Subgroups — C₄×C₄⋊C₄×D₁₀₁ #27173465

Order 12928.

928 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₄×D₁₀₁	C₂	1	2	yes	maximal
2	C₂×C₄²×D₁₀₁	C₂	1	2	yes	maximal
3	C₂×C₄²×D₁₀₁	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₄×Dic₁₀₁	C₂	1	2	yes	maximal, complemented
5	C₄×#13108053	C₂	1	2	yes	maximal, complemented
6	C₄×C₄⋊Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#20265028	C₂	1	2	yes	maximal, complemented
8	#95×D₁₀₁	C₂	1	2	yes	maximal
9	C₄×C₄⋊C₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
11	C₂²×C₄×D₁₀₁	C₂²	1	4	yes	—
12	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
15	C₂×#13108053	C₂²	1	4	yes	—
16	C₄⋊C₄×D₁₀₁	C₄	1	16	yes	direct product factor
17	C₂×C₄×Dic₁₀₁	C₂²	1	4	yes	—
18	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
19	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
22	C₄²×D₁₀₁	—	2	8	no	—
23	C₄²×D₁₀₁	C₄	1	8	yes	complemented
24	#15692162	C₂²	1	2	yes	—
25	#15692162	C₂²	1	4	yes	—
26	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
27	C₂×C₄²×C₁₀₁	C₂²	1	2	yes	—
28	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
29	#95×C₁₀₁	C₂²	1	2	yes	—
30	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
31	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
32	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
33	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
34	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
35	C₂×C₄×D₁₀₁	—	2	8	no	—
36	C₂×C₄×D₁₀₁	—	2	4	no	—
37	C₂×C₄×D₁₀₁	—	2	8	no	—
38	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
39	C₂×C₄×D₁₀₁	C₂×C₄	1	16	yes	—
40	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
41	C₂×C₄×D₁₀₁	C₂×C₄	1	8	yes	—
42	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
43	C₂×C₄×D₁₀₁	D₄	1	4	yes	—
44	C₂×C₄×D₁₀₁	Q₈	1	4	yes	—
45	C₄⋊Dic₁₀₁	C₂×C₄	1	8	yes	complemented
46	#13108053	C₂×C₄	1	16	yes	complemented
47	C₄×Dic₁₀₁	—	2	4	no	—
48	C₄×Dic₁₀₁	—	2	8	no	—
49	C₄×Dic₁₀₁	C₂×C₄	1	4	yes	complemented
50	C₄×Dic₁₀₁	C₂×C₄	1	4	yes	complemented
51	C₄×Dic₁₀₁	C₂×C₄	1	4	yes	complemented
52	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
54	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
56	C₄⋊C₄×C₁₀₁	C₂×C₄	1	8	yes	complemented
57	C₄²×C₁₀₁	—	2	4	no	—
58	C₄²×C₁₀₁	C₂×C₄	1	4	yes	complemented
59	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
60	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
61	C₂²×D₁₀₁	C₂²×C₄	1	4	yes	—
62	C₂²×D₁₀₁	C₂×D₄	1	2	yes	—
63	C₂²×D₁₀₁	C₂×Q₈	1	2	yes	—
64	C₂²×D₁₀₁	#41	1	4	yes	—
65	C₂×C₄×C₁₀₁	—	2	4	no	—
66	C₂×C₄×C₁₀₁	—	2	2	no	—
67	C₂×C₄×C₁₀₁	—	2	4	no	—
68	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
69	C₂×C₄×C₁₀₁	C₂²×C₄	1	8	yes	—
70	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
71	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
72	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
73	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
74	C₂×C₄×C₁₀₁	C₂×Q₈	1	2	yes	—
75	C₂×Dic₁₀₁	—	2	4	no	—
76	C₂×Dic₁₀₁	—	2	2	no	—
