Subgroups — #13354032

Order 1792.

778 conjugacy classes of subgroups.

The subgroup lattice has 200 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	#11739649	C₂	1	2	yes	maximal
3	#11729752	C₂	1	2	yes	maximal
4	C₂×#89347	C₂	1	2	yes	maximal
5	#11727595	C₂	1	4	yes	maximal, complemented
6	#11727545	C₂	1	4	yes	maximal, complemented
7	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
8	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₇×#117	C₂²	1	1	yes	characteristic
10	C₂²×#1136	C₂²	1	4	yes	—
11	C₇×#405	C₄	1	8	yes	—
12	C₂×C₇×#97	C₂²	1	2	yes	—
13	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
14	#89771	—	2	8	no	—
15	#89696	C₂²	1	8	yes	—
16	#89347	C₂²	1	4	yes	—
17	#89347	C₂²	1	4	yes	—
18	#89347	C₂²	1	8	yes	—
19	#20768	—	7	1	no	maximal, Sylow
20	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
21	C₂×C₇×M₁₆	—	2	4	no	—
22	C₂×C₇×M₁₆	C₂×C₄	1	4	yes	—
23	C₂²×C₇×C₈	—	2	4	no	—
24	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
25	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
26	C₂²×C₇×C₈	C₂³	1	2	yes	—
27	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
28	C₇×#117	C₂×C₄	1	4	yes	—
29	C₇×#117	D₄	1	4	yes	—
30	C₂×C₇×C₄⋊C₄	Q₈	1	2	yes	—
31	C₂×C₇×C₂²⋊C₄	Q₈	1	2	yes	—
32	C₂×C₄²×C₇	C₂³	1	2	yes	—
33	C₂×#1136	—	2	4	no	—
34	C₂×#1136	—	2	8	no	—
35	C₂×#1136	—	2	8	no	—
36	C₂×#1136	C₂³	1	8	yes	—
37	C₂²×C₇⋊C₈	—	2	4	no	—
38	C₇×#97	C₂×C₄	1	8	yes	—
39	C₄×C₇×C₈	C₂×C₄	1	8	yes	—
40	C₄⋊(C₇⋊C₈)	—	2	16	no	—
41	#2905	—	7	2	no	—
42	C₂×#405	D₇	1	1	yes	characteristic
43	#1838	—	7	4	no	—
44	#1837	—	7	4	no	—
45	#1738	—	7	2	no	—
46	C₂×#341	—	7	2	no	—
47	C₂⁴×C₇	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₇	C₂×Q₈	1	4	yes	—
51	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₇	#41	1	4	yes	—
53	C₂²×C₄×C₇	#41	1	4	yes	—
54	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
55	C₇×M₁₆	—	2	8	no	—
56	C₂×C₇×C₈	—	2	8	no	—
57	C₂×C₇×C₈	—	2	2	no	—
58	C₂×C₇×C₈	—	2	2	no	—
59	C₂×C₇×C₈	—	2	4	no	—
60	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
61	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
62	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
63	C₇×C₄⋊C₄	C₄⋊C₄	1	8	yes	—
64	C₇×C₂²⋊C₄	C₄⋊C₄	1	8	yes	—
65	C₄²×C₇	C₂²×C₄	1	4	yes	—
66	C₄²×C₇	C₂×D₄	1	4	yes	—
67	#1136	—	2	8	no	—
68	#1136	—	2	8	no	—
69	#1136	—	4	8	no	—
70	C₂×C₇⋊C₈	—	2	8	no	—
71	C₂×C₇⋊C₈	—	4	8	no	—
72	C₂²×M₁₆	—	7	4	no	—
73	C₂²×M₁₆	C₂×D₇	1	1	yes	characteristic
74	C₂³×C₈	C₂×D₇	1	1	yes	characteristic
75	C₂×#117	C₂×D₇	1	1	yes	characteristic
76	#424	—	14	8	no	—
77	#423	—	7	8	no	—
78	#405	Dic₇	1	8	yes	—
79	C₂×#97	C₂×D₇	1	2	yes	—
80	C₂×C₄×C₈	C₂×D₇	1	2	yes	—
81	#341	—	7	4	no	—
82	#341	—	7	8	no	—
83	#341	—	7	4	no	—
84	C₂³×C₇	—	4	2	no	—
85	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
86	C₂³×C₇	#125	1	4	yes	—
87	C₂³×C₇	C₂²×Q₈	1	1	yes	characteristic
88	C₂×C₄×C₇	—	4	4	no	—
89	C₂×C₄×C₇	C₂×C₄⋊C₄	1	8	yes	—
90	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
91	C₂×C₄×C₇	#117	1	4	yes	—
92	C₂×C₄×C₇	#117	1	4	yes	—
93	C₂×C₄×C₇	#124	1	8	yes	—
94	C₂×C₄×C₇	#125	1	4	yes	—
95	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
96	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₄×C₇	C₂×#41	1	2	yes	—
98	C₂×C₄×C₇	C₂×#41	1	2	yes	—
99	C₇×C₈	—	2	2	no	—
100	C₇×C₈	—	2	2	no	—
101	C₇×C₈	—	2	4	no	—
102	C₇⋊C₈	—	4	8	no	—
