Subgroups — C₄×#15692278 #27179904

Order 12928.

500 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692278	C₂	1	1	yes	maximal, characteristic
2	C₄×#13108075	C₂	1	2	yes	maximal, complemented
3	Q₁₆×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#13108061	C₂	1	2	yes	maximal, complemented
5	#20265262	C₂	1	2	yes	maximal, complemented
6	C₄×D₈₀₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#20265259	C₂	1	1	yes	maximal, characteristic, complemented
8	#20265214	C₂	1	1	yes	maximal, characteristic, complemented
9	#20265172	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×D₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Q₁₆×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13108075	C₂²	1	2	yes	—
13	#15692278	C₄	1	8	yes	direct product factor
14	C₂×#13108061	C₂²	1	2	yes	—
15	C₂×D₈₀₈	C₂²	1	1	yes	characteristic
16	C₂×C₈×D₁₀₁	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₁₀₁	C₂²	1	1	yes	characteristic
18	Q₈×Dic₁₀₁	C₂²	1	2	yes	—
19	C₄×D₄₀₄	—	2	2	no	—
20	C₄×D₄₀₄	C₂²	1	2	yes	—
21	#15692217	—	2	2	no	—
22	#15692217	C₂²	1	2	yes	—
23	#15692213	—	2	2	no	—
24	#15692213	C₂²	1	2	yes	—
25	C₄²×D₁₀₁	—	2	2	no	—
26	C₄²×D₁₀₁	C₂²	1	1	yes	characteristic
27	#15692189	C₂²	1	1	yes	characteristic
28	#15692186	C₂²	1	2	yes	—
29	#15692179	C₂²	1	2	yes	—
30	#15692177	C₂²	1	2	yes	—
31	#15692175	C₂²	1	2	yes	—
32	C₄×#11717579	C₂²	1	1	yes	characteristic
33	C₈×Dic₁₀₁	C₂²	1	1	yes	characteristic
34	C₈×Dic₁₀₁	C₂²	1	1	yes	characteristic
35	C₂×Q₁₆×C₁₀₁	C₂²	1	1	yes	characteristic
36	C₄×Q₈×C₁₀₁	C₂²	1	2	yes	—
37	#107×C₁₀₁	C₂²	1	1	yes	characteristic
38	#103×C₁₀₁	C₂²	1	2	yes	—
39	C₄×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
40	#13108075	—	2	4	no	—
41	#13108075	C₂×C₄	1	8	yes	complemented
42	C₂×D₄₀₄	—	2	2	no	—
43	C₂×D₄₀₄	C₂³	1	2	yes	—
44	C₂×C₄×D₁₀₁	—	2	2	no	—
45	C₂×C₄×D₁₀₁	—	2	2	no	—
46	C₂×C₄×D₁₀₁	—	2	2	no	—
47	C₂×C₄×D₁₀₁	D₄	1	1	yes	characteristic
48	C₂×C₄×D₁₀₁	D₄	1	1	yes	characteristic
49	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
50	#13108061	C₂×C₄	1	8	yes	complemented
51	D₈₀₈	C₂×C₄	1	4	yes	complemented
52	C₂×#11717579	C₂³	1	1	yes	characteristic
53	C₂×#11717579	C₂³	1	1	yes	characteristic
54	C₈×D₁₀₁	—	2	2	no	—
55	C₈×D₁₀₁	C₂×C₄	1	4	yes	complemented
