Subgroups — #27174497

Order 12928.

840 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₂₀₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15692234	C₂	1	2	yes	maximal, complemented
3	#20265097	C₂	1	2	yes	maximal, complemented
4	#20265089	C₂	1	2	yes	maximal, complemented
5	#20265083	C₂	1	2	yes	maximal, complemented
6	#20265057	C₂	1	2	yes	maximal
7	#20265049	C₂	1	2	yes	maximal, complemented
8	#20265037	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄×D₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₂₀₂	—	2	8	no	—
11	C₂²×Dic₂₀₂	—	2	4	no	—
12	C₂²×Dic₂₀₂	C₂²	1	2	yes	—
13	C₂²×Dic₂₀₂	C₂²	1	2	yes	—
14	C₂³×Dic₁₀₁	C₂²	1	2	yes	—
15	#15692234	—	2	8	no	—
16	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
17	C₂×#13108053	C₂²	1	4	yes	—
18	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
19	C₂×C₂²⋊Dic₁₀₁	C₂²	1	4	yes	—
20	#15692162	C₂²	1	2	yes	—
21	#15692162	C₂²	1	4	yes	—
22	#15692162	C₂²	1	4	yes	—
23	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
24	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
25	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₁₀₁	—	2	4	no	—
27	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
30	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
31	C₂×Dic₂₀₂	—	2	32	no	—
32	C₂×Dic₂₀₂	—	2	4	no	—
33	C₂×Dic₂₀₂	—	2	4	no	—
34	C₂×Dic₂₀₂	—	2	4	no	—
35	C₂×Dic₂₀₂	—	2	8	no	—
36	C₂×Dic₂₀₂	D₄	1	8	yes	complemented
37	C₂²×Dic₁₀₁	—	2	4	no	—
38	C₂²×Dic₁₀₁	—	2	8	no	—
39	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
40	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
41	C₄⋊Dic₁₀₁	—	2	4	no	—
42	#13108053	—	2	8	no	—
43	C₂²⋊Dic₁₀₁	—	2	4	no	—
44	C₂²⋊Dic₁₀₁	—	2	8	no	—
45	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
46	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
47	C₂×D₄×C₁₀₁	—	2	2	no	—
48	C₂×D₄×C₁₀₁	—	2	2	no	—
49	C₂×D₄×C₁₀₁	Q₈	1	4	yes	complemented
50	C₂²×C₄×C₁₀₁	—	2	4	no	—
51	C₂²×C₄×C₁₀₁	—	2	2	no	—
52	C₂²×C₄×C₁₀₁	—	2	4	no	—
53	C₂²×C₄×C₁₀₁	D₄	1	4	yes	—
54	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
59	C₄⋊C₄×C₁₀₁	—	2	2	no	—
60	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
61	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
62	C₄²×C₁₀₁	—	2	2	no	—
63	C₂³×C₁₀₁	—	2	2	no	—
64	C₂³×C₁₀₁	—	2	2	no	—
65	C₂³×C₁₀₁	—	2	1	no	—
66	C₂³×C₁₀₁	—	2	2	no	—
67	C₂³×C₁₀₁	—	2	1	no	—
68	C₂³×C₁₀₁	—	2	2	no	—
69	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
70	C₂³×C₁₀₁	C₂×Q₈	1	2	yes	—
71	C₂³×C₁₀₁	C₂×Q₈	1	2	yes	complemented
72	C₂³×C₁₀₁	#41	1	2	yes	—
73	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
74	D₄×C₁₀₁	—	2	8	no	—
75	C₂×C₄×C₁₀₁	—	2	1	no	—
76	C₂×C₄×C₁₀₁	—	2	2	no	—
77	C₂×C₄×C₁₀₁	—	2	1	no	—
78	C₂×C₄×C₁₀₁	—	2	8	no	—
79	C₂×C₄×C₁₀₁	—	2	1	no	—
80	C₂×C₄×C₁₀₁	—	2	1	no	—
81	C₂×C₄×C₁₀₁	—	2	2	no	—
82	C₂×C₄×C₁₀₁	—	2	2	no	—
83	C₂×C₄×C₁₀₁	—	4	4	no	—
84	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
85	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
