Subgroups — #27169385

Order 12928.

564 conjugacy classes of subgroups.

The subgroup lattice has 270 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	C₂×#15692162	C₂	1	2	yes	maximal
3	C₂×#15692162	C₂	1	2	yes	maximal
4	C₂×#15692162	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊C₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₁₀₁	C₂²	1	2	yes	—
7	C₂³×Dic₁₀₁	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₁₀₁	—	2	4	no	—
9	C₂×C₄⋊Dic₁₀₁	—	2	2	no	—
10	#15692162	—	2	4	no	—
11	#15692162	—	2	4	no	—
12	#15692162	—	2	2	no	—
13	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
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	2	no	—
24	C₂²×Dic₁₀₁	—	2	2	no	—
25	C₂²×Dic₁₀₁	—	2	2	no	—
26	C₂²×Dic₁₀₁	—	2	2	no	—
27	C₂²×Dic₁₀₁	—	2	2	no	—
28	C₂²×Dic₁₀₁	—	2	2	no	—
29	C₂²×Dic₁₀₁	—	2	2	no	—
30	C₂²×Dic₁₀₁	—	2	2	no	—
31	C₂²×Dic₁₀₁	D₄	1	4	yes	—
32	C₂²×Dic₁₀₁	Q₈	1	4	yes	—
33	C₄⋊Dic₁₀₁	—	2	8	no	—
34	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₁₀₁	—	2	1	no	—
36	C₂²×C₄×C₁₀₁	—	2	1	no	—
37	C₂²×C₄×C₁₀₁	—	2	2	no	—
38	C₂²×C₄×C₁₀₁	—	2	2	no	—
39	C₂²×C₄×C₁₀₁	—	2	1	no	—
40	C₂²×C₄×C₁₀₁	—	2	1	no	—
41	C₂²×C₄×C₁₀₁	—	2	1	no	—
42	C₂²×C₄×C₁₀₁	—	2	1	no	—
43	C₂²×C₄×C₁₀₁	—	2	2	no	—
44	C₂²×C₄×C₁₀₁	—	2	2	no	—
45	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
46	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
47	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
48	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
49	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
50	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
51	C₂²×C₄×C₁₀₁	Q₈	1	2	yes	—
52	C₄⋊C₄×C₁₀₁	—	2	8	no	—
53	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
54	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
55	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
56	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
57	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
58	C₂³×C₁₀₁	C₂×Q₈	1	2	yes	—
59	C₂³×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
60	C₂³×C₁₀₁	#41	1	2	yes	—
61	C₂³×C₁₀₁	#41	1	1	yes	characteristic
62	C₂³×C₁₀₁	#41	1	1	yes	characteristic
63	C₂³×C₁₀₁	#41	1	2	yes	—
64	C₂×C₄×C₁₀₁	—	2	2	no	—
65	C₂×C₄×C₁₀₁	—	2	2	no	—
66	C₂×C₄×C₁₀₁	—	2	4	no	—
67	C₂×C₄×C₁₀₁	—	2	2	no	—
68	C₂×C₄×C₁₀₁	—	2	4	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	—	2	2	no	—
73	C₂×C₄×C₁₀₁	—	2	2	no	—
74	C₂×C₄×C₁₀₁	—	2	2	no	—
75	C₂×C₄×C₁₀₁	—	2	4	no	—
76	C₂×C₄×C₁₀₁	—	4	1	no	—
77	C₂×C₄×C₁₀₁	—	4	2	no	—
78	C₂×C₄×C₁₀₁	—	4	1	no	—
79	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	4	yes	—
80	C₂×C₄×C₁₀₁	C₄⋊C₄	1	4	yes	complemented
81	C₂×Dic₁₀₁	—	2	4	no	—
82	C₂×Dic₁₀₁	—	2	4	no	—
83	C₂×Dic₁₀₁	—	2	4	no	—
84	C₂×Dic₁₀₁	—	2	4	no	—
85	C₂×Dic₁₀₁	—	2	4	no	—
86	C₂×Dic₁₀₁	—	2	4	no	—
87	C₂×Dic₁₀₁	—	4	2	no	—
