Subgroups — #13509305

Order 1792.

474 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11735991	C₂	1	2	yes	maximal
2	C₇×#2108	C₂	1	1	yes	maximal, characteristic
3	#11729182	C₂	1	2	yes	maximal
4	#11729158	C₂	1	1	yes	maximal, characteristic
5	C₂×#89299	C₂	1	1	yes	maximal, characteristic
6	#11726823	C₂	1	4	yes	maximal, complemented
7	#11726799	C₂	1	2	yes	maximal, complemented
8	#11726798	C₂	1	2	yes	maximal, complemented
9	#90210	—	2	2	no	—
10	C₂×C₇×M₃₂	C₂²	1	1	yes	characteristic
11	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
12	C₇×#405	C₂²	1	1	yes	characteristic
13	#89893	—	2	2	no	—
14	#89878	C₂²	1	2	yes	—
15	C₂×#6392	C₂²	1	2	yes	—
16	#89474	—	2	4	no	—
17	#89455	C₂²	1	4	yes	—
18	#89454	C₂²	1	4	yes	—
19	C₇×#346	C₂²	1	2	yes	—
20	C₄×C₇×C₁₆	C₂²	1	2	yes	—
21	#89300	C₂²	1	2	yes	—
22	#89300	C₂²	1	4	yes	—
23	#89299	C₂²	1	2	yes	—
24	#89299	C₂²	1	2	yes	—
25	#89299	C₂²	1	1	yes	characteristic
26	#89299	C₂²	1	1	yes	characteristic
27	#89299	C₂²	1	4	yes	—
28	#33729	—	7	1	no	maximal, Sylow
29	C₂×C₇×M₁₆	D₄	1	1	yes	characteristic
30	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
31	C₇×#117	D₄	1	1	yes	characteristic
32	#6489	—	2	2	no	—
33	C₂×C₄⋊Dic₇	—	2	2	no	—
34	#6453	—	4	2	no	—
35	#6451	—	4	2	no	—
36	#6441	—	4	4	no	—
37	#6433	—	2	4	no	—
38	#6432	—	2	4	no	—
39	C₇×M₃₂	Q₈	1	4	yes	complemented
40	C₂×C₇×C₁₆	Q₈	1	4	yes	complemented
41	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
42	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
43	C₂×C₇×C₁₆	C₂³	1	2	yes	—
44	C₇×#97	D₄	1	2	yes	—
45	C₄×C₇×C₈	C₂³	1	2	yes	—
46	#6392	—	2	2	no	—
47	#6392	—	2	4	no	—
48	#6392	C₂³	1	4	yes	—
49	#6392	C₂³	1	2	yes	—
50	#6392	C₂³	1	2	yes	—
51	#6391	—	2	2	no	—
52	#6391	—	2	4	no	—
53	#3164	—	7	2	no	—
54	#2255	—	7	4	no	—
55	#2253	—	7	2	no	—
56	#2252	—	7	2	no	—
57	#2159	—	7	2	no	—
58	#2158	—	7	1	no	—
59	C₂×#366	—	7	1	no	—
60	#2108	D₇	1	1	yes	characteristic
61	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×Dic₇	—	4	2	no	—
63	C₂×Dic₁₄	—	4	4	no	—
64	C₇×M₁₆	Q₁₆	1	4	yes	—
65	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
66	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
67	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
68	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
69	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
70	C₇×C₄⋊C₄	D₈	1	1	yes	characteristic
71	C₇×C₄⋊C₄	D₈	1	1	yes	characteristic
72	C₇×C₂²⋊C₄	D₈	1	2	yes	—
73	C₄²×C₇	C₂×D₄	1	2	yes	—
74	C₂²⋊Dic₇	—	4	2	no	—
75	C₄⋊Dic₇	—	2	2	no	—
