Subgroups — C₂²⋊C₄×Dic₂₀₂ #27173967

Order 12928.

820 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₂₀₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₂₀₂	C₂	1	2	yes	maximal, complemented
3	#20265083	C₂	1	2	yes	maximal, complemented
4	#20265081	C₂	1	4	yes	maximal, complemented
5	#20265037	C₂	1	1	yes	maximal, characteristic
6	#20265035	C₂	1	2	yes	maximal
7	C₂²⋊C₄×Dic₁₀₁	C₂	1	2	yes	maximal
8	C₄×C₂²⋊C₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₀₂	—	2	8	no	—
10	C₂²×Dic₂₀₂	C₄	1	8	yes	complemented
11	C₂²×Dic₂₀₂	C₂²	1	2	yes	—
12	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
13	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
14	C₂³×Dic₁₀₁	C₂²	1	2	yes	—
15	C₄×Dic₂₀₂	—	2	8	no	—
16	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
17	C₂×#13108053	C₂²	1	4	yes	—
18	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
19	C₂×C₄×Dic₁₀₁	C₂²	1	4	yes	—
20	#15692162	C₂²	1	4	yes	—
21	#15692162	C₂²	1	4	yes	—
22	#15692162	C₂²	1	2	yes	—
23	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
26	C₂×C₄²×C₁₀₁	C₂²	1	2	yes	—
27	#95×C₁₀₁	C₂²	1	2	yes	—
28	C₂×Dic₂₀₂	—	2	4	no	—
29	C₂×Dic₂₀₂	—	2	8	no	—
30	C₂×Dic₂₀₂	—	2	32	no	—
31	C₂×Dic₂₀₂	C₂×C₄	1	4	yes	—
32	C₂×Dic₂₀₂	C₂×C₄	1	8	yes	—
33	C₂×Dic₂₀₂	C₂×C₄	1	4	yes	—
34	C₂×Dic₂₀₂	D₄	1	8	yes	—
35	C₂²×Dic₁₀₁	—	2	8	no	—
36	C₂²×Dic₁₀₁	C₂×C₄	1	8	yes	—
37	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
38	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
39	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
40	C₄⋊Dic₁₀₁	—	2	4	no	—
41	#13108053	—	2	8	no	—
42	C₂²⋊Dic₁₀₁	—	2	4	no	—
43	C₄×Dic₁₀₁	—	2	8	no	—
44	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₁₀₁	—	2	4	no	—
46	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	—
47	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
48	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
50	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
51	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
52	C₂²⋊C₄×C₁₀₁	Q₈	1	4	yes	complemented
53	C₄²×C₁₀₁	—	2	4	no	—
54	C₂³×C₁₀₁	—	2	2	no	—
55	C₂³×C₁₀₁	—	2	2	no	—
56	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
57	C₂³×C₁₀₁	C₂×Q₈	1	2	yes	—
58	C₂³×C₁₀₁	#41	1	2	yes	—
59	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
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	8	no	—
64	C₂×C₄×C₁₀₁	—	2	2	no	—
65	C₂×C₄×C₁₀₁	—	2	2	no	—
66	C₂×C₄×C₁₀₁	—	2	2	no	—
67	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
69	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
