Subgroups — #27173604

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	C₂×C₄×Dic₂₀₂	C₂	1	1	yes	maximal, characteristic
2	#20265081	C₂	1	2	yes	maximal
3	#20265074	C₂	1	2	yes	maximal
4	#20265064	C₂	1	2	yes	maximal
5	#20265060	C₂	1	2	yes	maximal
6	C₄×#13108053	C₂	1	1	yes	maximal, characteristic
7	C₄×#13108053	C₂	1	1	yes	maximal, characteristic
8	C₄×C₄⋊Dic₁₀₁	C₂	1	1	yes	maximal, characteristic
9	#20265023	C₂	1	1	yes	maximal, characteristic
10	C₄²×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic
11	#376×C₁₀₁	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
13	C₄×Dic₂₀₂	C₄	1	8	yes	complemented
14	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
16	C₂×#13108053	C₂²	1	2	yes	—
17	C₂×#13108053	C₂²	1	2	yes	—
18	C₂×#13108053	C₂²	1	1	yes	characteristic
19	C₂×#13108053	C₂²	1	1	yes	characteristic
20	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
22	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
23	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
25	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
26	#15692162	C₂²	1	2	yes	—
27	#15692162	C₂²	1	2	yes	—
28	#15692162	C₂²	1	2	yes	—
29	#15692162	C₂²	1	2	yes	—
30	#15692162	C₂²	1	2	yes	—
31	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
32	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
33	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
34	#95×C₁₀₁	C₂²	1	2	yes	—
35	#95×C₁₀₁	C₂²	1	2	yes	—
36	C₂×Dic₂₀₂	—	2	4	no	—
37	C₂×Dic₂₀₂	C₂×C₄	1	4	yes	—
38	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
39	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
40	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
41	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
42	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
43	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
44	C₄⋊Dic₁₀₁	—	2	4	no	—
45	C₄⋊Dic₁₀₁	C₂×C₄	1	4	yes	—
46	#13108053	—	2	4	no	—
47	#13108053	—	2	4	no	—
48	#13108053	C₂×C₄	1	4	yes	—
49	#13108053	C₂×C₄	1	4	yes	—
50	C₄×Dic₁₀₁	—	2	2	no	—
51	C₄×Dic₁₀₁	—	2	2	no	—
52	C₄×Dic₁₀₁	—	2	4	no	—
53	C₄×Dic₁₀₁	—	2	4	no	—
54	C₄×Dic₁₀₁	C₂×C₄	1	4	yes	—
55	C₄×Dic₁₀₁	C₂×C₄	1	4	yes	—
56	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
57	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
60	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
61	C₄²×C₁₀₁	—	2	2	no	—
62	C₄²×C₁₀₁	C₂×C₄	1	4	yes	—
63	C₄²×C₁₀₁	Q₈	1	4	yes	complemented
64	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
65	C₂×C₄×C₁₀₁	—	2	2	no	—
66	C₂×C₄×C₁₀₁	—	2	2	no	—
67	C₂×C₄×C₁₀₁	—	2	2	no	—
68	C₂×C₄×C₁₀₁	—	2	2	no	—
69	C₂×C₄×C₁₀₁	—	2	2	no	—
70	C₂×C₄×C₁₀₁	—	2	2	no	—
71	C₂×C₄×C₁₀₁	—	2	2	no	—
72	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
73	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₁₀₁	C₂×Q₈	1	4	yes	—
76	C₂×C₄×C₁₀₁	C₂×Q₈	1	2	yes	—
77	C₂×C₄×C₁₀₁	#41	1	2	yes	—
78	C₂×Dic₁₀₁	—	2	2	no	—
79	C₂×Dic₁₀₁	—	2	2	no	—
80	C₂×Dic₁₀₁	—	2	2	no	—
81	C₂×Dic₁₀₁	—	2	2	no	—
82	C₂×Dic₁₀₁	—	2	2	no	—
83	C₂×Dic₁₀₁	—	2	2	no	—
84	C₂×Dic₁₀₁	—	2	2	no	—
85	C₂×Dic₁₀₁	—	2	2	no	—
86	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
87	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
88	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
89	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
90	C₂×Dic₁₀₁	#41	1	4	yes	—
91	C₂×Dic₁₀₁	#41	1	4	yes	—
92	Dic₂₀₂	—	2	8	no	—
93	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
94	C₂²×C₁₀₁	C₂²×Q₈	1	1	yes	characteristic
95	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
96	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
97	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
98	C₄×C₁₀₁	—	2	2	no	—
99	C₄×C₁₀₁	—	2	2	no	—
100	C₄×C₁₀₁	—	2	4	no	—
101	C₄×C₁₀₁	—	4	2	no	—
102	C₄×C₁₀₁	C₄×Q₈	1	4	yes	—
103	Dic₁₀₁	—	2	2	no	—
104	Dic₁₀₁	—	2	2	no	—
105	Dic₁₀₁	—	2	2	no	—
106	Dic₁₀₁	—	2	4	no	—
107	Dic₁₀₁	—	4	2	no	—
108	Dic₁₀₁	#117	1	4	yes	—
109	C₂×C₁₀₁	C₂×#117	1	1	yes	characteristic
110	C₂×C₁₀₁	C₂×C₄×Q₈	1	1	yes	characteristic
