Subgroups — #13272172

Order 1792.

438 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×Q₃₂	C₂	1	1	yes	maximal, characteristic
2	#11729820	C₂	1	2	yes	maximal
3	C₂×#89359	C₂	1	2	yes	maximal
4	C₂×#89299	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₆×Dic₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇×Q₁₆	C₂²	1	2	yes	—
7	C₂×C₇×Q₃₂	—	2	2	no	—
8	C₂×C₇×Q₃₂	—	2	2	no	—
9	C₂×C₇×Q₃₂	C₄	1	4	yes	complemented
10	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
11	#90002	—	2	2	no	—
12	C₂×#6409	—	2	2	no	—
13	C₂×#6392	C₂²	1	2	yes	—
14	C₂×C₈×Dic₇	C₂²	1	1	yes	characteristic
15	C₂²×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
16	#89359	—	2	4	no	—
17	#89299	—	2	2	no	—
18	C₁₆×Dic₇	—	2	2	no	—
19	#13236	—	7	1	no	maximal, Sylow
20	C₂²×C₇×Q₈	—	2	2	no	—
21	C₂×C₇×Q₁₆	—	2	2	no	—
22	C₂×C₇×Q₁₆	—	2	2	no	—
23	C₂×C₇×Q₁₆	—	2	2	no	—
24	C₂×C₇×Q₁₆	—	2	2	no	—
25	C₂×C₇×Q₁₆	—	2	2	no	—
26	C₂×C₇×Q₁₆	C₂×C₄	1	4	yes	—
27	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊Dic₇	—	2	2	no	—
29	C₂×C₄×Dic₇	D₄	1	1	yes	characteristic
30	C₂²×C₇⋊C₈	D₄	1	1	yes	characteristic
31	C₇×Q₃₂	—	2	8	no	—
32	C₂×C₇×C₁₆	C₂×C₄	1	2	yes	—
33	C₂×C₇×C₁₆	D₄	1	2	yes	—
34	C₂×C₇×C₁₆	D₄	1	2	yes	—
35	#6409	—	4	4	no	—
36	C₇⋊#95	—	4	4	no	—
37	#6392	—	2	2	no	—
38	#6392	—	2	2	no	—
39	C₈×Dic₇	—	2	1	no	—
40	C₈×Dic₇	—	2	1	no	—
41	C₂×C₇⋊C₁₆	—	2	1	no	—
42	C₂×C₇⋊C₁₆	—	2	1	no	—
43	C₂×C₇⋊C₁₆	D₄	1	2	yes	—
44	C₂²×Q₃₂	D₇	1	1	yes	characteristic
45	C₂×#366	—	7	1	no	—
46	C₂×#358	—	7	2	no	—
47	C₂×C₄×C₁₆	—	7	1	no	—
48	#1969	—	7	2	no	—
49	C₂×C₇×Q₈	—	2	4	no	—
50	C₂×C₇×Q₈	—	4	2	no	—
51	C₂×C₇×Q₈	—	4	2	no	—
52	C₂×C₇×Q₈	—	4	2	no	—
53	C₂×C₇×Q₈	—	4	2	no	—
54	C₂×C₇×Q₈	—	4	2	no	—
55	C₂²×C₄×C₇	—	4	2	no	—
56	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₇	—	4	2	no	—
58	C₂²×Dic₇	D₈	1	2	yes	—
59	C₇×Q₁₆	—	2	4	no	—
60	C₇×Q₁₆	—	2	4	no	—
61	C₇×Q₁₆	—	4	2	no	—
62	C₇×Q₁₆	—	4	2	no	—
63	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
65	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
66	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
67	C₂×C₇×C₈	#41	1	1	yes	characteristic
68	C₂×C₇×C₈	#41	1	1	yes	characteristic
69	C₄⋊Dic₇	—	4	2	no	—
70	C₄⋊Dic₇	—	4	2	no	—
71	C₄×Dic₇	—	2	1	no	—
72	C₄×Dic₇	—	2	1	no	—
73	C₂×C₇⋊C₈	—	2	1	no	—
74	C₂×C₇⋊C₈	—	2	1	no	—
75	C₂×C₇⋊C₈	D₈	1	2	yes	—
76	C₇×C₁₆	C₂²⋊C₄	1	4	yes	—
77	C₇⋊C₁₆	—	2	2	no	—
78	C₂²×Q₁₆	C₂×D₇	1	2	yes	—
79	C₂×Q₃₂	—	2	2	no	—
80	C₂×Q₃₂	—	2	2	no	—
81	C₂×Q₃₂	Dic₇	1	4	yes	complemented
82	C₂²×C₁₆	—	7	1	no	—
