Subgroups — #27179836

Order 12928.

492 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692262	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#13108058	C₂	1	1	yes	maximal, characteristic, complemented
3	#20265255	C₂	1	4	yes	maximal, complemented
4	#20265249	C₂	1	4	yes	maximal, complemented
5	#20265204	C₂	1	1	yes	maximal, characteristic, complemented
6	#20265144	C₂	1	1	yes	maximal, characteristic, complemented
7	#20265141	C₂	1	1	yes	maximal, characteristic, complemented
8	#20265133	C₂	1	1	yes	maximal, characteristic, complemented
9	#525×C₁₀₁	C₂	1	1	yes	maximal, characteristic
10	#15692262	—	2	4	no	—
11	C₂×#13108058	—	2	4	no	—
12	C₂×#13108058	C₂²	1	2	yes	—
13	C₂×#13108058	C₂²	1	2	yes	—
14	C₂×#13108058	C₂²	1	1	yes	characteristic
15	C₂×#13108058	C₂²	1	1	yes	characteristic
16	C₂×#13108058	C₂²	1	2	yes	complemented
17	C₂²×#11717579	C₂²	1	1	yes	characteristic
18	C₂²×#11717579	C₂²	1	1	yes	characteristic
19	C₂²×#11717579	C₂²	1	2	yes	complemented
20	C₄⋊#11717579	C₂²	1	4	yes	complemented
21	C₂²⋊#11717579	C₂²	1	4	yes	—
22	C₂²⋊#11717579	C₂²	1	4	yes	complemented
23	#15692166	C₂²	1	2	yes	complemented
24	C₄×#11717579	C₂²	1	2	yes	complemented
25	C₂×#41×C₁₀₁	C₂²	1	1	yes	characteristic
26	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
27	#123×C₁₀₁	C₄	1	2	yes	—
28	C₂²⋊Q₈×C₁₀₁	C₄	1	2	yes	—
29	#121×C₁₀₁	C₄	1	2	yes	—
30	#120×C₁₀₁	C₄	1	2	yes	—
31	C₄×D₄×C₁₀₁	C₂²	1	4	yes	—
32	#117×C₁₀₁	C₂²	1	1	yes	characteristic
33	#13108058	—	2	2	no	—
34	#13108058	—	2	2	no	—
35	#13108058	—	2	4	no	—
36	#13108058	—	2	4	no	—
37	#13108058	—	2	4	no	—
38	C₂×#11717579	—	2	2	no	—
39	C₂×#11717579	—	2	4	no	—
40	C₂×#11717579	D₄	1	4	yes	complemented
41	C₂×#11717579	C₂³	1	2	yes	—
42	C₂×#11717579	C₂³	1	2	yes	—
43	C₂×#11717579	C₂³	1	2	yes	—
44	C₂×#11717579	C₂³	1	1	yes	characteristic
45	C₂×#11717579	C₂³	1	1	yes	characteristic
46	C₂⁴×C₁₀₁	C₂×C₄	1	1	yes	characteristic
47	#41×C₁₀₁	—	2	2	no	—
48	#41×C₁₀₁	—	2	2	no	—
49	C₂×Q₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic
50	C₂×D₄×C₁₀₁	C₂×C₄	1	2	yes	—
51	C₂×D₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
52	C₂×D₄×C₁₀₁	C₂×C₄	1	4	yes	—
53	C₂²×C₄×C₁₀₁	—	2	2	no	—
54	C₂²×C₄×C₁₀₁	—	2	2	no	—
55	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
56	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
58	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
59	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	—
61	C₄⋊C₄×C₁₀₁	C₂×C₄	1	4	yes	—
62	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	—
63	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	4	yes	—
64	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	4	yes	—
65	C₄²×C₁₀₁	C₂³	1	2	yes	—
66	C₂³×C₁₀₁	—	2	2	no	—
67	C₂³×C₁₀₁	—	4	2	no	—
