Subgroups — C₄×#15689503 #26224715

Order 12032.

1086 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689503	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₁₈₈	C₂	1	4	yes	maximal
3	C₄×#13094024	C₂	1	2	yes	maximal
4	#20244249	C₂	1	4	yes	maximal
5	#20244190	C₂	1	2	yes	maximal
6	#20243548	C₂	1	1	yes	maximal, characteristic
7	C₄²×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₈₈	C₂²	1	4	yes	—
9	C₂×#13094024	C₂²	1	2	yes	—
10	C₂×C₄×Dic₉₄	C₂²	1	4	yes	—
11	#15689503	C₄	1	8	yes	direct product factor
12	C₄×Dic₁₈₈	—	2	16	no	—
13	C₂×#13093958	C₂²	1	4	yes	—
14	C₂×#13093949	C₂²	1	8	yes	—
15	#15689029	C₂²	1	4	yes	—
16	#15689021	—	2	4	no	—
17	#15688993	C₂²	1	2	yes	—
18	C₄×C₄⋊Dic₄₇	—	2	4	no	—
19	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
23	C₄³×C₄₇	C₂²	1	1	yes	characteristic
24	C₂²×Dic₉₄	C₂³	1	4	yes	—
25	C₂×Dic₁₈₈	—	2	8	no	—
26	C₂×Dic₁₈₈	—	2	8	no	—
27	C₂×Dic₁₈₈	C₂×C₄	1	16	yes	—
28	#13094024	—	2	4	no	—
29	#13094024	C₂×C₄	1	8	yes	—
30	C₄×Dic₉₄	—	2	16	no	—
31	C₂×C₄⋊Dic₄₇	—	2	4	no	—
32	C₂×C₄⋊Dic₄₇	—	2	4	no	—
33	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
34	C₂×Dic₄₇⋊C₄	—	2	4	no	—
35	C₂×C₄×Dic₄₇	—	2	4	no	—
36	#13093958	—	2	8	no	—
37	#13093949	—	2	16	no	—
38	C₄₇⋊#95	—	2	8	no	—
39	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
40	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
41	C₂×C₄²×C₄₇	D₄	1	2	yes	—
42	C₂×C₄²×C₄₇	D₄	1	2	yes	—
43	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
46	C₄×C₈×C₄₇	—	2	4	no	—
47	C₄×C₈×C₄₇	—	2	2	no	—
48	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
49	C₄×C₈×C₄₇	D₄	1	8	yes	—
50	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
55	C₂²×Dic₄₇	—	2	4	no	—
56	C₂²×Dic₄₇	—	2	4	no	—
57	C₂×Dic₉₄	—	2	8	no	—
58	C₂×Dic₉₄	—	2	8	no	—
59	C₂×Dic₉₄	—	2	4	no	—
60	C₂×Dic₉₄	C₂²×C₄	1	8	yes	—
61	C₂×C₈×C₄₇	—	2	2	no	—
62	C₂×C₈×C₄₇	—	2	2	no	—
63	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
65	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
66	C₂×C₈×C₄₇	#41	1	4	yes	—
67	C₄²×C₄₇	—	2	4	no	—
68	C₄²×C₄₇	—	2	2	no	—
69	C₄²×C₄₇	Q₁₆	1	8	yes	complemented
70	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
71	C₄²×C₄₇	C₂×D₄	1	4	yes	—
72	C₄²×C₄₇	#41	1	2	yes	—
73	C₄⋊Dic₄₇	—	2	16	no	—
74	C₄⋊Dic₄₇	—	2	8	no	—
75	C₄⋊Dic₄₇	—	4	8	no	—
76	Dic₄₇⋊C₄	—	4	8	no	—
77	C₄×Dic₄₇	—	4	8	no	—
78	Dic₁₈₈	—	2	32	no	—
79	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