83	C₂²×C₁₆	C₂×D₇	1	1	yes	characteristic
84	C₂×#107	—	7	2	no	—
85	C₂×#103	—	14	2	no	—
86	C₂×C₄×C₈	—	7	1	no	—
87	#391	—	14	2	no	—
88	#366	—	14	2	no	—
89	#358	—	14	4	no	—
90	C₄×C₁₆	—	14	2	no	—
91	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
92	C₇×Q₈	—	4	4	no	—
93	C₇×Q₈	—	4	4	no	—
94	C₇×Q₈	—	8	2	no	—
95	C₇×Q₈	—	8	2	no	—
96	C₂×C₄×C₇	—	4	4	no	—
97	C₂×C₄×C₇	—	8	2	no	—
98	C₂×C₄×C₇	—	8	2	no	—
99	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₇	#121	1	1	yes	characteristic
101	C₂×C₄×C₇	#127	1	1	yes	characteristic
102	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₇	#135	1	1	yes	characteristic
104	C₂×C₄×C₇	#135	1	1	yes	characteristic
105	C₂×Dic₇	—	2	2	no	—
106	C₂×Dic₇	—	2	2	no	—
107	C₂×Dic₇	—	8	2	no	—
108	C₂×Dic₇	—	8	4	no	—
109	C₂×Dic₇	Q₃₂	1	4	yes	complemented
110	C₇×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₇×C₈	C₄×D₄	1	1	yes	characteristic
112	C₇×C₈	#121	1	1	yes	characteristic
113	C₇×C₈	#124	1	1	yes	characteristic
114	C₇⋊C₈	—	2	2	no	—
115	C₂²×Q₈	—	2	2	no	—
116	C₂×Q₁₆	—	2	2	no	—
117	C₂×Q₁₆	—	2	2	no	—
118	C₂×Q₁₆	—	2	2	no	—
119	C₂×Q₁₆	—	2	2	no	—
120	C₂×Q₁₆	—	2	2	no	—
121	C₂×Q₁₆	C₂×Dic₇	1	4	yes	—
122	C₂²×C₈	—	7	1	no	—
123	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
124	C₂×C₄⋊C₄	—	14	2	no	—
125	C₂×C₄²	—	7	1	no	—
126	Q₃₂	—	2	8	no	—
127	C₂×C₁₆	—	7	2	no	—
128	C₂×C₁₆	—	14	1	no	—
129	C₂×C₁₆	—	14	1	no	—
130	C₂×C₁₆	C₂×Dic₇	1	2	yes	—
131	C₂×C₁₆	C₂²⋊D₇	1	2	yes	—
132	C₂×C₁₆	C₂²⋊D₇	1	2	yes	—
133	#107	—	14	2	no	—
134	#107	—	14	2	no	—
135	#103	—	28	4	no	—
136	C₄×C₈	—	14	1	no	—
137	C₄×C₈	—	14	1	no	—
138	#95	—	28	4	no	—
139	C₂²×C₇	C₄×D₈	1	1	yes	characteristic
140	C₂²×C₇	#466	1	1	yes	characteristic
141	C₂²×C₇	#493	1	1	yes	characteristic
142	C₂²×C₇	C₂×Q₃₂	1	2	yes	—
143	C₂²×C₇	#508	1	2	yes	—
144	C₄×C₇	—	8	4	no	—
145	C₄×C₇	#390	1	1	yes	characteristic
146	C₄×C₇	C₄×D₈	1	1	yes	characteristic
147	C₄×C₇	#466	1	1	yes	characteristic
148	C₄×C₇	#495	1	1	yes	characteristic
149	Dic₇	—	2	4	no	—
150	Dic₇	—	16	2	no	—
151	C₂×Q₈	—	2	4	no	—
152	C₂×Q₈	—	4	2	no	—
153	C₂×Q₈	—	4	2	no	—
154	C₂×Q₈	—	4	2	no	—
155	C₂×Q₈	—	4	2	no	—
156	C₂×Q₈	—	4	2	no	—
157	C₂²×C₄	—	4	2	no	—
158	C₂²×C₄	—	7	2	no	—
159	C₂²×C₄	—	28	2	no	—
160	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
161	Q₁₆	—	2	4	no	—
162	Q₁₆	—	2	4	no	—
163	Q₁₆	—	4	2	no	—
164	Q₁₆	—	4	2	no	—
165	C₂×C₈	—	7	2	no	—
166	C₂×C₈	—	14	1	no	—
167	C₂×C₈	—	14	1	no	—
168	C₂×C₈	D₄×D₇	1	1	yes	characteristic
169	C₂×C₈	#1159	1	1	yes	characteristic