111	C₂×C₁₀₁	C₄×#41	1	1	yes	characteristic
112	C₂×C₁₀₁	#529	1	2	yes	—
113	C₂×C₁₀₁	#533	1	2	yes	—
114	#2295	—	101	1	no	maximal, Sylow
115	C₁₀₁	#2295	1	1	yes	characteristic, complemented, Sylow
116	C₂×C₄×Q₈	—	101	1	no	—
117	#391	—	101	2	no	—
118	#389	—	101	2	no	—
119	#387	—	101	2	no	—
120	#387	—	101	2	no	—
121	C₄×C₄⋊C₄	—	101	1	no	—
122	C₄×C₄⋊C₄	—	101	1	no	—
123	C₄×C₄⋊C₄	—	101	1	no	—
124	#376	—	101	1	no	—
125	#376	D₁₀₁	1	1	yes	characteristic
126	C₄³	—	101	1	no	—
127	C₂²×Q₈	—	101	1	no	—
128	C₄×Q₈	—	101	8	no	—
129	C₂×C₄⋊C₄	—	101	1	no	—
130	C₂×C₄⋊C₄	—	101	1	no	—
131	C₂×C₄⋊C₄	—	101	1	no	—
132	C₂×C₄⋊C₄	—	101	2	no	—
133	C₂×C₄⋊C₄	—	101	2	no	—
134	C₂×C₄⋊C₄	—	101	2	no	—
135	C₂×C₄²	—	101	1	no	—
136	C₂×C₄²	—	101	1	no	—
137	C₂×C₄²	—	101	2	no	—
138	C₂×C₄²	—	101	2	no	—
139	C₂×C₄²	—	101	1	no	—
140	C₂×C₄²	—	101	1	no	—
141	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
142	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
143	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
144	#95	—	101	2	no	—
145	#95	—	101	2	no	—
146	#95	—	101	2	no	—
147	#95	—	101	2	no	—
148	#95	—	101	2	no	—
149	#95	C₂×D₁₀₁	1	2	yes	—
150	#95	C₂×D₁₀₁	1	2	yes	—
151	C₂×Q₈	—	101	4	no	—
152	C₂×Q₈	—	202	4	no	—
153	C₂²×C₄	—	101	1	no	—
154	C₂²×C₄	—	101	1	no	—
155	C₂²×C₄	—	101	1	no	—
156	C₂²×C₄	—	101	2	no	—
157	C₂²×C₄	—	101	1	no	—
158	C₂²×C₄	—	101	2	no	—
159	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
160	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
161	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
162	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
163	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
164	C₄⋊C₄	—	101	4	no	—
165	C₄⋊C₄	—	101	4	no	—
166	C₄⋊C₄	—	101	4	no	—
167	C₄⋊C₄	—	202	4	no	—
168	C₄⋊C₄	—	202	4	no	—
169	C₄⋊C₄	—	202	4	no	—
170	C₄²	—	2	2	no	—
171	C₄²	—	101	4	no	—
172	C₄²	—	101	4	no	—
173	C₄²	—	202	4	no	—
174	C₄²	—	202	4	no	—
175	C₄²	—	202	2	no	—
176	C₄²	—	202	2	no	—
177	C₄²	Dic₂₀₂	1	4	yes	complemented
178	C₄²	C₄×D₁₀₁	1	4	yes	—
179	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
180	Q₈	—	202	8	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	101	2	no	—
189	C₂×C₄	—	101	4	no	—
190	C₂×C₄	—	101	2	no	—
191	C₂×C₄	—	101	2	no	—
192	C₂×C₄	—	101	2	no	—
193	C₂×C₄	—	101	4	no	—
194	C₂×C₄	—	202	2	no	—
195	C₂×C₄	—	202	2	no	—
196	C₂×C₄	—	202	2	no	—
197	C₂×C₄	—	202	2	no	—
198	C₂×C₄	—	202	2	no	—
199	C₂×C₄	—	202	2	no	—
200	C₂×C₄	—	202	2	no	—
201	C₂×C₄	—	202	2	no	—
202	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
203	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
204	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
205	C₂×C₄	C₂×Dic₂₀₂	1	4	yes	—
206	C₂×C₄	C₂×Dic₂₀₂	1	2	yes	—
207	C₂×C₄	#13108074	1	2	yes	—
208	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
209	C₂²	C₂²×Dic₂₀₂	1	1	yes	characteristic
210	C₂²	#41×D₁₀₁	1	2	yes	—
211	C₂²	#41×D₁₀₁	1	2	yes	—
212	C₂²	C₂×#13108074	1	1	yes	characteristic
213	C₄	—	2	4	no	—
214	C₄	—	2	2	no	—
215	C₄	—	2	2	no	—
216	C₄	—	4	2	no	—
217	C₄	—	101	4	no	—
218	C₄	—	404	2	no	—
219	C₄	—	202	2	no	—
220	C₄	—	202	2	no	—
221	C₄	—	202	2	no	—
222	C₄	—	202	4	no	—
223	C₄	C₄×Dic₂₀₂	1	4	yes	—
224	C₂	#117×D₁₀₁	1	1	yes	characteristic
225	C₂	C₂×C₄×Dic₂₀₂	1	1	yes	characteristic
226	C₂	#20265645	1	1	yes	characteristic
227	C₂	#20265705	1	2	yes	—
228	C₂	#20265730	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Frattini	C₂³	C₂³×D₁₀₁
Fitting	#376×C₁₀₁	C₂
socle, prime radical, cosocle	C₂³×C₁₀₁	C₂⁴
hypercenter	#376	D₁₀₁
hypocenter	C₁₀₁	#2295
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂²×C₄ ◁ #376

Complements

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

N	H ≅ G/N	type
13. C₄×Dic₂₀₂	C₄	semidirect
63. C₄²×C₁₀₁	Q₈	semidirect
115. C₁₀₁	#2295	semidirect
177. C₄²	Dic₂₀₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₀₂	2
#20265081	2
#20265074	2
#20265064	2
#20265060	2
C₄×#13108053	2
C₄×C₄⋊Dic₁₀₁	2
#20265023	2
C₄²×Dic₁₀₁	2
#376×C₁₀₁	2
#2295	101