Subgroups — C₄×#14582716 #25011856

Order 11008.

434 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582716	C₂	1	2	yes	maximal
3	C₄×#13063261	C₂	1	1	yes	maximal, characteristic
4	#19132704	C₂	1	2	yes	maximal
5	C₄×M₃₂×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₇₂	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₈₆	—	2	1	no	—
8	C₄×Dic₁₇₂	—	2	1	no	—
9	C₄×Dic₁₇₂	—	2	1	no	—
10	C₄×Dic₁₇₂	C₄	1	2	yes	—
11	C₄×Dic₁₇₂	C₄	1	2	yes	—
12	C₂×#13063261	C₂²	1	2	yes	—
13	C₂×#13063259	C₂²	1	1	yes	characteristic
14	C₂×#13063250	—	2	1	no	—
15	C₄×#10621877	—	2	1	no	—
16	#14582716	C₄	1	8	yes	direct product factor
17	#14582664	—	2	1	no	—
18	C₂×M₃₂×C₄₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
20	C₄₃×#346	—	2	1	no	—
21	C₄×C₁₆×C₄₃	—	2	1	no	—
22	C₂²×Dic₈₆	—	2	1	no	—
23	C₂×Dic₁₇₂	—	2	1	no	—
24	C₂×Dic₁₇₂	—	2	2	no	—
25	C₂×Dic₁₇₂	C₂×C₄	1	4	yes	—
26	C₂×Dic₁₇₂	C₂×C₄	1	2	yes	—
27	C₂×#10621877	—	2	2	no	—
28	C₄×Dic₈₆	—	2	2	no	—
29	C₄×Dic₈₆	—	2	2	no	—
30	C₄×Dic₈₆	—	4	1	no	—
31	C₄×Dic₈₆	—	4	1	no	—
32	C₂×C₄⋊Dic₄₃	—	2	1	no	—
33	C₂×Dic₄₃⋊C₄	—	4	1	no	—
34	C₂×C₄×Dic₄₃	—	4	1	no	—
35	#13063261	—	2	2	no	—
36	#13063261	C₂×C₄	1	4	yes	—
37	#13063259	—	2	1	no	—
38	#13063259	C₂×C₄	1	2	yes	—
39	#13063250	—	2	2	no	—
40	#13063250	—	2	2	no	—
41	#13063236	—	4	1	no	—
42	C₄×C₄₃⋊C₈	—	4	1	no	—
43	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
46	M₃₂×C₄₃	—	2	2	no	—
47	M₃₂×C₄₃	C₂×C₄	1	4	yes	—
48	C₂×C₁₆×C₄₃	—	2	2	no	—
49	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
52	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₃	SD₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₄₃	SD₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×Dic₄₃	—	4	1	no	—
57	C₂²×Dic₄₃	—	4	1	no	—
58	C₂×Dic₈₆	—	2	2	no	—
59	C₂×Dic₈₆	—	2	4	no	—
60	C₂×Dic₈₆	—	4	1	no	—
61	C₂×Dic₈₆	—	4	2	no	—
62	C₂×C₈×C₄₃	—	2	2	no	—
63	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₄₃	#41	1	2	yes	—
69	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₄₃	D₈	1	1	yes	characteristic
72	C₄²×C₄₃	D₈	1	1	yes	characteristic
73	C₄⋊Dic₄₃	—	2	2	no	—
74	C₄⋊Dic₄₃	—	4	1	no	—
75	Dic₄₃⋊C₄	—	4	2	no	—
76	Dic₄₃⋊C₄	—	4	2	no	—
77	C₄×Dic₄₃	—	4	2	no	—
78	C₄×Dic₄₃	—	4	2	no	—
79	#10621877	—	2	4	no	—
80	#10621877	—	4	2	no	—
81	C₂×C₄₃⋊C₈	—	4	2	no	—
82	Dic₁₇₂	—	2	2	no	—
83	Dic₁₇₂	—	2	4	no	—
84	Dic₁₇₂	C₄²	1	4	yes	—
85	C₁₆×C₄₃	—	2	4	no	—
86	C₂³×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×C₄×C₄₃	—	2	2	no	—
88	C₂×C₄×C₄₃	—	4	1	no	—
89	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
90	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
91	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
92	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
93	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
95	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₄₃	#135	1	2	yes	—
97	C₂×Dic₄₃	—	4	2	no	—