70	C₂×C₄×C₁₀₁	C₂×Q₈	1	4	yes	complemented
71	C₂×Dic₁₀₁	—	2	4	no	—
72	C₂×Dic₁₀₁	—	2	16	no	—
73	C₂×Dic₁₀₁	—	2	4	no	—
74	C₂×Dic₁₀₁	—	2	4	no	—
75	C₂×Dic₁₀₁	—	2	4	no	—
76	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
77	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
78	C₂×Dic₁₀₁	C₂×D₄	1	8	yes	—
79	Dic₂₀₂	—	2	8	no	—
80	Dic₂₀₂	—	2	16	no	—
81	Dic₂₀₂	C₂²⋊C₄	1	16	yes	direct product factor
82	C₂²×C₁₀₁	—	2	2	no	—
83	C₂²×C₁₀₁	—	2	2	no	—
84	C₂²×C₁₀₁	—	2	4	no	—
85	C₂²×C₁₀₁	—	2	4	no	—
86	C₂²×C₁₀₁	C₄×Q₈	1	4	yes	complemented
87	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
88	C₂²×C₁₀₁	C₂²×D₄	1	2	yes	—
89	C₂²×C₁₀₁	C₂²×Q₈	1	1	yes	characteristic
90	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
91	C₂²×C₁₀₁	2₋¹⁺⁴	1	2	yes	—
92	C₄×C₁₀₁	—	2	2	no	—
93	C₄×C₁₀₁	—	2	4	no	—
94	C₄×C₁₀₁	—	4	2	no	—
95	C₄×C₁₀₁	C₂×C₂²⋊C₄	1	4	yes	—
96	Dic₁₀₁	—	2	4	no	—
97	Dic₁₀₁	—	4	4	no	—
98	Dic₁₀₁	C₂×C₂²⋊C₄	1	8	yes	—
99	C₂×C₁₀₁	—	2	4	no	—
100	C₂×C₁₀₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₁₀₁	C₂×C₄×Q₈	1	1	yes	characteristic
102	C₂×C₁₀₁	#519	1	1	yes	characteristic
103	C₂×C₁₀₁	#548	1	2	yes	—
104	C₂×C₁₀₁	D₄×Q₈	1	2	yes	—
105	Q₈×C₂²⋊C₄	—	101	1	no	maximal, Sylow
106	C₁₀₁	Q₈×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
107	C₂³×Q₈	—	101	1	no	—
108	C₂×C₄×Q₈	—	101	2	no	—
109	#391	—	101	2	no	—
110	#391	—	101	4	no	—
111	#380	—	101	1	no	—
112	#380	—	101	2	no	—
113	C₄×C₂²⋊C₄	—	101	2	no	—
114	C₄×C₂²⋊C₄	D₁₀₁	1	1	yes	characteristic
115	C₂²×Q₈	—	101	8	no	—
116	C₂²×Q₈	—	101	1	no	—
117	C₂²×Q₈	—	101	2	no	—
118	C₂²×Q₈	—	101	1	no	—
119	C₂²×Q₈	—	202	8	no	—
120	C₂³×C₄	—	101	2	no	—
121	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
122	C₄×Q₈	—	202	8	no	—
123	C₂×C₄⋊C₄	—	101	4	no	—
124	C₂×C₄⋊C₄	—	101	2	no	—
125	C₂×C₂²⋊C₄	—	101	2	no	—
126	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
127	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
128	C₂×C₄²	—	101	4	no	—
129	C₂×C₄²	C₂×D₁₀₁	1	2	yes	—
130	#95	—	101	4	no	—
131	#95	—	101	4	no	—
132	#95	—	101	2	no	—
133	#95	C₂×D₁₀₁	1	2	yes	—
134	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
135	C₂×Q₈	—	101	4	no	—
136	C₂×Q₈	—	101	8	no	—
137	C₂×Q₈	—	101	4	no	—
138	C₂×Q₈	—	101	8	no	—
139	C₂×Q₈	—	202	8	no	—
140	C₂×Q₈	—	202	32	no	—
141	C₂×Q₈	—	202	4	no	—
142	C₂²×C₄	—	2	4	no	—
143	C₂²×C₄	—	101	2	no	—
144	C₂²×C₄	—	101	8	no	—
145	C₂²×C₄	—	101	2	no	—
146	C₂²×C₄	—	101	4	no	—
147	C₂²×C₄	—	202	8	no	—
148	C₂²×C₄	C₄×D₁₀₁	1	4	yes	—
149	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
