Subgroups — C₂×C₂²⋊C₈×D₁₀₁ #27179562

Order 12928.

1072 conjugacy classes of subgroups.

The subgroup lattice has 248 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	C₂²×C₈×D₁₀₁	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊#11717579	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15692167	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₈×D₁₀₁	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₂²⋊C₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁴×D₁₀₁	C₄	1	1	yes	characteristic
8	C₂³×Dic₁₀₁	C₄	1	1	yes	characteristic
9	C₂²×C₄×D₁₀₁	—	2	8	no	—
10	C₂²×C₄×D₁₀₁	C₄	1	1	yes	characteristic
11	C₂²×C₄×D₁₀₁	C₄	1	1	yes	characteristic
12	C₂²×C₄×D₁₀₁	C₄	1	4	yes	—
13	C₂²×C₄×D₁₀₁	C₂²	1	4	yes	—
14	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
15	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
16	C₂²×#11717579	C₂²	1	2	yes	complemented
17	C₂×C₈×D₁₀₁	—	2	8	no	—
18	C₂×C₈×D₁₀₁	C₂²	1	8	yes	complemented
19	C₂²⋊#11717579	C₂²	1	4	yes	complemented
20	#15692167	C₂²	1	8	yes	complemented
21	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
22	C₂²×C₈×C₁₀₁	C₂²	1	2	yes	complemented
23	C₂²⋊C₈×C₁₀₁	C₂²	1	4	yes	complemented
24	C₂³×D₁₀₁	—	2	4	no	—
25	C₂³×D₁₀₁	—	2	4	no	—
26	C₂³×D₁₀₁	C₈	1	8	yes	complemented
27	C₂³×D₁₀₁	C₂×C₄	1	1	yes	characteristic
28	C₂³×D₁₀₁	C₂×C₄	1	4	yes	—
29	C₂³×D₁₀₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×Dic₁₀₁	—	2	4	no	—
31	C₂²×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×Dic₁₀₁	C₂×C₄	1	4	yes	—
33	C₂²×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×D₁₀₁	—	2	4	no	—
35	C₂×C₄×D₁₀₁	—	2	4	no	—
36	C₂×C₄×D₁₀₁	—	2	4	no	—
37	C₂×C₄×D₁₀₁	—	2	16	no	—
38	C₂×C₄×D₁₀₁	—	2	16	no	—
39	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
40	C₂×C₄×D₁₀₁	C₂×C₄	1	2	yes	—
41	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
42	C₂×C₄×D₁₀₁	C₂×C₄	1	2	yes	—
43	C₂×C₄×D₁₀₁	D₄	1	8	yes	—
44	C₂×C₄×D₁₀₁	C₂³	1	2	yes	—
45	C₂×C₄×D₁₀₁	C₂³	1	2	yes	—
46	C₂×#11717579	—	2	4	no	—
47	C₂×#11717579	C₂³	1	4	yes	complemented
48	C₈×D₁₀₁	—	2	16	no	—
49	C₂⁴×C₁₀₁	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₀₁	—	2	4	no	—
51	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
53	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
54	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
55	C₂×C₈×C₁₀₁	—	2	4	no	—
56	C₂×C₈×C₁₀₁	C₂³	1	4	yes	complemented
57	C₂³×C₁₀₁	—	2	2	no	—
