Subgroups — C₄×C₂²⋊C₄×D₁₀₁ #27173460

Order 12928.

1360 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	C₂³×C₄×D₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊C₄×D₁₀₁	C₂	1	2	yes	maximal
3	C₂×C₄²×D₁₀₁	C₂	1	2	yes	maximal, complemented
4	C₂²⋊C₄×Dic₁₀₁	C₂	1	2	yes	maximal, complemented
5	C₄×#13108051	C₂	1	2	yes	maximal, complemented
6	C₄×C₂²⋊Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#20265024	C₂	1	2	yes	maximal, complemented
8	#95×D₁₀₁	C₂	1	2	yes	maximal, complemented
9	C₄×C₂²⋊C₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁴×D₁₀₁	C₂²	1	1	yes	characteristic
11	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
12	C₂²×C₄×D₁₀₁	—	2	8	no	—
13	C₂²×C₄×D₁₀₁	C₄	1	8	yes	complemented
14	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
15	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
16	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₁₀₁	C₂²	1	4	yes	—
18	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
20	C₂×#13108051	C₂²	1	4	yes	—
21	C₂²⋊C₄×D₁₀₁	C₄	1	16	yes	direct product factor
22	C₂×C₄×Dic₁₀₁	C₂²	1	4	yes	complemented
23	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	complemented
24	C₄²×D₁₀₁	—	2	8	no	—
25	#15692162	C₂²	1	2	yes	complemented
26	#15692162	C₂²	1	4	yes	complemented
27	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
29	C₂×C₄²×C₁₀₁	C₂²	1	2	yes	complemented
30	#95×C₁₀₁	C₂²	1	2	yes	complemented
31	C₂³×D₁₀₁	—	2	4	no	—
32	C₂³×D₁₀₁	—	2	4	no	—
33	C₂³×D₁₀₁	C₂×C₄	1	4	yes	—
34	C₂³×D₁₀₁	C₂×C₄	1	8	yes	—
35	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
36	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
37	C₂²×Dic₁₀₁	—	2	4	no	—
38	C₂²×Dic₁₀₁	C₂×C₄	1	4	yes	complemented
39	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
40	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
41	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
42	C₂×C₄×D₁₀₁	—	2	4	no	—
43	C₂×C₄×D₁₀₁	—	2	8	no	—
44	C₂×C₄×D₁₀₁	—	2	8	no	—
45	C₂×C₄×D₁₀₁	—	2	4	no	—
46	C₂×C₄×D₁₀₁	—	2	4	no	—
47	C₂×C₄×D₁₀₁	—	2	32	no	—
48	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
49	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
50	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	complemented
51	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	complemented
52	C₂×C₄×D₁₀₁	C₂×C₄	1	16	yes	complemented
53	C₂×C₄×D₁₀₁	D₄	1	8	yes	—
54	C₂²⋊Dic₁₀₁	C₂×C₄	1	8	yes	complemented
55	#13108051	C₂×C₄	1	16	yes	complemented
56	C₄×Dic₁₀₁	—	2	4	no	—
57	C₄×Dic₁₀₁	—	2	8	no	—
58	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₁₀₁	—	2	4	no	—
60	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	complemented
