Subgroups — C₄×#15692275 #27179902

Order 12928.

500 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692275	C₂	1	1	yes	maximal, characteristic
2	C₄×#13108074	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#13108064	C₂	1	2	yes	maximal, complemented
4	C₄×#13108061	C₂	1	2	yes	maximal, complemented
5	C₄×#13108062	C₂	1	2	yes	maximal, complemented
6	C₄×#13108059	C₂	1	2	yes	maximal, complemented
7	#20265213	C₂	1	1	yes	maximal, characteristic, complemented
8	#20265173	C₂	1	1	yes	maximal, characteristic, complemented
9	#20265171	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄×#11717579	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#41×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13108074	C₂²	1	1	yes	characteristic
13	#15692275	C₄	1	8	yes	direct product factor
14	C₂×#13108064	C₂²	1	1	yes	characteristic
15	C₂×#13108062	C₂²	1	1	yes	characteristic
16	C₂×#13108061	C₂²	1	1	yes	characteristic
17	C₂×#13108059	C₂²	1	1	yes	characteristic
18	C₂²×#11717579	C₂²	1	1	yes	characteristic
19	C₂²×#11717579	C₂²	1	1	yes	characteristic
20	C₄×Dic₂₀₂	C₂²	1	2	yes	—
21	C₄×D₄₀₄	C₂²	1	2	yes	—
22	C₄×C₂²⋊D₁₀₁	—	2	2	no	—
23	#15692214	C₂²	1	1	yes	characteristic
24	#15692212	—	2	1	no	—
25	C₄²×D₁₀₁	—	2	1	no	—
26	#15692188	C₂²	1	2	yes	—
27	#15692186	C₂²	1	2	yes	—
28	#15692179	C₂²	1	2	yes	—
29	#15692178	C₂²	1	2	yes	—
30	#15692176	C₂²	1	2	yes	—
31	#15692175	C₂²	1	2	yes	—
32	C₄×#11717579	C₂²	1	2	yes	—
33	C₄×#11717579	C₂²	1	2	yes	—
34	C₂×#41×C₁₀₁	C₂²	1	1	yes	characteristic
35	C₄×Q₈×C₁₀₁	C₂²	1	2	yes	—
36	C₄×D₄×C₁₀₁	—	2	2	no	—
37	C₄×D₄×C₁₀₁	C₂²	1	2	yes	—
38	#117×C₁₀₁	—	2	1	no	—
39	#117×C₁₀₁	C₂²	1	1	yes	characteristic
40	C₂×C₄²×C₁₀₁	—	2	1	no	—
41	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
42	#13108074	—	2	2	no	—
43	#13108074	C₂×C₄	1	4	yes	complemented
44	C₂×Dic₂₀₂	C₂³	1	1	yes	characteristic
45	C₂×D₄₀₄	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₁₀₁	—	2	1	no	—
47	C₂×C₄×D₁₀₁	—	2	1	no	—
48	C₂×C₄×D₁₀₁	—	2	2	no	—
49	#13108064	C₂×C₄	1	4	yes	complemented
50	#13108062	C₂×C₄	1	4	yes	complemented
51	#13108061	C₂×C₄	1	4	yes	complemented
52	#13108059	C₂×C₄	1	4	yes	complemented
53	C₂×#11717579	—	2	2	no	—
54	C₂×#11717579	C₂×C₄	1	4	yes	complemented
55	C₂×#11717579	C₂³	1	2	yes	—
56	C₂×#11717579	C₂³	1	1	yes	characteristic
57	C₂×#11717579	C₂³	1	1	yes	characteristic
58	C₄⋊Dic₁₀₁	C₂³	1	2	yes	—
59	#13108053	—	2	2	no	—
60	C₂²⋊Dic₁₀₁	—	2	1	no	—
61	#13108051	—	2	2	no	—
62	C₄×Dic₁₀₁	—	2	1	no	—
63	C₄×Dic₁₀₁	—	2	2	no	—
64	#41×C₁₀₁	—	2	2	no	—
65	#41×C₁₀₁	C₂×C₄	1	4	yes	complemented
66	C₂×Q₈×C₁₀₁	C₂³	1	1	yes	characteristic
67	C₂×D₄×C₁₀₁	—	2	1	no	—
68	C₂×D₄×C₁₀₁	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₁₀₁	—	2	1	no	—
70	C₂²×C₄×C₁₀₁	—	2	2	no	—
71	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
72	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₀₁	—	2	2	no	—
74	C₄⋊C₄×C₁₀₁	C₂³	1	2	yes	—
75	C₂²⋊C₄×C₁₀₁	—	2	1	no	—
76	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
77	C₄²×C₁₀₁	—	2	1	no	—
78	C₄²×C₁₀₁	—	2	2	no	—
79	C₄²×C₁₀₁	D₄	1	1	yes	characteristic
80	C₄²×C₁₀₁	D₄	1	1	yes	characteristic
81	C₄²×C₁₀₁	C₂³	1	2	yes	—
82	C₂³×C₁₀₁	—	2	1	no	—
83	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
84	C₂²×D₁₀₁	—	2	1	no	—
