Subgroups — #27169361

Order 12928.

656 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Dic₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#15692162	C₂	1	4	yes	maximal
4	C₂²×C₄⋊C₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₁₀₁	C₂²	1	2	yes	—
6	C₂³×Dic₁₀₁	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₀₁	—	2	2	no	—
8	C₂×C₄⋊Dic₁₀₁	—	2	2	no	—
9	C₂×C₄⋊Dic₁₀₁	C₄	1	4	yes	complemented
10	C₂×C₄×Dic₁₀₁	—	2	2	no	—
11	C₂×C₄×Dic₁₀₁	—	2	2	no	—
12	C₂×C₄×Dic₁₀₁	C₄	1	4	yes	complemented
13	#15692162	—	2	8	no	—
14	C₂³×C₄×C₁₀₁	C₂²	1	2	yes	—
15	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₁₀₁	—	2	2	no	—
17	C₂×C₄⋊C₄×C₁₀₁	—	2	2	no	—
18	C₂×C₄⋊C₄×C₁₀₁	C₄	1	4	yes	complemented
19	C₂²×Dic₁₀₁	—	2	2	no	—
20	C₂²×Dic₁₀₁	—	2	2	no	—
21	C₂²×Dic₁₀₁	—	2	2	no	—
22	C₂²×Dic₁₀₁	—	2	2	no	—
23	C₂²×Dic₁₀₁	—	2	4	no	—
24	C₂²×Dic₁₀₁	—	2	2	no	—
25	C₂²×Dic₁₀₁	—	2	2	no	—
26	C₂²×Dic₁₀₁	—	2	4	no	—
27	C₂²×Dic₁₀₁	C₂×C₄	1	4	yes	—
28	C₂²×Dic₁₀₁	C₂×C₄	1	4	yes	—
29	C₂²×Dic₁₀₁	D₄	1	4	yes	—
30	C₂²×Dic₁₀₁	Q₈	1	4	yes	—
31	C₄⋊Dic₁₀₁	—	2	8	no	—
32	C₄×Dic₁₀₁	—	2	8	no	—
33	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₁₀₁	—	2	2	no	—
35	C₂²×C₄×C₁₀₁	—	2	2	no	—
36	C₂²×C₄×C₁₀₁	—	2	4	no	—
37	C₂²×C₄×C₁₀₁	—	2	4	no	—
38	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
40	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	—
41	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
42	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
43	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
44	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
45	C₂²×C₄×C₁₀₁	Q₈	1	2	yes	—
46	C₄⋊C₄×C₁₀₁	—	2	8	no	—
47	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
48	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
49	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
50	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
51	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
52	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
53	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
54	C₂³×C₁₀₁	C₂×Q₈	1	2	yes	—
55	C₂³×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
56	C₂³×C₁₀₁	#41	1	2	yes	—
57	C₂³×C₁₀₁	#41	1	2	yes	—
58	C₂×C₄×C₁₀₁	—	2	4	no	—
59	C₂×C₄×C₁₀₁	—	2	8	no	—
60	C₂×C₄×C₁₀₁	—	4	4	no	—
61	C₂×C₄×C₁₀₁	—	4	2	no	—
62	C₂×C₄×C₁₀₁	—	4	2	no	—
63	C₂×C₄×C₁₀₁	C₄²	1	4	yes	—
64	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	4	yes	—
65	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	4	yes	—
66	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	4	yes	—
67	C₂×C₄×C₁₀₁	C₄⋊C₄	1	4	yes	—
68	C₂×C₄×C₁₀₁	C₄⋊C₄	1	4	yes	—
69	C₂×C₄×C₁₀₁	C₄⋊C₄	1	4	yes	—
70	C₂×Dic₁₀₁	—	2	4	no	—
71	C₂×Dic₁₀₁	—	2	4	no	—
72	C₂×Dic₁₀₁	—	2	4	no	—
73	C₂×Dic₁₀₁	—	2	4	no	—
74	C₂×Dic₁₀₁	—	2	4	no	—
