Subgroups — #11737166

Order 896.

484 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#6482	C₂	1	1	yes	maximal, characteristic
2	#90265	C₂	1	2	yes	maximal, complemented
3	C₄×C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₁₆×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#89798	C₂	1	2	yes	maximal, complemented
6	D₇×#107	C₂	1	1	yes	maximal, characteristic, complemented
7	#89644	C₂	1	1	yes	maximal, characteristic, complemented
8	#89589	C₂	1	2	yes	maximal, complemented
9	#89575	C₂	1	2	yes	maximal, complemented
10	C₄×D₅₆	C₂	1	1	yes	maximal, characteristic, complemented
11	#89461	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#1161	C₂²	1	2	yes	—
13	C₂×C₇×Q₁₆	C₂²	1	1	yes	characteristic
14	C₄×C₇×Q₈	C₂²	1	2	yes	—
15	Q₈×Dic₇	C₂²	1	2	yes	—
16	C₂×#1143	C₂²	1	2	yes	—
17	#6482	C₄	1	8	yes	complemented
18	C₂×D₅₆	C₂²	1	1	yes	characteristic
19	C₂×#1131	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₇	C₂²	1	1	yes	characteristic
21	#6456	—	2	2	no	—
22	#6456	C₂²	1	2	yes	—
23	D₇×C₄⋊C₄	—	2	2	no	—
24	D₇×C₄⋊C₄	C₂²	1	2	yes	—
25	C₄×D₂₈	—	2	2	no	—
26	C₄×D₂₈	C₂²	1	2	yes	—
27	#6435	—	2	2	no	—
28	#6435	C₂²	1	1	yes	characteristic
29	C₇×#107	C₂²	1	1	yes	characteristic
30	C₇×#103	C₂²	1	2	yes	—
31	C₄×C₇×C₈	C₂²	1	1	yes	characteristic
32	#6409	C₂²	1	2	yes	—
33	#6395	C₂²	1	2	yes	—
34	#6392	C₂²	1	1	yes	characteristic
35	#6389	C₂²	1	1	yes	characteristic
36	C₈×Dic₇	C₂²	1	1	yes	characteristic
37	#6383	C₂²	1	2	yes	—
38	#6381	C₂²	1	2	yes	—
39	#6377	C₂²	1	1	yes	characteristic
40	#2951	—	7	1	no	maximal, Sylow
41	C₂×C₇×Q₈	C₂³	1	2	yes	—
42	#1161	—	2	4	no	—
43	#1161	C₂×C₄	1	8	yes	complemented
44	C₂×D₂₈	—	2	2	no	—
45	C₂×D₂₈	C₂³	1	2	yes	—
46	C₂×C₄×D₇	—	2	2	no	—
47	C₂×C₄×D₇	—	2	2	no	—
48	C₂×C₄×D₇	—	2	2	no	—
49	C₂×C₄×D₇	—	2	2	no	—
50	C₂×C₄×D₇	C₂³	1	1	yes	characteristic
51	C₇×Q₁₆	C₂×C₄	1	4	yes	complemented
52	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
53	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
54	C₇×C₄⋊C₄	—	2	2	no	—
55	C₇×C₄⋊C₄	C₂³	1	2	yes	—
56	C₄²×C₇	—	2	2	no	—
57	C₄²×C₇	C₂³	1	1	yes	characteristic
58	#1143	C₂×C₄	1	8	yes	complemented
59	(C₂×D₇)⋊C₄	—	2	2	no	—
60	(C₂×D₇)⋊C₄	—	2	2	no	—
61	(C₂×D₇)⋊C₄	D₄	1	1	yes	characteristic
62	(C₂×D₇)⋊C₄	D₄	1	1	yes	characteristic
63	C₄⋊Dic₇	—	2	2	no	—
64	C₄⋊Dic₇	C₂³	1	2	yes	—
65	Dic₇⋊C₄	—	2	2	no	—
66	Dic₇⋊C₄	D₄	1	1	yes	characteristic
67	Dic₇⋊C₄	D₄	1	1	yes	characteristic
68	C₄×Dic₇	—	2	2	no	—
69	C₄×Dic₇	C₂³	1	1	yes	characteristic
70	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
71	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
72	D₅₆	C₂×C₄	1	4	yes	complemented
73	#1131	—	2	2	no	—
