Subgroups — #27187980

Order 12928.

500 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	#20266205	C₂	1	2	yes	maximal, complemented
2	#20266204	C₂	1	4	yes	maximal, complemented
3	D₄×#11717581	C₂	1	2	yes	maximal, complemented
4	#20266196	C₂	1	1	yes	maximal, characteristic, complemented
5	#20266195	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15692312	C₂	1	2	yes	maximal, complemented
7	D₁₀₁⋊(C₂²⋊C₈)	C₂	1	2	yes	maximal, complemented
8	C₄×D₄×D₁₀₁	C₂	1	1	yes	maximal, characteristic
9	C₂×D₁₀₁⋊C₈	C₂²	1	4	yes	complemented
10	C₂²×#11717581	C₂²	1	4	yes	complemented
11	#15692322	C₂²	1	2	yes	complemented
12	C₄⋊#11717581	C₂²	1	2	yes	complemented
13	#15692312	C₂²	1	1	yes	characteristic
14	#15692312	C₂²	1	2	yes	—
15	#15692312	C₂²	1	1	yes	characteristic
16	#15692312	C₂²	1	4	yes	complemented
17	C₂²⋊#11717581	C₂²	1	4	yes	complemented
18	C₄×#11717581	C₂²	1	2	yes	complemented
19	C₄×#11717581	C₂²	1	2	yes	complemented
20	C₂×D₄×D₁₀₁	C₄	1	1	yes	characteristic
21	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
22	C₄×D₄₀₄	C₄	1	1	yes	characteristic
23	D₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
24	#15692217	C₂²	1	1	yes	characteristic
25	C₄×C₂²⋊D₁₀₁	C₄	1	2	yes	—
26	#15692215	C₂²	1	2	yes	—
27	C₄⋊C₄×D₁₀₁	C₄	1	1	yes	characteristic
28	C₂²⋊C₄×D₁₀₁	C₄	1	2	yes	—
29	C₄²×D₁₀₁	C₂²	1	1	yes	characteristic
30	C₄×D₄×C₁₀₁	C₄	1	1	yes	characteristic
31	D₁₀₁⋊C₈	—	4	4	no	—
32	C₂×#11717581	—	4	4	no	—
33	C₂×#11717581	C₂³	1	2	yes	—
34	C₂×#11717581	C₂³	1	4	yes	—
35	C₂×#11717581	C₂³	1	2	yes	—
36	C₂³×D₁₀₁	C₂×C₄	1	2	yes	—
37	C₂×D₄₀₄	C₂×C₄	1	1	yes	characteristic
38	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	2	yes	—
39	D₄×D₁₀₁	C₈	1	8	yes	complemented
40	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
41	C₂×C₄×D₁₀₁	—	4	4	no	—
42	C₂×C₄×D₁₀₁	C₂×C₄	1	2	yes	—
43	C₂×C₄×D₁₀₁	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
47	C₂×C₄×D₁₀₁	C₂³	1	2	yes	—
48	C₄⋊Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
49	#13108053	C₂×C₄	1	2	yes	—
50	C₂²⋊Dic₁₀₁	C₂×C₄	1	2	yes	—
51	#13108051	C₂×C₄	1	2	yes	—
52	#13108051	C₂×C₄	1	2	yes	—
53	C₄×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
54	C₄×Dic₁₀₁	C₂³	1	1	yes	characteristic
55	C₄×Dic₁₀₁	C₂³	1	1	yes	characteristic
56	C₂×D₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
58	C₄⋊C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
59	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	—
60	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
61	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
62	C₂²×D₁₀₁	—	4	2	no	—
63	C₂²×D₁₀₁	—	4	2	no	—
64	C₂²×D₁₀₁	C₂×C₈	1	8	yes	complemented
65	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
66	C₂²×D₁₀₁	C₂²×C₄	1	1	yes	characteristic
67	D₄×C₁₀₁	C₂×C₈	1	4	yes	complemented