150	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
151	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
152	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
153	C₄⋊C₄	—	202	8	no	—
154	C₄⋊C₄	—	202	4	no	—
155	C₂²⋊C₄	—	2	2	no	—
156	C₂²⋊C₄	—	202	4	no	—
157	C₂²⋊C₄	Dic₂₀₂	1	4	yes	direct product factor
158	C₄²	—	2	4	no	—
159	C₄²	—	202	8	no	—
160	C₂³	—	2	2	no	—
161	C₂³	—	2	2	no	—
162	C₂³	C₂×C₄×D₁₀₁	1	2	yes	—
163	C₂³	C₂×Dic₂₀₂	1	2	yes	—
164	C₂³	#13108074	1	2	yes	—
165	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
166	Q₈	—	101	16	no	—
167	Q₈	—	202	16	no	—
168	Q₈	—	202	8	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	8	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	101	4	no	—
177	C₂×C₄	—	101	8	no	—
178	C₂×C₄	—	101	4	no	—
179	C₂×C₄	—	202	16	no	—
180	C₂×C₄	—	202	4	no	—
181	C₂×C₄	—	202	4	no	—
182	C₂×C₄	—	202	4	no	—
183	C₂×C₄	—	202	4	no	—
184	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
185	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
186	C₂×C₄	D₄×D₁₀₁	1	4	yes	—
187	C₂×C₄	C₂×Dic₂₀₂	1	4	yes	complemented
188	C₂²	—	2	4	no	—
189	C₂²	—	2	4	no	—
190	C₂²	—	2	2	no	—
191	C₂²	—	2	2	no	—
192	C₂²	C₄×Dic₂₀₂	1	4	yes	complemented
193	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
194	C₂²	C₂×D₄×D₁₀₁	1	2	yes	—
195	C₂²	C₂²×Dic₂₀₂	1	1	yes	characteristic
196	C₂²	C₂×#13108074	1	1	yes	characteristic
197	C₂²	#15692302	1	2	yes	—
198	C₄	—	2	4	no	—
199	C₄	—	2	2	no	—
200	C₄	—	4	2	no	—
201	C₄	—	101	8	no	—
202	C₄	—	404	4	no	—
203	C₄	—	202	4	no	—
204	C₄	C₂²⋊C₄×D₁₀₁	1	4	yes	—
205	C₂	—	2	4	no	—
206	C₂	C₂×C₂²⋊C₄×D₁₀₁	1	1	yes	characteristic
207	C₂	C₂×C₄×Dic₂₀₂	1	1	yes	characteristic
208	C₂	#20265654	1	1	yes	characteristic
209	C₂	#20265836	1	2	yes	—
210	C₂	D₄×Dic₂₀₂	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Dic₂₀₂	C₂	semidirect
3. #20265083	C₂	semidirect
4. #20265081	C₂	semidirect
10. C₂²×Dic₂₀₂	C₄	semidirect
52. C₂²⋊C₄×C₁₀₁	Q₈	semidirect
70. C₂×C₄×C₁₀₁	C₂×Q₈	semidirect
81. Dic₂₀₂	C₂²⋊C₄	direct (N × H)
86. C₂²×C₁₀₁	C₄×Q₈	semidirect
106. C₁₀₁	Q₈×C₂²⋊C₄	semidirect
157. C₂²⋊C₄	Dic₂₀₂	direct (N × H)
187. C₂×C₄	C₂×Dic₂₀₂	semidirect
192. C₂²	C₄×Dic₂₀₂	semidirect

Maximal subgroups

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

subgroup	index
C₂³×Dic₂₀₂	2
C₂×C₄×Dic₂₀₂	2
#20265083	2
#20265081	2
#20265037	2
#20265035	2
C₂²⋊C₄×Dic₁₀₁	2
C₄×C₂²⋊C₄×C₁₀₁	2
Q₈×C₂²⋊C₄	101

Subgroups — C22⋊C4×Dic202 #27173967

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²⋊C₄×Dic₂₀₂ #27173967