Subgroups — C₄×#15692253 #27173700

Order 12928.

588 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692253	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₂₀₂	C₂	1	1	yes	maximal, characteristic
3	#20265084	C₂	1	1	yes	maximal, characteristic
4	#20265082	C₂	1	1	yes	maximal, characteristic
5	#20265072	C₂	1	4	yes	maximal
6	#20265046	C₂	1	1	yes	maximal, characteristic
7	C₄×#13108053	C₂	1	4	yes	maximal
8	C₄²×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Q₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
11	#15692253	C₄	1	8	yes	direct product factor
12	C₄×Dic₂₀₂	—	2	4	no	—
13	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
14	C₂×#13108053	C₂²	1	4	yes	—
15	C₂×#13108053	C₂²	1	2	yes	—
16	C₂×#13108053	C₂²	1	4	yes	—
17	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
18	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
20	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
22	#15692162	C₂²	1	4	yes	—
23	#15692162	C₂²	1	4	yes	—
24	C₂²×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
25	C₄×Q₈×C₁₀₁	—	2	4	no	—
26	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
28	C₂×C₄²×C₁₀₁	C₂²	1	2	yes	—
29	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
30	C₂×Dic₂₀₂	—	2	2	no	—
31	C₂×Dic₂₀₂	—	2	2	no	—
32	C₂×Dic₂₀₂	C₂×C₄	1	4	yes	—
33	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
34	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
35	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
36	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
37	C₄⋊Dic₁₀₁	—	2	2	no	—
38	#13108053	—	2	4	no	—
39	#13108053	—	2	8	no	—
40	#13108053	C₂×C₄	1	16	yes	—
41	C₄×Dic₁₀₁	—	2	4	no	—
42	C₄×Dic₁₀₁	—	2	2	no	—
43	C₄×Dic₁₀₁	—	2	4	no	—
44	C₄×Dic₁₀₁	C₂×C₄	1	4	yes	—
45	C₄×Dic₁₀₁	Q₈	1	8	yes	complemented
46	C₂×Q₈×C₁₀₁	—	2	2	no	—
47	C₂×Q₈×C₁₀₁	—	2	2	no	—
48	C₂×Q₈×C₁₀₁	C₂×C₄	1	4	yes	—
49	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
50	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
51	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₁₀₁	—	2	4	no	—
55	C₄⋊C₄×C₁₀₁	—	2	2	no	—
56	C₄²×C₁₀₁	—	2	4	no	—
57	C₄²×C₁₀₁	D₄	1	4	yes	—
58	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
59	Q₈×C₁₀₁	—	2	8	no	—
60	C₂×C₄×C₁₀₁	—	2	2	no	—
61	C₂×C₄×C₁₀₁	—	2	2	no	—
62	C₂×C₄×C₁₀₁	—	2	2	no	—
63	C₂×C₄×C₁₀₁	—	2	2	no	—
64	C₂×C₄×C₁₀₁	—	2	4	no	—
65	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
66	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
67	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
68	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
70	C₂×C₄×C₁₀₁	C₂×Q₈	1	4	yes	—
71	C₂×C₄×C₁₀₁	#41	1	2	yes	—
72	C₂×Dic₁₀₁	—	2	2	no	—
73	C₂×Dic₁₀₁	—	2	2	no	—
74	C₂×Dic₁₀₁	—	2	2	no	—
75	C₂×Dic₁₀₁	—	2	2	no	—
76	C₂×Dic₁₀₁	—	2	2	no	—
77	C₂×Dic₁₀₁	—	2	4	no	—
78	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
79	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
80	C₂×Dic₁₀₁	C₂×Q₈	1	4	yes	—
81	C₂×Dic₁₀₁	C₂×Q₈	1	4	yes	—
82	C₂×Dic₁₀₁	#41	1	4	yes	—
83	Dic₂₀₂	—	2	8	no	—
84	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
85	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
86	C₂²×C₁₀₁	C₂²×Q₈	1	2	yes	—
87	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
88	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
89	C₄×C₁₀₁	—	2	4	no	—
90	C₄×C₁₀₁	—	2	2	no	—
91	C₄×C₁₀₁	—	4	2	no	—
92	C₄×C₁₀₁	C₄×D₄	1	4	yes	—
93	C₄×C₁₀₁	C₄⋊Q₈	1	4	yes	complemented
94	Dic₁₀₁	—	2	4	no	—
95	Dic₁₀₁	—	2	4	no	—
96	Dic₁₀₁	—	4	2	no	—
97	Dic₁₀₁	C₄×Q₈	1	8	yes	complemented
98	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
99	C₂×C₁₀₁	C₂×C₄×Q₈	1	2	yes	—
100	C₂×C₁₀₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
101	C₂×C₁₀₁	#532	1	1	yes	characteristic
102	C₂×C₁₀₁	#533	1	2	yes	—
103	C₄×C₄⋊Q₈	—	101	1	no	maximal, Sylow
