Subgroups — C₄×#15692213 #27173472

Order 12928.

752 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692213	C₂	1	2	yes	maximal
2	C₂×C₄²×D₁₀₁	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×Dic₁₀₁	C₂	1	2	yes	maximal, complemented
4	C₄×C₄⋊Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#13108051	C₂	1	2	yes	maximal
6	#20265024	C₂	1	2	yes	maximal
7	#20265022	C₂	1	2	yes	maximal, complemented
8	C₄²×Dic₁₀₁	C₂	1	2	yes	maximal, complemented
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	1	yes	characteristic
12	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
13	#15692213	C₄	1	16	yes	direct product factor
14	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
15	C₂×#13108051	C₂²	1	4	yes	—
16	C₂×C₄×Dic₁₀₁	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	1	yes	characteristic
20	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₁₀₁	C₂²	1	4	yes	—
22	C₄²×D₁₀₁	C₄	1	8	yes	complemented
23	#15692162	C₂²	1	2	yes	—
24	#15692162	C₂²	1	4	yes	—
25	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
26	C₂×C₄²×C₁₀₁	C₂²	1	2	yes	—
27	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
28	#95×C₁₀₁	C₂²	1	2	yes	—
29	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
30	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
31	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
32	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
33	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
34	C₂×C₄×D₁₀₁	—	2	2	no	—
35	C₂×C₄×D₁₀₁	—	2	2	no	—
36	C₂×C₄×D₁₀₁	—	2	4	no	—
37	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
38	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
39	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
40	C₂×C₄×D₁₀₁	C₂×C₄	1	8	yes	—
41	C₄⋊Dic₁₀₁	C₂×C₄	1	8	yes	complemented
42	#13108051	C₂×C₄	1	16	yes	—
43	C₄×Dic₁₀₁	—	2	4	no	—
44	C₄×Dic₁₀₁	—	2	8	no	—
45	C₄×Dic₁₀₁	C₂×C₄	1	4	yes	—
46	C₄×Dic₁₀₁	C₂×C₄	1	4	yes	—
47	C₄×Dic₁₀₁	C₂×C₄	1	4	yes	—
48	C₄×Dic₁₀₁	C₂×C₄	1	16	yes	complemented
49	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
51	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
53	C₄⋊C₄×C₁₀₁	C₂×C₄	1	8	yes	complemented
54	C₄²×C₁₀₁	—	2	4	no	—
55	C₄²×C₁₀₁	C₂×C₄	1	4	yes	—
56	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
57	C₂²×D₁₀₁	—	2	2	no	—
58	C₂²×D₁₀₁	—	2	1	no	—
59	C₂²×D₁₀₁	—	2	1	no	—
60	C₂²×D₁₀₁	C₂²×C₄	1	4	yes	—
61	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₁₀₁	—	2	4	no	—
63	C₂×C₄×C₁₀₁	—	2	4	no	—
64	C₂×C₄×C₁₀₁	—	2	2	no	—
65	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₁₀₁	C₂²×C₄	1	8	yes	—
67	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
69	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
70	C₂×C₄×C₁₀₁	#41	1	2	yes	—
