Subgroups — #11736732

Order 896.

464 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₂₈	C₂	1	1	yes	maximal, characteristic
2	C₇×#431	C₂	1	1	yes	maximal, characteristic, complemented
3	#89698	C₂	1	1	yes	maximal, characteristic, complemented
4	#89608	C₂	1	4	yes	maximal, complemented
5	#89498	C₂	1	2	yes	maximal, complemented
6	D₇×C₂²⋊C₈	C₂	1	2	yes	maximal, complemented
7	#89460	C₂	1	2	yes	maximal, complemented
8	C₈×D₂₈	C₂	1	2	yes	maximal, complemented
9	C₂²×D₂₈	C₄	1	1	yes	characteristic
10	C₂²×C₄×D₇	C₂²	1	2	yes	—
11	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
12	C₂×(C₂×D₇)⋊C₄	C₄	1	2	yes	—
13	C₂×C₄⋊Dic₇	C₄	1	1	yes	characteristic
14	C₂×#1131	C₂²	1	4	yes	complemented
15	C₂×C₈×D₇	C₂²	1	4	yes	complemented
16	C₄×D₂₈	C₄	1	2	yes	—
17	C₄×D₂₈	C₄	1	1	yes	characteristic
18	C₄×D₂₈	C₄	1	1	yes	characteristic
19	C₄×D₂₈	C₂²	1	2	yes	—
20	C₄×D₂₈	C₂²	1	1	yes	characteristic
21	C₄×D₂₈	C₂²	1	1	yes	characteristic
22	C₇×C₄⋊C₈	C₂²	1	2	yes	complemented
23	C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
24	C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
25	C₄×C₇×C₈	C₂²	1	2	yes	complemented
26	C₂²⋊(C₇⋊C₈)	C₂²	1	2	yes	—
27	(C₂×D₇)⋊C₈	C₂²	1	4	yes	complemented
28	(C₂×D₇)⋊C₈	C₂²	1	4	yes	complemented
29	C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	complemented
30	#6377	C₂²	1	2	yes	complemented
31	#2974	—	7	1	no	maximal, Sylow
32	C₂³×D₇	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
36	C₂²×Dic₇	C₂×C₄	1	2	yes	—
37	C₂×D₂₈	—	2	4	no	—
38	C₂×D₂₈	C₂×C₄	1	2	yes	—
39	C₂×D₂₈	C₂×C₄	1	1	yes	characteristic
40	C₂×D₂₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×D₇	—	4	4	no	—
42	C₂×C₄×D₇	C₂×C₄	1	2	yes	—
43	C₂×C₄×D₇	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₇	C₂³	1	2	yes	—
45	C₂×C₄×D₇	C₂³	1	2	yes	—
46	C₂×C₇×C₈	C₂³	1	2	yes	—
47	C₂×C₇×C₈	C₂³	1	2	yes	—
48	C₄²×C₇	C₂×C₄	1	1	yes	characteristic
49	C₄²×C₇	C₂×C₄	1	1	yes	characteristic
50	C₄²×C₇	C₂³	1	1	yes	characteristic
51	C₄²×C₇	C₂³	1	1	yes	characteristic
52	(C₂×D₇)⋊C₄	C₂×C₄	1	2	yes	—
53	(C₂×D₇)⋊C₄	C₂×C₄	1	2	yes	—
54	(C₂×D₇)⋊C₄	C₂×C₄	1	2	yes	—
55	(C₂×D₇)⋊C₄	C₂×C₄	1	2	yes	—
56	C₄⋊Dic₇	C₂×C₄	1	2	yes	—
57	C₄⋊Dic₇	C₂×C₄	1	1	yes	characteristic
58	C₄⋊Dic₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₇⋊C₈	C₂³	1	4	yes	—
60	#1131	—	4	4	no	—
61	C₈×D₇	—	4	4	no	—
62	C₂×C₄×D₄	—	7	1	no	—
63	#436	—	7	2	no	—
64	#435	—	7	4	no	—
65	C₈×D₄	—	7	2	no	—
66	#432	—	7	1	no	—
67	#431	D₇	1	1	yes	characteristic, complemented
68	#407	—	7	2	no	—