104	C₁₀₁	C₄×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
105	C₂×C₄⋊Q₈	—	101	1	no	—
106	C₂×C₄×Q₈	—	101	1	no	—
107	C₂×C₄×Q₈	D₁₀₁	1	1	yes	characteristic
108	#391	—	101	1	no	—
109	#391	—	101	1	no	—
110	#389	—	101	4	no	—
111	#384	—	101	1	no	—
112	C₄×C₄⋊C₄	—	101	4	no	—
113	C₄³	—	101	1	no	—
114	C₂²×Q₈	—	101	1	no	—
115	C₂²×Q₈	C₂×D₁₀₁	1	1	yes	characteristic
116	C₄⋊Q₈	—	101	8	no	—
117	C₄×Q₈	—	2	4	no	—
118	C₄×Q₈	—	202	4	no	—
119	C₂×C₄⋊C₄	—	101	4	no	—
120	C₂×C₄⋊C₄	—	101	1	no	—
121	C₂×C₄⋊C₄	—	101	2	no	—
122	C₂×C₄⋊C₄	—	101	4	no	—
123	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
124	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
125	C₂×C₄²	—	101	1	no	—
126	C₂×C₄²	—	101	2	no	—
127	C₂×C₄²	—	101	2	no	—
128	C₂×C₄²	—	101	1	no	—
129	C₂×C₄²	—	101	2	no	—
130	C₂×C₄²	C₂×D₁₀₁	1	2	yes	—
131	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
132	#95	—	101	4	no	—
133	#95	—	101	4	no	—
134	C₂×Q₈	—	2	2	no	—
135	C₂×Q₈	—	2	2	no	—
136	C₂×Q₈	—	101	4	no	—
137	C₂×Q₈	—	202	2	no	—
138	C₂×Q₈	—	202	2	no	—
139	C₂×Q₈	C₄×D₁₀₁	1	4	yes	—
140	C₂²×C₄	—	101	2	no	—
141	C₂²×C₄	—	101	2	no	—
142	C₂²×C₄	—	101	2	no	—
143	C₂²×C₄	—	101	2	no	—
144	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
145	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
146	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
147	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
148	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	—	101	16	no	—
152	C₄⋊C₄	—	202	8	no	—
153	C₄⋊C₄	—	202	4	no	—
154	C₄⋊C₄	—	202	2	no	—
155	C₄²	—	2	4	no	—
156	C₄²	—	101	4	no	—
157	C₄²	—	101	8	no	—
158	C₄²	—	202	4	no	—
159	C₄²	—	202	4	no	—
160	C₄²	—	202	2	no	—
161	C₄²	C₂²⋊D₁₀₁	1	4	yes	—
162	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
163	Q₈	—	2	8	no	—
164	Q₈	—	202	8	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	101	4	no	—
171	C₂×C₄	—	101	4	no	—
172	C₂×C₄	—	101	4	no	—
173	C₂×C₄	—	101	4	no	—
174	C₂×C₄	—	101	4	no	—
175	C₂×C₄	—	202	2	no	—
176	C₂×C₄	—	202	4	no	—
177	C₂×C₄	—	202	2	no	—
178	C₂×C₄	—	202	2	no	—
179	C₂×C₄	—	202	2	no	—
180	C₂×C₄	—	202	2	no	—
181	C₂×C₄	C₂×C₄×D₁₀₁	1	4	yes	—
182	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
183	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
184	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
185	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
186	C₂×C₄	Q₈×D₁₀₁	1	4	yes	—
187	C₂×C₄	#13108074	1	2	yes	—
188	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
189	C₂²	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
190	C₂²	C₂×Q₈×D₁₀₁	1	2	yes	—
191	C₂²	#41×D₁₀₁	1	2	yes	—
192	C₂²	C₂×#13108074	1	1	yes	characteristic
193	C₄	—	2	4	no	—
194	C₄	—	2	2	no	—
195	C₄	—	4	2	no	—
196	C₄	—	101	8	no	—
197	C₄	—	404	2	no	—
198	C₄	—	202	4	no	—
199	C₄	—	202	4	no	—
200	C₄	C₄×C₂²⋊D₁₀₁	1	4	yes	—
201	C₄	#15692253	1	4	yes	direct product factor
202	C₂	C₂×C₄×C₂²⋊D₁₀₁	1	1	yes	characteristic
203	C₂	C₄×Q₈×D₁₀₁	1	2	yes	—
204	C₂	C₂×#15692253	1	1	yes	characteristic
205	C₂	#20265723	1	1	yes	characteristic
206	C₂	#20265731	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Frattini	C₂³	C₂³×D₁₀₁
Fitting	C₂×C₄×Q₈×C₁₀₁	C₂
socle, prime radical, cosocle	C₂³×C₁₀₁	C₂⁴
hypercenter	C₂×C₄×Q₈	D₁₀₁
hypocenter	C₁₀₁	C₄×C₄⋊Q₈
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
11. #15692253	C₄	direct (N × H)
45. C₄×Dic₁₀₁	Q₈	semidirect
93. C₄×C₁₀₁	C₄⋊Q₈	semidirect
97. Dic₁₀₁	C₄×Q₈	semidirect
104. C₁₀₁	C₄×C₄⋊Q₈	semidirect
201. C₄	#15692253	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×#15692253	2
C₂×C₄×Dic₂₀₂	2
#20265084	2
#20265082	2
#20265072	2
#20265046	2
C₄×#13108053	2
C₄²×Dic₁₀₁	2
C₂×C₄×Q₈×C₁₀₁	2
C₄×C₄⋊Q₈	101

Subgroups — C4×#15692253 #27173700

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15692253 #27173700