Subgroups — #13509314

Order 1792.

474 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	#11736618	C₂	1	2	yes	maximal
2	C₇×#2158	C₂	1	1	yes	maximal, characteristic
3	C₂×#89300	C₂	1	1	yes	maximal, characteristic
4	#11729153	C₂	1	1	yes	maximal, characteristic
5	#11728403	C₂	1	2	yes	maximal
6	#11727097	C₂	1	2	yes	maximal, complemented
7	#11727094	C₂	1	2	yes	maximal, complemented
8	#11727076	C₂	1	4	yes	maximal, complemented
9	#90210	—	2	2	no	—
10	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
11	C₇×#427	C₂²	1	2	yes	—
12	C₂×#6392	C₂²	1	2	yes	—
13	#89873	C₂²	1	1	yes	characteristic
14	C₂×#6386	C₂²	1	1	yes	characteristic
15	#89767	—	2	2	no	—
16	#89650	C₂²	1	4	yes	—
17	#89646	C₂²	1	4	yes	—
18	#89628	—	2	4	no	—
19	C₇×#367	C₂²	1	1	yes	characteristic
20	C₇×#367	C₂²	1	1	yes	characteristic
21	C₇×#366	C₂²	1	2	yes	—
22	#89300	C₂²	1	2	yes	—
23	#89300	C₂²	1	2	yes	—
24	#89297	C₂²	1	2	yes	—
25	C₁₆×Dic₇	C₂²	1	2	yes	—
26	#89285	C₂²	1	4	yes	—
27	#89284	C₂²	1	4	yes	—
28	#33730	—	7	1	no	maximal, Sylow
29	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
30	C₇×#117	—	2	2	no	—
31	#6489	D₄	1	1	yes	characteristic
32	C₂×C₄⋊Dic₇	—	2	2	no	—
33	C₂×#1136	D₄	1	1	yes	characteristic
34	#6453	—	2	4	no	—
35	#6451	—	2	4	no	—
36	#6441	—	4	4	no	—
37	#6433	—	4	2	no	—
38	#6432	—	4	2	no	—
39	C₂×C₇×C₁₆	Q₈	1	4	yes	complemented
40	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
41	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
42	C₇×#107	C₂³	1	2	yes	—
43	C₇×#107	C₂³	1	2	yes	—
44	C₇×#106	—	2	2	no	—
45	#6392	—	2	2	no	—
46	#6392	C₂³	1	4	yes	—
47	#6389	D₄	1	2	yes	—
48	C₈×Dic₇	C₂³	1	2	yes	—
49	#6386	Q₈	1	4	yes	complemented
50	C₂×C₇⋊C₁₆	C₂³	1	2	yes	—
51	#6382	—	2	4	no	—
52	#6381	—	2	4	no	—
53	#3164	—	7	2	no	—
54	#2255	—	7	4	no	—
55	#2254	—	7	2	no	—
56	#2253	—	7	2	no	—
57	#2159	—	7	2	no	—
58	#2158	D₇	1	1	yes	characteristic
59	C₂×#367	—	7	1	no	—
60	#2108	—	7	1	no	—
61	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×Dic₇	—	4	2	no	—
63	C₂×Dic₁₄	—	4	4	no	—
64	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
65	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
66	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
67	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
68	C₇×C₄⋊C₄	—	2	2	no	—
69	C₇×C₄⋊C₄	—	2	2	no	—
70	C₇×C₂²⋊C₄	—	4	2	no	—
71	C₄²×C₇	—	4	2	no	—
72	C₂²⋊Dic₇	D₈	1	2	yes	—
73	C₄⋊Dic₇	—	2	4	no	—
74	C₄⋊Dic₇	—	4	2	no	—
75	C₄⋊Dic₇	—	4	4	no	—
76	C₄⋊Dic₇	D₈	1	2	yes	—
77	Dic₇⋊C₄	—	4	4	no	—
78	Dic₇⋊C₄	—	4	4	no	—
79	C₄×Dic₇	C₂×D₄	1	2	yes	—
80	#1136	Q₁₆	1	4	yes	—
81	C₂×C₇⋊C₈	C₂×D₄	1	2	yes	—
82	C₇×C₁₆	C₂×Q₈	1	2	yes	—
83	C₇×C₁₆	C₂×Q₈	1	2	yes	—
84	C₇⋊C₁₆	C₂×Q₈	1	4	yes	—
85	#544	—	14	2	no	—
86	C₂×M₃₂	—	7	1	no	—
87	C₂²×C₁₆	C₂×D₇	1	1	yes	characteristic
88	#501	—	14	4	no	—
89	#500	—	7	4	no	—
90	#499	—	7	4	no	—
91	#428	—	14	2	no	—
92	#427	C₂×D₇	1	2	yes	—
93	C₂×#107	—	7	2	no	—
94	#405	—	7	1	no	—
95	#367	—	7	2	no	—
96	#367	—	7	2	no	—
97	#367	—	7	4	no	—
98	#367	C₂×D₇	1	1	yes	characteristic
99	#367	C₂×D₇	1	1	yes	characteristic
100	#366	—	7	4	no	—
101	#366	C₂×D₇	1	2	yes	—
102	#346	—	7	2	no	—
103	C₄×C₁₆	—	7	2	no	—
104	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₇	—	4	2	no	—
106	C₂×C₄×C₇	—	4	2	no	—
107	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
108	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
109	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
110	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
111	C₂×Dic₇	—	4	4	no	—
