Subgroups — #13240708

Order 1792.

262 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#346	C₂	1	1	yes	maximal, characteristic
2	#11727511	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₇×C₁₆	—	2	1	no	—
5	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
6	C₂×#6389	—	2	1	no	—
7	C₂²×C₇⋊C₁₆	—	2	1	no	—
8	C₄×C₄⋊Dic₇	C₄	1	1	yes	characteristic
9	C₂×C₄⋊(C₇⋊C₈)	C₄	1	1	yes	characteristic
10	C₇×#346	—	2	1	no	—
11	C₇×#346	C₄	1	2	yes	complemented
12	#89345	—	2	1	no	—
13	C₄⋊(C₇⋊C₁₆)	—	2	1	no	—
14	C₄⋊(C₇⋊C₁₆)	C₄	1	2	yes	complemented
15	#7170	—	7	1	no	maximal, Sylow
16	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
17	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
18	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄⋊Dic₇	C₈	1	2	yes	—
20	C₂×C₄×Dic₇	—	2	1	no	—
21	C₂²×C₇⋊C₈	—	2	1	no	—
22	C₂×C₇×C₁₆	—	2	2	no	—
23	C₂×C₇×C₁₆	—	2	2	no	—
24	C₂×C₇×C₁₆	—	2	2	no	—
25	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₇×C₈	D₄	1	1	yes	characteristic
28	C₄×C₇×C₈	Q₈	1	1	yes	characteristic
29	C₇⋊#95	—	2	1	no	—
30	#6389	—	4	1	no	—
31	#6389	—	4	1	no	—
32	C₂×C₇⋊C₁₆	—	2	2	no	—
33	C₂×C₇⋊C₁₆	—	4	1	no	—
34	C₂×C₇⋊C₁₆	—	4	1	no	—
35	C₄⋊(C₇⋊C₈)	—	2	1	no	—
36	C₄⋊(C₇⋊C₈)	C₈	1	2	yes	—
37	C₂×C₄⋊C₁₆	—	7	1	no	—
38	C₂×#346	D₇	1	1	yes	characteristic
39	#1770	—	7	1	no	—
40	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
43	C₂²×Dic₇	—	2	2	no	—
44	C₂×C₇×C₈	—	2	1	no	—
45	C₂×C₇×C₈	—	2	1	no	—
46	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₇×C₈	D₈	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	Q₁₆	1	1	yes	characteristic
54	C₄²×C₇	C₄²	1	1	yes	characteristic
55	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
56	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
57	C₄²×C₇	M₁₆	1	1	yes	characteristic
58	C₄⋊Dic₇	—	4	2	no	—
59	C₄×Dic₇	—	4	1	no	—
60	C₄×Dic₇	—	4	1	no	—
61	C₂×C₇⋊C₈	—	2	2	no	—
62	C₂×C₇⋊C₈	—	4	1	no	—
63	C₂×C₇⋊C₈	—	4	1	no	—
64	C₇×C₁₆	—	2	4	no	—
65	C₇⋊C₁₆	—	4	2	no	—
66	C₂²×C₁₆	—	2	1	no	—
67	C₂²×C₁₆	—	14	1	no	—
68	C₂×C₄⋊C₈	—	7	1	no	—
69	C₂×#97	—	14	1	no	—
70	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
71	C₄×C₄⋊C₄	—	7	1	no	—
72	C₄⋊C₁₆	—	7	2	no	—
73	C₄⋊C₁₆	—	14	1	no	—
74	#346	—	2	1	no	—
75	#346	Dic₇	1	2	yes	complemented
76	#336	—	14	1	no	—
77	C₂³×C₇	C₂²⋊C₈	1	1	yes	characteristic
78	C₂×C₄×C₇	—	2	2	no	—
79	C₂×C₄×C₇	—	2	2	no	—
80	C₂×C₄×C₇	#95	1	1	yes	characteristic
81	C₂×C₄×C₇	C₄×C₈	1	1	yes	characteristic
82	C₂×C₄×C₇	#97	1	1	yes	characteristic
83	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
84	C₂×C₄×C₇	#102	1	1	yes	characteristic
85	C₂×C₄×C₇	#103	1	1	yes	characteristic
86	C₂×C₄×C₇	#104	1	1	yes	characteristic
87	C₂×C₄×C₇	#104	1	1	yes	characteristic
88	C₂×C₄×C₇	C₄⋊C₈	1	1	yes	characteristic
89	C₂×C₄×C₇	C₄⋊C₈	1	1	yes	characteristic
90	C₂×Dic₇	—	4	2	no	—
91	C₂×Dic₇	—	4	2	no	—
92	C₂×Dic₇	—	4	2	no	—
93	C₇×C₈	—	2	1	no	—
94	C₇×C₈	—	2	1	no	—
95	C₇×C₈	#102	1	1	yes	characteristic
96	C₇×C₈	#103	1	1	yes	characteristic
97	C₇×C₈	#106	1	1	yes	characteristic
98	C₇×C₈	#107	1	1	yes	characteristic
99	C₇⋊C₈	—	4	2	no	—
100	C₂²×C₈	—	14	1	no	—
101	C₂²×C₈	D₂₈	1	1	yes	characteristic
102	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
103	C₂×C₄⋊C₄	—	7	2	no	—
104	C₂×C₄²	—	14	1	no	—
105	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
106	C₂×C₁₆	—	2	2	no	—
107	C₂×C₁₆	—	2	2	no	—
108	C₂×C₁₆	—	2	2	no	—
109	C₂×C₁₆	—	14	2	no	—
110	C₂×C₁₆	—	28	1	no	—
111	C₂×C₁₆	—	28	1	no	—
112	C₄⋊C₈	—	7	2	no	—
113	C₄⋊C₈	—	14	1	no	—
114	#97	—	28	1	no	—
115	#97	—	28	1	no	—
116	C₄×C₈	Dic₁₄	1	1	yes	characteristic