56	C₄⋊Dic₁₀₁	—	2	2	no	—
57	C₄⋊Dic₁₀₁	C₂³	1	2	yes	—
58	#13108051	—	2	2	no	—
59	#13108051	—	2	2	no	—
60	#13108051	—	2	2	no	—
61	C₄×Dic₁₀₁	—	2	2	no	—
62	C₄×Dic₁₀₁	—	2	2	no	—
63	C₄×Dic₁₀₁	D₄	1	1	yes	characteristic
64	C₄×Dic₁₀₁	D₄	1	1	yes	characteristic
65	C₄×Dic₁₀₁	C₂³	1	1	yes	characteristic
66	C₂×Q₈×C₁₀₁	C₂³	1	2	yes	—
67	Q₁₆×C₁₀₁	C₂×C₄	1	4	yes	complemented
68	C₂×C₈×C₁₀₁	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₁₀₁	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₁₀₁	—	2	2	no	—
71	C₄⋊C₄×C₁₀₁	C₂³	1	2	yes	—
72	C₄²×C₁₀₁	—	2	2	no	—
73	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
74	C₂²×D₁₀₁	—	2	2	no	—
75	C₂²×D₁₀₁	C₂×D₄	1	1	yes	characteristic
76	Q₈×C₁₀₁	—	2	2	no	—
77	Q₈×C₁₀₁	C₂²×C₄	1	4	yes	—
78	C₂×C₄×C₁₀₁	—	2	2	no	—
79	C₂×C₄×C₁₀₁	—	2	2	no	—
80	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
83	C₂×Dic₁₀₁	—	2	2	no	—
84	C₂×Dic₁₀₁	C₂×D₄	1	1	yes	characteristic
85	C₂×Dic₁₀₁	C₂×D₄	1	1	yes	characteristic
86	C₂×Dic₁₀₁	C₂×D₄	1	1	yes	characteristic
87	D₄₀₄	—	2	8	no	—
88	D₄₀₄	—	2	2	no	—
89	D₄₀₄	C₂²×C₄	1	4	yes	—
90	C₄×D₁₀₁	—	2	2	no	—
91	C₄×D₁₀₁	—	2	8	no	—
92	C₄×D₁₀₁	—	2	2	no	—
93	C₄×D₁₀₁	—	4	4	no	—
94	C₄×D₁₀₁	—	4	4	no	—
95	C₄×D₁₀₁	C₂²×C₄	1	2	yes	—
96	C₄×D₁₀₁	#41	1	2	yes	—
97	C₈×C₁₀₁	—	2	1	no	—
98	C₈×C₁₀₁	C₂²×C₄	1	2	yes	—
99	#11717579	—	2	1	no	—
100	#11717579	C₂²×C₄	1	2	yes	—
101	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
102	C₂×D₁₀₁	—	2	1	no	—
103	C₂×D₁₀₁	—	2	1	no	—
104	C₂×D₁₀₁	—	2	4	no	—
105	C₂×D₁₀₁	—	4	2	no	—
106	C₂×D₁₀₁	—	4	2	no	—
107	C₂×D₁₀₁	C₄×D₄	1	2	yes	—
108	C₄×C₁₀₁	—	2	4	no	—
109	C₄×C₁₀₁	—	2	1	no	—
110	C₄×C₁₀₁	—	4	2	no	—
111	C₄×C₁₀₁	#135	1	2	yes	complemented
112	C₄×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
113	C₄×C₁₀₁	C₂×#41	1	1	yes	characteristic
114	Dic₁₀₁	—	2	1	no	—
115	Dic₁₀₁	—	4	2	no	—
116	Dic₁₀₁	C₄×D₄	1	2	yes	—
117	Dic₁₀₁	#135	1	2	yes	complemented
118	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
119	C₂×C₁₀₁	C₂×#135	1	1	yes	characteristic
120	C₂×C₁₀₁	C₂×#135	1	1	yes	characteristic
121	D₁₀₁	—	2	2	no	—
122	D₁₀₁	—	4	4	no	—
123	C₄×#135	—	101	1	no	maximal, Sylow
124	C₁₀₁	C₄×#135	1	1	yes	characteristic, complemented, Sylow
125	C₂×#135	—	101	1	no	—
126	C₄×#41	—	101	2	no	—
127	C₄×Q₁₆	—	101	1	no	—