77	C₂×Dic₁₀₁	—	2	4	no	—
78	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
79	C₂×Dic₁₀₁	C₂²×C₄	1	8	yes	—
80	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
81	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
82	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
83	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
84	C₂×Dic₁₀₁	C₂×Q₈	1	2	yes	—
85	C₄×D₁₀₁	—	2	8	no	—
86	C₄×D₁₀₁	—	2	32	no	—
87	C₄×D₁₀₁	C₄²	1	16	yes	complemented
88	C₄×D₁₀₁	C₄⋊C₄	1	16	yes	direct product factor
89	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
90	C₂²×C₁₀₁	C₂³×C₄	1	2	yes	—
91	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
92	C₂²×C₁₀₁	C₂²×Q₈	1	1	yes	characteristic
93	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
94	C₂×D₁₀₁	C₂×C₄²	1	4	yes	—
95	C₂×D₁₀₁	C₂×C₄⋊C₄	1	4	yes	—
96	C₂×D₁₀₁	#117	1	4	yes	—
97	C₂×D₁₀₁	C₄×D₄	1	8	yes	—
98	C₂×D₁₀₁	C₄×Q₈	1	8	yes	—
99	C₄×C₁₀₁	—	2	2	no	—
100	C₄×C₁₀₁	—	2	8	no	—
101	C₄×C₁₀₁	C₂×C₄²	1	4	yes	complemented
102	C₄×C₁₀₁	C₂×C₄⋊C₄	1	4	yes	complemented
103	Dic₁₀₁	—	2	2	no	—
104	Dic₁₀₁	—	2	8	no	—
105	Dic₁₀₁	C₂×C₄²	1	4	yes	complemented
106	Dic₁₀₁	C₂×C₄⋊C₄	1	4	yes	complemented
107	C₂×C₁₀₁	C₂²×C₄²	1	1	yes	characteristic
108	C₂×C₁₀₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₁₀₁	C₂×#117	1	1	yes	characteristic
110	C₂×C₁₀₁	C₂×C₄×D₄	1	2	yes	—
111	C₂×C₁₀₁	C₂×C₄×Q₈	1	2	yes	—
112	D₁₀₁	C₄×C₄⋊C₄	1	8	yes	direct product factor
113	C₂×C₄×C₄⋊C₄	—	101	1	no	maximal, Sylow
114	C₁₀₁	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
115	C₂²×C₄⋊C₄	—	101	2	no	—
116	C₂²×C₄²	—	101	1	no	—
117	C₂²×C₄²	—	101	2	no	—
118	C₄×C₄⋊C₄	—	101	2	no	—
119	C₄×C₄⋊C₄	—	101	1	no	—
120	C₄×C₄⋊C₄	—	101	2	no	—
121	C₄×C₄⋊C₄	—	101	2	no	—
122	C₄×C₄⋊C₄	D₁₀₁	1	1	yes	characteristic, direct product factor
123	C₂×#95	—	101	2	no	—
124	C₂³×C₄	—	101	1	no	—
125	C₂³×C₄	—	101	1	no	—
126	C₂³×C₄	—	101	4	no	—
127	C₂³×C₄	—	101	1	no	—
128	C₂×C₄⋊C₄	—	101	4	no	—
129	C₂×C₄⋊C₄	—	101	2	no	—
130	C₂×C₄⋊C₄	—	101	16	no	—
131	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
132	C₂×C₄²	—	101	1	no	—
133	C₂×C₄²	—	101	1	no	—
134	C₂×C₄²	—	101	1	no	—
135	C₂×C₄²	—	101	8	no	—
136	C₂×C₄²	—	101	4	no	—
137	C₂×C₄²	—	101	2	no	—
138	C₂×C₄²	—	202	8	no	—
139	C₂×C₄²	C₂×D₁₀₁	1	2	yes	—
140	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
141	#95	—	101	4	no	—
142	#95	—	101	2	no	—
143	#95	C₂×D₁₀₁	1	2	yes	—
144	C₂⁴	—	101	1	no	—
145	C₂²×C₄	—	101	1	no	—
146	C₂²×C₄	—	101	1	no	—
147	C₂²×C₄	—	101	4	no	—
148	C₂²×C₄	—	101	4	no	—
149	C₂²×C₄	—	101	8	no	—
150	C₂²×C₄	—	101	4	no	—
151	C₂²×C₄	—	101	4	no	—
152	C₂²×C₄	—	101	1	no	—
153	C₂²×C₄	—	101	4	no	—
154	C₂²×C₄	—	101	4	no	—
155	C₂²×C₄	—	101	16	no	—
156	C₂²×C₄	—	202	8	no	—
157	C₂²×C₄	—	202	8	no	—
158	C₂²×C₄	—	202	4	no	—
159	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
160	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
161	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
162	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
