Subgroups — #27169357

Order 12928.

568 conjugacy classes of subgroups.

The subgroup lattice has 256 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	C₂×#95×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₁₀₁	C₂²	1	2	yes	—
7	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
8	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
9	C₂×C₄×Dic₁₀₁	—	2	4	no	—
10	C₂×C₄×Dic₁₀₁	C₄	1	4	yes	complemented
11	#15692162	—	2	4	no	—
12	#15692162	—	2	4	no	—
13	#15692162	C₄	1	4	yes	complemented
14	C₂³×C₄×C₁₀₁	C₂²	1	2	yes	—
15	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
16	#95×C₁₀₁	C₄	1	4	yes	complemented
17	C₂²×Dic₁₀₁	—	2	2	no	—
18	C₂²×Dic₁₀₁	—	2	1	no	—
19	C₂²×Dic₁₀₁	—	2	2	no	—
20	C₂²×Dic₁₀₁	—	2	2	no	—
21	C₂²×Dic₁₀₁	—	2	1	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	C₂×C₄	1	4	yes	—
29	C₂²×Dic₁₀₁	C₂×C₄	1	2	yes	—
30	C₂²×Dic₁₀₁	C₂×C₄	1	2	yes	—
31	C₂²×Dic₁₀₁	D₄	1	4	yes	—
32	C₂²×Dic₁₀₁	D₄	1	2	yes	—
33	C₂²×Dic₁₀₁	Q₈	1	2	yes	—
34	C₄×Dic₁₀₁	—	2	8	no	—
35	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₁₀₁	—	2	2	no	—
37	C₂²×C₄×C₁₀₁	—	2	2	no	—
38	C₂²×C₄×C₁₀₁	—	2	2	no	—
39	C₂²×C₄×C₁₀₁	—	2	2	no	—
40	C₂²×C₄×C₁₀₁	—	2	2	no	—
41	C₂²×C₄×C₁₀₁	—	2	2	no	—
42	C₂²×C₄×C₁₀₁	—	2	2	no	—
43	C₂²×C₄×C₁₀₁	—	2	2	no	—
44	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
45	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
46	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	—
47	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
48	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
49	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
50	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
51	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
52	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
53	C₂³×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
54	C₂³×C₁₀₁	#41	1	2	yes	—
55	C₂³×C₁₀₁	#41	1	1	yes	characteristic
56	C₂³×C₁₀₁	#41	1	2	yes	—
57	C₂³×C₁₀₁	#41	1	2	yes	—
58	C₂³×C₁₀₁	#41	1	1	yes	characteristic
59	C₂×C₄×C₁₀₁	—	2	2	no	—
60	C₂×C₄×C₁₀₁	—	2	2	no	—
61	C₂×C₄×C₁₀₁	—	2	4	no	—
62	C₂×C₄×C₁₀₁	—	2	4	no	—
63	C₂×C₄×C₁₀₁	—	2	4	no	—
64	C₂×C₄×C₁₀₁	—	2	4	no	—
65	C₂×C₄×C₁₀₁	—	2	4	no	—
66	C₂×C₄×C₁₀₁	—	4	2	no	—
67	C₂×C₄×C₁₀₁	—	4	2	no	—
68	C₂×C₄×C₁₀₁	—	4	2	no	—
69	C₂×C₄×C₁₀₁	—	4	2	no	—
70	C₂×C₄×C₁₀₁	C₄²	1	4	yes	—
71	C₂×Dic₁₀₁	—	2	2	no	—
72	C₂×Dic₁₀₁	—	2	2	no	—
73	C₂×Dic₁₀₁	—	2	2	no	—
74	C₂×Dic₁₀₁	—	2	2	no	—
75	C₂×Dic₁₀₁	—	2	4	no	—
76	C₂×Dic₁₀₁	—	2	4	no	—
77	C₂×Dic₁₀₁	—	2	4	no	—
78	C₂×Dic₁₀₁	—	2	4	no	—
79	C₂×Dic₁₀₁	—	2	4	no	—
80	C₂×Dic₁₀₁	—	2	4	no	—
81	C₂×Dic₁₀₁	—	2	4	no	—
82	C₂×Dic₁₀₁	—	4	2	no	—
83	C₂×Dic₁₀₁	—	4	2	no	—
84	C₂×Dic₁₀₁	—	4	2	no	—
85	C₂×Dic₁₀₁	—	4	2	no	—