128	C₄×Q₁₆	D₁₀₁	1	1	yes	characteristic, complemented
129	C₄×SD₁₆	—	101	2	no	—
130	C₄×SD₁₆	—	101	2	no	—
131	C₄×D₈	—	101	1	no	—
132	C₄×D₈	—	101	1	no	—
133	#427	—	101	1	no	—
134	#416	—	101	2	no	—
135	C₂×C₄×C₈	—	101	1	no	—
136	C₂×#41	—	101	2	no	—
137	#135	—	101	8	no	—
138	C₂×Q₁₆	C₂×D₁₀₁	1	1	yes	characteristic
139	C₂×SD₁₆	—	101	2	no	—
140	C₂×D₈	—	101	1	no	—
141	C₂²×C₈	—	101	1	no	—
142	C₂²×C₈	—	101	1	no	—
143	C₄×Q₈	—	101	2	no	—
144	C₄×Q₈	C₂×D₁₀₁	1	2	yes	—
145	C₄×D₄	—	101	2	no	—
146	C₄×D₄	—	101	2	no	—
147	C₄×D₄	—	202	2	no	—
148	C₄×D₄	—	202	2	no	—
149	#117	—	101	2	no	—
150	#117	—	202	2	no	—
151	C₂×C₄²	—	101	1	no	—
152	C₂×C₄²	—	202	2	no	—
153	#107	—	101	1	no	—
154	#107	C₂×D₁₀₁	1	1	yes	characteristic
155	#106	—	101	2	no	—
156	#103	—	101	2	no	—
157	#103	C₂×D₁₀₁	1	2	yes	—
158	#102	—	101	2	no	—
159	#102	—	101	2	no	—
160	C₄×C₈	—	101	1	no	—
161	C₄×C₈	—	101	1	no	—
162	C₄×C₈	—	101	1	no	—
163	C₄×C₈	C₂×D₁₀₁	1	1	yes	characteristic
164	#41	—	101	8	no	—
165	#41	—	202	4	no	—
166	C₂×Q₈	C₂²×D₁₀₁	1	2	yes	—
167	C₂×D₄	—	101	2	no	—
168	C₂×D₄	—	202	2	no	—
169	C₂²×C₄	—	101	1	no	—
170	C₂²×C₄	—	101	1	no	—
171	C₂²×C₄	—	101	1	no	—
172	C₂²×C₄	—	202	2	no	—
173	C₂²×C₄	—	202	2	no	—
174	C₂²×C₄	—	202	2	no	—
175	Q₁₆	C₄×D₁₀₁	1	4	yes	complemented
176	SD₁₆	—	101	8	no	—
177	D₈	—	101	4	no	—
178	C₂×C₈	—	101	1	no	—
179	C₂×C₈	—	101	4	no	—
180	C₂×C₈	—	101	1	no	—
181	C₂×C₈	—	202	2	no	—
182	C₂×C₈	C₂²×D₁₀₁	1	1	yes	characteristic
183	C₂×C₈	C₂²×D₁₀₁	1	1	yes	characteristic
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	101	2	no	—
186	C₄⋊C₄	—	202	2	no	—
187	C₄⋊C₄	C₂²×D₁₀₁	1	2	yes	—
188	C₂²⋊C₄	—	202	2	no	—
189	C₂²⋊C₄	—	202	2	no	—
190	C₂²⋊C₄	—	202	2	no	—
191	C₄²	—	2	2	no	—
192	C₄²	—	101	1	no	—
193	C₄²	—	101	1	no	—
194	C₄²	—	101	1	no	—
195	C₄²	—	202	2	no	—
196	C₄²	—	202	2	no	—
197	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
198	C₂³	—	101	1	no	—
199	C₂³	—	202	2	no	—
200	Q₈	—	2	2	no	—
201	Q₈	C₂×C₄×D₁₀₁	1	4	yes	—
202	D₄	—	101	4	no	—
203	D₄	—	202	8	no	—
204	D₄	—	202	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	101	1	no	—
208	C₂×C₄	—	101	1	no	—
209	C₂×C₄	—	101	2	no	—
210	C₂×C₄	—	101	2	no	—
211	C₂×C₄	—	101	1	no	—
212	C₂×C₄	—	404	4	no	—
213	C₂×C₄	—	404	4	no	—
214	C₂×C₄	—	202	2	no	—