74	C₈×D₇	C₂×C₄	1	4	yes	complemented
75	C₂×#135	—	7	1	no	—
76	#520	—	7	2	no	—
77	#440	—	7	2	no	—
78	#440	—	7	2	no	—
79	C₄×Q₁₆	—	7	1	no	—
80	C₄×Q₁₆	D₇	1	1	yes	characteristic, complemented
81	C₄×D₈	—	7	1	no	—
82	C₄×D₈	—	7	1	no	—
83	C₂×#107	—	7	1	no	—
84	#417	—	7	2	no	—
85	#405	—	7	1	no	—
86	C₂²×D₇	—	2	2	no	—
87	C₂²×D₇	C₂×D₄	1	1	yes	characteristic
88	C₇×Q₈	—	2	2	no	—
89	C₇×Q₈	C₂²×C₄	1	4	yes	—
90	C₂×C₄×C₇	—	2	2	no	—
91	C₂×C₄×C₇	—	2	2	no	—
92	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
95	C₂×Dic₇	—	2	2	no	—
96	C₂×Dic₇	C₂×D₄	1	1	yes	characteristic
97	C₂×Dic₇	C₂×D₄	1	1	yes	characteristic
98	C₂×Dic₇	C₂×D₄	1	1	yes	characteristic
99	D₂₈	—	2	8	no	—
100	D₂₈	—	2	2	no	—
101	D₂₈	C₂²×C₄	1	4	yes	—
102	C₄×D₇	—	2	8	no	—
103	C₄×D₇	—	8	2	no	—
104	C₄×D₇	—	8	2	no	—
105	C₄×D₇	—	8	2	no	—
106	C₄×D₇	C₂²×C₄	1	2	yes	—
107	C₄×D₇	#41	1	2	yes	—
108	C₇×C₈	—	2	1	no	—
109	C₇×C₈	C₂²×C₄	1	2	yes	—
110	C₇⋊C₈	—	2	1	no	—
111	C₇⋊C₈	C₂²×C₄	1	2	yes	—
112	C₂×#41	—	7	2	no	—
113	#135	—	7	8	no	—
114	C₂×Q₁₆	C₂×D₇	1	1	yes	characteristic
115	C₂×SD₁₆	—	7	2	no	—
116	C₂×D₈	—	7	1	no	—
117	C₂×M₁₆	—	7	1	no	—
118	C₂²×C₈	—	7	1	no	—
119	C₄×Q₈	—	7	2	no	—
120	C₄×Q₈	C₂×D₇	1	2	yes	—
121	C₄×D₄	—	7	2	no	—
122	C₄×D₄	—	7	2	no	—
123	C₄×D₄	—	14	2	no	—
124	C₄×D₄	—	14	2	no	—
125	#117	—	7	1	no	—
126	#117	—	14	2	no	—
127	C₂×C₄⋊C₄	—	7	2	no	—
128	C₂×C₄⋊C₄	—	14	2	no	—
129	#107	—	7	1	no	—
130	#107	—	7	2	no	—
131	#107	C₂×D₇	1	1	yes	characteristic
132	#103	—	7	2	no	—
133	#103	C₂×D₇	1	2	yes	—
134	#102	—	7	2	no	—
135	#102	—	7	2	no	—
136	#97	—	7	1	no	—
137	#97	—	7	1	no	—
138	C₄×C₈	—	7	1	no	—
139	C₄×C₈	C₂×D₇	1	1	yes	characteristic
140	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
141	C₂×D₇	—	2	1	no	—
142	C₂×D₇	—	2	1	no	—
143	C₂×D₇	—	2	4	no	—
144	C₂×D₇	—	4	2	no	—
145	C₂×D₇	—	4	2	no	—
146	C₂×D₇	C₄×D₄	1	2	yes	—
147	C₄×C₇	—	2	4	no	—
148	C₄×C₇	—	2	1	no	—
149	C₄×C₇	—	2	1	no	—
150	C₄×C₇	—	4	2	no	—
151	C₄×C₇	C₂³×C₄	1	1	yes	characteristic
152	C₄×C₇	C₂×#41	1	1	yes	characteristic
153	Dic₇	—	2	1	no	—
154	Dic₇	—	2	1	no	—
155	Dic₇	—	4	2	no	—
156	Dic₇	C₄×D₄	1	2	yes	—
157	#41	—	7	8	no	—
158	#41	—	14	4	no	—
159	C₂×Q₈	C₂²×D₇	1	2	yes	—
160	C₂×D₄	—	7	2	no	—
161	C₂×D₄	—	14	2	no	—
162	C₂²×C₄	—	7	1	no	—
163	C₂²×C₄	—	14	2	no	—
164	C₂²×C₄	—	14	2	no	—
165	C₂²×C₄	—	14	2	no	—
166	C₂²×C₄	—	14	2	no	—
167	Q₁₆	C₄×D₇	1	4	yes	complemented
168	SD₁₆	—	7	8	no	—
169	D₈	—	7	4	no	—
170	M₁₆	—	14	2	no	—
171	C₂×C₈	—	7	1	no	—
172	C₂×C₈	—	7	4	no	—
173	C₂×C₈	—	7	1	no	—
174	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
175	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	7	1	no	—
178	C₄⋊C₄	—	7	1	no	—
179	C₄⋊C₄	—	7	2	no	—