103	C₂³×C₄	C₂²×D₇	1	1	yes	characteristic
104	C₂×M₁₆	—	2	4	no	—
105	C₂×M₁₆	—	7	8	no	—
106	C₂×M₁₆	—	14	4	no	—
107	C₂×M₁₆	—	14	8	no	—
108	C₂×M₁₆	—	14	8	no	—
109	C₂×M₁₆	C₂×Dic₇	1	4	yes	—
110	C₂²×C₈	—	2	4	no	—
111	C₂²×C₈	—	14	4	no	—
112	C₂²×C₈	C₂×Dic₇	1	4	yes	—
113	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
114	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
115	C₂²×C₈	C₂²×D₇	1	2	yes	—
116	#117	D₂₈	1	4	yes	—
117	#117	C₂×Dic₇	1	4	yes	—
118	C₂×C₄⋊C₄	Dic₁₄	1	2	yes	—
119	C₂×C₂²⋊C₄	Dic₁₄	1	2	yes	—
120	C₂×C₄²	C₂²×D₇	1	2	yes	—
121	C₄⋊C₈	—	14	16	no	—
122	#97	C₂×Dic₇	1	8	yes	—
123	C₄×C₈	C₂×Dic₇	1	8	yes	—
124	C₂²×C₇	—	2	2	no	—
125	C₂²×C₇	—	2	4	no	—
126	C₂²×C₇	—	4	4	no	—
127	C₂²×C₇	#382	1	4	yes	—
128	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
129	C₂²×C₇	C₂×#125	1	2	yes	—
130	C₄×C₇	—	2	8	no	—
131	C₄×C₇	#382	1	4	yes	—
132	C₄×C₇	C₂×#117	1	1	yes	characteristic
133	C₄×C₇	C₂×#117	1	1	yes	characteristic
134	C₄×C₇	C₂×#124	1	2	yes	—
135	C₂⁴	C₂×Dic₁₄	1	1	yes	characteristic
136	C₂²×C₄	C₂×Dic₁₄	1	4	yes	—
137	C₂²×C₄	C₂×Dic₁₄	1	1	yes	characteristic
138	C₂²×C₄	C₂×D₂₈	1	2	yes	—
139	C₂²×C₄	#1157	1	4	yes	—
140	C₂²×C₄	#1157	1	4	yes	—
141	C₂²×C₄	C₂²×Dic₇	1	2	yes	—
142	C₂²×C₄	C₂³×D₇	1	1	yes	characteristic
143	M₁₆	—	2	8	no	—
144	M₁₆	—	14	8	no	—
145	M₁₆	—	14	8	no	—
146	M₁₆	—	28	8	no	—
147	C₂×C₈	—	2	8	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	—	14	8	no	—
152	C₂×C₈	—	28	8	no	—
153	C₂×C₈	C₂²×Dic₇	1	2	yes	—
154	C₂×C₈	C₂²×Dic₇	1	4	yes	—
155	C₂×C₈	C₂²×Dic₇	1	2	yes	—
156	C₄⋊C₄	C₄⋊Dic₇	1	8	yes	—
157	C₂²⋊C₄	C₄⋊Dic₇	1	8	yes	—
158	C₄²	C₂×D₂₈	1	4	yes	—
159	C₄²	C₂²×Dic₇	1	4	yes	—
160	C₂×C₇	—	2	2	no	—
161	C₂×C₇	—	2	4	no	—
162	C₂×C₇	C₂×#382	1	1	yes	characteristic
163	C₂×C₇	#2916	1	2	yes	—
164	C₂³	—	4	2	no	—
165	C₂³	#6433	1	4	yes	—
166	C₂³	C₂×C₄⋊Dic₇	1	2	yes	—
167	C₂³	C₂²×Dic₁₄	1	1	yes	characteristic
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	#6433	1	4	yes	—
170	C₂×C₄	#6438	1	8	yes	—
171	C₂×C₄	C₂×C₄⋊Dic₇	1	8	yes	—
172	C₂×C₄	C₂×C₄⋊Dic₇	1	2	yes	—
173	C₂×C₄	#6489	1	4	yes	—
174	C₂×C₄	#6489	1	4	yes	—
175	C₂×C₄	C₂²×D₂₈	1	1	yes	characteristic
176	C₂×C₄	C₂×#1157	1	2	yes	—
177	C₂×C₄	C₂×#1157	1	2	yes	—
178	C₂×C₄	C₂³×Dic₇	1	1	yes	characteristic
179	C₈	—	2	2	no	—
180	C₈	—	2	2	no	—
181	C₈	—	2	4	no	—
182	C₈	—	28	8	no	—
183	C₇	#20768	1	1	yes	characteristic, complemented, Sylow
184	C₂²	—	2	2	no	—
185	C₂²	—	2	4	no	—
186	C₂²	—	4	4	no	—
187	C₂²	#89708	1	4	yes	—
188	C₂²	C₂×#6433	1	2	yes	—
189	C₂²	C₂²×C₄⋊Dic₇	1	1	yes	characteristic
190	C₄	—	2	8	no	—
191	C₄	#89708	1	4	yes	—
192	C₄	C₂×#6438	1	2	yes	—
193	C₄	C₂×#6489	1	1	yes	characteristic
194	C₄	C₂×#6489	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	—	2	4	no	—
197	C₂	C₂×#89708	1	1	yes	characteristic
198	C₂	#11741234	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₈
commutator	C₂×C₄×C₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇
Fitting	C₂×C₇×#405	C₂
socle	C₂²×C₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₇	C₂×Q₈
cosocle	C₂²×C₄×C₇	C₂⁴
hypercenter	C₂×#405	D₇
hypocenter	C₇	#20768
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
5. #11727595	C₂	semidirect
6. #11727545	C₂	semidirect
183. C₇	#20768	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
#11739649	2
#11729752	2
C₂×#89347	2
#11727595	2
#11727545	2
#20768	7