75	C₂×Dic₁₀₁	—	2	4	no	—
76	C₂×Dic₁₀₁	—	2	4	no	—
77	C₂×Dic₁₀₁	—	2	8	no	—
78	C₂×Dic₁₀₁	—	4	4	no	—
79	C₂×Dic₁₀₁	—	4	2	no	—
80	C₂×Dic₁₀₁	—	4	2	no	—
81	C₂×Dic₁₀₁	C₄⋊C₄	1	8	yes	complemented
82	C₂²×C₁₀₁	C₂×C₄²	1	1	yes	characteristic
83	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	2	yes	—
87	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₁₀₁	#117	1	2	yes	—
91	C₂²×C₁₀₁	C₄×D₄	1	2	yes	—
92	C₂²×C₁₀₁	C₄×D₄	1	2	yes	—
93	C₂²×C₁₀₁	C₄×Q₈	1	2	yes	—
94	C₂²×C₁₀₁	C₄×Q₈	1	2	yes	—
95	C₂²×C₁₀₁	#121	1	2	yes	—
96	C₂²×C₁₀₁	#121	1	2	yes	—
97	C₂²×C₁₀₁	C₂²⋊Q₈	1	2	yes	—
98	C₂²×C₁₀₁	C₂²⋊Q₈	1	2	yes	—
99	C₂²×C₁₀₁	#124	1	1	yes	characteristic
100	C₂²×C₁₀₁	#124	1	1	yes	characteristic
101	C₂²×C₁₀₁	#125	1	2	yes	—
102	C₂²×C₁₀₁	#127	1	1	yes	characteristic
103	C₂²×C₁₀₁	C₄⋊Q₈	1	1	yes	characteristic
104	C₂²×C₁₀₁	C₄⋊Q₈	1	2	yes	—
105	C₄×C₁₀₁	—	4	4	no	—
106	C₄×C₁₀₁	#95	1	8	yes	complemented
107	Dic₁₀₁	—	2	8	no	—
108	Dic₁₀₁	—	4	4	no	—
109	C₂×C₁₀₁	C₂×#95	1	1	yes	characteristic
110	C₂×C₁₀₁	C₄×C₄⋊C₄	1	2	yes	—
111	C₂×C₁₀₁	#380	1	2	yes	—
112	C₂×C₁₀₁	#382	1	1	yes	characteristic
113	C₂×C₁₀₁	#384	1	1	yes	characteristic
114	C₂×C₁₀₁	#389	1	2	yes	—
115	C₂×C₁₀₁	#389	1	2	yes	—
116	C₂×C₁₀₁	#390	1	1	yes	characteristic
117	C₂×C₁₀₁	#390	1	1	yes	characteristic
118	C₂×C₁₀₁	#391	1	1	yes	characteristic
119	C₂×C₁₀₁	#391	1	1	yes	characteristic
120	C₄⋊#95	—	101	1	no	maximal, Sylow
121	C₁₀₁	C₄⋊#95	1	1	yes	characteristic, complemented, Sylow
122	C₂²×C₄⋊C₄	—	101	1	no	—
123	C₂²×C₄⋊C₄	D₁₀₁	1	1	yes	characteristic
124	C₂²×C₄²	—	101	1	no	—
125	C₂×#95	—	101	4	no	—
126	C₂³×C₄	—	101	2	no	—
127	C₂³×C₄	—	101	2	no	—
128	C₂³×C₄	C₂×D₁₀₁	1	2	yes	—
129	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₄⋊C₄	—	101	4	no	—
133	C₂×C₄⋊C₄	—	202	2	no	—
134	C₂×C₄⋊C₄	—	202	2	no	—
135	C₂×C₄⋊C₄	Dic₁₀₁	1	4	yes	complemented
136	C₂×C₄²	—	101	4	no	—
137	C₂×C₄²	—	202	2	no	—
138	C₂×C₄²	—	202	2	no	—
139	#95	—	202	8	no	—
140	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	—	2	4	no	—
145	C₂²×C₄	—	101	4	no	—
146	C₂²×C₄	—	101	4	no	—
147	C₂²×C₄	—	101	4	no	—
148	C₂²×C₄	—	101	4	no	—
149	C₂²×C₄	—	202	4	no	—
150	C₂²×C₄	—	202	2	no	—
151	C₂²×C₄	—	202	2	no	—
152	C₂²×C₄	—	202	4	no	—
153	C₂²×C₄	—	202	2	no	—
154	C₂²×C₄	—	202	2	no	—
155	C₂²×C₄	—	202	2	no	—
156	C₂²×C₄	—	202	2	no	—
157	C₂²×C₄	Dic₂₀₂	1	2	yes	—
158	C₂²×C₄	C₄×D₁₀₁	1	2	yes	—
159	C₂²×C₄	C₄×D₁₀₁	1	2	yes	—
160	C₂²×C₄	D₄₀₄	1	2	yes	—
161	C₂²×C₄	C₂×Dic₁₀₁	1	4	yes	—
162	C₂²×C₄	C₂×Dic₁₀₁	1	2	yes	—
163	C₂²×C₄	C₂²⋊D₁₀₁	1	2	yes	—
164	C₂²×C₄	C₂²⋊D₁₀₁	1	2	yes	—
165	C₄⋊C₄	—	2	8	no	—
166	C₄⋊C₄	—	202	8	no	—
167	C₄²	—	202	8	no	—
168	C₂³	C₂×C₄×D₁₀₁	1	1	yes	characteristic
169	C₂³	C₂×C₄×D₁₀₁	1	1	yes	characteristic
170	C₂³	C₂²×Dic₁₀₁	1	1	yes	characteristic
171	C₂³	D₄×D₁₀₁	1	2	yes	—
172	C₂³	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