71	C₂×C₄×C₁₀₁	#41	1	2	yes	—
72	C₂×Dic₁₀₁	—	2	4	no	—
73	C₂×Dic₁₀₁	—	2	4	no	—
74	C₂×Dic₁₀₁	—	2	1	no	—
75	C₂×Dic₁₀₁	—	2	1	no	—
76	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
77	C₂×Dic₁₀₁	C₂²×C₄	1	8	yes	—
78	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
79	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
80	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
81	C₂×Dic₁₀₁	#41	1	4	yes	—
82	C₄×D₁₀₁	—	2	8	no	—
83	C₄×D₁₀₁	—	2	8	no	—
84	C₄×D₁₀₁	C₄²	1	16	yes	complemented
85	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
86	C₂²×C₁₀₁	C₂³×C₄	1	2	yes	—
87	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
88	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
89	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
90	C₂×D₁₀₁	—	2	2	no	—
91	C₂×D₁₀₁	—	2	2	no	—
92	C₂×D₁₀₁	—	2	4	no	—
93	C₂×D₁₀₁	—	2	4	no	—
94	C₂×D₁₀₁	C₂×C₄²	1	4	yes	—
95	C₄×C₁₀₁	—	2	2	no	—
96	C₄×C₁₀₁	—	2	8	no	—
97	C₄×C₁₀₁	C₂×C₄²	1	4	yes	—
98	C₄×C₁₀₁	#117	1	4	yes	complemented
99	Dic₁₀₁	—	2	2	no	—
100	Dic₁₀₁	—	2	8	no	—
101	Dic₁₀₁	C₂×C₄²	1	4	yes	—
102	Dic₁₀₁	#117	1	4	yes	complemented
103	C₂×C₁₀₁	C₂²×C₄²	1	1	yes	characteristic
104	C₂×C₁₀₁	C₂×#117	1	1	yes	characteristic
105	C₂×C₁₀₁	C₂×#117	1	1	yes	characteristic
106	C₂×C₁₀₁	C₄×#41	1	2	yes	—
107	C₂×C₁₀₁	C₄×#41	1	2	yes	—
108	D₁₀₁	—	2	4	no	—
109	C₄×#117	—	101	1	no	maximal, Sylow
110	C₁₀₁	C₄×#117	1	1	yes	characteristic, complemented, Sylow
111	C₂×#117	—	101	2	no	—
112	C₂²×C₄²	—	101	1	no	—
113	C₄×C₄⋊C₄	—	101	2	no	—
114	C₄×C₄⋊C₄	—	101	1	no	—
115	C₄×C₄⋊C₄	D₁₀₁	1	1	yes	characteristic, complemented
116	C₄×C₂²⋊C₄	—	101	2	no	—
117	C₄×C₂²⋊C₄	—	101	2	no	—
118	#376	—	101	2	no	—
119	C₄³	—	101	2	no	—
120	C₂³×C₄	—	101	1	no	—
121	C₂³×C₄	—	101	1	no	—
122	C₂³×C₄	—	101	1	no	—
123	#117	—	101	16	no	—
124	C₂×C₄⋊C₄	—	101	2	no	—
125	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
126	C₂×C₂²⋊C₄	—	101	4	no	—
127	C₂×C₄²	—	101	4	no	—
128	C₂×C₄²	—	101	1	no	—
129	C₂×C₄²	—	101	1	no	—
130	C₂×C₄²	—	101	1	no	—
131	C₂×C₄²	—	101	8	no	—
132	C₂×C₄²	—	101	4	no	—
133	C₂×C₄²	—	101	2	no	—
134	C₂×C₄²	C₂×D₁₀₁	1	2	yes	—
135	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
136	#95	—	101	4	no	—
137	#95	—	101	2	no	—
138	#95	C₂×D₁₀₁	1	2	yes	—
139	C₂⁴	—	101	1	no	—
140	C₂²×C₄	—	101	1	no	—
141	C₂²×C₄	—	101	1	no	—
142	C₂²×C₄	—	101	4	no	—
143	C₂²×C₄	—	101	8	no	—
144	C₂²×C₄	—	101	1	no	—
145	C₂²×C₄	—	101	4	no	—
146	C₂²×C₄	—	101	4	no	—
147	C₂²×C₄	—	101	4	no	—
148	C₂²×C₄	—	202	4	no	—
149	C₂²×C₄	—	202	2	no	—
150	C₂²×C₄	—	202	2	no	—
151	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
152	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
153	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