112	C₂×Dic₇	—	16	2	no	—
113	C₂×Dic₇	C₂×D₈	1	4	yes	—
114	Dic₁₄	—	16	4	no	—
115	C₇×C₈	C₄⋊Q₈	1	2	yes	—
116	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
117	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
118	C₇⋊C₈	C₂×Q₁₆	1	4	yes	—
119	C₂×M₁₆	—	7	1	no	—
120	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
121	C₄⋊Q₈	—	14	4	no	—
122	C₄⋊Q₈	—	28	2	no	—
123	#125	—	14	4	no	—
124	#125	—	28	2	no	—
125	C₂²⋊Q₈	—	28	4	no	—
126	#117	—	2	2	no	—
127	#117	—	7	1	no	—
128	C₂×C₄⋊C₄	—	14	2	no	—
129	M₃₂	—	7	4	no	—
130	C₂×C₁₆	—	7	2	no	—
131	C₂×C₁₆	Dic₁₄	1	4	yes	complemented
132	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
133	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
134	#107	—	7	4	no	—
135	#107	—	14	4	no	—
136	#107	—	14	2	no	—
137	#107	C₂²×D₇	1	2	yes	—
138	#107	C₂²×D₇	1	2	yes	—
139	#106	—	2	2	no	—
140	#106	—	14	4	no	—
141	#97	—	7	2	no	—
142	C₄×C₈	—	7	2	no	—
143	C₂²×C₇	—	2	2	no	—
144	C₂²×C₇	#500	1	2	yes	—
145	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
146	C₄×C₇	—	8	2	no	—
147	C₄×C₇	—	8	2	no	—
148	C₄×C₇	#500	1	2	yes	—
149	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
150	C₄×C₇	C₂²×Q₁₆	1	1	yes	characteristic
151	Dic₇	—	2	4	no	—
152	Dic₇	—	8	4	no	—
153	C₂×Q₈	—	28	4	no	—
154	C₂²×C₄	—	28	2	no	—
155	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
156	M₁₆	—	7	4	no	—
157	C₂×C₈	—	7	2	no	—
158	C₂×C₈	C₂×Dic₁₄	1	2	yes	—
159	C₂×C₈	D₄×D₇	1	1	yes	characteristic
160	C₂×C₈	Q₈×D₇	1	2	yes	—
161	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
162	C₄⋊C₄	—	2	2	no	—
163	C₄⋊C₄	—	2	2	no	—
164	C₄⋊C₄	—	7	2	no	—
165	C₄⋊C₄	—	14	4	no	—
166	C₄⋊C₄	—	28	4	no	—
167	C₄⋊C₄	—	28	4	no	—
168	C₄⋊C₄	—	28	4	no	—
169	C₄⋊C₄	—	28	2	no	—
170	C₂²⋊C₄	—	4	2	no	—
171	C₂²⋊C₄	—	7	2	no	—
172	C₄²	—	4	2	no	—
173	C₄²	—	7	2	no	—
174	C₁₆	—	7	4	no	—
175	C₁₆	C₂×Dic₁₄	1	2	yes	—
176	C₁₆	C₂×Dic₁₄	1	2	yes	—
177	C₂×C₇	—	2	2	no	—
178	C₂×C₇	C₂×#500	1	1	yes	characteristic
179	C₂×C₇	#3416	1	2	yes	—
180	C₂³	D₇×D₈	1	1	yes	characteristic
181	Q₈	—	112	4	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	7	4	no	—
185	C₂×C₄	—	28	4	no	—
186	C₂×C₄	—	112	2	no	—
187	C₂×C₄	#6451	1	2	yes	—
188	C₂×C₄	D₇×D₈	1	1	yes	characteristic
189	C₂×C₄	D₇×Q₁₆	1	2	yes	—
190	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
191	C₈	—	7	4	no	—
192	C₈	#6451	1	2	yes	—
193	C₈	C₂²×Dic₁₄	1	1	yes	characteristic
194	C₈	C₂×Q₈×D₇	1	1	yes	characteristic
195	C₇	#33730	1	1	yes	characteristic, complemented, Sylow
196	C₂²	—	2	2	no	—
197	C₂²	#89646	1	2	yes	—
198	C₂²	C₂×D₇×D₈	1	1	yes	characteristic
199	C₄	—	8	2	no	—
200	C₄	—	8	2	no	—
201	C₄	—	14	4	no	—
202	C₄	—	56	4	no	—
203	C₄	#89646	1	2	yes	—
204	C₄	C₂×#6451	1	1	yes	characteristic
205	C₄	C₂×D₇×Q₁₆	1	1	yes	characteristic
206	C₂	—	2	2	no	—
207	C₂	C₂×#89646	1	1	yes	characteristic
208	C₂	#11739477	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₇×D₈
commutator, cosocle	C₂×C₇×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₇×#2158	C₂
socle	C₂²×C₇	C₂²×D₈
prime radical	C₂³×C₇	C₂×D₈
hypercenter	#2158	D₇
hypocenter	C₇	#33730
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
6. #11727097	C₂	semidirect
7. #11727094	C₂	semidirect
8. #11727076	C₂	semidirect
39. C₂×C₇×C₁₆	Q₈	semidirect
49. #6386	Q₈	semidirect
131. C₂×C₁₆	Dic₁₄	semidirect
195. C₇	#33730	semidirect

Maximal subgroups

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

subgroup	index
#11736618	2
C₇×#2158	2
C₂×#89300	2
#11729153	2
#11728403	2
#11727097	2
#11727094	2
#11727076	2
#33730	7