98	C₂×Dic₄₃	—	4	4	no	—
99	C₂×Dic₄₃	—	4	2	no	—
100	C₂×Dic₄₃	—	8	2	no	—
101	Dic₈₆	—	2	4	no	—
102	Dic₈₆	—	4	2	no	—
103	Dic₈₆	—	4	4	no	—
104	C₈×C₄₃	—	2	1	no	—
105	C₈×C₄₃	—	2	1	no	—
106	C₈×C₄₃	C₂×C₄²	1	1	yes	characteristic
107	C₈×C₄₃	#117	1	1	yes	characteristic
108	C₈×C₄₃	C₄×D₄	1	2	yes	—
109	C₄₃⋊C₈	—	4	4	no	—
110	C₄×#362	—	43	1	no	maximal, Sylow
111	C₂²×C₄₃	—	2	1	no	—
112	C₂²×C₄₃	—	2	1	no	—
113	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
114	C₂²×C₄₃	C₄×SD₁₆	1	2	yes	—
115	C₄×C₄₃	—	2	1	no	—
116	C₄×C₄₃	—	2	1	no	—
117	C₄×C₄₃	—	2	1	no	—
118	C₄×C₄₃	#362	1	2	yes	complemented
119	C₄×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
120	C₄×C₄₃	#416	1	1	yes	characteristic
121	C₄×C₄₃	C₄×D₈	1	2	yes	—
122	Dic₄₃	—	4	4	no	—
123	Dic₄₃	—	8	2	no	—
124	C₂×C₄×Q₁₆	—	43	1	no	—
125	C₂×#362	—	43	2	no	—
126	C₄×M₃₂	D₄₃	1	1	yes	characteristic
127	C₄×#108	—	43	1	no	—
128	#1387	—	43	2	no	—
129	C₂×C₄₃	—	2	2	no	—
130	C₂×C₄₃	C₄×#102	1	1	yes	characteristic
131	C₂×C₄₃	C₂×#362	1	1	yes	characteristic
132	C₂×C₄₃	#2148	1	1	yes	characteristic
133	C₂²×Q₁₆	—	43	1	no	—
134	C₂×C₄×Q₈	—	86	1	no	—
135	C₂×M₃₂	C₂×D₄₃	1	2	yes	—
136	C₄×Q₁₆	—	43	2	no	—
137	C₄×Q₁₆	—	43	2	no	—
138	C₄×Q₁₆	—	86	1	no	—
139	C₄×Q₁₆	—	86	1	no	—
140	C₂×#108	—	43	2	no	—
141	C₂×#107	—	43	1	no	—
142	C₂×#103	—	86	1	no	—
143	C₄×M₁₆	—	86	1	no	—
144	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
145	#362	—	43	8	no	—
146	#346	—	2	1	no	—
147	C₄×C₁₆	—	2	1	no	—
148	#341	—	86	1	no	—
149	C₄₃	C₄×#362	1	1	yes	characteristic, complemented, Sylow
150	C₂²×Q₈	—	86	1	no	—
151	C₂×Q₁₆	—	43	4	no	—
152	C₂×Q₁₆	—	43	2	no	—
153	C₂×Q₁₆	—	86	2	no	—
154	C₂×Q₁₆	—	86	1	no	—
155	C₂×M₁₆	—	86	2	no	—
156	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
157	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
158	C₄×Q₈	—	86	2	no	—
159	C₄×Q₈	—	86	2	no	—
160	C₄×Q₈	—	172	1	no	—
161	C₄×Q₈	—	172	1	no	—
162	C₂×C₄⋊C₄	—	86	1	no	—
163	C₂×C₄⋊C₄	—	172	1	no	—
164	C₂×C₄²	—	172	1	no	—
165	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
166	M₃₂	—	2	2	no	—
167	M₃₂	C₄×D₄₃	1	4	yes	—
168	C₂×C₁₆	—	2	2	no	—
169	#108	—	43	4	no	—
170	#108	—	86	2	no	—
171	#107	—	43	2	no	—
172	#107	—	86	1	no	—
173	#103	—	86	2	no	—
174	#103	—	86	2	no	—
175	#97	—	172	1	no	—
176	C₄×C₈	—	172	1	no	—
177	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
178	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
179	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
180	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
181	C₂×Q₈	—	86	2	no	—
182	C₂×Q₈	—	86	4	no	—
183	C₂×Q₈	—	172	1	no	—
184	C₂×Q₈	—	172	2	no	—
185	C₂²×C₄	—	172	1	no	—
186	C₂²×C₄	—	172	1	no	—
187	C₂²×C₄	#10621873	1	1	yes	characteristic
188	C₂²×C₄	#10621873	1	1	yes	characteristic
189	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
190	Q₁₆	—	43	4	no	—
191	Q₁₆	—	86	2	no	—
