C₂×#1144	1	1	yes	characteristic
187	C₇	#11956	1	1	yes	characteristic, complemented, Sylow
188	C₂²	—	4	1	no	—
189	C₂²	#89276	1	1	yes	characteristic
190	C₂²	#89276	1	1	yes	characteristic
191	C₂²	C₂×#6397	1	1	yes	characteristic
192	C₄	—	2	2	no	—
193	C₄	—	4	1	no	—
194	C₄	—	56	4	no	—
195	C₄	#89276	1	1	yes	characteristic
196	C₄	#89276	1	1	yes	characteristic
197	C₄	C₂×#6384	1	1	yes	characteristic
198	C₄	C₂×#6390	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	—	2	1	no	—
202	C₂	C₂×#89276	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#89276
commutator	C₄²×C₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₂×C₇×#364	C₂
socle	C₂×C₇	C₂×#333
prime radical	C₂³×C₇	#100
cosocle	C₄×C₇×C₈	C₂³
hypercenter	C₂×#364	D₇
hypocenter	C₇	#11956
LCS	G ▷ C₄²×C₇ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
4. #11725379	C₂	semidirect
187. C₇	#11956	semidirect

Maximal subgroups

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

subgroup	index
#11735887	2
C₂×C₇×#364	2
C₂×#89258	2
#11725379	2
#11956	7