180	C₄⋊C₄	—	14	2	no	—
181	C₄⋊C₄	—	14	2	no	—
182	C₄⋊C₄	C₂²×D₇	1	2	yes	—
183	C₂²⋊C₄	—	7	1	no	—
184	C₂²⋊C₄	—	7	1	no	—
185	C₂²⋊C₄	—	14	2	no	—
186	C₂²⋊C₄	—	14	2	no	—
187	C₄²	—	2	2	no	—
188	C₄²	—	7	1	no	—
189	C₄²	—	14	2	no	—
190	C₄²	C₂²×D₇	1	1	yes	characteristic
191	C₂×C₇	C₂×C₄×D₄	1	1	yes	characteristic
192	C₂×C₇	#576	1	1	yes	characteristic
193	C₂×C₇	#578	1	1	yes	characteristic
194	D₇	—	2	2	no	—
195	D₇	—	4	4	no	—
196	C₂³	—	7	1	no	—
197	C₂³	—	14	2	no	—
198	Q₈	—	2	2	no	—
199	Q₈	C₂×C₄×D₇	1	4	yes	—
200	D₄	—	7	4	no	—
201	D₄	—	14	8	no	—
202	D₄	—	14	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	7	1	no	—
206	C₂×C₄	—	7	2	no	—
207	C₂×C₄	—	7	2	no	—
208	C₂×C₄	—	7	1	no	—
209	C₂×C₄	—	7	1	no	—
210	C₂×C₄	—	14	8	no	—
211	C₂×C₄	—	14	2	no	—
212	C₂×C₄	—	56	2	no	—
213	C₂×C₄	—	56	2	no	—
214	C₂×C₄	—	56	2	no	—
215	C₂×C₄	D₄×D₇	1	1	yes	characteristic
216	C₂×C₄	D₄×D₇	1	1	yes	characteristic
217	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
218	C₈	—	2	1	no	—
219	C₈	—	7	2	no	—
220	C₈	—	14	1	no	—
221	C₈	C₂×C₄×D₇	1	2	yes	—
222	C₇	#2951	1	1	yes	characteristic, complemented, Sylow
223	C₂²	—	7	2	no	—
224	C₂²	—	14	1	no	—
225	C₂²	—	14	1	no	—
226	C₂²	—	14	4	no	—
227	C₂²	—	28	2	no	—
228	C₂²	—	28	2	no	—
229	C₂²	C₂×D₄×D₇	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	4	no	—
233	C₄	—	4	2	no	—
234	C₄	—	7	2	no	—
235	C₄	—	14	1	no	—
236	C₄	—	14	1	no	—
237	C₄	—	28	2	no	—
238	C₄	C₂²×C₄×D₇	1	1	yes	characteristic
239	C₄	D₇×#41	1	1	yes	characteristic
240	C₂	—	14	2	no	—
241	C₂	—	28	4	no	—
242	C₂	C₄×D₄×D₇	1	1	yes	characteristic
243	C₂	#90457	1	1	yes	characteristic
244	C₂	#90460	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇
commutator	C₄×C₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₇
Fitting	C₄×C₇×Q₁₆	C₂
socle	C₂²×C₇	C₂²×D₄
prime radical	C₂×#6482	C₂
cosocle	C₂×C₄×C₇	C₂⁴
hypercenter	C₄×Q₁₆	D₇
hypocenter	C₇	#2951
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #90265	C₂	semidirect
3. C₄×C₇×Q₁₆	C₂	semidirect
4. Q₁₆×Dic₇	C₂	semidirect
5. #89798	C₂	semidirect
6. D₇×#107	C₂	semidirect
7. #89644	C₂	semidirect
8. #89589	C₂	semidirect
9. #89575	C₂	semidirect
10. C₄×D₅₆	C₂	semidirect
11. #89461	C₂	semidirect
17. #6482	C₄	semidirect
43. #1161	C₂×C₄	semidirect
51. C₇×Q₁₆	C₂×C₄	semidirect
58. #1143	C₂×C₄	semidirect
72. D₅₆	C₂×C₄	semidirect
74. C₈×D₇	C₂×C₄	semidirect
80. C₄×Q₁₆	D₇	semidirect
167. Q₁₆	C₄×D₇	semidirect
222. C₇	#2951	semidirect

Maximal subgroups

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

subgroup	index
C₂×#6482	2
#90265	2
C₄×C₇×Q₁₆	2
Q₁₆×Dic₇	2
#89798	2
D₇×#107	2
#89644	2
#89589	2
#89575	2
C₄×D₅₆	2
#89461	2
#2951	7