215	C₂×C₄	—	202	8	no	—
216	C₂×C₄	—	202	2	no	—
217	C₂×C₄	—	202	2	no	—
218	C₂×C₄	D₄×D₁₀₁	1	1	yes	characteristic
219	C₂×C₄	D₄×D₁₀₁	1	1	yes	characteristic
220	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
221	C₈	—	2	1	no	—
222	C₈	—	101	2	no	—
223	C₈	—	202	1	no	—
224	C₈	C₂×C₄×D₁₀₁	1	2	yes	—
225	C₂²	—	101	2	no	—
226	C₂²	—	404	2	no	—
227	C₂²	—	404	2	no	—
228	C₂²	—	202	4	no	—
229	C₂²	—	202	1	no	—
230	C₂²	—	202	1	no	—
231	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
232	C₄	—	2	4	no	—
233	C₄	—	2	1	no	—
234	C₄	—	4	2	no	—
235	C₄	—	101	2	no	—
236	C₄	—	101	2	no	—
237	C₄	—	404	2	no	—
238	C₄	—	202	1	no	—
239	C₄	#15692278	1	2	yes	direct product factor
240	C₄	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
241	C₄	#41×D₁₀₁	1	1	yes	characteristic
242	C₂	—	404	4	no	—
243	C₂	—	202	2	no	—
244	C₂	C₄×D₄×D₁₀₁	1	1	yes	characteristic
245	C₂	#135×D₁₀₁	1	1	yes	characteristic
246	C₂	C₂×#15692278	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₀₁
commutator	C₄×C₁₀₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₁
Fitting	C₄×Q₁₆×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂²×D₄
prime radical	C₂×#15692278	C₂
cosocle	C₂×C₄×C₁₀₁	C₂⁴
hypercenter	C₄×Q₁₆	D₁₀₁
hypocenter	C₁₀₁	C₄×#135
LCS	G ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. C₄×#13108075	C₂	semidirect
3. Q₁₆×Dic₁₀₁	C₂	semidirect
4. C₄×#13108061	C₂	semidirect
5. #20265262	C₂	semidirect
6. C₄×D₈₀₈	C₂	semidirect
7. #20265259	C₂	semidirect
8. #20265214	C₂	semidirect
9. #20265172	C₂	semidirect
10. C₄×C₈×D₁₀₁	C₂	semidirect
11. C₄×Q₁₆×C₁₀₁	C₂	semidirect
13. #15692278	C₄	direct (N × H)
41. #13108075	C₂×C₄	semidirect
50. #13108061	C₂×C₄	semidirect
51. D₈₀₈	C₂×C₄	semidirect
55. C₈×D₁₀₁	C₂×C₄	semidirect
67. Q₁₆×C₁₀₁	C₂×C₄	semidirect
111. C₄×C₁₀₁	#135	semidirect
117. Dic₁₀₁	#135	semidirect
124. C₁₀₁	C₄×#135	semidirect
128. C₄×Q₁₆	D₁₀₁	semidirect
175. Q₁₆	C₄×D₁₀₁	semidirect
239. C₄	#15692278	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×#15692278	2
C₄×#13108075	2
Q₁₆×Dic₁₀₁	2
C₄×#13108061	2
#20265262	2
C₄×D₈₀₈	2
#20265259	2
#20265214	2
#20265172	2
C₄×C₈×D₁₀₁	2
C₄×Q₁₆×C₁₀₁	2
C₄×#135	101

Subgroups — C4×#15692278 #27179904

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15692278 #27179904