163	C₄⋊C₄	—	101	16	no	—
164	C₄⋊C₄	—	101	8	no	—
165	C₄⋊C₄	C₄×D₁₀₁	1	8	yes	direct product factor
166	C₄²	—	2	4	no	—
167	C₄²	—	101	4	no	—
168	C₄²	—	101	4	no	—
169	C₄²	—	101	4	no	—
170	C₄²	—	202	8	no	—
171	C₄²	—	202	4	no	—
172	C₄²	C₄×D₁₀₁	1	4	yes	complemented
173	C₂³	—	101	2	no	—
174	C₂³	—	101	4	no	—
175	C₂³	—	101	2	no	—
176	C₂³	—	101	2	no	—
177	C₂³	—	101	4	no	—
178	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	101	2	no	—
183	C₂×C₄	—	101	2	no	—
184	C₂×C₄	—	101	4	no	—
185	C₂×C₄	—	101	2	no	—
186	C₂×C₄	—	101	2	no	—
187	C₂×C₄	—	101	16	no	—
188	C₂×C₄	—	101	16	no	—
189	C₂×C₄	—	101	2	no	—
190	C₂×C₄	—	101	8	no	—
191	C₂×C₄	—	202	8	no	—
192	C₂×C₄	—	202	32	no	—
193	C₂×C₄	—	202	4	no	—
194	C₂×C₄	—	202	2	no	—
195	C₂×C₄	—	202	4	no	—
196	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
197	C₂×C₄	C₂×C₄×D₁₀₁	1	8	yes	—
198	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
199	C₂×C₄	C₂×C₄×D₁₀₁	1	4	yes	—
200	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
201	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
202	C₂×C₄	Q₈×D₁₀₁	1	2	yes	—
203	C₂²	—	101	4	no	—
204	C₂²	—	101	4	no	—
205	C₂²	—	101	4	no	—
206	C₂²	—	101	8	no	—
207	C₂²	—	101	8	no	—
208	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
209	C₂²	C₂²×C₄×D₁₀₁	1	2	yes	—
210	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
211	C₂²	C₂×Q₈×D₁₀₁	1	1	yes	characteristic
212	C₂²	#41×D₁₀₁	1	2	yes	—
213	C₄	—	2	2	no	—
214	C₄	—	2	8	no	—
215	C₄	—	101	4	no	—
216	C₄	—	101	4	no	—
217	C₄	—	202	2	no	—
218	C₄	—	202	8	no	—
219	C₄	C₄²×D₁₀₁	1	4	yes	complemented
220	C₄	C₄⋊C₄×D₁₀₁	1	4	yes	direct product factor
221	C₂	—	101	8	no	—
222	C₂	C₂×C₄²×D₁₀₁	1	1	yes	characteristic
223	C₂	C₂×C₄⋊C₄×D₁₀₁	1	1	yes	characteristic
224	C₂	#117×D₁₀₁	1	1	yes	characteristic
225	C₂	C₄×D₄×D₁₀₁	1	2	yes	—
226	C₂	C₄×Q₈×D₁₀₁	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₄⋊C₄×Dic₁₀₁	C₂	semidirect
5. C₄×#13108053	C₂	semidirect
6. C₄×C₄⋊Dic₁₀₁	C₂	semidirect
7. #20265028	C₂	semidirect
9. C₄×C₄⋊C₄×C₁₀₁	C₂	semidirect
16. C₄⋊C₄×D₁₀₁	C₄	direct (N × H)
23. C₄²×D₁₀₁	C₄	semidirect
45. C₄⋊Dic₁₀₁	C₂×C₄	semidirect
46. #13108053	C₂×C₄	semidirect
49. C₄×Dic₁₀₁	C₂×C₄	semidirect
50. C₄×Dic₁₀₁	C₂×C₄	semidirect
51. C₄×Dic₁₀₁	C₂×C₄	semidirect
56. C₄⋊C₄×C₁₀₁	C₂×C₄	semidirect
58. C₄²×C₁₀₁	C₂×C₄	semidirect
87. C₄×D₁₀₁	C₄²	semidirect
88. C₄×D₁₀₁	C₄⋊C₄	direct (N × H)
101. C₄×C₁₀₁	C₂×C₄²	semidirect
102. C₄×C₁₀₁	C₂×C₄⋊C₄	semidirect
105. Dic₁₀₁	C₂×C₄²	semidirect
106. Dic₁₀₁	C₂×C₄⋊C₄	semidirect
112. D₁₀₁	C₄×C₄⋊C₄	direct (N × H)
114. C₁₀₁	C₂×C₄×C₄⋊C₄	semidirect
122. C₄×C₄⋊C₄	D₁₀₁	direct (N × H)
165. C₄⋊C₄	C₄×D₁₀₁	direct (N × H)
172. C₄²	C₄×D₁₀₁	semidirect
219. C₄	C₄²×D₁₀₁	semidirect
220. C₄	C₄⋊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
C₂×C₄⋊C₄×D₁₀₁	2
C₂×C₄²×D₁₀₁	2
C₄⋊C₄×Dic₁₀₁	2
C₄×#13108053	2
C₄×C₄⋊Dic₁₀₁	2
#20265028	2
#95×D₁₀₁	2
C₄×C₄⋊C₄×C₁₀₁	2
C₂×C₄×C₄⋊C₄	101

Subgroups — C4×C4⋊C4×D101 #27173465

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₄⋊C₄×D₁₀₁ #27173465