68	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
69	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
70	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
71	Q₈×C₁₀₁	—	2	2	no	—
72	D₄×C₁₀₁	—	2	4	no	—
73	D₄×C₁₀₁	—	2	2	no	—
74	D₄×C₁₀₁	—	4	4	no	—
75	C₂×C₄×C₁₀₁	—	2	2	no	—
76	C₂×C₄×C₁₀₁	—	2	4	no	—
77	C₂×C₄×C₁₀₁	—	2	2	no	—
78	C₂×C₄×C₁₀₁	—	2	2	no	—
79	C₂×C₄×C₁₀₁	—	2	2	no	—
80	C₂×C₄×C₁₀₁	—	2	2	no	—
81	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
82	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
86	C₂×C₄×C₁₀₁	#41	1	2	yes	—
87	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
88	#11717579	—	2	2	no	—
89	#11717579	—	2	2	no	—
90	#11717579	—	2	2	no	—
91	#11717579	C₂×D₄	1	2	yes	—
92	#11717579	C₂×D₄	1	2	yes	—
93	C₂²×C₁₀₁	—	2	2	no	—
94	C₂²×C₁₀₁	—	2	2	no	—
95	C₂²×C₁₀₁	—	2	2	no	—
96	C₂²×C₁₀₁	—	2	2	no	—
97	C₂²×C₁₀₁	—	2	4	no	—
98	C₂²×C₁₀₁	—	2	2	no	—
99	C₂²×C₁₀₁	—	4	4	no	—
100	C₂²×C₁₀₁	C₄×D₄	1	2	yes	—
101	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
102	C₄×C₁₀₁	—	2	2	no	—
103	C₄×C₁₀₁	—	2	2	no	—
104	C₄×C₁₀₁	—	2	4	no	—
105	C₄×C₁₀₁	C₄×D₄	1	2	yes	—
106	C₄×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
107	C₄×C₁₀₁	C₂×#41	1	1	yes	characteristic
108	C₂×C₁₀₁	—	2	2	no	—
109	C₂×C₁₀₁	—	4	2	no	—
110	C₂×C₁₀₁	—	4	2	no	—
111	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
112	C₂×C₁₀₁	#568	1	2	yes	—
113	#2930	—	101	1	no	maximal, Sylow
114	C₁₀₁	#2930	1	1	yes	characteristic, complemented, Sylow
115	C₂×#131	—	101	1	no	—
116	C₂²×M₁₆	—	101	1	no	—
117	#525	D₁₀₁	1	1	yes	characteristic
118	#436	—	101	4	no	—
119	#435	—	101	4	no	—
120	#424	—	101	1	no	—
121	#408	—	101	1	no	—
122	#407	—	101	1	no	—
123	#405	—	101	1	no	—
124	C₂×#41	C₂×D₁₀₁	1	1	yes	characteristic
125	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
126	#131	—	202	4	no	—
127	C₂×M₁₆	—	101	2	no	—
128	C₂×M₁₆	—	101	2	no	—
129	C₂×M₁₆	—	101	1	no	—
130	C₂×M₁₆	—	101	2	no	—
131	C₂×M₁₆	—	101	1	no	—
132	C₂×M₁₆	—	202	4	no	—
133	C₂²×C₈	—	101	1	no	—
134	C₂²×C₈	—	101	1	no	—
135	C₂²×C₈	—	101	2	no	—
136	#123	Dic₁₀₁	1	2	yes	—
137	C₂²⋊Q₈	Dic₁₀₁	1	2	yes	—
138	#121	Dic₁₀₁	1	2	yes	—
139	#120	Dic₁₀₁	1	2	yes	—
140	C₄×D₄	C₂×D₁₀₁	1	4	yes	—
141	#117	C₂×D₁₀₁	1	1	yes	characteristic
142	C₄⋊C₈	—	101	4	no	—
143	C₂²⋊C₈	—	101	4	no	—
144	C₂²⋊C₈	—	101	4	no	—
145	#97	—	101	2	no	—
146	C₄×C₈	—	101	2	no	—
147	C₂⁴	C₂×Dic₁₀₁	1	1	yes	characteristic
148	#41	—	2	2	no	—
149	#41	—	2	2	no	—
150	C₂×Q₈	C₂×Dic₁₀₁	1	1	yes	characteristic
151	C₂×D₄	C₂×Dic₁₀₁	1	2	yes	—
152	C₂×D₄	C₂×Dic₁₀₁	1	1	yes	characteristic
153	C₂×D₄	C₂×Dic₁₀₁	1	4	yes	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	C₂×Dic₁₀₁	1	2	yes	—
157	C₂²×C₄	C₂×Dic₁₀₁	1	1	yes	characteristic
158	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