86	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
87	C₂×C₄×C₁₀₁	C₂×Q₈	1	2	yes	complemented
88	C₂×Dic₁₀₁	—	2	16	no	—
89	C₂×Dic₁₀₁	—	2	4	no	—
90	C₂×Dic₁₀₁	—	2	4	no	—
91	C₂×Dic₁₀₁	—	2	4	no	—
92	C₂×Dic₁₀₁	—	2	4	no	—
93	C₂×Dic₁₀₁	—	2	4	no	—
94	C₂×Dic₁₀₁	C₂×D₄	1	8	yes	—
95	Dic₂₀₂	—	2	16	no	—
96	Dic₂₀₂	—	2	16	no	—
97	C₂²×C₁₀₁	—	2	2	no	—
98	C₂²×C₁₀₁	—	2	2	no	—
99	C₂²×C₁₀₁	—	2	4	no	—
100	C₂²×C₁₀₁	—	2	4	no	—
101	C₂²×C₁₀₁	—	2	2	no	—
102	C₂²×C₁₀₁	—	2	4	no	—
103	C₂²×C₁₀₁	—	4	1	no	—
104	C₂²×C₁₀₁	—	4	1	no	—
105	C₂²×C₁₀₁	—	4	2	no	—
106	C₂²×C₁₀₁	C₂²⋊Q₈	1	4	yes	complemented
107	C₂²×C₁₀₁	C₂²×D₄	1	2	yes	—
108	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
109	C₂²×C₁₀₁	C₂²×Q₈	1	1	yes	characteristic
110	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
111	C₂²×C₁₀₁	2₋¹⁺⁴	1	2	yes	—
112	C₄×C₁₀₁	—	2	2	no	—
113	C₄×C₁₀₁	—	2	2	no	—
114	C₄×C₁₀₁	—	4	1	no	—
115	C₄×C₁₀₁	—	4	1	no	—
116	C₄×C₁₀₁	#120	1	4	yes	—
117	Dic₁₀₁	—	2	8	no	—
118	Dic₁₀₁	—	4	4	no	—
119	C₂×C₁₀₁	—	2	4	no	—
120	C₂×C₁₀₁	—	4	2	no	—
121	C₂×C₁₀₁	C₂×#120	1	1	yes	characteristic
122	C₂×C₁₀₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
123	C₂×C₁₀₁	#536	1	1	yes	characteristic
124	C₂×C₁₀₁	#548	1	2	yes	—
125	C₂×C₁₀₁	D₄×Q₈	1	2	yes	—
126	#2409	—	101	1	no	maximal, Sylow
127	C₁₀₁	#2409	1	1	yes	characteristic, complemented, Sylow
128	C₂³×Q₈	—	101	1	no	—
129	C₂×C₂²⋊Q₈	—	101	2	no	—
130	C₂×C₄×D₄	D₁₀₁	1	1	yes	characteristic
131	#395	—	101	2	no	—
132	#393	—	101	2	no	—
133	#391	—	101	2	no	—
134	#386	—	101	2	no	—
135	#385	—	101	2	no	—
136	#380	—	101	1	no	—
137	C₂²×Q₈	—	101	2	no	—
138	C₂²×Q₈	—	101	2	no	—
139	C₂²×Q₈	—	202	8	no	—
140	C₂²×Q₈	—	202	4	no	—
141	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
142	C₂³×C₄	—	101	2	no	—
143	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
144	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
145	C₂²⋊Q₈	—	202	8	no	—
146	C₄×D₄	—	2	4	no	—
147	C₂×C₄⋊C₄	—	101	2	no	—
148	C₂×C₄⋊C₄	—	101	4	no	—
149	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
150	C₂×C₂²⋊C₄	—	101	4	no	—
151	C₂×C₂²⋊C₄	—	101	2	no	—
152	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
153	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
154	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
155	#95	—	101	2	no	—
156	#95	—	101	4	no	—
157	#95	—	101	4	no	—
158	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
159	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
160	C₂×Q₈	—	101	8	no	—
161	C₂×Q₈	—	202	8	no	—
162	C₂×Q₈	—	202	32	no	—
163	C₂×Q₈	—	202	4	no	—
164	C₂×Q₈	—	202	4	no	—
165	C₂×Q₈	—	202	4	no	—
166	C₂×D₄	—	2	2	no	—
167	C₂×D₄	—	2	2	no	—
168	C₂×D₄	Dic₂₀₂	1	4	yes	complemented
169	C₂²×C₄	—	2	4	no	—
170	C₂²×C₄	—	2	4	no	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	101	4	no	—
173	C₂²×C₄	—	101	4	no	—
174	C₂²×C₄	—	202	8	no	—
175	C₂²×C₄	—	202	4	no	—
176	C₂²×C₄	C₂²⋊D₁₀₁	1	4	yes	—
177	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