85	Q₈×C₁₀₁	—	2	1	no	—
86	Q₈×C₁₀₁	C₂²×C₄	1	2	yes	—
87	D₄×C₁₀₁	—	2	4	no	—
88	D₄×C₁₀₁	—	2	1	no	—
89	D₄×C₁₀₁	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₁₀₁	—	2	1	no	—
91	C₂×C₄×C₁₀₁	—	2	4	no	—
92	C₂×C₄×C₁₀₁	—	2	4	no	—
93	C₂×C₄×C₁₀₁	—	2	2	no	—
94	C₂×C₄×C₁₀₁	—	4	4	no	—
95	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
99	C₂×C₄×C₁₀₁	#41	1	2	yes	—
100	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
101	C₂²⋊D₁₀₁	—	2	4	no	—
102	C₂×Dic₁₀₁	—	2	1	no	—
103	C₂×Dic₁₀₁	—	2	2	no	—
104	D₄₀₄	—	2	1	no	—
105	D₄₀₄	C₂²×C₄	1	2	yes	—
106	C₄×D₁₀₁	—	2	4	no	—
107	C₄×D₁₀₁	—	4	4	no	—
108	Dic₂₀₂	—	2	1	no	—
109	Dic₂₀₂	C₂²×C₄	1	2	yes	—
110	#11717579	—	2	2	no	—
111	#11717579	C₂²×C₄	1	2	yes	—
112	#11717579	C₂²×C₄	1	2	yes	—
113	C₂²×C₁₀₁	—	2	2	no	—
114	C₂²×C₁₀₁	—	2	2	no	—
115	C₂²×C₁₀₁	—	4	2	no	—
116	C₂²×C₁₀₁	C₄×D₄	1	2	yes	—
117	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
118	C₂×D₁₀₁	—	2	2	no	—
119	C₂×D₁₀₁	—	4	2	no	—
120	C₄×C₁₀₁	—	2	2	no	—
121	C₄×C₁₀₁	—	2	2	no	—
122	C₄×C₁₀₁	—	4	2	no	—
123	C₄×C₁₀₁	C₄×D₄	1	2	yes	—
124	C₄×C₁₀₁	#135	1	4	yes	complemented
125	C₄×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
126	C₄×C₁₀₁	C₂×#41	1	1	yes	characteristic
127	Dic₁₀₁	—	2	2	no	—
128	Dic₁₀₁	—	4	2	no	—
129	C₂×C₁₀₁	—	2	2	no	—
130	C₂×C₁₀₁	—	4	2	no	—
131	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
132	C₂×C₁₀₁	C₂×#135	1	2	yes	—
133	D₁₀₁	—	4	2	no	—
134	C₄×#135	—	101	1	no	maximal, Sylow
135	C₁₀₁	C₄×#135	1	1	yes	characteristic, complemented, Sylow
136	C₂×#135	—	101	1	no	—
137	C₄×#41	—	101	1	no	—
138	C₄×#41	D₁₀₁	1	1	yes	characteristic, complemented
139	C₄×Q₁₆	—	101	2	no	—
140	C₄×SD₁₆	—	101	2	no	—
141	C₄×SD₁₆	—	101	2	no	—
142	C₄×D₈	—	101	2	no	—
143	#427	—	101	1	no	—
144	#416	—	101	1	no	—
145	#416	—	101	1	no	—
146	C₂×C₄×C₈	—	101	1	no	—
147	C₂×#41	—	101	1	no	—
148	C₂×#41	C₂×D₁₀₁	1	1	yes	characteristic
149	#135	—	101	8	no	—
150	C₂×Q₁₆	—	101	1	no	—
151	C₂×SD₁₆	—	101	1	no	—
152	C₂×SD₁₆	—	101	1	no	—
153	C₂×D₈	—	101	1	no	—
154	C₂²×C₈	—	101	1	no	—
155	C₂²×C₈	—	101	1	no	—
156	C₄×Q₈	—	101	2	no	—
157	C₄×Q₈	C₂×D₁₀₁	1	2	yes	—
158	C₄×D₄	—	2	2	no	—
159	C₄×D₄	—	101	2	no	—
160	C₄×D₄	—	202	2	no	—
161	C₄×D₄	C₂×D₁₀₁	1	2	yes	—
162	#117	—	2	1	no	—
163	#117	—	101	1	no	—
164	#117	—	202	1	no	—
165	#117	C₂×D₁₀₁	1	1	yes	characteristic
166	C₂×C₄²	—	2	1	no	—
167	C₂×C₄²	—	202	1	no	—
168	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
169	#107	—	101	2	no	—
170	#106	—	101	2	no	—
171	#103	—	101	2	no	—
172	#103	—	101	2	no	—
173	#102	—	101	2	no	—
174	#102	—	101	2	no	—
175	C₄×C₈	—	101	2	no	—
176	C₄×C₈	—	101	2	no	—
177	#41	—	2	2	no	—
178	#41	—	101	4	no	—
179	#41	—	202	2	no	—
180	#41	C₄×D₁₀₁	1	4	yes	complemented
181	C₂×Q₈	—	101	1	no	—
182	C₂×Q₈	C₂²×D₁₀₁	1	1	yes	characteristic
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	—	101	1	no	—
185	C₂×D₄	—	202	1	no	—
186	C₂×D₄	C₂²×D₁₀₁	1	1	yes	characteristic
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	202	2	no	—
190	C₂²×C₄	—	202	1	no	—
191	C₂²×C₄	C₂²⋊D₁₀₁	1	2	yes	—
192	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
193	Q₁₆	—	101	4	no	—
194	SD₁₆	—	101	4	no	—
195	SD₁₆	—	101	4	no	—
196	D₈	—	101	4	no	—
197	C₂×C₈	—	101	4	no	—
198	C₂×C₈	—	101	1	no	—
199	C₂×C₈	—	101	1	no	—