69	C₂×C₂²⋊C₈	—	7	2	no	—
70	C₂³×C₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×D₇	—	2	4	no	—
72	C₂²×D₇	—	2	4	no	—
73	C₂²×D₇	—	4	2	no	—
74	C₂²×D₇	C₂²×C₄	1	2	yes	—
75	C₂²×D₇	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₇	—	2	2	no	—
77	C₂×C₄×C₇	—	2	2	no	—
78	C₂×C₄×C₇	—	2	2	no	—
79	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
85	C₂×Dic₇	—	4	2	no	—
86	C₂×Dic₇	C₂²×C₄	1	2	yes	—
87	C₂×Dic₇	C₂²×C₄	1	2	yes	—
88	D₂₈	—	2	4	no	—
89	D₂₈	—	2	2	no	—
90	D₂₈	M₁₆	1	4	yes	complemented
91	C₄×D₇	—	2	4	no	—
92	C₄×D₇	—	2	4	no	—
93	C₄×D₇	—	4	2	no	—
94	C₄×D₇	—	4	2	no	—
95	C₇×C₈	—	2	2	no	—
96	C₇×C₈	—	2	2	no	—
97	C₇⋊C₈	—	2	4	no	—
98	C₂²×D₄	—	7	1	no	—
99	C₂³×C₄	—	7	2	no	—
100	C₂×M₁₆	—	7	4	no	—
101	C₂²×C₈	—	7	4	no	—
102	C₄×D₄	—	7	1	no	—
103	C₄×D₄	—	7	2	no	—
104	C₄×D₄	—	7	1	no	—
105	C₄×D₄	—	7	1	no	—
106	C₄×D₄	—	7	2	no	—
107	C₄×D₄	—	7	1	no	—
108	C₂×C₄⋊C₄	—	7	1	no	—
109	C₂×C₂²⋊C₄	—	7	2	no	—
110	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
111	C₄⋊C₈	—	7	2	no	—
112	C₄⋊C₈	C₂×D₇	1	2	yes	complemented
113	C₂²⋊C₈	—	7	2	no	—
114	C₂²⋊C₈	—	7	4	no	—
115	C₂²⋊C₈	—	7	4	no	—
116	C₂²⋊C₈	C₂×D₇	1	1	yes	characteristic
117	C₂²⋊C₈	C₂×D₇	1	1	yes	characteristic
118	#97	—	7	2	no	—
119	C₄×C₈	C₂×D₇	1	2	yes	complemented
120	C₂²×C₇	—	2	1	no	—
121	C₂²×C₇	—	2	1	no	—
122	C₂²×C₇	#131	1	2	yes	—
123	C₂²×C₇	C₂³×C₄	1	1	yes	characteristic
124	C₂×D₇	—	2	2	no	—
125	C₂×D₇	—	2	2	no	—
126	C₂×D₇	—	2	2	no	—
127	C₂×D₇	—	2	2	no	—
128	C₂×D₇	—	2	2	no	—
129	C₂×D₇	—	4	8	no	—
130	C₂×D₇	C₂×M₁₆	1	4	yes	—
131	C₄×C₇	—	2	1	no	—
132	C₄×C₇	—	2	1	no	—
133	C₄×C₇	—	2	1	no	—
134	C₄×C₇	—	2	1	no	—
135	C₄×C₇	C₂×M₁₆	1	2	yes	—
136	C₄×C₇	H(5,𝔽₂)	1	2	yes	—
137	Dic₇	—	2	2	no	—
138	Dic₇	—	2	2	no	—
139	C₂⁴	—	7	2	no	—
140	C₂×D₄	—	7	1	no	—
141	C₂×D₄	—	7	2	no	—
142	C₂×D₄	—	7	1	no	—
143	C₂×D₄	—	14	4	no	—
144	C₂²×C₄	—	7	2	no	—
145	C₂²×C₄	—	7	2	no	—
146	C₂²×C₄	—	7	2	no	—
147	C₂²×C₄	—	7	2	no	—
148	C₂²×C₄	—	7	2	no	—
149	C₂²×C₄	—	28	4	no	—
150	C₂²×C₄	C₄×D₇	1	1	yes	characteristic
151	C₂²×C₄	C₄×D₇	1	1	yes	characteristic
152	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
153	M₁₆	—	28	4	no	—
154	C₂×C₈	—	7	4	no	—
155	C₂×C₈	—	28	4	no	—
156	C₂×C₈	C₂²×D₇	1	2	yes	—
157	C₂×C₈	C₂²×D₇	1	2	yes	—
158	C₄⋊C₄	—	7	2	no	—
159	C₄⋊C₄	—	7	1	no	—
160	C₄⋊C₄	—	7	1	no	—
161	C₂²⋊C₄	—	7	2	no	—
162	C₂²⋊C₄	—	7	2	no	—
163	C₂²⋊C₄	—	7	2	no	—
164	C₂²⋊C₄	—	7	2	no	—
165	C₄²	C₄×D₇	1	1	yes	characteristic
166	C₄²	C₄×D₇	1	1	yes	characteristic
167	C₄²	C₂²×D₇	1	1	yes	characteristic