68	C₂×C₄×C₁₀₁	—	4	2	no	—
69	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
72	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
73	C₂²⋊D₁₀₁	C₂×C₈	1	8	yes	complemented
74	C₂×Dic₁₀₁	—	2	4	no	—
75	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
76	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
77	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
78	C₂×Dic₁₀₁	C₂⁴	1	1	yes	characteristic
79	D₄₀₄	C₂×C₈	1	4	yes	complemented
80	C₄×D₁₀₁	—	2	2	no	—
81	C₄×D₁₀₁	—	2	4	no	—
82	C₄×D₁₀₁	—	4	2	no	—
83	C₄×D₁₀₁	—	4	2	no	—
84	C₄×D₁₀₁	—	4	1	no	—
85	C₄×D₁₀₁	C₂×C₈	1	4	yes	complemented
86	#11717581	—	2	2	no	—
87	#11717581	—	2	4	no	—
88	#11717581	—	2	2	no	—
89	C₂²×C₁₀₁	—	2	2	no	—
90	C₂²×C₁₀₁	—	2	2	no	—
91	C₂²×C₁₀₁	C₂²×C₈	1	4	yes	—
92	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
93	C₂×D₁₀₁	—	2	1	no	—
94	C₂×D₁₀₁	—	2	2	no	—
95	C₂×D₁₀₁	—	2	1	no	—
96	C₂×D₁₀₁	—	2	2	no	—
97	C₂×D₁₀₁	—	4	8	no	—
98	C₂×D₁₀₁	C₂²×C₈	1	2	yes	—
99	C₂×D₁₀₁	C₂²×C₈	1	4	yes	—
100	C₄×C₁₀₁	—	2	1	no	—
101	C₄×C₁₀₁	—	2	2	no	—
102	C₄×C₁₀₁	—	2	1	no	—
103	C₄×C₁₀₁	C₂²×C₈	1	2	yes	—
104	Dic₁₀₁	—	2	2	no	—
105	Dic₁₀₁	—	2	1	no	—
106	Dic₁₀₁	C₂²×C₈	1	2	yes	—
107	Dic₁₀₁	H(5,𝔽₂)	1	2	yes	—
108	C₂×C₁₀₁	—	2	4	no	—
109	C₂×C₁₀₁	#518	1	1	yes	characteristic
110	C₂×C₁₀₁	C₂³×C₈	1	1	yes	characteristic
111	C₂×C₁₀₁	#568	1	1	yes	characteristic
112	D₁₀₁	—	2	2	no	—
113	D₁₀₁	—	2	4	no	—
114	#2972	—	101	1	no	maximal, Sylow
115	C₁₀₁	#2972	1	1	yes	characteristic, complemented, Sylow
116	C₂×C₄×D₄	—	101	1	no	—
117	C₈×D₄	—	101	2	no	—
118	C₈×D₄	—	101	4	no	—
119	C₈×D₄	—	101	2	no	—
120	#431	—	101	1	no	—
121	#431	—	101	1	no	—
122	C₂×C₂²⋊C₈	—	101	2	no	—
123	C₂×C₂²⋊C₈	—	101	2	no	—
124	C₂²×D₄	—	101	1	no	—
125	C₂³×C₄	—	101	2	no	—
126	C₂²×C₈	—	101	4	no	—
127	C₂²×C₈	—	101	4	no	—
128	C₄×D₄	—	101	1	no	—
129	C₄×D₄	—	101	2	no	—
130	C₄×D₄	—	101	1	no	—
131	C₄×D₄	—	101	2	no	—
132	C₄×D₄	—	101	1	no	—
133	C₄×D₄	#87938	1	1	yes	characteristic
134	C₂×C₄⋊C₄	—	101	1	no	—
135	C₂×C₂²⋊C₄	—	101	2	no	—
136	C₂×C₄²	—	101	1	no	—
137	C₄⋊C₈	—	101	2	no	—
138	C₄⋊C₈	—	101	2	no	—
139	C₂²⋊C₈	—	101	4	no	—
140	C₂²⋊C₈	—	101	1	no	—
141	C₂²⋊C₈	—	101	2	no	—
142	C₂²⋊C₈	—	101	1	no	—
143	C₂²⋊C₈	—	101	4	no	—
144	C₄×C₈	—	101	2	no	—
145	C₄×C₈	—	101	2	no	—
146	C₂⁴	—	101	2	no	—
147	C₂×D₄	—	101	2	no	—
148	C₂×D₄	—	101	8	no	—
149	C₂×D₄	—	101	1	no	—
150	C₂×D₄	C₂×#87938	1	1	yes	characteristic
151	C₂²×C₄	—	101	1	no	—
152	C₂²×C₄	—	101	2	no	—
153	C₂²×C₄	—	101	1	no	—
154	C₂²×C₄	—	101	2	no	—
155	C₂²×C₄	—	101	2	no	—
156	C₂²×C₄	—	101	1	no	—
157	C₂²×C₄	—	101	2	no	—
158	C₂²×C₄	—	404	4	no	—
159	C₂²×C₄	C₂×#87938	1	2	yes	—
160	C₂×C₈	—	101	2	no	—
161	C₂×C₈	—	101	4	no	—
162	C₂×C₈	—	101	2	no	—
163	C₂×C₈	—	404	4	no	—
164	C₂×C₈	—	404	4	no	—
165	C₄⋊C₄	—	101	2	no	—
166	C₄⋊C₄	—	101	1	no	—
