Subgroups — #13252829

Order 1792.

570 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11730756	C₂	1	1	yes	maximal, characteristic
2	C₇×#1842	C₂	1	1	yes	maximal, characteristic
3	C₂×#89277	C₂	1	4	yes	maximal
4	C₄×C₄⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	C₂×#6432	—	2	2	no	—
6	C₂×#6432	C₂²	1	2	yes	—
7	C₂×C₇×C₄⋊C₈	—	2	2	no	—
8	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
9	C₄³×C₇	C₂²	1	1	yes	characteristic
10	#89707	—	2	1	no	—
11	#89707	—	2	1	no	—
12	#89707	C₄	1	4	yes	—
13	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₄×C₇⋊C₈	—	2	1	no	—
15	#89319	—	2	1	no	—
16	#89277	—	2	8	no	—
17	#9692	—	7	1	no	maximal, Sylow
18	C₂²×Dic₁₄	—	2	2	no	—
19	C₂²×C₇×C₈	—	2	2	no	—
20	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
22	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₇	D₄	1	2	yes	—
24	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
25	C₂×C₄⋊Dic₇	—	2	4	no	—
26	C₂×C₄⋊Dic₇	—	2	4	no	—
27	C₂×C₄⋊Dic₇	—	2	2	no	—
28	C₂×C₄⋊Dic₇	C₂×C₄	1	4	yes	—
29	C₂²×C₇⋊C₈	—	2	2	no	—
30	#6432	—	2	4	no	—
31	#6432	—	2	4	no	—
32	#6432	—	4	4	no	—
33	#6432	—	4	4	no	—
34	C₇×C₄⋊C₈	—	2	4	no	—
35	C₇×C₄⋊C₈	—	4	4	no	—
36	C₄⋊(C₇⋊C₈)	—	2	2	no	—
37	C₄⋊(C₇⋊C₈)	D₄	1	4	yes	—
38	C₄×C₇⋊C₈	—	4	2	no	—
39	#2868	—	7	1	no	—
40	#1842	D₇	1	1	yes	characteristic
41	C₄×C₄⋊C₈	—	7	1	no	—
42	C₂×#332	—	7	4	no	—
43	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×Dic₇	—	2	4	no	—
49	C₂×Dic₁₄	—	4	2	no	—
50	C₂×Dic₁₄	—	4	2	no	—
51	C₂×Dic₁₄	—	4	4	no	—
52	C₂×Dic₁₄	—	4	4	no	—
53	C₂×C₇×C₈	—	4	2	no	—
54	C₂×C₇×C₈	—	4	4	no	—
55	C₄²×C₇	—	2	4	no	—
56	C₄²×C₇	—	2	2	no	—
57	C₄²×C₇	D₈	1	4	yes	—
58	C₄²×C₇	SD₁₆	1	2	yes	—
59	C₄²×C₇	SD₁₆	1	4	yes	—
60	C₄²×C₇	Q₁₆	1	2	yes	—
61	C₄²×C₇	C₂×D₄	1	2	yes	—
62	C₄²×C₇	#41	1	2	yes	—
63	C₄⋊Dic₇	—	2	8	no	—
64	C₄⋊Dic₇	—	4	8	no	—
65	C₄⋊Dic₇	—	4	8	no	—
66	C₄⋊Dic₇	—	4	4	no	—
67	C₂×C₇⋊C₈	—	2	4	no	—
68	C₂×C₇⋊C₈	—	4	2	no	—
69	C₂×C₇⋊C₈	—	4	2	no	—
70	C₂×C₄⋊Q₈	—	7	2	no	—
71	C₂×C₄⋊Q₈	—	14	2	no	—
72	C₂×C₄⋊C₈	—	2	2	no	—
73	C₂×C₄⋊C₈	—	7	2	no	—
74	C₂×C₄⋊C₈	C₂×D₇	1	2	yes	—
75	C₂×C₄×C₈	—	14	1	no	—
76	#384	—	7	4	no	—
77	#384	—	14	1	no	—
78	#384	—	14	1	no	—
79	C₄³	C₂×D₇	1	1	yes	characteristic
80	#336	—	14	1	no	—
81	#332	—	14	8	no	—
82	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₇	—	2	2	no	—