117	C₄×C₈	C₄×D₇	1	1	yes	characteristic
118	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
119	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
120	#95	—	14	1	no	—
121	C₂²×C₇	D₄⋊C₈	1	1	yes	characteristic
122	C₂²×C₇	Q₈⋊C₈	1	1	yes	characteristic
123	C₂²×C₇	#336	1	1	yes	characteristic
124	C₂²×C₇	#349	1	2	yes	—
125	C₂²×C₇	#350	1	2	yes	—
126	C₄×C₇	—	2	1	no	—
127	C₄×C₇	—	2	1	no	—
128	C₄×C₇	—	2	1	no	—
129	C₄×C₇	—	2	1	no	—
130	C₄×C₇	D₄⋊C₈	1	1	yes	characteristic
131	C₄×C₇	Q₈⋊C₈	1	1	yes	characteristic
132	C₄×C₇	#339	1	1	yes	characteristic
133	C₄×C₇	#340	1	1	yes	characteristic
134	Dic₇	—	8	2	no	—
135	C₂²×C₄	—	14	2	no	—
136	C₂²×C₄	C₈×D₇	1	1	yes	characteristic
137	C₂²×C₄	#1131	1	1	yes	characteristic
138	C₂²×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	14	2	no	—
142	C₂×C₈	—	28	1	no	—
143	C₂×C₈	—	28	1	no	—
144	C₂×C₈	#1132	1	1	yes	characteristic
145	C₂×C₈	#1132	1	1	yes	characteristic
146	C₂×C₈	D₅₆	1	1	yes	characteristic
147	C₂×C₈	Dic₂₈	1	1	yes	characteristic
148	C₂×C₈	Dic₇⋊C₄	1	1	yes	characteristic
149	C₂×C₈	C₄⋊Dic₇	1	1	yes	characteristic
150	C₂×C₈	(C₂×D₇)⋊C₄	1	1	yes	characteristic
151	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
152	C₄⋊C₄	—	28	2	no	—
153	C₄²	—	28	1	no	—
154	C₄²	—	28	1	no	—
155	C₄²	C₈×D₇	1	1	yes	characteristic
156	C₄²	#1131	1	1	yes	characteristic
157	C₄²	C₄×Dic₇	1	1	yes	characteristic
158	C₄²	Dic₇⋊C₄	1	1	yes	characteristic
159	C₁₆	—	2	4	no	—
160	C₁₆	—	28	2	no	—
161	C₂×C₇	#1279	1	1	yes	characteristic
162	C₂×C₇	#1330	1	2	yes	—
163	C₂×C₇	#1338	1	2	yes	—
164	C₂×C₇	#1377	1	1	yes	characteristic
165	C₂×C₇	#1379	1	1	yes	characteristic
166	C₂³	(C₂×D₇)⋊C₈	1	1	yes	characteristic
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	28	2	no	—
170	C₂×C₄	—	28	2	no	—
171	C₂×C₄	—	28	2	no	—
172	C₂×C₄	#6379	1	1	yes	characteristic
173	C₂×C₄	#6379	1	1	yes	characteristic
174	C₂×C₄	C₈×Dic₇	1	1	yes	characteristic
175	C₂×C₄	Dic₇⋊C₈	1	1	yes	characteristic
176	C₂×C₄	Dic₇⋊C₈	1	1	yes	characteristic
177	C₂×C₄	#6389	1	1	yes	characteristic
178	C₂×C₄	#6390	1	1	yes	characteristic
179	C₂×C₄	(C₂×D₇)⋊C₈	1	1	yes	characteristic
180	C₂×C₄	#6395	1	1	yes	characteristic
181	C₂×C₄	C₇⋊#95	1	1	yes	characteristic
182	C₈	—	2	1	no	—
183	C₈	—	2	1	no	—
184	C₈	—	28	2	no	—
185	C₈	#6390	1	1	yes	characteristic
186	C₈	#6391	1	1	yes	characteristic
187	C₈	#6392	1	1	yes	characteristic
188	C₈	#6395	1	1	yes	characteristic
189	C₇	#7170	1	1	yes	characteristic, complemented, Sylow
190	C₂²	#89251	1	1	yes	characteristic
191	C₂²	#89254	1	1	yes	characteristic
192	C₂²	#89309	1	2	yes	—
193	C₂²	#89310	1	2	yes	—
194	C₂²	#89345	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	2	1	no	—
199	C₄	—	56	2	no	—
200	C₄	#89251	1	1	yes	characteristic
201	C₄	#89254	1	1	yes	characteristic
202	C₄	#89318	1	1	yes	characteristic
203	C₄	#89346	1	1	yes	characteristic
204	C₂	#11725321	1	2	yes	—
205	C₂	#11725324	1	2	yes	—
206	C₂	#11725461	1	1	yes	characteristic
207	C₂	#11725618	1	1	yes	characteristic
208	C₂	#11725621	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	(C₂×D₇)⋊C₄
commutator	C₂×C₄×C₇	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₇
Fitting	C₂×C₇×#346	C₂
socle, prime radical	C₂³×C₇	C₂²⋊C₈
cosocle	C₂×C₄×C₇×C₈	C₂²
hypercenter	C₂×#346	D₇
hypocenter	C₇	#7170
LCS	G ▷ C₂×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
11. C₇×#346	C₄	semidirect
14. C₄⋊(C₇⋊C₁₆)	C₄	semidirect
75. #346	Dic₇	semidirect
189. C₇	#7170	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#346	2
#11727511	2
C₂×C₄⋊(C₇⋊C₁₆)	2
#7170	7