Subgroups — #13272013

Order 1792.

510 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#89471	C₂	1	1	yes	maximal, characteristic
2	C₇×#1845	C₂	1	1	yes	maximal, characteristic
3	C₂×#89287	C₂	1	4	yes	maximal
4	C₂×C₄×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₈	—	2	1	no	—
6	C₂²×Dic₂₈	C₂²	1	2	yes	—
7	C₂×#6432	—	2	1	no	—
8	C₂×C₇×#107	—	2	1	no	—
9	C₂×C₇×#107	C₂²	1	2	yes	—
10	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₇×#384	—	2	1	no	—
12	C₂²×C₇⋊C₁₆	C₂²	1	2	yes	—
13	#89471	—	2	2	no	—
14	#89471	C₄	1	4	yes	complemented
15	#89287	—	2	8	no	—
16	C₄×C₇⋊C₁₆	—	2	2	no	—
17	#13203	—	7	1	no	maximal, Sylow
18	C₂²×Dic₁₄	—	2	2	no	—
19	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
20	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
21	C₂×C₇×C₄⋊C₄	—	2	2	no	—
22	C₂×C₇×C₄⋊C₄	—	4	2	no	—
23	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
24	C₂×C₄⋊Dic₇	—	4	2	no	—
25	C₂×Dic₂₈	—	2	4	no	—
26	C₂×Dic₂₈	—	2	4	no	—
27	C₂×Dic₂₈	—	2	4	no	—
28	C₂×Dic₂₈	—	2	2	no	—
29	C₂×Dic₂₈	—	4	2	no	—
30	C₂×Dic₂₈	—	4	2	no	—
31	C₂×Dic₂₈	C₂×C₄	1	4	yes	—
32	#6432	—	2	4	no	—
33	#6432	—	4	2	no	—
34	C₇×#107	—	2	4	no	—
35	C₇×#107	—	4	2	no	—
36	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
37	C₄×C₇×C₈	D₄	1	2	yes	—
38	C₂×C₇⋊C₁₆	—	2	2	no	—
39	C₂×C₇⋊C₁₆	—	2	2	no	—
40	C₂×C₇⋊C₁₆	D₄	1	4	yes	—
41	C₂×#494	—	7	1	no	—
42	C₂×#358	—	7	4	no	—
43	C₂×C₄×C₁₆	—	7	1	no	—
44	#1845	D₇	1	1	yes	characteristic
45	C₂²×C₄×C₇	—	4	2	no	—
46	C₂²×C₄×C₇	D₈	1	2	yes	—
47	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×Dic₇	—	4	2	no	—
49	C₂×Dic₁₄	—	2	4	no	—
50	C₂×Dic₁₄	—	4	2	no	—
51	C₂×Dic₁₄	—	4	2	no	—
52	C₂×Dic₁₄	—	4	2	no	—
53	C₂×Dic₁₄	—	4	2	no	—
54	C₂×Dic₁₄	—	4	2	no	—
55	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
56	C₂×C₇×C₈	D₈	1	2	yes	—
57	C₂×C₇×C₈	SD₁₆	1	2	yes	—
58	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
60	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
61	C₂×C₇×C₈	#41	1	2	yes	—
62	C₇×C₄⋊C₄	—	4	2	no	—
63	C₇×C₄⋊C₄	—	4	2	no	—
64	C₇×C₄⋊C₄	—	8	4	no	—
65	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
66	C₄²×C₇	SD₁₆	1	2	yes	—
67	C₄⋊Dic₇	—	4	8	no	—
68	Dic₂₈	—	2	8	no	—
69	Dic₂₈	—	4	8	no	—
70	Dic₂₈	—	4	4	no	—
71	C₇⋊C₁₆	—	2	4	no	—
72	C₂²×Q₁₆	—	7	2	no	—
73	C₂²×Q₁₆	—	14	1	no	—
74	C₂×C₄⋊Q₈	—	14	1	no	—
75	C₂²×C₁₆	—	7	2	no	—
76	#494	—	7	4	no	—
77	#494	—	14	2	no	—
78	C₂×#107	—	2	1	no	—
79	C₂×#107	C₂×D₇	1	2	yes	—
80	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
81	#384	—	2	1	no	—
82	#358	—	14	8	no	—
83	C₄×C₁₆	—	14	2	no	—
84	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₇	—	8	2	no	—
86	C₂×C₄×C₇	—	8	4	no	—
87	C₂×C₄×C₇	#102	1	4	yes	—
88	C₂×C₄×C₇	SD₃₂	1	4	yes	—
89	C₂×C₄×C₇	Q₃₂	1	4	yes	—
90	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₇	#124	1	1	yes	characteristic
92	C₂×C₄×C₇	#127	1	1	yes	characteristic
93	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
94	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
95	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
96	C₂×Dic₇	—	4	4	no	—
97	C₂×Dic₇	—	8	2	no	—
98	C₂×Dic₇	—	8	2	no	—
99	Dic₁₄	—	4	4	no	—
100	Dic₁₄	—	4	4	no	—
101	Dic₁₄	—	8	2	no	—
102	Dic₁₄	—	8	2	no	—
103	C₇×C₈	#102	1	4	yes	—
104	C₇×C₈	C₄×D₄	1	2	yes	—
105	C₇×C₈	#121	1	2	yes	—
106	C₂²×Q₈	—	14	2	no	—
107	C₂×Q₁₆	—	7	4	no	—
108	C₂×Q₁₆	—	14	4	no	—
109	C₂×Q₁₆	—	14	2	no	—
110	C₂×Q₁₆	—	14	4	no	—
111	C₂×Q₁₆	—	14	4	no	—
112	C₂×Q₁₆	—	28	2	no	—
113	C₂×Q₁₆	—	28	2	no	—
114	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
115	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
116	C₄⋊Q₈	—	14	4	no	—