86	C₂×Dic₁₀₁	C₂²⋊C₄	1	4	yes	—
87	C₂×Dic₁₀₁	C₄⋊C₄	1	4	yes	—
88	C₂²×C₁₀₁	C₂×C₄²	1	1	yes	characteristic
89	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂²×C₁₀₁	#117	1	1	yes	characteristic
92	C₂²×C₁₀₁	#117	1	1	yes	characteristic
93	C₂²×C₁₀₁	#117	1	1	yes	characteristic
94	C₂²×C₁₀₁	#117	1	2	yes	—
95	C₂²×C₁₀₁	#117	1	1	yes	characteristic
96	C₂²×C₁₀₁	#117	1	2	yes	—
97	C₂²×C₁₀₁	C₄×D₄	1	2	yes	—
98	C₂²×C₁₀₁	C₄×D₄	1	1	yes	characteristic
99	C₂²×C₁₀₁	C₄×D₄	1	2	yes	—
100	C₂²×C₁₀₁	C₄×D₄	1	1	yes	characteristic
101	C₂²×C₁₀₁	C₄×Q₈	1	1	yes	characteristic
102	C₂²×C₁₀₁	C₄×Q₈	1	1	yes	characteristic
103	C₂²×C₁₀₁	#121	1	2	yes	—
104	C₂²×C₁₀₁	C₂²⋊Q₈	1	2	yes	—
105	C₂²×C₁₀₁	#123	1	2	yes	—
106	C₂²×C₁₀₁	#123	1	2	yes	—
107	C₂²×C₁₀₁	#124	1	2	yes	—
108	C₂²×C₁₀₁	#125	1	2	yes	—
109	C₂²×C₁₀₁	#126	1	2	yes	—
110	C₂²×C₁₀₁	#126	1	2	yes	—
111	C₄×C₁₀₁	—	2	4	no	—
112	C₄×C₁₀₁	—	4	4	no	—
113	Dic₁₀₁	—	2	4	no	—
114	Dic₁₀₁	—	2	4	no	—
115	Dic₁₀₁	—	4	4	no	—
116	C₂×C₁₀₁	#376	1	1	yes	characteristic
117	C₂×C₁₀₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₁₀₁	C₄×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₁₀₁	#380	1	1	yes	characteristic
120	C₂×C₁₀₁	#381	1	1	yes	characteristic
121	C₂×C₁₀₁	#382	1	1	yes	characteristic
122	C₂×C₁₀₁	#383	1	1	yes	characteristic
123	C₂×C₁₀₁	#387	1	2	yes	—
124	C₂×C₁₀₁	#387	1	2	yes	—
125	C₂×C₁₀₁	#388	1	2	yes	—
126	C₂×C₁₀₁	#388	1	2	yes	—
127	#1434	—	101	1	no	maximal, Sylow
128	C₁₀₁	#1434	1	1	yes	characteristic, complemented, Sylow
129	C₂²×C₄²	—	101	1	no	—
130	C₂×#95	—	101	1	no	—
131	C₂×#95	—	101	2	no	—
132	C₂×#95	—	101	2	no	—
133	C₂×#95	D₁₀₁	1	1	yes	characteristic
134	C₂³×C₄	—	101	1	no	—
135	C₂³×C₄	—	101	1	no	—
136	C₂³×C₄	—	101	2	no	—
137	C₂³×C₄	C₂×D₁₀₁	1	2	yes	—
138	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
139	C₂×C₄²	—	101	4	no	—
140	C₂×C₄²	—	202	4	no	—
141	#95	—	101	4	no	—
142	#95	—	202	4	no	—
143	#95	—	202	4	no	—
144	#95	Dic₁₀₁	1	4	yes	complemented
145	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	—	2	2	no	—
152	C₂²×C₄	—	2	2	no	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	101	2	no	—
155	C₂²×C₄	—	101	2	no	—
156	C₂²×C₄	—	101	2	no	—
157	C₂²×C₄	—	101	4	no	—
158	C₂²×C₄	—	101	2	no	—
159	C₂²×C₄	—	101	4	no	—
160	C₂²×C₄	—	202	2	no	—
161	C₂²×C₄	—	202	2	no	—
162	C₂²×C₄	—	202	2	no	—
163	C₂²×C₄	—	202	2	no	—
164	C₂²×C₄	—	202	2	no	—
165	C₂²×C₄	—	202	2	no	—
166	C₂²×C₄	—	202	2	no	—
167	C₂²×C₄	—	202	1	no	—
168	C₂²×C₄	—	202	2	no	—
169	C₂²×C₄	—	202	2	no	—
170	C₂²×C₄	—	202	1	no	—
171	C₂²×C₄	C₄×D₁₀₁	1	2	yes	—
172	C₂²×C₄	C₄×D₁₀₁	1	2	yes	—
173	C₂²×C₄	C₂×Dic₁₀₁	1	4	yes	—
174	C₂²×C₄	C₂×Dic₁₀₁	1	2	yes	—
175	C₄²	—	202	8	no	—
176	C₂³	C₂×C₄×D₁₀₁	1	1	yes	characteristic
