Subgroups — #27174098

Order 12928.

464 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×C₄×Dic₂₀₂	C₂	1	1	yes	maximal, characteristic
2	#20265081	C₂	1	2	yes	maximal
3	#20265062	C₂	1	2	yes	maximal
4	#20265061	C₂	1	2	yes	maximal
5	#20265045	C₂	1	1	yes	maximal, characteristic
6	#20265044	C₂	1	1	yes	maximal, characteristic
7	#20265041	C₂	1	1	yes	maximal, characteristic
8	C₄×#13108053	C₂	1	2	yes	maximal
9	#20265028	C₂	1	1	yes	maximal, characteristic
10	#20265022	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	1	yes	characteristic
15	C₂×#13108053	C₂²	1	2	yes	—
16	C₂×#13108053	C₂²	1	2	yes	—
17	C₂×#13108053	C₂²	1	1	yes	characteristic
18	C₂×#13108053	C₂²	1	1	yes	characteristic
19	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
20	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
21	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
23	#15692162	C₂²	1	1	yes	characteristic
24	#15692162	C₂²	1	2	yes	—
25	#15692162	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	2	yes	—
32	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
33	#95×C₁₀₁	C₂²	1	1	yes	characteristic
34	#95×C₁₀₁	C₂²	1	1	yes	characteristic
35	#95×C₁₀₁	C₂²	1	2	yes	—
36	C₂×Dic₂₀₂	—	2	2	no	—
37	C₂×Dic₂₀₂	—	2	2	no	—
38	C₂×Dic₂₀₂	C₂×C₄	1	4	yes	—
39	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
40	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
41	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
42	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
43	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
44	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
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	4	no	—
51	C₄×Dic₁₀₁	—	2	4	no	—
52	C₄×Dic₁₀₁	C₂×C₄	1	4	yes	—
53	C₄×Dic₁₀₁	C₂×C₄	1	4	yes	—
54	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
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	2	yes	—
59	C₄²×C₁₀₁	—	2	4	no	—
60	C₄²×C₁₀₁	C₂×C₄	1	4	yes	—
61	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
62	C₂×C₄×C₁₀₁	—	2	2	no	—
63	C₂×C₄×C₁₀₁	—	2	2	no	—
64	C₂×C₄×C₁₀₁	—	2	1	no	—
65	C₂×C₄×C₁₀₁	—	2	1	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	C₂²×C₄	1	2	yes	—
72	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
73	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₁₀₁	#41	1	4	yes	—
75	C₂×Dic₁₀₁	—	2	2	no	—
76	C₂×Dic₁₀₁	—	2	2	no	—
77	C₂×Dic₁₀₁	—	2	1	no	—
78	C₂×Dic₁₀₁	—	2	1	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	C₂²×C₄	1	2	yes	—
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	#41	1	2	yes	—
90	C₂×Dic₁₀₁	#41	1	2	yes	—
91	Dic₂₀₂	—	2	8	no	—
92	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
93	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
94	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
95	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
96	C₂²×C₁₀₁	2₋¹⁺⁴	1	2	yes	—
97	C₄×C₁₀₁	—	2	2	no	—
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	Dic₁₀₁	—	2	2	no	—
103	Dic₁₀₁	—	2	2	no	—
104	Dic₁₀₁	—	2	2	no	—
105	Dic₁₀₁	—	4	2	no	—
106	Dic₁₀₁	—	4	2	no	—
107	Dic₁₀₁	#117	1	4	yes	—
108	C₂×C₁₀₁	C₂×#117	1	1	yes	characteristic
109	C₂×C₁₀₁	C₄×#41	1	1	yes	characteristic
110	C₂×C₁₀₁	#519	1	1	yes	characteristic
111	C₂×C₁₀₁	#552	1	2	yes	—