173	C₂³	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
174	C₂³	C₂×D₄₀₄	1	1	yes	characteristic
175	C₂³	Q₈×D₁₀₁	1	2	yes	—
176	C₂³	C₂×Dic₂₀₂	1	1	yes	characteristic
177	C₂³	#13108073	1	2	yes	—
178	C₂³	#13108075	1	2	yes	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	8	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	101	8	no	—
185	C₂×C₄	—	404	4	no	—
186	C₂×C₄	—	404	2	no	—
187	C₂×C₄	—	404	2	no	—
188	C₂×C₄	—	202	4	no	—
189	C₂×C₄	—	202	8	no	—
190	C₂×C₄	—	202	4	no	—
191	C₂×C₄	—	202	4	no	—
192	C₂×C₄	—	202	4	no	—
193	C₂×C₄	—	202	4	no	—
194	C₂×C₄	—	202	4	no	—
195	C₂×C₄	—	202	4	no	—
196	C₂×C₄	C₄×Dic₁₀₁	1	4	yes	—
197	C₂×C₄	#13108051	1	4	yes	—
198	C₂×C₄	#13108051	1	4	yes	—
199	C₂×C₄	C₂²⋊Dic₁₀₁	1	4	yes	—
200	C₂×C₄	#13108053	1	4	yes	—
201	C₂×C₄	#13108053	1	4	yes	—
202	C₂×C₄	C₄⋊Dic₁₀₁	1	4	yes	—
203	C₂²	C₂×C₄×Dic₁₀₁	1	1	yes	characteristic
204	C₂²	C₂×#13108051	1	1	yes	characteristic
205	C₂²	C₂×#13108051	1	1	yes	characteristic
206	C₂²	C₂×C₂²⋊Dic₁₀₁	1	1	yes	characteristic
207	C₂²	C₄⋊C₄×D₁₀₁	1	2	yes	—
208	C₂²	C₂×#13108053	1	1	yes	characteristic
209	C₂²	C₂×#13108053	1	1	yes	characteristic
210	C₂²	C₂×C₄⋊Dic₁₀₁	1	1	yes	characteristic
211	C₂²	#15692213	1	2	yes	—
212	C₂²	#15692217	1	2	yes	—
213	C₂²	D₄×Dic₁₀₁	1	2	yes	—
214	C₂²	#15692220	1	2	yes	—
215	C₂²	Q₈×Dic₁₀₁	1	2	yes	—
216	C₂²	#15692226	1	2	yes	—
217	C₂²	#15692230	1	2	yes	—
218	C₂²	#15692231	1	2	yes	—
219	C₂²	#15692235	1	2	yes	—
220	C₂²	#15692242	1	1	yes	characteristic
221	C₂²	#15692244	1	1	yes	characteristic
222	C₂²	#15692246	1	2	yes	—
223	C₂²	#15692251	1	1	yes	characteristic
224	C₂²	#15692253	1	1	yes	characteristic
225	C₂²	#15692254	1	2	yes	—
226	C₄	—	4	4	no	—
227	C₄	—	404	4	no	—
228	C₄	—	202	8	no	—
229	C₄	#15692162	1	8	yes	complemented
230	C₂	C₂×#15692162	1	1	yes	characteristic
231	C₂	C₄⋊C₄×Dic₁₀₁	1	2	yes	—
232	C₂	#20265034	1	2	yes	—
233	C₂	#20265042	1	1	yes	characteristic
234	C₂	#20265046	1	1	yes	characteristic
235	C₂	#20265073	1	2	yes	—
236	C₂	#20265076	1	2	yes	—
237	C₂	#20265078	1	1	yes	characteristic
238	C₂	#20265080	1	1	yes	characteristic
239	C₂	#20265082	1	1	yes	characteristic
240	C₂	#20265084	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂⁴	C₂²×D₁₀₁
commutator	C₂²×C₁₀₁	C₂×C₄²
Fitting	C₂²×C₄⋊C₄×C₁₀₁	C₂
socle, prime radical, cosocle	C₂⁴×C₁₀₁	C₂³
hypercenter	C₂²×C₄⋊C₄	D₁₀₁
hypocenter	C₁₀₁	C₄⋊#95
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
9. C₂×C₄⋊Dic₁₀₁	C₄	semidirect
12. C₂×C₄×Dic₁₀₁	C₄	semidirect
18. C₂×C₄⋊C₄×C₁₀₁	C₄	semidirect
81. C₂×Dic₁₀₁	C₄⋊C₄	semidirect
106. C₄×C₁₀₁	#95	semidirect
121. C₁₀₁	C₄⋊#95	semidirect
135. C₂×C₄⋊C₄	Dic₁₀₁	semidirect
229. C₄	#15692162	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Dic₁₀₁	2
C₂²×C₄×Dic₁₀₁	2
C₂×#15692162	2
C₂²×C₄⋊C₄×C₁₀₁	2
C₄⋊#95	101