88	C₂×Dic₁₀₁	—	4	2	no	—
89	C₂×Dic₁₀₁	—	4	2	no	—
90	C₂×Dic₁₀₁	—	4	2	no	—
91	C₂×Dic₁₀₁	—	4	2	no	—
92	C₂×Dic₁₀₁	—	4	2	no	—
93	C₂×Dic₁₀₁	—	4	2	no	—
94	C₂×Dic₁₀₁	—	4	2	no	—
95	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂²×C₁₀₁	#117	1	1	yes	characteristic
98	C₂²×C₁₀₁	C₄×D₄	1	2	yes	—
99	C₂²×C₁₀₁	C₄×Q₈	1	2	yes	—
100	C₂²×C₁₀₁	#120	1	1	yes	characteristic
101	C₂²×C₁₀₁	#121	1	2	yes	—
102	C₂²×C₁₀₁	#121	1	1	yes	characteristic
103	C₂²×C₁₀₁	#121	1	1	yes	characteristic
104	C₂²×C₁₀₁	#121	1	2	yes	—
105	C₂²×C₁₀₁	C₂²⋊Q₈	1	2	yes	—
106	C₂²×C₁₀₁	C₂²⋊Q₈	1	2	yes	—
107	C₂²×C₁₀₁	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂²×C₁₀₁	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂²×C₁₀₁	C₂²⋊Q₈	1	2	yes	—
110	C₂²×C₁₀₁	#123	1	1	yes	characteristic
111	C₂²×C₁₀₁	#123	1	2	yes	—
112	C₂²×C₁₀₁	#124	1	1	yes	characteristic
113	C₂²×C₁₀₁	#125	1	2	yes	—
114	C₂²×C₁₀₁	#125	1	2	yes	—
115	C₂²×C₁₀₁	#126	1	2	yes	—
116	C₂²×C₁₀₁	C₄⋊Q₈	1	1	yes	characteristic
117	C₂²×C₁₀₁	C₄⋊Q₈	1	2	yes	—
118	C₄×C₁₀₁	—	2	4	no	—
119	C₄×C₁₀₁	—	2	4	no	—
120	C₄×C₁₀₁	—	4	2	no	—
121	Dic₁₀₁	—	4	4	no	—
122	Dic₁₀₁	—	4	4	no	—
123	C₂×C₁₀₁	#380	1	1	yes	characteristic
124	C₂×C₁₀₁	#386	1	1	yes	characteristic
125	C₂×C₁₀₁	#387	1	2	yes	—
126	C₂×C₁₀₁	#389	1	2	yes	—
127	C₂×C₁₀₁	#391	1	1	yes	characteristic
128	C₂×C₁₀₁	#394	1	1	yes	characteristic
129	C₂×C₁₀₁	#395	1	1	yes	characteristic
130	C₂×C₁₀₁	#396	1	2	yes	—
131	C₂×C₁₀₁	#398	1	1	yes	characteristic
132	C₂×C₁₀₁	#398	1	2	yes	—
133	C₂×C₁₀₁	#400	1	1	yes	characteristic
134	#1441	—	101	1	no	maximal, Sylow
135	C₁₀₁	#1441	1	1	yes	characteristic, complemented, Sylow
136	C₂²×C₄⋊C₄	—	101	1	no	—
137	C₂²×C₄⋊C₄	D₁₀₁	1	1	yes	characteristic
138	C₂×#95	—	101	2	no	—
139	C₂×#95	—	101	2	no	—
140	C₂×#95	—	101	1	no	—
141	C₂³×C₄	—	101	2	no	—
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	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
146	C₂×C₄⋊C₄	—	2	2	no	—
147	C₂×C₄⋊C₄	—	2	2	no	—
148	C₂×C₄⋊C₄	—	202	4	no	—
149	C₂×C₄⋊C₄	—	202	2	no	—
150	C₂×C₄⋊C₄	Dic₁₀₁	1	4	yes	complemented
151	#95	—	202	4	no	—
152	#95	—	202	4	no	—
153	#95	—	202	2	no	—
154	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	2	1	no	—
157	C₂²×C₄	—	2	1	no	—
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	—	101	4	no	—
166	C₂²×C₄	—	101	4	no	—
167	C₂²×C₄	—	202	2	no	—
168	C₂²×C₄	—	202	2	no	—
169	C₂²×C₄	—	202	2	no	—
170	C₂²×C₄	—	202	2	no	—
171	C₂²×C₄	—	202	2	no	—
172	C₂²×C₄	—	202	2	no	—
173	C₂²×C₄	—	202	2	no	—
174	C₂²×C₄	—	202	2	no	—
175	C₂²×C₄	—	202	2	no	—
176	C₂²×C₄	—	202	2	no	—
177	C₂²×C₄	—	202	2	no	—
178	C₂²×C₄	—	202	2	no	—
179	C₂²×C₄	Dic₂₀₂	1	2	yes	—
180	C₂²×C₄	D₄₀₄	1	2	yes	—
181	C₂²×C₄	C₂×Dic₁₀₁	1	2	yes	—
182	C₂²×C₄	C₂×Dic₁₀₁	1	2	yes	—
183	C₂²×C₄	C₂×Dic₁₀₁	1	2	yes	—
184	C₂²×C₄	C₂²⋊D₁₀₁	1	2	yes	—