200	C₂×C₈	—	101	2	no	—
201	C₂×C₈	—	202	2	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	101	2	no	—
204	C₄⋊C₄	—	202	2	no	—
205	C₄⋊C₄	C₂²×D₁₀₁	1	2	yes	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₂²⋊C₄	—	202	1	no	—
209	C₂²⋊C₄	—	202	2	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	202	1	no	—
213	C₄²	—	202	2	no	—
214	C₄²	C₂²⋊D₁₀₁	1	1	yes	characteristic
215	C₄²	C₂²⋊D₁₀₁	1	1	yes	characteristic
216	C₄²	C₂²×D₁₀₁	1	2	yes	—
217	C₂³	—	2	1	no	—
218	C₂³	—	202	1	no	—
219	C₂³	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
220	Q₈	—	2	1	no	—
221	Q₈	—	101	2	no	—
222	Q₈	—	202	1	no	—
223	Q₈	C₂×C₄×D₁₀₁	1	2	yes	—
224	D₄	—	2	4	no	—
225	D₄	—	2	1	no	—
226	D₄	—	101	2	no	—
227	D₄	—	202	4	no	—
228	D₄	—	202	1	no	—
229	D₄	C₂×C₄×D₁₀₁	1	2	yes	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	404	4	no	—
236	C₂×C₄	—	202	2	no	—
237	C₂×C₄	—	202	1	no	—
238	C₂×C₄	—	202	4	no	—
239	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
240	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
241	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
242	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
243	C₂×C₄	#13108074	1	2	yes	—
244	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
245	C₈	—	101	2	no	—
246	C₈	—	101	2	no	—
247	C₈	—	202	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	404	2	no	—
252	C₂²	—	202	2	no	—
253	C₂²	C₄×C₂²⋊D₁₀₁	1	2	yes	—
254	C₂²	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	4	2	no	—
258	C₄	—	404	2	no	—
259	C₄	—	202	2	no	—
260	C₄	C₄×C₂²⋊D₁₀₁	1	2	yes	—
261	C₄	#15692275	1	4	yes	direct product factor
262	C₄	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
263	C₄	C₂×#13108074	1	1	yes	characteristic
264	C₂	—	2	2	no	—
265	C₂	—	4	2	no	—
266	C₂	—	404	2	no	—
267	C₂	C₂×C₄×C₂²⋊D₁₀₁	1	1	yes	characteristic
268	C₂	C₂×#15692275	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂²⋊D₁₀₁
commutator	C₄×C₁₀₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₁
Fitting	C₄×#41×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂²×D₄
prime radical	C₂×#15692275	C₂
cosocle	C₂×C₄×C₁₀₁	C₂⁴
hypercenter	C₄×#41	D₁₀₁
hypocenter	C₁₀₁	C₄×#135
LCS	G ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
2. C₄×#13108074	C₂	semidirect
3. C₄×#13108064	C₂	semidirect
4. C₄×#13108061	C₂	semidirect
5. C₄×#13108062	C₂	semidirect
6. C₄×#13108059	C₂	semidirect
7. #20265213	C₂	semidirect
8. #20265173	C₂	semidirect
9. #20265171	C₂	semidirect
10. C₂×C₄×#11717579	C₂	semidirect
11. C₄×#41×C₁₀₁	C₂	semidirect
13. #15692275	C₄	direct (N × H)
43. #13108074	C₂×C₄	semidirect
49. #13108064	C₂×C₄	semidirect
50. #13108062	C₂×C₄	semidirect
51. #13108061	C₂×C₄	semidirect
52. #13108059	C₂×C₄	semidirect
54. C₂×#11717579	C₂×C₄	semidirect
65. #41×C₁₀₁	C₂×C₄	semidirect
124. C₄×C₁₀₁	#135	semidirect
135. C₁₀₁	C₄×#135	semidirect
138. C₄×#41	D₁₀₁	semidirect
180. #41	C₄×D₁₀₁	semidirect
261. C₄	#15692275	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×#15692275	2
C₄×#13108074	2
C₄×#13108064	2
C₄×#13108061	2
C₄×#13108062	2
C₄×#13108059	2
#20265213	2
#20265173	2
#20265171	2
C₂×C₄×#11717579	2
C₄×#41×C₁₀₁	2
C₄×#135	101

Subgroups — C4×#15692275 #27179902

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15692275 #27179902