168	C₄²	C₂²×D₇	1	1	yes	characteristic
169	C₂×C₇	—	2	2	no	—
170	C₂×C₇	#518	1	1	yes	characteristic
171	C₂×C₇	C₂²×M₁₆	1	1	yes	characteristic
172	C₂×C₇	C₂×#131	1	1	yes	characteristic
173	D₇	—	2	4	no	—
174	D₇	—	4	2	no	—
175	C₂³	—	7	2	no	—
176	C₂³	—	7	2	no	—
177	C₂³	—	14	4	no	—
178	C₂³	—	14	4	no	—
179	C₂³	—	28	2	no	—
180	C₂³	C₂×C₄×D₇	1	1	yes	characteristic
181	D₄	—	7	4	no	—
182	D₄	—	14	4	no	—
183	D₄	—	14	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	7	2	no	—
188	C₂×C₄	—	7	2	no	—
189	C₂×C₄	—	14	4	no	—
190	C₂×C₄	—	14	4	no	—
191	C₂×C₄	—	28	2	no	—
192	C₂×C₄	—	28	2	no	—
193	C₂×C₄	—	28	2	no	—
194	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
195	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
196	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
197	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
199	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
200	C₈	—	2	2	no	—
201	C₈	—	2	2	no	—
202	C₈	—	14	4	no	—
203	C₇	#2974	1	1	yes	characteristic, complemented, Sylow
204	C₂²	—	2	1	no	—
205	C₂²	—	2	1	no	—
206	C₂²	—	7	4	no	—
207	C₂²	—	14	2	no	—
208	C₂²	—	14	2	no	—
209	C₂²	—	14	2	no	—
210	C₂²	—	14	2	no	—
211	C₂²	—	14	2	no	—
212	C₂²	—	28	8	no	—
213	C₂²	#6464	1	2	yes	—
214	C₂²	C₂²×C₄×D₇	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	14	2	no	—
220	C₄	—	14	2	no	—
221	C₄	D₇×M₁₆	1	2	yes	—
222	C₄	#6553	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	—	14	4	no	—
225	C₂	—	28	2	no	—
226	C₂	#90212	1	1	yes	characteristic
227	C₂	C₂×#6464	1	1	yes	characteristic
228	C₂	C₂×D₇×M₁₆	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₇
commutator, socle	C₂²×C₇	C₂³×C₄
Fitting	C₇×#431	C₂
prime radical	C₂²×D₂₈	C₄
cosocle	C₂×C₄×C₇	C₂⁴
hypercenter	#431	D₇
hypocenter	C₇	#2974
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
2. C₇×#431	C₂	semidirect
3. #89698	C₂	semidirect
4. #89608	C₂	semidirect
5. #89498	C₂	semidirect
6. D₇×C₂²⋊C₈	C₂	semidirect
7. #89460	C₂	semidirect
8. C₈×D₂₈	C₂	semidirect
14. C₂×#1131	C₂²	semidirect
15. C₂×C₈×D₇	C₂²	semidirect
22. C₇×C₄⋊C₈	C₂²	semidirect
25. C₄×C₇×C₈	C₂²	semidirect
27. (C₂×D₇)⋊C₈	C₂²	semidirect
28. (C₂×D₇)⋊C₈	C₂²	semidirect
29. C₄⋊(C₇⋊C₈)	C₂²	semidirect
30. #6377	C₂²	semidirect
67. #431	D₇	semidirect
90. D₂₈	M₁₆	semidirect
112. C₄⋊C₈	C₂×D₇	semidirect
119. C₄×C₈	C₂×D₇	semidirect
203. C₇	#2974	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₂₈	2
C₇×#431	2
#89698	2
#89608	2
#89498	2
D₇×C₂²⋊C₈	2
#89460	2
C₈×D₂₈	2
#2974	7