58	C₂³×C₁₀₁	—	2	2	no	—
59	C₂³×C₁₀₁	C₂×C₈	1	4	yes	complemented
60	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
61	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
62	C₂²×D₁₀₁	—	2	2	no	—
63	C₂²×D₁₀₁	—	2	2	no	—
64	C₂²×D₁₀₁	—	2	32	no	—
65	C₂²×D₁₀₁	—	2	8	no	—
66	C₂²×D₁₀₁	—	2	8	no	—
67	C₂²×D₁₀₁	C₂²⋊C₄	1	4	yes	—
68	C₂²×D₁₀₁	C₂×C₈	1	4	yes	—
69	C₂²×D₁₀₁	C₂×C₈	1	16	yes	complemented
70	C₂²×D₁₀₁	C₂×C₈	1	4	yes	complemented
71	C₂²×D₁₀₁	M₁₆	1	4	yes	—
72	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
73	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₁₀₁	—	2	2	no	—
75	C₂×C₄×C₁₀₁	—	2	4	no	—
76	C₂×C₄×C₁₀₁	—	2	4	no	—
77	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
80	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
81	C₂×Dic₁₀₁	—	2	2	no	—
82	C₂×Dic₁₀₁	—	2	8	no	—
83	C₂×Dic₁₀₁	C₂²⋊C₄	1	4	yes	—
84	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
85	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
86	C₄×D₁₀₁	—	2	8	no	—
87	C₄×D₁₀₁	—	2	4	no	—
88	C₄×D₁₀₁	—	2	4	no	—
89	C₄×D₁₀₁	—	2	8	no	—
90	C₄×D₁₀₁	C₂²⋊C₄	1	8	yes	—
91	C₄×D₁₀₁	C₂×D₄	1	8	yes	—
92	C₈×C₁₀₁	—	2	4	no	—
93	#11717579	—	2	4	no	—
94	C₂²×C₁₀₁	—	2	2	no	—
95	C₂²×C₁₀₁	—	2	2	no	—
96	C₂²×C₁₀₁	—	2	8	no	—
97	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
98	C₂²×C₁₀₁	C₂²×C₈	1	2	yes	—
99	C₂²×C₁₀₁	C₂²×C₈	1	4	yes	complemented
100	C₂²×C₁₀₁	C₂×M₁₆	1	2	yes	—
101	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
102	C₂×D₁₀₁	—	2	2	no	—
103	C₂×D₁₀₁	—	2	2	no	—
104	C₂×D₁₀₁	—	2	8	no	—
105	C₂×D₁₀₁	—	2	32	no	—
106	C₂×D₁₀₁	C₂²⋊C₈	1	16	yes	direct product factor
107	C₂×D₁₀₁	C₂×C₂²⋊C₄	1	4	yes	—
108	C₂×D₁₀₁	C₂²×C₈	1	4	yes	—
109	C₂×D₁₀₁	C₂²×C₈	1	4	yes	complemented
110	C₂×D₁₀₁	C₂×M₁₆	1	4	yes	—
111	C₄×C₁₀₁	—	2	2	no	—
112	C₄×C₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
113	C₄×C₁₀₁	C₂²×D₄	1	2	yes	—
114	Dic₁₀₁	—	2	2	no	—
115	Dic₁₀₁	C₂×C₂²⋊C₄	1	4	yes	—
116	C₂×C₁₀₁	—	2	4	no	—
117	C₂×C₁₀₁	C₂×C₂²⋊C₈	1	4	yes	complemented
118	C₂×C₁₀₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₁₀₁	C₂³×C₈	1	1	yes	characteristic
120	C₂×C₁₀₁	C₂²×M₁₆	1	1	yes	characteristic
121	D₁₀₁	—	2	4	no	—
122	D₁₀₁	C₂×C₂²⋊C₈	1	8	yes	direct product factor
123	C₂²×C₂²⋊C₈	—	101	1	no	maximal, Sylow
124	C₁₀₁	C₂²×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
125	C₂⁴×C₄	—	101	1	no	—
126	C₂³×C₈	—	101	2	no	—
127	C₂×C₂²⋊C₈	—	101	1	no	—
128	C₂×C₂²⋊C₈	—	101	2	no	—
