Subgroups — C₄×#15692244 #27173678

Order 12928.

652 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692244	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₄₀₄	C₂	1	1	yes	maximal, characteristic
3	#20265084	C₂	1	1	yes	maximal, characteristic, complemented
4	#20265078	C₂	1	1	yes	maximal, characteristic
5	#20265067	C₂	1	4	yes	maximal, complemented
6	#20265042	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#13108051	C₂	1	4	yes	maximal, complemented
8	C₄²×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄×Q₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₄₀₄	C₂²	1	1	yes	characteristic
11	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
12	#15692244	C₄	1	8	yes	direct product factor
13	C₄×D₄₀₄	—	2	4	no	—
14	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
15	C₂×#13108051	C₂²	1	4	yes	—
16	C₂×#13108051	C₂²	1	2	yes	—
17	C₂×#13108051	C₂²	1	4	yes	—
18	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
19	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
20	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
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	C₂³×D₁₀₁	C₂³	1	2	yes	—
31	C₂×D₄₀₄	—	2	2	no	—
32	C₂×D₄₀₄	—	2	2	no	—
33	C₂×D₄₀₄	C₂×C₄	1	4	yes	—
34	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
35	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
36	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
37	C₂×C₄×D₁₀₁	—	2	4	no	—
38	C₂×C₄×D₁₀₁	—	2	8	no	—
39	C₄⋊Dic₁₀₁	—	2	2	no	—
40	#13108051	—	2	4	no	—
41	#13108051	—	2	8	no	—
42	#13108051	C₂×C₄	1	16	yes	complemented
43	C₄×Dic₁₀₁	—	2	2	no	—
44	C₄×Dic₁₀₁	—	2	4	no	—
45	C₄×Dic₁₀₁	—	2	4	no	—
46	C₄×Dic₁₀₁	C₂×C₄	1	4	yes	complemented
47	C₂×Q₈×C₁₀₁	—	2	2	no	—
48	C₂×Q₈×C₁₀₁	—	2	2	no	—
49	C₂×Q₈×C₁₀₁	C₂×C₄	1	4	yes	complemented
50	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
51	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
52	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
55	C₄⋊C₄×C₁₀₁	—	2	4	no	—
56	C₄⋊C₄×C₁₀₁	—	2	2	no	—
57	C₄²×C₁₀₁	—	2	4	no	—
58	C₄²×C₁₀₁	D₄	1	4	yes	—
59	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
60	C₂²×D₁₀₁	—	2	2	no	—
61	C₂²×D₁₀₁	—	2	4	no	—
62	C₂²×D₁₀₁	—	2	4	no	—
63	C₂²×D₁₀₁	—	2	2	no	—
64	C₂²×D₁₀₁	C₂²×C₄	1	4	yes	—
65	Q₈×C₁₀₁	—	2	8	no	—
66	C₂×C₄×C₁₀₁	—	2	2	no	—
67	C₂×C₄×C₁₀₁	—	2	2	no	—
68	C₂×C₄×C₁₀₁	—	2	2	no	—
69	C₂×C₄×C₁₀₁	—	2	2	no	—
70	C₂×C₄×C₁₀₁	—	2	4	no	—
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×D₄	1	2	yes	—
75	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
76	C₂×C₄×C₁₀₁	#41	1	2	yes	—
77	C₂×C₄×C₁₀₁	#41	1	4	yes	—
78	C₂×Dic₁₀₁	—	2	2	no	—
79	C₂×Dic₁₀₁	—	2	2	no	—
80	C₂×Dic₁₀₁	—	2	2	no	—
81	C₂×Dic₁₀₁	—	2	2	no	—
82	C₂×Dic₁₀₁	—	2	2	no	—
83	C₂×Dic₁₀₁	—	2	4	no	—
84	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
85	C₂×Dic₁₀₁	#41	1	4	yes	—
86	D₄₀₄	—	2	8	no	—
87	C₄×D₁₀₁	—	4	8	no	—
88	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
89	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
90	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
91	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
92	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
93	C₂×D₁₀₁	—	2	4	no	—
94	C₂×D₁₀₁	—	2	8	no	—
95	C₂×D₁₀₁	—	4	2	no	—
96	C₂×D₁₀₁	—	4	4	no	—
97	C₂×D₁₀₁	—	4	2	no	—
98	C₄×C₁₀₁	—	2	4	no	—
99	C₄×C₁₀₁	—	2	2	no	—
100	C₄×C₁₀₁	—	4	2	no	—
101	C₄×C₁₀₁	C₄×D₄	1	4	yes	—
102	C₄×C₁₀₁	#124	1	4	yes	complemented
103	Dic₁₀₁	—	2	4	no	—
104	Dic₁₀₁	—	2	4	no	—
105	Dic₁₀₁	—	4	2	no	—
106	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
107	C₂×C₁₀₁	C₄×#41	1	2	yes	—
108	C₂×C₁₀₁	C₂×#124	1	1	yes	characteristic
109	C₂×C₁₀₁	#529	1	2	yes	—
110	C₂×C₁₀₁	#532	1	1	yes	characteristic
111	D₁₀₁	—	4	4	no	—
112	C₄×#124	—	101	1	no	maximal, Sylow
113	C₁₀₁	C₄×#124	1	1	yes	characteristic, complemented, Sylow
114	C₂×#124	—	101	1	no	—
115	C₂×C₄×Q₈	D₁₀₁	1	1	yes	characteristic, complemented
116	C₂×C₄×D₄	—	101	1	no	—
117	#391	—	101	1	no	—
118	#390	—	101	1	no	—
119	#388	—	101	4	no	—