84	C₂×C₄×C₇	—	2	2	no	—
85	C₂×C₄×C₇	—	2	2	no	—
86	C₂×C₄×C₇	#101	1	4	yes	—
87	C₂×C₄×C₇	#102	1	4	yes	—
88	C₂×C₄×C₇	#102	1	4	yes	—
89	C₂×C₄×C₇	#103	1	4	yes	—
90	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
91	C₂×C₄×C₇	#121	1	2	yes	—
92	C₂×C₄×C₇	#124	1	1	yes	characteristic
93	C₂×C₄×C₇	#127	1	1	yes	characteristic
94	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
95	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
96	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
97	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
98	C₂×C₄×C₇	#137	1	2	yes	—
99	C₂×Dic₇	—	4	4	no	—
100	C₂×Dic₇	—	4	8	no	—
101	Dic₁₄	—	8	4	no	—
102	Dic₁₄	—	8	2	no	—
103	Dic₁₄	—	8	2	no	—
104	C₇×C₈	—	8	2	no	—
105	C₇⋊C₈	—	4	4	no	—
106	C₂²×Q₈	—	14	2	no	—
107	C₂²×C₈	—	2	2	no	—
108	C₂²×C₈	—	14	2	no	—
109	C₄⋊Q₈	—	14	4	no	—
110	C₄⋊Q₈	—	14	4	no	—
111	C₄⋊Q₈	—	28	4	no	—
112	C₄⋊Q₈	—	28	4	no	—
113	C₂×C₄⋊C₄	—	7	4	no	—
114	C₂×C₄⋊C₄	—	14	4	no	—
115	C₂×C₄⋊C₄	—	14	4	no	—
116	C₂×C₄⋊C₄	—	14	2	no	—
117	C₂×C₄²	C₄×D₇	1	2	yes	—
118	C₂×C₄²	D₂₈	1	1	yes	characteristic
119	C₂×C₄²	D₂₈	1	1	yes	characteristic
120	C₂×C₄²	C₂²⋊D₇	1	2	yes	—
121	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
122	C₄⋊C₈	—	2	4	no	—
123	C₄⋊C₈	—	4	4	no	—
124	C₄⋊C₈	—	7	4	no	—
125	C₄⋊C₈	—	14	2	no	—
126	C₄×C₈	—	28	2	no	—
127	C₂²×C₇	#390	1	1	yes	characteristic
128	C₂²×C₇	C₂×#101	1	2	yes	—
129	C₂²×C₇	C₂×#102	1	2	yes	—
130	C₂²×C₇	C₂×#103	1	1	yes	characteristic
131	C₂²×C₇	#419	1	1	yes	characteristic
132	C₄×C₇	—	2	4	no	—
133	C₄×C₇	—	2	2	no	—
134	C₄×C₇	#332	1	8	yes	complemented
135	C₄×C₇	#461	1	2	yes	—
136	C₄×C₇	#462	1	2	yes	—
137	C₄×C₇	#486	1	2	yes	—
138	C₄×C₇	#492	1	1	yes	characteristic
139	C₄×C₇	#493	1	1	yes	characteristic
140	Dic₇	—	8	4	no	—
141	C₂×Q₈	—	28	4	no	—
142	C₂×Q₈	—	28	2	no	—
143	C₂×Q₈	—	28	4	no	—
144	C₂×Q₈	—	28	2	no	—
145	C₂²×C₄	—	14	4	no	—
146	C₂²×C₄	(C₂×D₇)⋊C₄	1	2	yes	—
147	C₂²×C₄	(C₂×D₇)⋊C₄	1	2	yes	—
148	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
149	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
150	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
151	C₂×C₈	—	4	4	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	14	4	no	—
154	C₂×C₈	—	28	2	no	—
155	C₂×C₈	—	28	2	no	—
156	C₄⋊C₄	—	14	8	no	—