177	C₂³	C₂×C₄×D₁₀₁	1	1	yes	characteristic
178	C₂³	C₂²×Dic₁₀₁	1	1	yes	characteristic
179	C₂³	D₄×D₁₀₁	1	2	yes	—
180	C₂³	D₄×D₁₀₁	1	1	yes	characteristic
181	C₂³	Q₈×D₁₀₁	1	1	yes	characteristic
182	C₂³	#13108073	1	2	yes	—
183	C₂³	#13108073	1	1	yes	characteristic
184	C₂³	#13108074	1	2	yes	—
185	C₂³	#13108074	1	2	yes	—
186	C₂³	#13108075	1	1	yes	characteristic
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	101	4	no	—
199	C₂×C₄	—	101	4	no	—
200	C₂×C₄	—	404	2	no	—
201	C₂×C₄	—	404	2	no	—
202	C₂×C₄	—	404	2	no	—
203	C₂×C₄	—	404	2	no	—
204	C₂×C₄	—	202	4	no	—
205	C₂×C₄	—	202	4	no	—
206	C₂×C₄	—	202	4	no	—
207	C₂×C₄	—	202	2	no	—
208	C₂×C₄	—	202	2	no	—
209	C₂×C₄	—	202	2	no	—
210	C₂×C₄	—	202	2	no	—
211	C₂×C₄	—	202	4	no	—
212	C₂×C₄	—	202	4	no	—
213	C₂×C₄	—	202	4	no	—
214	C₂×C₄	—	202	4	no	—
215	C₂×C₄	C₄×Dic₁₀₁	1	4	yes	—
216	C₂²	C₂×C₄×Dic₁₀₁	1	1	yes	characteristic
217	C₂²	C₂²⋊C₄×D₁₀₁	1	1	yes	characteristic
218	C₂²	C₄⋊C₄×D₁₀₁	1	1	yes	characteristic
219	C₂²	#15692211	1	1	yes	characteristic
220	C₂²	#15692212	1	1	yes	characteristic
221	C₂²	#15692212	1	1	yes	characteristic
222	C₂²	#15692212	1	2	yes	—
223	C₂²	#15692213	1	1	yes	characteristic
224	C₂²	#15692214	1	2	yes	—
225	C₂²	#15692215	1	2	yes	—
226	C₂²	#15692217	1	1	yes	characteristic
227	C₂²	D₄×Dic₁₀₁	1	2	yes	—
228	C₂²	D₄×Dic₁₀₁	1	1	yes	characteristic
229	C₂²	#15692220	1	1	yes	characteristic
230	C₂²	Q₈×Dic₁₀₁	1	1	yes	characteristic
231	C₂²	#15692228	1	2	yes	—
232	C₂²	#15692233	1	2	yes	—
233	C₂²	#15692237	1	2	yes	—
234	C₂²	#15692240	1	2	yes	—
235	C₂²	#15692241	1	2	yes	—
236	C₂²	#15692245	1	2	yes	—
237	C₂²	#15692248	1	2	yes	—
238	C₂²	#15692250	1	2	yes	—
239	C₄	—	2	4	no	—
240	C₄	—	4	4	no	—
241	C₄	—	404	4	no	—
242	C₄	—	202	4	no	—
243	C₄	—	202	4	no	—
244	C₂	#20265023	1	1	yes	characteristic
245	C₂	C₂²⋊C₄×Dic₁₀₁	1	1	yes	characteristic
246	C₂	C₄⋊C₄×Dic₁₀₁	1	1	yes	characteristic
247	C₂	#20265036	1	1	yes	characteristic
248	C₂	#20265039	1	1	yes	characteristic
249	C₂	#20265041	1	1	yes	characteristic
250	C₂	#20265044	1	1	yes	characteristic
251	C₂	#20265059	1	2	yes	—
252	C₂	#20265063	1	2	yes	—
253	C₂	#20265066	1	2	yes	—
254	C₂	#20265070	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂⁴	C₂²×D₁₀₁
commutator	C₂²×C₁₀₁	C₂×C₄²
Fitting	C₂×#95×C₁₀₁	C₂
socle, prime radical, cosocle	C₂⁴×C₁₀₁	C₂³
hypercenter	C₂×#95	D₁₀₁
hypocenter	C₁₀₁	#1434
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#95

Complements

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

N	H ≅ G/N	type
10. C₂×C₄×Dic₁₀₁	C₄	semidirect
13. #15692162	C₄	semidirect
16. #95×C₁₀₁	C₄	semidirect
128. C₁₀₁	#1434	semidirect
144. #95	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
C₂×#95×C₁₀₁	2
#1434	101