76	C₄⋊Dic₇	—	2	4	no	—
77	C₄⋊Dic₇	—	2	2	no	—
78	C₄⋊Dic₇	—	4	4	no	—
79	C₄⋊Dic₇	—	4	2	no	—
80	Dic₇⋊C₄	—	4	4	no	—
81	Dic₇⋊C₄	—	4	4	no	—
82	C₄×Dic₇	—	4	2	no	—
83	C₇×C₁₆	C₂×Q₈	1	2	yes	—
84	C₇×C₁₆	C₂×Q₈	1	2	yes	—
85	C₇×C₁₆	C₂×Q₈	1	4	yes	—
86	#544	—	14	2	no	—
87	C₂×M₃₂	C₂×D₇	1	1	yes	characteristic
88	C₂²×C₁₆	C₂×D₇	1	1	yes	characteristic
89	#501	—	14	4	no	—
90	#500	—	7	4	no	—
91	#499	—	7	4	no	—
92	#428	—	14	2	no	—
93	#427	—	7	2	no	—
94	C₂×#107	—	7	2	no	—
95	#405	C₂×D₇	1	1	yes	characteristic
96	#367	—	7	2	no	—
97	#367	—	7	4	no	—
98	#366	—	7	2	no	—
99	#366	—	7	2	no	—
100	#366	—	7	1	no	—
101	#366	—	7	1	no	—
102	#366	—	7	4	no	—
103	#346	C₂×D₇	1	2	yes	—
104	C₄×C₁₆	C₂×D₇	1	2	yes	—
105	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
107	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
108	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
109	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
110	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
111	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×Dic₇	—	4	2	no	—
113	C₂×Dic₇	—	4	2	no	—
114	C₂×Dic₇	—	4	4	no	—
115	C₂×Dic₇	—	16	2	no	—
116	Dic₁₄	—	16	4	no	—
117	C₇×C₈	C₄⋊Q₈	1	2	yes	—
118	C₇×C₈	C₂×Q₁₆	1	4	yes	—
119	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
120	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
121	C₂×M₁₆	D₂₈	1	1	yes	characteristic
122	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
123	C₄⋊Q₈	—	14	4	no	—
124	C₄⋊Q₈	—	28	2	no	—
125	#125	—	14	4	no	—
126	#125	—	28	2	no	—
127	C₂²⋊Q₈	—	28	4	no	—
128	#117	—	14	2	no	—
129	#117	D₂₈	1	1	yes	characteristic
130	C₂×C₄⋊C₄	—	14	2	no	—
131	M₃₂	Dic₁₄	1	4	yes	complemented
132	C₂×C₁₆	Dic₁₄	1	4	yes	complemented
133	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
134	C₂×C₁₆	C₂²×D₇	1	2	yes	—
135	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
136	#107	—	7	4	no	—
137	#107	—	7	2	no	—
138	#107	—	7	2	no	—
139	#107	—	14	2	no	—
140	#107	—	14	4	no	—
141	#106	—	14	2	no	—
142	#106	—	14	4	no	—
143	#97	D₂₈	1	2	yes	—
144	C₄×C₈	C₂²×D₇	1	2	yes	—
145	C₂²×C₇	—	2	1	no	—
146	C₂²×C₇	—	2	1	no	—
147	C₂²×C₇	#500	1	2	yes	—
148	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
149	C₄×C₇	—	2	2	no	—
150	C₄×C₇	—	2	2	no	—
151	C₄×C₇	#500	1	2	yes	—
152	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
153	C₄×C₇	C₂²×Q₁₆	1	1	yes	characteristic
154	Dic₇	—	8	2	no	—
155	Dic₇	—	8	4	no	—
156	Dic₇	—	8	2	no	—
157	C₂×Q₈	—	28	4	no	—
158	C₂²×C₄	—	28	2	no	—
159	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