129	C₂×C₂²⋊C₈	—	101	8	no	—
130	C₂×C₂²⋊C₈	D₁₀₁	1	1	yes	characteristic, direct product factor
131	C₂⁵	—	101	1	no	—
132	C₂³×C₄	—	101	1	no	—
133	C₂³×C₄	—	101	1	no	—
134	C₂³×C₄	—	101	1	no	—
135	C₂³×C₄	—	101	4	no	—
136	C₂³×C₄	—	101	1	no	—
137	C₂³×C₄	—	101	4	no	—
138	C₂³×C₄	—	101	1	no	—
139	C₂³×C₄	—	202	8	no	—
140	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
141	C₂²×C₈	—	101	2	no	—
142	C₂²×C₈	—	101	8	no	—
143	C₂²×C₈	—	202	8	no	—
144	C₂²×C₈	C₂×D₁₀₁	1	2	yes	complemented
145	C₂²⋊C₈	—	101	4	no	—
146	C₂²⋊C₈	—	101	8	no	—
147	C₂²⋊C₈	C₂×D₁₀₁	1	4	yes	direct product factor
148	C₂⁴	—	101	1	no	—
149	C₂⁴	—	101	8	no	—
150	C₂⁴	—	101	4	no	—
151	C₂⁴	—	101	1	no	—
152	C₂⁴	—	202	4	no	—
153	C₂⁴	—	202	4	no	—
154	C₂⁴	C₄×D₁₀₁	1	1	yes	characteristic
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	—	101	1	no	—
157	C₂²×C₄	—	101	2	no	—
158	C₂²×C₄	—	101	8	no	—
159	C₂²×C₄	—	101	2	no	—
160	C₂²×C₄	—	101	4	no	—
161	C₂²×C₄	—	101	1	no	—
162	C₂²×C₄	—	101	4	no	—
163	C₂²×C₄	—	101	4	no	—
164	C₂²×C₄	—	101	2	no	—
165	C₂²×C₄	—	101	2	no	—
166	C₂²×C₄	—	202	4	no	—
167	C₂²×C₄	—	202	16	no	—
168	C₂²×C₄	—	202	16	no	—
169	C₂²×C₄	—	202	4	no	—
170	C₂²×C₄	—	202	4	no	—
171	C₂²×C₄	—	202	4	no	—
172	C₂²×C₄	C₄×D₁₀₁	1	1	yes	characteristic
173	C₂²×C₄	C₄×D₁₀₁	1	2	yes	—
174	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
175	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	101	4	no	—
178	C₂×C₈	—	202	16	no	—
179	C₂×C₈	—	202	4	no	—
180	C₂×C₈	C₂²×D₁₀₁	1	4	yes	complemented
181	C₂³	—	2	2	no	—
182	C₂³	—	2	2	no	—
183	C₂³	—	101	2	no	—
184	C₂³	—	101	4	no	—
185	C₂³	—	101	4	no	—
186	C₂³	—	101	4	no	—
187	C₂³	—	101	2	no	—
188	C₂³	—	101	4	no	—
189	C₂³	—	101	16	no	—
190	C₂³	—	202	8	no	—
191	C₂³	—	202	32	no	—
192	C₂³	—	202	8	no	—
193	C₂³	—	202	2	no	—
194	C₂³	—	202	2	no	—
195	C₂³	C₈×D₁₀₁	1	4	yes	complemented
196	C₂³	C₂×C₄×D₁₀₁	1	2	yes	—
197	C₂³	C₂×C₄×D₁₀₁	1	1	yes	characteristic
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	101	2	no	—
202	C₂×C₄	—	101	4	no	—
203	C₂×C₄	—	101	8	no	—
204	C₂×C₄	—	101	8	no	—
205	C₂×C₄	—	101	2	no	—
206	C₂×C₄	—	202	8	no	—
207	C₂×C₄	—	202	8	no	—
208	C₂×C₄	—	202	8	no	—
209	C₂×C₄	—	202	2	no	—
210	C₂×C₄	—	202	4	no	—
211	C₂×C₄	—	202	4	no	—
212	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
213	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
214	C₂×C₄	D₄×D₁₀₁	1	4	yes	—
215	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