61	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
63	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
64	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	8	yes	complemented
65	C₄²×C₁₀₁	—	2	4	no	—
66	C₂³×C₁₀₁	—	2	2	no	—
67	C₂³×C₁₀₁	—	2	2	no	—
68	C₂³×C₁₀₁	C₂²×C₄	1	4	yes	—
69	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
70	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
71	C₂²×D₁₀₁	—	2	2	no	—
72	C₂²×D₁₀₁	—	2	2	no	—
73	C₂²×D₁₀₁	—	2	8	no	—
74	C₂²×D₁₀₁	—	2	8	no	—
75	C₂²×D₁₀₁	—	2	32	no	—
76	C₂²×D₁₀₁	C₄²	1	16	yes	complemented
77	C₂²×D₁₀₁	C₂²×C₄	1	4	yes	—
78	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
79	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
80	C₂²×D₁₀₁	C₂²×C₄	1	4	yes	—
81	C₂²×D₁₀₁	C₂×D₄	1	4	yes	—
82	C₂²×D₁₀₁	#41	1	4	yes	—
83	C₂×C₄×C₁₀₁	—	2	4	no	—
84	C₂×C₄×C₁₀₁	—	2	2	no	—
85	C₂×C₄×C₁₀₁	—	2	4	no	—
86	C₂×C₄×C₁₀₁	—	2	8	no	—
87	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₁₀₁	C₂²×C₄	1	8	yes	complemented
90	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
91	C₂×Dic₁₀₁	—	2	4	no	—
92	C₂×Dic₁₀₁	—	2	2	no	—
93	C₂×Dic₁₀₁	—	2	4	no	—
94	C₂×Dic₁₀₁	—	2	8	no	—
95	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
96	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
97	C₂×Dic₁₀₁	C₂²×C₄	1	8	yes	complemented
98	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	—
99	C₄×D₁₀₁	—	2	8	no	—
100	C₄×D₁₀₁	—	2	32	no	—
101	C₄×D₁₀₁	—	2	8	no	—
102	C₄×D₁₀₁	—	2	8	no	—
103	C₄×D₁₀₁	C₂²⋊C₄	1	16	yes	direct product factor
104	C₂²×C₁₀₁	—	2	2	no	—
105	C₂²×C₁₀₁	—	2	2	no	—
106	C₂²×C₁₀₁	—	2	8	no	—
107	C₂²×C₁₀₁	C₂×C₄²	1	4	yes	complemented
108	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
109	C₂²×C₁₀₁	C₂³×C₄	1	2	yes	—
110	C₂²×C₁₀₁	C₂²×D₄	1	2	yes	—
111	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
112	C₂×D₁₀₁	—	2	2	no	—
113	C₂×D₁₀₁	—	2	2	no	—
114	C₂×D₁₀₁	—	2	8	no	—
115	C₂×D₁₀₁	—	2	32	no	—
116	C₂×D₁₀₁	C₂×C₄²	1	4	yes	—
117	C₂×D₁₀₁	C₂×C₄²	1	4	yes	complemented
118	C₂×D₁₀₁	C₂×C₂²⋊C₄	1	4	yes	—
119	C₂×D₁₀₁	#117	1	4	yes	—
120	C₂×D₁₀₁	C₄×D₄	1	16	yes	—
121	C₄×C₁₀₁	—	2	2	no	—
122	C₄×C₁₀₁	—	2	8	no	—
123	C₄×C₁₀₁	C₂×C₂²⋊C₄	1	4	yes	complemented
124	Dic₁₀₁	—	2	2	no	—
125	Dic₁₀₁	—	2	8	no	—
126	Dic₁₀₁	C₂×C₂²⋊C₄	1	4	yes	complemented
127	C₂×C₁₀₁	—	2	4	no	—
128	C₂×C₁₀₁	C₂²×C₄²	1	1	yes	characteristic
129	C₂×C₁₀₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₁₀₁	C₂×#117	1	1	yes	characteristic
131	C₂×C₁₀₁	C₂×C₄×D₄	1	4	yes	—
132	D₁₀₁	—	2	4	no	—
133	D₁₀₁	C₄×C₂²⋊C₄	1	8	yes	direct product factor
134	C₂×C₄×C₂²⋊C₄	—	101	1	no	maximal, Sylow
135	C₁₀₁	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