160	M₁₆	Dic₂₈	1	4	yes	—
161	C₂×C₈	C₂×Dic₁₄	1	2	yes	—
162	C₂×C₈	C₂×Dic₁₄	1	2	yes	—
163	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
164	C₂×C₈	C₂×D₂₈	1	2	yes	—
165	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
166	C₄⋊C₄	—	14	2	no	—
167	C₄⋊C₄	—	14	4	no	—
168	C₄⋊C₄	—	14	2	no	—
169	C₄⋊C₄	—	28	4	no	—
170	C₄⋊C₄	—	28	4	no	—
171	C₄⋊C₄	—	28	2	no	—
172	C₄⋊C₄	—	28	4	no	—
173	C₄⋊C₄	D₅₆	1	1	yes	characteristic
174	C₄⋊C₄	D₅₆	1	1	yes	characteristic
175	C₂²⋊C₄	—	28	2	no	—
176	C₂²⋊C₄	D₅₆	1	2	yes	—
177	C₄²	—	28	2	no	—
178	C₄²	C₂×D₂₈	1	2	yes	—
179	C₁₆	C₂×Dic₁₄	1	2	yes	—
180	C₁₆	C₂×Dic₁₄	1	2	yes	—
181	C₁₆	C₂×Dic₁₄	1	4	yes	—
182	C₂×C₇	—	2	2	no	—
183	C₂×C₇	C₂×#500	1	1	yes	characteristic
184	C₂×C₇	#3415	1	1	yes	characteristic
185	C₂×C₇	#3417	1	1	yes	characteristic
186	C₂³	C₂×D₅₆	1	1	yes	characteristic
187	Q₈	—	112	4	no	—
188	C₂×C₄	—	28	2	no	—
189	C₂×C₄	—	28	2	no	—
190	C₂×C₄	—	28	4	no	—
191	C₂×C₄	—	112	2	no	—
192	C₂×C₄	#6432	1	2	yes	—
193	C₂×C₄	C₂×D₅₆	1	2	yes	—
194	C₂×C₄	C₂×D₅₆	1	2	yes	—
195	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
196	C₂×C₄	C₂×Dic₂₈	1	2	yes	—
197	C₂×C₄	C₂²×D₂₈	1	1	yes	characteristic
198	C₈	#6432	1	2	yes	—
199	C₈	C₂×Dic₂₈	1	4	yes	—
200	C₈	C₂²×Dic₁₄	1	1	yes	characteristic
201	C₈	C₂²×Dic₁₄	1	1	yes	characteristic
202	C₇	#33729	1	1	yes	characteristic, complemented, Sylow
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	#89454	1	2	yes	—
206	C₂²	C₂²×D₅₆	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	56	4	no	—
210	C₄	—	56	2	no	—
211	C₄	—	56	2	no	—
212	C₄	#89454	1	2	yes	—
213	C₄	C₂×#6432	1	1	yes	characteristic
214	C₄	C₂²×Dic₂₈	1	1	yes	characteristic
215	C₂	—	2	2	no	—
216	C₂	C₂×#89454	1	1	yes	characteristic
217	C₂	#11739458	1	1	yes	characteristic
218	C₂	#11739460	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₅₆
commutator, cosocle	C₂×C₇×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₇×#2108	C₂
socle	C₂²×C₇	C₂²×D₈
prime radical	C₂³×C₇	C₂×D₈
hypercenter	#2108	D₇
hypocenter	C₇	#33729
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
6. #11726823	C₂	semidirect
7. #11726799	C₂	semidirect
8. #11726798	C₂	semidirect
39. C₇×M₃₂	Q₈	semidirect
40. C₂×C₇×C₁₆	Q₈	semidirect
131. M₃₂	Dic₁₄	semidirect
132. C₂×C₁₆	Dic₁₄	semidirect
202. C₇	#33729	semidirect

Maximal subgroups

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

subgroup	index
#11735991	2
C₇×#2108	2
#11729182	2
#11729158	2
C₂×#89299	2
#11726823	2
#11726799	2
#11726798	2
#33729	7