170	C₂×C₈	#1159	1	1	yes	characteristic
171	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
172	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
173	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
174	C₄⋊C₄	—	28	2	no	—
175	C₄⋊C₄	—	28	2	no	—
176	C₄²	—	14	1	no	—
177	C₄²	—	14	1	no	—
178	C₁₆	—	14	2	no	—
179	C₁₆	C₂²⋊Dic₇	1	4	yes	—
180	C₂×C₇	#1970	1	1	yes	characteristic
181	C₂×C₇	C₄×Q₃₂	1	2	yes	—
182	C₂×C₇	#2216	1	2	yes	—
183	C₂×C₇	#2247	1	1	yes	characteristic
184	C₂×C₇	#2249	1	1	yes	characteristic
185	C₂³	D₇×D₈	1	1	yes	characteristic
186	Q₈	—	4	4	no	—
187	Q₈	—	4	4	no	—
188	Q₈	—	8	2	no	—
189	Q₈	—	8	2	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	7	4	no	—
192	C₂×C₄	—	8	2	no	—
193	C₂×C₄	—	8	2	no	—
194	C₂×C₄	—	14	2	no	—
195	C₂×C₄	—	14	2	no	—
196	C₂×C₄	—	56	2	no	—
197	C₂×C₄	—	56	4	no	—
198	C₂×C₄	D₇×D₈	1	1	yes	characteristic
199	C₂×C₄	#6475	1	1	yes	characteristic
200	C₂×C₄	#6475	1	1	yes	characteristic
201	C₂×C₄	D₄×Dic₇	1	1	yes	characteristic
202	C₂×C₄	#6501	1	1	yes	characteristic
203	C₂×C₄	#6503	1	1	yes	characteristic
204	C₈	—	14	2	no	—
205	C₈	D₄×Dic₇	1	1	yes	characteristic
206	C₈	#6499	1	1	yes	characteristic
207	C₈	#6501	1	1	yes	characteristic
208	C₈	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
209	C₇	#13236	1	1	yes	characteristic, complemented, Sylow
210	C₂²	D₇×Q₃₂	1	2	yes	—
211	C₂²	#89691	1	2	yes	—
212	C₂²	D₈×Dic₇	1	1	yes	characteristic
213	C₂²	#89923	1	1	yes	characteristic
214	C₂²	#89929	1	1	yes	characteristic
215	C₄	—	8	4	no	—
216	C₄	—	14	4	no	—
217	C₄	—	112	2	no	—
218	C₄	D₈×Dic₇	1	1	yes	characteristic
219	C₄	#89927	1	1	yes	characteristic
220	C₄	#89929	1	1	yes	characteristic
221	C₄	#89993	1	1	yes	characteristic
222	C₂	Dic₇×Q₃₂	1	2	yes	—
223	C₂	#11729253	1	1	yes	characteristic
224	C₂	#11729254	1	1	yes	characteristic
225	C₂	#11729258	1	2	yes	—
226	C₂	#11729792	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₇×D₈
commutator	C₂×C₇×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₂²×C₇×Q₃₂	C₂
socle, prime radical	C₂³×C₇	C₂×D₈
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₂²×Q₃₂	D₇
hypocenter	C₇	#13236
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₃₂

Complements

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

N	H ≅ G/N	type
9. C₂×C₇×Q₃₂	C₄	semidirect
81. C₂×Q₃₂	Dic₇	semidirect
109. C₂×Dic₇	Q₃₂	semidirect
209. C₇	#13236	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₇×Q₃₂	2
#11729820	2
C₂×#89359	2
C₂×#89299	2
C₂×C₁₆×Dic₇	2
#13236	7