216	C₈	—	2	4	no	—
217	C₈	—	202	4	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	8	no	—
220	C₂²	—	2	2	no	—
221	C₂²	—	101	4	no	—
222	C₂²	—	101	16	no	—
223	C₂²	—	101	4	no	—
224	C₂²	—	101	4	no	—
225	C₂²	—	101	4	no	—
226	C₂²	—	202	32	no	—
227	C₂²	—	202	2	no	—
228	C₂²	—	202	8	no	—
229	C₂²	—	202	2	no	—
230	C₂²	C₂²⋊C₄×D₁₀₁	1	2	yes	—
231	C₂²	C₂×C₈×D₁₀₁	1	2	yes	—
232	C₂²	C₂×C₈×D₁₀₁	1	4	yes	complemented
233	C₂²	M₁₆×D₁₀₁	1	2	yes	—
234	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	101	4	no	—
237	C₄	—	202	2	no	—
238	C₄	C₂²⋊C₄×D₁₀₁	1	2	yes	—
239	C₄	C₂×D₄×D₁₀₁	1	2	yes	—
240	C₂	—	2	4	no	—
241	C₂	—	101	8	no	—
242	C₂	—	202	4	no	—
243	C₂	C₂²⋊C₈×D₁₀₁	1	4	yes	direct product factor
244	C₂	C₂×C₂²⋊C₄×D₁₀₁	1	1	yes	characteristic
245	C₂	C₂²×C₈×D₁₀₁	1	1	yes	characteristic
246	C₂	C₂×M₁₆×D₁₀₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₀₁
commutator	C₂×C₁₀₁	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₁₀₁
Fitting	C₂×C₂²⋊C₈×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂²×C₄
prime radical	C₂⁴×D₁₀₁	C₄
cosocle	C₂×C₄×C₁₀₁	C₂⁴
hypercenter	C₂×C₂²⋊C₈	D₁₀₁
hypocenter	C₁₀₁	C₂²×C₂²⋊C₈
LCS	G ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×C₈×D₁₀₁	C₂	semidirect
3. C₂×C₂²⋊#11717579	C₂	semidirect
4. C₂×#15692167	C₂	semidirect
5. C₂²⋊C₈×D₁₀₁	C₂	direct (N × H)
6. C₂×C₂²⋊C₈×C₁₀₁	C₂	semidirect
16. C₂²×#11717579	C₂²	semidirect
18. C₂×C₈×D₁₀₁	C₂²	semidirect
19. C₂²⋊#11717579	C₂²	semidirect
20. #15692167	C₂²	semidirect
22. C₂²×C₈×C₁₀₁	C₂²	semidirect
23. C₂²⋊C₈×C₁₀₁	C₂²	semidirect
26. C₂³×D₁₀₁	C₈	semidirect
47. C₂×#11717579	C₂³	semidirect
56. C₂×C₈×C₁₀₁	C₂³	semidirect
59. C₂³×C₁₀₁	C₂×C₈	semidirect
69. C₂²×D₁₀₁	C₂×C₈	semidirect
70. C₂²×D₁₀₁	C₂×C₈	semidirect
99. C₂²×C₁₀₁	C₂²×C₈	semidirect
106. C₂×D₁₀₁	C₂²⋊C₈	direct (N × H)
109. C₂×D₁₀₁	C₂²×C₈	semidirect
117. C₂×C₁₀₁	C₂×C₂²⋊C₈	semidirect
122. D₁₀₁	C₂×C₂²⋊C₈	direct (N × H)
124. C₁₀₁	C₂²×C₂²⋊C₈	semidirect
130. C₂×C₂²⋊C₈	D₁₀₁	direct (N × H)
144. C₂²×C₈	C₂×D₁₀₁	semidirect
147. C₂²⋊C₈	C₂×D₁₀₁	direct (N × H)
180. C₂×C₈	C₂²×D₁₀₁	semidirect
195. C₂³	C₈×D₁₀₁	semidirect
232. C₂²	C₂×C₈×D₁₀₁	semidirect
243. C₂	C₂²⋊C₈×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
C₂²×C₈×D₁₀₁	2
C₂×C₂²⋊#11717579	2
C₂×#15692167	2
C₂²⋊C₈×D₁₀₁	2
C₂×C₂²⋊C₈×C₁₀₁	2
C₂²×C₂²⋊C₈	101

Subgroups — C2×C22⋊C8×D101 #27179562

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂²⋊C₈×D₁₀₁ #27179562