154	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
155	C₄⋊C₄	—	101	8	no	—
156	C₄⋊C₄	C₄×D₁₀₁	1	8	yes	complemented
157	C₂²⋊C₄	—	101	16	no	—
158	C₄²	—	2	4	no	—
159	C₄²	—	101	4	no	—
160	C₄²	—	101	4	no	—
161	C₄²	—	101	4	no	—
162	C₄²	—	101	16	no	—
163	C₄²	—	202	8	no	—
164	C₄²	—	202	4	no	—
165	C₄²	C₄×D₁₀₁	1	4	yes	—
166	C₂³	—	101	4	no	—
167	C₂³	—	101	2	no	—
168	C₂³	—	202	2	no	—
169	C₂³	—	202	1	no	—
170	C₂³	—	202	1	no	—
171	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	101	2	no	—
176	C₂×C₄	—	101	4	no	—
177	C₂×C₄	—	101	2	no	—
178	C₂×C₄	—	101	4	no	—
179	C₂×C₄	—	101	16	no	—
180	C₂×C₄	—	101	2	no	—
181	C₂×C₄	—	101	8	no	—
182	C₂×C₄	—	202	8	no	—
183	C₂×C₄	—	202	4	no	—
184	C₂×C₄	—	202	1	no	—
185	C₂×C₄	—	202	4	no	—
186	C₂×C₄	—	202	1	no	—
187	C₂×C₄	—	202	8	no	—
188	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
189	C₂×C₄	C₂×C₄×D₁₀₁	1	8	yes	—
190	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
191	C₂×C₄	C₂×C₄×D₁₀₁	1	4	yes	—
192	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
193	C₂×C₄	#13108073	1	2	yes	—
194	C₂×C₄	#13108075	1	2	yes	—
195	C₂²	—	101	4	no	—
196	C₂²	—	202	2	no	—
197	C₂²	—	202	2	no	—
198	C₂²	—	202	4	no	—
199	C₂²	—	202	4	no	—
200	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
201	C₂²	C₂²×C₄×D₁₀₁	1	2	yes	—
202	C₂²	#41×D₁₀₁	1	2	yes	—
203	C₂²	C₂×#13108073	1	1	yes	characteristic
204	C₂²	C₂×#13108075	1	1	yes	characteristic
205	C₄	—	2	2	no	—
206	C₄	—	2	8	no	—
207	C₄	—	101	4	no	—
208	C₄	—	101	4	no	—
209	C₄	—	202	2	no	—
210	C₄	—	202	8	no	—
211	C₄	C₄²×D₁₀₁	1	4	yes	—
212	C₄	#15692213	1	4	yes	direct product factor
213	C₂	—	202	4	no	—
214	C₂	C₂×C₄²×D₁₀₁	1	1	yes	characteristic
215	C₂	#117×D₁₀₁	1	1	yes	characteristic
216	C₂	C₂×#15692213	1	1	yes	characteristic
217	C₂	C₄×#13108073	1	2	yes	—
218	C₂	C₄×#13108075	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	C₄×#117
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
3. C₄⋊C₄×Dic₁₀₁	C₂	semidirect
4. C₄×C₄⋊Dic₁₀₁	C₂	semidirect
7. #20265022	C₂	semidirect
8. C₄²×Dic₁₀₁	C₂	semidirect
9. C₄×C₄⋊C₄×C₁₀₁	C₂	semidirect
13. #15692213	C₄	direct (N × H)
22. C₄²×D₁₀₁	C₄	semidirect
41. C₄⋊Dic₁₀₁	C₂×C₄	semidirect
48. C₄×Dic₁₀₁	C₂×C₄	semidirect
53. C₄⋊C₄×C₁₀₁	C₂×C₄	semidirect
84. C₄×D₁₀₁	C₄²	semidirect
98. C₄×C₁₀₁	#117	semidirect
102. Dic₁₀₁	#117	semidirect
110. C₁₀₁	C₄×#117	semidirect
115. C₄×C₄⋊C₄	D₁₀₁	semidirect
156. C₄⋊C₄	C₄×D₁₀₁	semidirect
212. C₄	#15692213	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×#15692213	2
C₂×C₄²×D₁₀₁	2
C₄⋊C₄×Dic₁₀₁	2
C₄×C₄⋊Dic₁₀₁	2
C₄×#13108051	2
#20265024	2
#20265022	2
C₄²×Dic₁₀₁	2
C₄×C₄⋊C₄×C₁₀₁	2
C₄×#117	101

Subgroups — C4×#15692213 #27173472

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15692213 #27173472