167	C₄⋊C₄	C₂×#87938	1	1	yes	characteristic
168	C₂²⋊C₄	—	101	2	no	—
169	C₂²⋊C₄	—	101	2	no	—
170	C₂²⋊C₄	—	101	2	no	—
171	C₂²⋊C₄	C₂×#87938	1	2	yes	—
172	C₄²	—	101	1	no	—
173	C₄²	—	101	1	no	—
174	C₄²	—	101	1	no	—
175	C₄²	C₂×#87938	1	1	yes	characteristic
176	C₂³	—	101	8	no	—
177	C₂³	—	101	2	no	—
178	C₂³	—	101	1	no	—
179	C₂³	—	404	2	no	—
180	C₂³	—	404	2	no	—
181	C₂³	C₂²×#87938	1	2	yes	—
182	D₄	—	101	8	no	—
183	D₄	—	101	4	no	—
184	D₄	D₁₀₁⋊C₈	1	4	yes	complemented
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	—	101	1	no	—
187	C₂×C₄	—	101	1	no	—
188	C₂×C₄	—	101	4	no	—
189	C₂×C₄	—	101	1	no	—
190	C₂×C₄	—	101	2	no	—
191	C₂×C₄	—	404	2	no	—
192	C₂×C₄	—	404	2	no	—
193	C₂×C₄	—	404	1	no	—
194	C₂×C₄	—	202	4	no	—
195	C₂×C₄	—	202	2	no	—
196	C₂×C₄	—	202	4	no	—
197	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
198	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
199	C₂×C₄	C₂²×#87938	1	2	yes	—
200	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
201	C₈	—	202	2	no	—
202	C₈	—	202	4	no	—
203	C₈	—	202	2	no	—
204	C₂²	—	2	2	no	—
205	C₂²	—	2	2	no	—
206	C₂²	—	101	2	no	—
207	C₂²	—	101	4	no	—
208	C₂²	—	404	8	no	—
209	C₂²	—	202	1	no	—
210	C₂²	—	202	2	no	—
211	C₂²	—	202	1	no	—
212	C₂²	—	202	2	no	—
213	C₂²	C₂×D₁₀₁⋊C₈	1	4	yes	—
214	C₂²	C₂³×#87938	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	2	2	no	—
217	C₄	—	2	1	no	—
218	C₄	—	101	2	no	—
219	C₄	—	101	2	no	—
220	C₄	—	202	2	no	—
221	C₄	—	202	1	no	—
222	C₄	C₂×D₁₀₁⋊C₈	1	2	yes	—
223	C₂	—	2	4	no	—
224	C₂	—	202	2	no	—
225	C₂	—	202	4	no	—
226	C₂	#20266244	1	1	yes	characteristic
227	C₂	C₂²×D₁₀₁⋊C₈	1	1	yes	characteristic
228	C₂	#20266256	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#87938
commutator	C₂×C₁₀₁	C₂³×C₈
Fitting	C₄×D₄×C₁₀₁	C₄
socle	C₂²×C₁₀₁	C₂³×C₄
prime radical	C₂×D₄×D₁₀₁	C₄
cosocle	C₂×Dic₁₀₁	C₂⁴
hypercenter	C₄×D₄	#87938
hypocenter	C₁₀₁	#2972
LCS	G ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #20266205	C₂	semidirect
2. #20266204	C₂	semidirect
3. D₄×#11717581	C₂	semidirect
4. #20266196	C₂	semidirect
5. #20266195	C₂	semidirect
6. C₂×#15692312	C₂	semidirect
7. D₁₀₁⋊(C₂²⋊C₈)	C₂	semidirect
9. C₂×D₁₀₁⋊C₈	C₂²	semidirect
10. C₂²×#11717581	C₂²	semidirect
11. #15692322	C₂²	semidirect
12. C₄⋊#11717581	C₂²	semidirect
16. #15692312	C₂²	semidirect
17. C₂²⋊#11717581	C₂²	semidirect
18. C₄×#11717581	C₂²	semidirect
19. C₄×#11717581	C₂²	semidirect
39. D₄×D₁₀₁	C₈	semidirect
64. C₂²×D₁₀₁	C₂×C₈	semidirect
67. D₄×C₁₀₁	C₂×C₈	semidirect
73. C₂²⋊D₁₀₁	C₂×C₈	semidirect
79. D₄₀₄	C₂×C₈	semidirect
85. C₄×D₁₀₁	C₂×C₈	semidirect
115. C₁₀₁	#2972	semidirect
184. D₄	D₁₀₁⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#20266205	2
#20266204	2
D₄×#11717581	2
#20266196	2
#20266195	2
C₂×#15692312	2
D₁₀₁⋊(C₂²⋊C₈)	2
C₄×D₄×D₁₀₁	2
#2972	101