159	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
160	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
161	M₁₆	—	202	4	no	—
162	M₁₆	—	202	4	no	—
163	M₁₆	—	202	4	no	—
164	M₁₆	—	202	2	no	—
165	M₁₆	—	202	2	no	—
166	C₂×C₈	—	101	4	no	—
167	C₂×C₈	—	101	1	no	—
168	C₂×C₈	—	101	1	no	—
169	C₂×C₈	—	101	2	no	—
170	C₂×C₈	—	101	2	no	—
171	C₂×C₈	—	101	2	no	—
172	C₂×C₈	—	202	4	no	—
173	C₂×C₈	—	202	2	no	—
174	C₄⋊C₄	C₂×Dic₁₀₁	1	2	yes	—
175	C₄⋊C₄	C₂×Dic₁₀₁	1	4	yes	—
176	C₂²⋊C₄	C₂×Dic₁₀₁	1	2	yes	—
177	C₂²⋊C₄	C₂×Dic₁₀₁	1	4	yes	—
178	C₂²⋊C₄	C₂×Dic₁₀₁	1	4	yes	—
179	C₄²	C₂²×D₁₀₁	1	2	yes	—
180	C₂³	—	2	2	no	—
181	C₂³	—	4	2	no	—
182	C₂³	C₂²×Dic₁₀₁	1	2	yes	—
183	C₂³	C₂²×Dic₁₀₁	1	2	yes	—
184	C₂³	C₂²×Dic₁₀₁	1	1	yes	characteristic
185	Q₈	—	2	2	no	—
186	D₄	—	2	4	no	—
187	D₄	—	2	2	no	—
188	D₄	—	4	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	C₂²×Dic₁₀₁	1	4	yes	—
196	C₂×C₄	C₂²×Dic₁₀₁	1	2	yes	—
197	C₂×C₄	C₂²×Dic₁₀₁	1	2	yes	—
198	C₂×C₄	C₂²×Dic₁₀₁	1	1	yes	characteristic
199	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
200	C₂×C₄	#13108073	1	2	yes	—
201	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
202	C₈	—	101	2	no	—
203	C₈	—	101	2	no	—
204	C₈	—	202	2	no	—
205	C₈	—	202	2	no	—
206	C₈	—	202	2	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	4	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	4	4	no	—
214	C₂²	D₄×Dic₁₀₁	1	2	yes	—
215	C₂²	C₂³×Dic₁₀₁	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	4	no	—
219	C₄	D₄×Dic₁₀₁	1	2	yes	—
220	C₄	C₂×D₄×D₁₀₁	1	1	yes	characteristic
221	C₄	C₂×#13108073	1	1	yes	characteristic
222	C₂	—	2	2	no	—
223	C₂	—	4	2	no	—
224	C₂	—	4	2	no	—
225	C₂	C₂×D₄×Dic₁₀₁	1	1	yes	characteristic
226	C₂	#20265944	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₁
commutator, socle	C₂²×C₁₀₁	C₂³×C₄
Fitting, prime radical	#525×C₁₀₁	C₂
cosocle	C₂×C₄×C₁₀₁	C₂⁴
hypercenter	#525	D₁₀₁
hypocenter	C₁₀₁	#2930
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ #525

Complements

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

N	H ≅ G/N	type
1. C₂×#15692262	C₂	semidirect
2. C₂²×#13108058	C₂	semidirect
3. #20265255	C₂	semidirect
4. #20265249	C₂	semidirect
5. #20265204	C₂	semidirect
6. #20265144	C₂	semidirect
7. #20265141	C₂	semidirect
8. #20265133	C₂	semidirect
16. C₂×#13108058	C₂²	semidirect
19. C₂²×#11717579	C₂²	semidirect
20. C₄⋊#11717579	C₂²	semidirect
22. C₂²⋊#11717579	C₂²	semidirect
23. #15692166	C₂²	semidirect
24. C₄×#11717579	C₂²	semidirect
40. C₂×#11717579	D₄	semidirect
114. C₁₀₁	#2930	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15692262	2
C₂²×#13108058	2
#20265255	2
#20265249	2
#20265204	2
#20265144	2
#20265141	2
#20265133	2
#525×C₁₀₁	2
#2930	101