157	C₄⋊C₄	—	28	4	no	—
158	C₄⋊C₄	—	28	8	no	—
159	C₄⋊C₄	—	28	8	no	—
160	C₄²	—	2	2	no	—
161	C₄²	—	2	4	no	—
162	C₄²	#1132	1	2	yes	—
163	C₄²	Dic₂₈	1	2	yes	—
164	C₄²	#1141	1	4	yes	—
165	C₄²	#1143	1	4	yes	—
166	C₄²	D₄×D₇	1	2	yes	—
167	C₄²	#1161	1	2	yes	—
168	C₂×C₇	C₂×#332	1	2	yes	—
169	C₂×C₇	#1546	1	2	yes	—
170	C₂×C₇	#1957	1	1	yes	characteristic
171	C₂×C₇	#1966	1	1	yes	characteristic
172	C₂×C₇	#1981	1	1	yes	characteristic
173	C₂³	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
174	Q₈	—	56	2	no	—
175	Q₈	—	56	4	no	—
176	Q₈	—	56	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	28	8	no	—
181	C₂×C₄	—	28	4	no	—
182	C₂×C₄	#6383	1	4	yes	—
183	C₂×C₄	#6383	1	4	yes	—
184	C₂×C₄	#6390	1	4	yes	—
185	C₂×C₄	#6398	1	4	yes	—
186	C₂×C₄	#6456	1	2	yes	—
187	C₂×C₄	#6458	1	2	yes	—
188	C₂×C₄	C₂×#1132	1	1	yes	characteristic
189	C₂×C₄	C₂×Dic₂₈	1	1	yes	characteristic
190	C₂×C₄	#6472	1	2	yes	—
191	C₂×C₄	C₂×#1141	1	2	yes	—
192	C₂×C₄	#6503	1	1	yes	characteristic
193	C₂×C₄	C₂×#1143	1	2	yes	—
194	C₂×C₄	#6510	1	1	yes	characteristic
195	C₈	—	8	2	no	—
196	C₈	—	28	4	no	—
197	C₇	#9692	1	1	yes	characteristic, complemented, Sylow
198	C₂²	C₂×#6383	1	2	yes	—
199	C₂²	#89760	1	1	yes	characteristic
200	C₂²	C₂×#6390	1	1	yes	characteristic
201	C₂²	#89892	1	1	yes	characteristic
202	C₂²	C₂×#6398	1	2	yes	—
203	C₄	—	2	4	no	—
204	C₄	—	2	2	no	—
205	C₄	—	56	4	no	—
206	C₄	#89277	1	8	yes	complemented
207	C₄	#89613	1	2	yes	—
208	C₄	#89621	1	2	yes	—
209	C₄	#89845	1	1	yes	characteristic
210	C₄	#89857	1	1	yes	characteristic
211	C₄	#89860	1	2	yes	—
212	C₂	#11727334	1	1	yes	characteristic
213	C₂	C₂×#89277	1	2	yes	—
214	C₂	#11728210	1	1	yes	characteristic
215	C₂	#11728260	1	2	yes	—
216	C₂	#11728276	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₇)⋊C₄
commutator	C₂²×C₄×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1842	C₂
socle, prime radical	C₂³×C₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#1842	D₇
hypocenter	C₇	#9692
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
134. C₄×C₇	#332	semidirect
197. C₇	#9692	semidirect
206. C₄	#89277	semidirect

Maximal subgroups

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

subgroup	index
#11730756	2
C₇×#1842	2
C₂×#89277	2
C₄×C₄⋊(C₇⋊C₈)	2
#9692	7