120	#382	—	101	1	no	—
121	C₄×C₂²⋊C₄	—	101	4	no	—
122	C₄³	—	101	1	no	—
123	C₂²×Q₈	C₂×D₁₀₁	1	1	yes	characteristic
124	C₂²×D₄	—	101	1	no	—
125	C₂³×C₄	—	101	2	no	—
126	#124	—	101	8	no	—
127	C₄×Q₈	—	2	4	no	—
128	C₄×D₄	—	202	4	no	—
129	C₂×C₄⋊C₄	—	101	1	no	—
130	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
131	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
132	C₂×C₂²⋊C₄	—	101	4	no	—
133	C₂×C₂²⋊C₄	—	101	2	no	—
134	C₂×C₂²⋊C₄	—	101	4	no	—
135	C₂×C₄²	—	101	1	no	—
136	C₂×C₄²	—	101	2	no	—
137	C₂×C₄²	—	101	1	no	—
138	C₂×C₄²	—	101	2	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	4	no	—
143	C₂⁴	—	101	2	no	—
144	C₂×Q₈	—	2	2	no	—
145	C₂×Q₈	—	2	2	no	—
146	C₂×Q₈	C₄×D₁₀₁	1	4	yes	complemented
147	C₂×D₄	—	101	4	no	—
148	C₂×D₄	—	202	2	no	—
149	C₂×D₄	—	202	2	no	—
150	C₂²×C₄	—	101	2	no	—
151	C₂²×C₄	—	101	2	no	—
152	C₂²×C₄	—	101	2	no	—
153	C₂²×C₄	—	202	4	no	—
154	C₂²×C₄	—	202	8	no	—
155	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
156	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
157	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
158	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
159	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	—	202	2	no	—
163	C₂²⋊C₄	—	101	16	no	—
164	C₂²⋊C₄	—	202	8	no	—
165	C₂²⋊C₄	—	202	4	no	—
166	C₄²	—	2	4	no	—
167	C₄²	—	101	4	no	—
168	C₄²	—	202	4	no	—
169	C₄²	—	202	2	no	—
170	C₄²	—	202	4	no	—
171	C₄²	C₂²⋊D₁₀₁	1	4	yes	—
172	C₂³	—	101	4	no	—
173	C₂³	—	202	2	no	—
174	C₂³	—	202	4	no	—
175	C₂³	—	202	2	no	—
176	C₂³	—	202	4	no	—
177	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
178	Q₈	—	2	8	no	—
179	D₄	—	202	8	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	101	4	no	—
186	C₂×C₄	—	101	4	no	—
187	C₂×C₄	—	404	8	no	—
188	C₂×C₄	—	202	2	no	—
189	C₂×C₄	—	202	4	no	—
190	C₂×C₄	—	202	2	no	—
191	C₂×C₄	—	202	2	no	—
192	C₂×C₄	—	202	2	no	—
193	C₂×C₄	—	202	2	no	—
194	C₂×C₄	C₂×C₄×D₁₀₁	1	4	yes	—
195	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
196	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
197	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
198	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
199	C₂×C₄	#13108074	1	2	yes	—
200	C₂×C₄	#13108075	1	4	yes	—
201	C₂²	—	404	2	no	—
202	C₂²	—	404	4	no	—
203	C₂²	—	404	2	no	—
204	C₂²	—	202	4	no	—
205	C₂²	—	202	8	no	—
206	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
207	C₂²	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
208	C₂²	#41×D₁₀₁	1	2	yes	—
209	C₂²	C₂×#13108074	1	1	yes	characteristic
210	C₂²	C₂×#13108075	1	2	yes	—
211	C₄	—	2	4	no	—
212	C₄	—	2	2	no	—
213	C₄	—	4	2	no	—
214	C₄	—	404	2	no	—
215	C₄	—	202	4	no	—
216	C₄	—	202	4	no	—
217	C₄	C₄×C₂²⋊D₁₀₁	1	4	yes	—
218	C₄	#15692244	1	4	yes	direct product factor
219	C₂	—	404	4	no	—
220	C₂	C₂×C₄×C₂²⋊D₁₀₁	1	1	yes	characteristic
221	C₂	C₄×#13108075	1	2	yes	—
222	C₂	C₂×#15692244	1	1	yes	characteristic
223	C₂	#20265714	1	2	yes	—
224	C₂	#20265723	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Frattini	C₂³	C₂³×D₁₀₁
Fitting	C₂×C₄×Q₈×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂²×D₄₀₄	C₂²
hypercenter	C₂×C₄×Q₈	D₁₀₁
hypocenter	C₁₀₁	C₄×#124
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
3. #20265084	C₂	semidirect
5. #20265067	C₂	semidirect
6. #20265042	C₂	semidirect
7. C₄×#13108051	C₂	semidirect
8. C₄²×Dic₁₀₁	C₂	semidirect
9. C₂×C₄×Q₈×C₁₀₁	C₂	semidirect
12. #15692244	C₄	direct (N × H)
42. #13108051	C₂×C₄	semidirect
46. C₄×Dic₁₀₁	C₂×C₄	semidirect
49. C₂×Q₈×C₁₀₁	C₂×C₄	semidirect
102. C₄×C₁₀₁	#124	semidirect
113. C₁₀₁	C₄×#124	semidirect
115. C₂×C₄×Q₈	D₁₀₁	semidirect
146. C₂×Q₈	C₄×D₁₀₁	semidirect
218. C₄	#15692244	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×#15692244	2
C₂×C₄×D₄₀₄	2
#20265084	2
#20265078	2
#20265067	2
#20265042	2
C₄×#13108051	2
C₄²×Dic₁₀₁	2
C₂×C₄×Q₈×C₁₀₁	2
C₄×#124	101

Subgroups — C4×#15692244 #27173678

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15692244 #27173678