117	C₄⋊Q₈	—	28	2	no	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₄⋊C₄	—	4	2	no	—
120	C₂×C₄⋊C₄	—	28	2	no	—
121	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
122	C₂×C₁₆	—	7	4	no	—
123	C₂×C₁₆	—	14	2	no	—
124	C₂×C₁₆	—	14	2	no	—
125	#107	—	2	4	no	—
126	#107	—	4	2	no	—
127	C₄×C₈	C₄×D₇	1	2	yes	—
128	C₄×C₈	D₂₈	1	2	yes	—
129	C₂²×C₇	C₂×#102	1	1	yes	characteristic
130	C₂²×C₇	#486	1	1	yes	characteristic
131	C₂²×C₇	#493	1	1	yes	characteristic
132	C₂²×C₇	C₂×SD₃₂	1	2	yes	—
133	C₂²×C₇	C₂×Q₃₂	1	2	yes	—
134	C₄×C₇	—	16	2	no	—
135	C₄×C₇	#358	1	8	yes	complemented
136	C₄×C₇	#390	1	1	yes	characteristic
137	C₄×C₇	C₂×#102	1	1	yes	characteristic
138	C₄×C₇	#486	1	1	yes	characteristic
139	C₄×C₇	#492	1	1	yes	characteristic
140	Dic₇	—	8	4	no	—
141	C₂×Q₈	—	14	4	no	—
142	C₂×Q₈	—	28	2	no	—
143	C₂×Q₈	—	28	2	no	—
144	C₂×Q₈	—	28	2	no	—
145	C₂×Q₈	—	28	2	no	—
146	C₂×Q₈	—	28	2	no	—
147	C₂²×C₄	—	4	2	no	—
148	C₂²×C₄	—	28	2	no	—
149	C₂²×C₄	#1141	1	2	yes	—
150	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
151	Q₁₆	—	14	8	no	—
152	Q₁₆	—	28	4	no	—
153	Q₁₆	—	28	8	no	—
154	C₂×C₈	(C₂×D₇)⋊C₄	1	2	yes	—
155	C₂×C₈	#1141	1	2	yes	—
156	C₂×C₈	#1143	1	2	yes	—
157	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
158	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
159	C₂×C₈	D₄×D₇	1	2	yes	—
160	C₂×C₈	#1161	1	2	yes	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	8	4	no	—
164	C₄⋊C₄	—	28	8	no	—
165	C₄²	(C₂×D₇)⋊C₄	1	2	yes	—
166	C₄²	#1143	1	2	yes	—
167	C₁₆	—	14	4	no	—
168	C₂×C₇	#1957	1	1	yes	characteristic
169	C₂×C₇	C₂×#358	1	2	yes	—
170	C₂×C₇	#2241	1	2	yes	—
171	C₂×C₇	#2247	1	1	yes	characteristic
172	C₂×C₇	#2248	1	1	yes	characteristic
173	C₂³	C₂×#1141	1	1	yes	characteristic
174	Q₈	—	28	4	no	—
175	Q₈	—	28	4	no	—
176	Q₈	—	56	2	no	—
177	Q₈	—	56	2	no	—
178	C₂×C₄	—	8	4	no	—
179	C₂×C₄	—	8	2	no	—
180	C₂×C₄	—	28	4	no	—
181	C₂×C₄	—	56	2	no	—
182	C₂×C₄	—	56	2	no	—
183	C₂×C₄	#6383	1	4	yes	—
184	C₂×C₄	#6401	1	4	yes	—
185	C₂×C₄	#6403	1	4	yes	—
186	C₂×C₄	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
187	C₂×C₄	C₂×#1141	1	1	yes	characteristic
188	C₂×C₄	#6503	1	1	yes	characteristic
189	C₂×C₄	C₂×#1143	1	1	yes	characteristic
190	C₂×C₄	C₂×#1143	1	1	yes	characteristic
191	C₂×C₄	#6510	1	1	yes	characteristic
192	C₈	#6383	1	4	yes	—
193	C₈	#6456	1	2	yes	—
194	C₈	#6458	1	2	yes	—
195	C₇	#13203	1	1	yes	characteristic, complemented, Sylow
196	C₂²	C₂×#6383	1	1	yes	characteristic
197	C₂²	#89845	1	1	yes	characteristic
198	C₂²	#89860	1	1	yes	characteristic
199	C₂²	C₂×#6401	1	2	yes	—
200	C₂²	C₂×#6403	1	2	yes	—
201	C₄	—	16	2	no	—
202	C₄	—	56	4	no	—
203	C₄	#89287	1	8	yes	complemented
204	C₄	C₂×#6383	1	1	yes	characteristic
205	C₄	#89760	1	1	yes	characteristic
206	C₄	#89857	1	1	yes	characteristic
207	C₄	#89860	1	1	yes	characteristic
208	C₂	#11727334	1	1	yes	characteristic
209	C₂	C₂×#89287	1	2	yes	—
210	C₂	#11729045	1	1	yes	characteristic
211	C₂	#11729066	1	1	yes	characteristic
212	C₂	#11729136	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#1141
commutator	C₂×C₇×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₇×#1845	C₂
socle, prime radical	C₂³×C₇	C₂×D₈
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	#1845	D₇
hypocenter	C₇	#13203
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

Complements

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

N	H ≅ G/N	type
14. #89471	C₄	semidirect
135. C₄×C₇	#358	semidirect
195. C₇	#13203	semidirect
203. C₄	#89287	semidirect

Maximal subgroups

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

subgroup	index
C₂×#89471	2
C₇×#1845	2
C₂×#89287	2
C₂×C₄×C₇⋊C₁₆	2
#13203	7