136	C₂⁴×C₄	—	101	1	no	—
137	C₂²×C₂²⋊C₄	—	101	2	no	—
138	C₂²×C₄²	—	101	2	no	—
139	C₄×C₂²⋊C₄	—	101	1	no	—
140	C₄×C₂²⋊C₄	—	101	2	no	—
141	C₄×C₂²⋊C₄	—	101	2	no	—
142	C₄×C₂²⋊C₄	—	101	2	no	—
143	C₄×C₂²⋊C₄	D₁₀₁	1	1	yes	characteristic, direct product factor
144	C₂×#95	—	101	2	no	—
145	C₂⁵	—	101	1	no	—
146	C₂³×C₄	—	101	1	no	—
147	C₂³×C₄	—	101	1	no	—
148	C₂³×C₄	—	101	8	no	—
149	C₂³×C₄	—	101	4	no	—
150	C₂³×C₄	—	101	1	no	—
151	C₂³×C₄	—	101	1	no	—
152	C₂³×C₄	—	101	1	no	—
153	C₂³×C₄	—	202	8	no	—
154	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
155	C₂×C₂²⋊C₄	—	101	2	no	—
156	C₂×C₂²⋊C₄	—	101	4	no	—
157	C₂×C₂²⋊C₄	—	101	16	no	—
158	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
159	C₂×C₄²	—	101	4	no	—
160	C₂×C₄²	—	101	2	no	—
161	C₂×C₄²	—	202	8	no	—
162	C₂×C₄²	C₂×D₁₀₁	1	2	yes	complemented
163	#95	—	101	4	no	—
164	#95	—	101	2	no	—
165	#95	C₂×D₁₀₁	1	2	yes	complemented
166	C₂⁴	—	101	1	no	—
167	C₂⁴	—	101	8	no	—
168	C₂⁴	—	101	4	no	—
169	C₂⁴	—	101	1	no	—
170	C₂⁴	—	202	4	no	—
171	C₂⁴	—	202	4	no	—
172	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	—	101	1	no	—
175	C₂²×C₄	—	101	4	no	—
176	C₂²×C₄	—	101	8	no	—
177	C₂²×C₄	—	101	4	no	—
178	C₂²×C₄	—	101	1	no	—
179	C₂²×C₄	—	101	4	no	—
180	C₂²×C₄	—	101	4	no	—
181	C₂²×C₄	—	101	4	no	—
182	C₂²×C₄	—	101	4	no	—
183	C₂²×C₄	—	101	16	no	—
184	C₂²×C₄	—	202	4	no	—
185	C₂²×C₄	—	202	32	no	—
186	C₂²×C₄	—	202	8	no	—
187	C₂²×C₄	—	202	8	no	—
188	C₂²×C₄	—	202	4	no	—
189	C₂²×C₄	—	202	4	no	—
190	C₂²×C₄	—	202	4	no	—
191	C₂²×C₄	C₄×D₁₀₁	1	4	yes	complemented
192	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
193	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
194	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
195	C₂²⋊C₄	—	101	8	no	—
196	C₂²⋊C₄	—	101	16	no	—
197	C₂²⋊C₄	C₄×D₁₀₁	1	8	yes	direct product factor
198	C₄²	—	2	4	no	—
199	C₄²	—	202	8	no	—
200	C₄²	—	202	4	no	—
201	C₂³	—	2	2	no	—
202	C₂³	—	2	2	no	—
203	C₂³	—	101	2	no	—
204	C₂³	—	101	4	no	—
205	C₂³	—	101	4	no	—
206	C₂³	—	101	4	no	—
207	C₂³	—	101	4	no	—
208	C₂³	—	101	2	no	—
209	C₂³	—	101	16	no	—
210	C₂³	—	202	8	no	—
211	C₂³	—	202	8	no	—
212	C₂³	—	202	32	no	—
213	C₂³	—	202	2	no	—
214	C₂³	—	202	2	no	—
215	C₂³	C₂×C₄×D₁₀₁	1	4	yes	—
216	C₂³	C₂×C₄×D₁₀₁	1	2	yes	—
217	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	101	2	no	—
223	C₂×C₄	—	101	4	no	—
224	C₂×C₄	—	101	16	no	—
225	C₂×C₄	—	101	8	no	—
226	C₂×C₄	—	101	2	no	—
227	C₂×C₄	—	202	8	no	—
228	C₂×C₄	—	202	8	no	—
229	C₂×C₄	—	202	32	no	—
230	C₂×C₄	—	202	8	no	—
231	C₂×C₄	—	202	4	no	—
232	C₂×C₄	—	202	2	no	—
233	C₂×C₄	—	202	4	no	—
234	C₂×C₄	—	202	8	no	—
235	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