112	C₂×C₁₀₁	#556	1	2	yes	—
113	#2356	—	101	1	no	maximal, Sylow
114	C₁₀₁	#2356	1	1	yes	characteristic, complemented, Sylow
115	C₂×C₄×Q₈	—	101	1	no	—
116	#391	—	101	2	no	—
117	#387	—	101	2	no	—
118	#387	—	101	2	no	—
119	#383	—	101	1	no	—
120	#383	—	101	1	no	—
121	#382	—	101	1	no	—
122	C₄×C₄⋊C₄	—	101	2	no	—
123	C₄×C₄⋊C₄	—	101	1	no	—
124	#376	—	101	1	no	—
125	#376	D₁₀₁	1	1	yes	characteristic
126	C₂²×Q₈	—	101	1	no	—
127	C₄×Q₈	—	101	8	no	—
128	C₂×C₄⋊C₄	—	101	1	no	—
129	C₂×C₄⋊C₄	—	101	1	no	—
130	C₂×C₄⋊C₄	—	101	1	no	—
131	C₂×C₄⋊C₄	—	101	2	no	—
132	C₂×C₄⋊C₄	—	101	2	no	—
133	C₂×C₄²	—	101	1	no	—
134	C₂×C₄²	—	101	1	no	—
135	C₂×C₄²	—	101	2	no	—
136	C₂×C₄²	—	101	2	no	—
137	C₂×C₄²	C₂×D₁₀₁	1	2	yes	—
138	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
139	#95	—	101	1	no	—
140	#95	—	101	2	no	—
141	#95	—	101	1	no	—
142	#95	—	101	2	no	—
143	#95	—	101	2	no	—
144	#95	—	101	2	no	—
145	#95	—	101	2	no	—
146	#95	—	101	2	no	—
147	#95	C₂×D₁₀₁	1	1	yes	characteristic
148	#95	C₂×D₁₀₁	1	1	yes	characteristic
149	#95	C₂×D₁₀₁	1	2	yes	—
150	C₂×Q₈	—	101	4	no	—
151	C₂×Q₈	—	202	2	no	—
152	C₂×Q₈	—	202	2	no	—
153	C₂²×C₄	—	101	1	no	—
154	C₂²×C₄	—	101	1	no	—
155	C₂²×C₄	—	101	1	no	—
156	C₂²×C₄	—	101	1	no	—
157	C₂²×C₄	—	101	2	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	1	yes	characteristic
163	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
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	C₄²	—	2	4	no	—
170	C₄²	—	101	4	no	—
171	C₄²	—	101	4	no	—
172	C₄²	—	202	4	no	—
173	C₄²	—	202	4	no	—
174	C₄²	C₄×D₁₀₁	1	4	yes	—
175	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
176	Q₈	—	202	8	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	2	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	—	101	2	no	—
187	C₂×C₄	—	101	2	no	—
188	C₂×C₄	—	101	2	no	—
189	C₂×C₄	—	101	2	no	—
190	C₂×C₄	—	101	2	no	—
191	C₂×C₄	—	101	2	no	—
192	C₂×C₄	—	202	2	no	—
193	C₂×C₄	—	202	2	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	1	no	—
199	C₂×C₄	—	202	1	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	#13108074	1	4	yes	—
206	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
207	C₂²	#41×D₁₀₁	1	1	yes	characteristic
208	C₂²	#41×D₁₀₁	1	1	yes	characteristic
209	C₂²	C₂×#13108074	1	2	yes	—
210	C₂²	#15692302	1	2	yes	—
211	C₄	—	2	4	no	—
212	C₄	—	2	2	no	—
213	C₄	—	2	2	no	—
214	C₄	—	2	2	no	—
215	C₄	—	4	2	no	—
216	C₄	—	101	4	no	—
217	C₄	—	404	2	no	—
218	C₄	—	404	2	no	—
219	C₄	—	202	2	no	—
220	C₄	—	202	2	no	—
221	C₄	—	202	2	no	—
222	C₂	#117×D₁₀₁	1	1	yes	characteristic
223	C₂	C₄×#13108074	1	1	yes	characteristic
224	C₂	#20265654	1	1	yes	characteristic
225	C₂	#20265860	1	2	yes	—
226	C₂	#20265895	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Fitting	#376×C₁₀₁	C₂
socle, prime radical, cosocle	C₂³×C₁₀₁	C₂⁴
hypercenter	#376	D₁₀₁
hypocenter	C₁₀₁	#2356
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ #376

Complements

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

N	H ≅ G/N	type
13. C₄×Dic₂₀₂	C₄	semidirect
114. C₁₀₁	#2356	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
#20265062	2
#20265061	2
#20265045	2
#20265044	2
#20265041	2
C₄×#13108053	2
#20265028	2
#20265022	2
#376×C₁₀₁	2
#2356	101