178	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
179	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
180	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
181	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	202	8	no	—
184	C₄⋊C₄	—	202	4	no	—
185	C₂²⋊C₄	—	2	2	no	—
186	C₂²⋊C₄	—	2	2	no	—
187	C₂²⋊C₄	—	202	8	no	—
188	C₂²⋊C₄	—	202	4	no	—
189	C₄²	—	2	2	no	—
190	C₂³	—	2	2	no	—
191	C₂³	—	2	2	no	—
192	C₂³	—	2	2	no	—
193	C₂³	—	2	1	no	—
194	C₂³	—	2	1	no	—
195	C₂³	—	2	2	no	—
196	C₂³	C₂×C₂²⋊D₁₀₁	1	2	yes	—
197	C₂³	C₂×Dic₂₀₂	1	2	yes	—
198	C₂³	C₂×Dic₂₀₂	1	2	yes	complemented
199	C₂³	#13108074	1	2	yes	—
200	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
201	Q₈	—	202	16	no	—
202	Q₈	—	202	16	no	—
203	D₄	—	2	8	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	8	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	—	101	8	no	—
214	C₂×C₄	—	202	4	no	—
215	C₂×C₄	—	202	4	no	—
216	C₂×C₄	—	202	4	no	—
217	C₂×C₄	—	202	16	no	—
218	C₂×C₄	—	202	4	no	—
219	C₂×C₄	—	202	4	no	—
220	C₂×C₄	D₄×D₁₀₁	1	4	yes	—
221	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
222	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
223	C₂×C₄	C₂×Dic₂₀₂	1	2	yes	complemented
224	C₂²	—	2	2	no	—
225	C₂²	—	2	4	no	—
226	C₂²	—	2	4	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	4	no	—
229	C₂²	—	2	2	no	—
230	C₂²	—	4	1	no	—
231	C₂²	—	4	2	no	—
232	C₂²	—	4	1	no	—
233	C₂²	#15692234	1	4	yes	complemented
234	C₂²	C₂×D₄×D₁₀₁	1	2	yes	—
235	C₂²	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
236	C₂²	C₂²×Dic₂₀₂	1	1	yes	characteristic
237	C₂²	C₂×#13108074	1	1	yes	characteristic
238	C₂²	#15692302	1	2	yes	—
239	C₄	—	2	2	no	—
240	C₄	—	2	2	no	—
241	C₄	—	4	1	no	—
242	C₄	—	4	1	no	—
243	C₄	—	404	4	no	—
244	C₄	—	202	8	no	—
245	C₄	#15692223	1	4	yes	—
246	C₂	—	2	4	no	—
247	C₂	—	4	2	no	—
248	C₂	C₂×#15692223	1	1	yes	characteristic
249	C₂	C₂×#15692234	1	1	yes	characteristic
250	C₂	#20265748	1	1	yes	characteristic
251	C₂	#20265836	1	2	yes	—
252	C₂	D₄×Dic₂₀₂	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator, socle, cosocle	C₂³×C₁₀₁	C₂⁴
Fitting	C₂×C₄×D₄×C₁₀₁	C₂
prime radical	C₂²×D₄×C₁₀₁	C₂²
hypercenter	C₂×C₄×D₄	D₁₀₁
hypocenter	C₁₀₁	#2409
LCS	G ▷ C₂³×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×Dic₂₀₂	C₂	semidirect
2. C₂×#15692234	C₂	semidirect
3. #20265097	C₂	semidirect
4. #20265089	C₂	semidirect
5. #20265083	C₂	semidirect
7. #20265049	C₂	semidirect
8. #20265037	C₂	semidirect
36. C₂×Dic₂₀₂	D₄	semidirect
49. C₂×D₄×C₁₀₁	Q₈	semidirect
71. C₂³×C₁₀₁	C₂×Q₈	semidirect
87. C₂×C₄×C₁₀₁	C₂×Q₈	semidirect
106. C₂²×C₁₀₁	C₂²⋊Q₈	semidirect
127. C₁₀₁	#2409	semidirect
168. C₂×D₄	Dic₂₀₂	semidirect
198. C₂³	C₂×Dic₂₀₂	semidirect
223. C₂×C₄	C₂×Dic₂₀₂	semidirect
233. C₂²	#15692234	semidirect

Maximal subgroups

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

subgroup	index
C₂³×Dic₂₀₂	2
C₂×#15692234	2
#20265097	2
#20265089	2
#20265083	2
#20265057	2
#20265049	2
#20265037	2
C₂×C₄×D₄×C₁₀₁	2
#2409	101