236	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
237	C₂×C₄	C₂×C₄×D₁₀₁	1	8	yes	complemented
238	C₂×C₄	D₄×D₁₀₁	1	4	yes	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	8	no	—
242	C₂²	—	101	4	no	—
243	C₂²	—	101	4	no	—
244	C₂²	—	101	4	no	—
245	C₂²	—	101	16	no	—
246	C₂²	—	101	4	no	—
247	C₂²	—	202	32	no	—
248	C₂²	—	202	2	no	—
249	C₂²	—	202	2	no	—
250	C₂²	—	202	8	no	—
251	C₂²	C₄²×D₁₀₁	1	4	yes	complemented
252	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
253	C₂²	C₂²×C₄×D₁₀₁	1	2	yes	—
254	C₂²	C₂×D₄×D₁₀₁	1	2	yes	—
255	C₂²	#41×D₁₀₁	1	2	yes	—
256	C₄	—	2	2	no	—
257	C₄	—	2	8	no	—
258	C₄	—	101	4	no	—
259	C₄	—	202	2	no	—
260	C₄	—	202	8	no	—
261	C₄	C₂²⋊C₄×D₁₀₁	1	4	yes	direct product factor
262	C₂	—	2	4	no	—
263	C₂	—	101	8	no	—
264	C₂	—	202	4	no	—
265	C₂	C₂×C₄²×D₁₀₁	1	1	yes	characteristic
266	C₂	C₂×C₂²⋊C₄×D₁₀₁	1	1	yes	characteristic
267	C₂	#117×D₁₀₁	1	1	yes	characteristic
268	C₂	C₄×D₄×D₁₀₁	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₂×C₄²×D₁₀₁	C₂	semidirect
4. C₂²⋊C₄×Dic₁₀₁	C₂	semidirect
5. C₄×#13108051	C₂	semidirect
6. C₄×C₂²⋊Dic₁₀₁	C₂	semidirect
7. #20265024	C₂	semidirect
8. #95×D₁₀₁	C₂	semidirect
9. C₄×C₂²⋊C₄×C₁₀₁	C₂	semidirect
13. C₂²×C₄×D₁₀₁	C₄	semidirect
21. C₂²⋊C₄×D₁₀₁	C₄	direct (N × H)
22. C₂×C₄×Dic₁₀₁	C₂²	semidirect
23. C₂×C₄×Dic₁₀₁	C₂²	semidirect
25. #15692162	C₂²	semidirect
26. #15692162	C₂²	semidirect
29. C₂×C₄²×C₁₀₁	C₂²	semidirect
30. #95×C₁₀₁	C₂²	semidirect
38. C₂²×Dic₁₀₁	C₂×C₄	semidirect
50. C₂×C₄×D₁₀₁	C₂×C₄	semidirect
51. C₂×C₄×D₁₀₁	C₂×C₄	semidirect
52. C₂×C₄×D₁₀₁	C₂×C₄	semidirect
54. C₂²⋊Dic₁₀₁	C₂×C₄	semidirect
55. #13108051	C₂×C₄	semidirect
60. C₂²×C₄×C₁₀₁	C₂×C₄	semidirect
64. C₂²⋊C₄×C₁₀₁	C₂×C₄	semidirect
76. C₂²×D₁₀₁	C₄²	semidirect
89. C₂×C₄×C₁₀₁	C₂²×C₄	semidirect
97. C₂×Dic₁₀₁	C₂²×C₄	semidirect
103. C₄×D₁₀₁	C₂²⋊C₄	direct (N × H)
107. C₂²×C₁₀₁	C₂×C₄²	semidirect
117. C₂×D₁₀₁	C₂×C₄²	semidirect
123. C₄×C₁₀₁	C₂×C₂²⋊C₄	semidirect
126. Dic₁₀₁	C₂×C₂²⋊C₄	semidirect
133. D₁₀₁	C₄×C₂²⋊C₄	direct (N × H)
135. C₁₀₁	C₂×C₄×C₂²⋊C₄	semidirect
143. C₄×C₂²⋊C₄	D₁₀₁	direct (N × H)
162. C₂×C₄²	C₂×D₁₀₁	semidirect
165. #95	C₂×D₁₀₁	semidirect
191. C₂²×C₄	C₄×D₁₀₁	semidirect
197. C₂²⋊C₄	C₄×D₁₀₁	direct (N × H)
237. C₂×C₄	C₂×C₄×D₁₀₁	semidirect
251. C₂²	C₄²×D₁₀₁	semidirect
261. C₄	C₂²⋊C₄×D₁₀₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄×D₁₀₁	2
C₂×C₂²⋊C₄×D₁₀₁	2
C₂×C₄²×D₁₀₁	2
C₂²⋊C₄×Dic₁₀₁	2
C₄×#13108051	2
C₄×C₂²⋊Dic₁₀₁	2
#20265024	2
#95×D₁₀₁	2
C₄×C₂²⋊C₄×C₁₀₁	2
C₂×C₄×C₂²⋊C₄	101

Subgroups — C4×C22⋊C4×D101 #27173460

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₂²⋊C₄×D₁₀₁ #27173460