185	C₂²×C₄	C₂²⋊D₁₀₁	1	2	yes	—
186	C₄⋊C₄	—	2	8	no	—
187	C₄⋊C₄	—	202	8	no	—
188	C₂³	C₂²×Dic₁₀₁	1	1	yes	characteristic
189	C₂³	D₄×D₁₀₁	1	2	yes	—
190	C₂³	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
191	C₂³	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
192	C₂³	C₂×D₄₀₄	1	1	yes	characteristic
193	C₂³	Q₈×D₁₀₁	1	2	yes	—
194	C₂³	C₂×Dic₂₀₂	1	1	yes	characteristic
195	C₂³	#13108073	1	2	yes	—
196	C₂³	#13108074	1	1	yes	characteristic
197	C₂³	#13108074	1	1	yes	characteristic
198	C₂³	#13108075	1	2	yes	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	1	no	—
214	C₂×C₄	—	404	2	no	—
215	C₂×C₄	—	404	2	no	—
216	C₂×C₄	—	404	2	no	—
217	C₂×C₄	—	404	2	no	—
218	C₂×C₄	—	404	2	no	—
219	C₂×C₄	—	404	2	no	—
220	C₂×C₄	—	404	2	no	—
221	C₂×C₄	—	404	2	no	—
222	C₂×C₄	—	202	4	no	—
223	C₂×C₄	—	202	4	no	—
224	C₂×C₄	—	202	4	no	—
225	C₂×C₄	—	202	4	no	—
226	C₂×C₄	—	202	4	no	—
227	C₂×C₄	—	202	4	no	—
228	C₂×C₄	C₂²⋊Dic₁₀₁	1	4	yes	—
229	C₂×C₄	C₄⋊Dic₁₀₁	1	4	yes	complemented
230	C₂²	C₂×C₂²⋊Dic₁₀₁	1	1	yes	characteristic
231	C₂²	C₂×C₄⋊Dic₁₀₁	1	1	yes	characteristic
232	C₂²	#15692214	1	1	yes	characteristic
233	C₂²	D₄×Dic₁₀₁	1	2	yes	—
234	C₂²	Q₈×Dic₁₀₁	1	2	yes	—
235	C₂²	#15692223	1	1	yes	characteristic
236	C₂²	#15692227	1	2	yes	—
237	C₂²	#15692229	1	1	yes	characteristic
238	C₂²	#15692229	1	1	yes	characteristic
239	C₂²	#15692230	1	2	yes	—
240	C₂²	#15692231	1	2	yes	—
241	C₂²	#15692233	1	2	yes	—
242	C₂²	#15692234	1	1	yes	characteristic
243	C₂²	#15692234	1	1	yes	characteristic
244	C₂²	#15692235	1	2	yes	—
245	C₂²	#15692236	1	1	yes	characteristic
246	C₂²	#15692240	1	2	yes	—
247	C₂²	#15692244	1	1	yes	characteristic
248	C₂²	#15692245	1	2	yes	—
249	C₂²	#15692246	1	2	yes	—
250	C₂²	#15692250	1	2	yes	—
251	C₂²	#15692253	1	1	yes	characteristic
252	C₂²	#15692254	1	2	yes	—
253	C₄	—	2	4	no	—
254	C₄	—	2	4	no	—
255	C₄	—	4	2	no	—
256	C₄	—	404	4	no	—
257	C₄	—	404	4	no	—
258	C₂	#20265037	1	1	yes	characteristic
259	C₂	#20265057	1	1	yes	characteristic
260	C₂	#20265063	1	2	yes	—
261	C₂	#20265076	1	2	yes	—
262	C₂	#20265084	1	1	yes	characteristic
263	C₂	#20265094	1	1	yes	characteristic
264	C₂	#20265097	1	1	yes	characteristic
265	C₂	#20265099	1	2	yes	—
266	C₂	#20265107	1	1	yes	characteristic
267	C₂	#20265110	1	2	yes	—
268	C₂	#20265116	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂⁴	C₂²×D₁₀₁
commutator	C₂³×C₁₀₁	C₂²×C₄
Fitting	C₂²×C₄⋊C₄×C₁₀₁	C₂
socle, prime radical, cosocle	C₂⁴×C₁₀₁	C₂³
hypercenter	C₂²×C₄⋊C₄	D₁₀₁
hypocenter	C₁₀₁	#1441
LCS	G ▷ C₂³×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
18. C₂×C₄⋊C₄×C₁₀₁	C₄	semidirect
80. C₂×C₄×C₁₀₁	C₄⋊C₄	semidirect
135. C₁₀₁	#1441	semidirect
150. C₂×C₄⋊C₄	Dic₁₀₁	semidirect
229. C₂×C₄	C₄⋊Dic₁₀₁	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
C₂×#15692162	2
C₂²×C₄⋊C₄×C₁₀₁	2
#1441	101