192	Q₁₆	—	86	4	no	—
193	M₁₆	—	86	4	no	—
194	M₁₆	—	172	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	172	2	no	—
197	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
198	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
199	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
200	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
201	C₂×C₈	#10621898	1	2	yes	—
202	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
203	C₄⋊C₄	—	86	2	no	—
204	C₄⋊C₄	—	172	2	no	—
205	C₄⋊C₄	—	172	2	no	—
206	C₄⋊C₄	—	172	1	no	—
207	C₄²	—	172	2	no	—
208	C₄²	—	172	2	no	—
209	C₄²	D₃₄₄	1	1	yes	characteristic
210	C₄²	D₃₄₄	1	1	yes	characteristic
211	C₄²	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
212	C₄²	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
213	C₁₆	—	2	4	no	—
214	C₂³	C₂×#10621873	1	1	yes	characteristic
215	Q₈	—	86	4	no	—
216	Q₈	—	172	2	no	—
217	Q₈	—	172	4	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	4	1	no	—
220	C₂×C₄	—	344	2	no	—
221	C₂×C₄	—	172	2	no	—
222	C₂×C₄	—	172	2	no	—
223	C₂×C₄	—	172	4	no	—
224	C₂×C₄	#13063247	1	1	yes	characteristic
225	C₂×C₄	#13063247	1	1	yes	characteristic
226	C₂×C₄	#13063247	1	1	yes	characteristic
227	C₂×C₄	#13063247	1	1	yes	characteristic
228	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
229	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
230	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
231	C₂×C₄	#13063347	1	2	yes	—
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₈	—	172	4	no	—
235	C₈	C₄²×D₄₃	1	1	yes	characteristic
236	C₈	#13063282	1	1	yes	characteristic
237	C₈	C₄×C₂²⋊D₄₃	1	2	yes	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	C₂×#13063247	1	1	yes	characteristic
241	C₂²	C₄×#10621873	1	2	yes	—
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	344	2	no	—
246	C₄	—	172	4	no	—
247	C₄	#14582716	1	2	yes	direct product factor
248	C₄	C₄×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
249	C₄	#14582896	1	1	yes	characteristic
250	C₄	C₄×D₃₄₄	1	2	yes	—
251	C₂	—	2	2	no	—
252	C₂	C₄×#13063247	1	1	yes	characteristic
253	C₂	C₂×#14582716	1	1	yes	characteristic
254	C₂	#19136345	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13063247
commutator	C₈×C₄₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄×M₃₂×C₄₃	C₂
socle	C₂²×C₄₃	C₂×#102
prime radical	C₂³×C₄₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₄×M₃₂	D₄₃
hypocenter	C₄₃	C₄×#362
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
16. #14582716	C₄	direct (N × H)
118. C₄×C₄₃	#362	semidirect
149. C₄₃	C₄×#362	semidirect
247. C₄	#14582716	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×C₄×Dic₁₇₂	2
C₂×#14582716	2
C₄×#13063261	2
#19132704	2
C₄×M₃₂×C₄₃	2
C₄×#362	43

Subgroups — C4×#14582716 #25011856

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#14582716 #25011856