80	C₂×C₄×C₄₇	—	2	2	no	—
81	C₂×C₄×C₄₇	—	2	2	no	—
82	C₂×C₄×C₄₇	—	2	2	no	—
83	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
84	C₂×C₄×C₄₇	#127	1	2	yes	—
85	C₂×C₄×C₄₇	#127	1	2	yes	—
86	C₂×C₄×C₄₇	C₂×Q₁₆	1	4	yes	—
87	C₂×C₄×C₄₇	C₂×Q₁₆	1	4	yes	—
88	C₂×C₄×C₄₇	C₂×Q₁₆	1	4	yes	—
89	C₂×C₄×C₄₇	#135	1	4	yes	—
90	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
92	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
93	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
94	C₂×Dic₄₇	—	2	8	no	—
95	C₂×Dic₄₇	—	4	4	no	—
96	C₂×Dic₄₇	—	4	4	no	—
97	C₂×Dic₄₇	—	4	4	no	—
98	C₂×Dic₄₇	—	4	8	no	—
99	Dic₉₄	—	2	16	no	—
100	Dic₉₄	—	4	8	no	—
101	C₈×C₄₇	—	2	4	no	—
102	C₈×C₄₇	C₄×D₄	1	8	yes	—
103	C₄×#494	—	47	1	no	maximal, Sylow
104	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
105	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
106	C₂²×C₄₇	#532	1	1	yes	characteristic
107	C₂²×C₄₇	C₂²×Q₁₆	1	2	yes	—
108	C₂²×C₄₇	C₂×#135	1	2	yes	—
109	C₄×C₄₇	—	2	2	no	—
110	C₄×C₄₇	—	2	4	no	—
111	C₄×C₄₇	C₄×Q₁₆	1	8	yes	complemented
112	C₄×C₄₇	#494	1	4	yes	complemented
113	C₄×C₄₇	C₂×C₄×D₄	1	2	yes	—
114	C₄×C₄₇	#532	1	2	yes	—
115	Dic₄₇	—	4	8	no	—
116	Dic₄₇	—	8	4	no	—
117	C₂×#494	—	47	1	no	—
118	C₂×C₄×Q₁₆	—	47	4	no	—
119	C₄×C₄⋊Q₈	—	47	2	no	—
120	#1969	—	47	4	no	—
121	#1956	—	47	2	no	—
122	#1845	—	47	1	no	—
123	C₄²×C₈	D₄₇	1	1	yes	characteristic
124	C₂×C₄₇	C₄×#127	1	1	yes	characteristic
125	C₂×C₄₇	C₂×C₄×Q₁₆	1	2	yes	—
126	C₂×C₄₇	C₂×#494	1	1	yes	characteristic
127	C₂×C₄₇	#3147	1	1	yes	characteristic
128	C₂×C₄₇	#3170	1	2	yes	—
129	C₂²×Q₁₆	—	47	4	no	—
130	C₂×C₄⋊Q₈	—	47	2	no	—
131	C₂×C₄×Q₈	—	47	4	no	—
132	#494	—	47	8	no	—
133	C₄×Q₁₆	—	94	16	no	—
134	C₂×#107	—	47	4	no	—
135	C₂×#103	—	47	8	no	—
136	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
137	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
138	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
139	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
140	#391	—	47	4	no	—
141	#389	—	94	4	no	—
142	#384	—	47	2	no	—
143	C₄×C₄⋊C₄	—	94	4	no	—
144	C₄³	C₂×D₄₇	1	1	yes	characteristic
145	C₄₇	C₄×#494	1	1	yes	characteristic, complemented, Sylow
146	C₂²×Q₈	—	47	4	no	—
147	C₂×Q₁₆	—	47	16	no	—
148	C₂×Q₁₆	—	94	8	no	—
149	C₂×Q₁₆	—	94	8	no	—
150	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
151	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
152	C₄⋊Q₈	—	47	8	no	—
153	C₄⋊Q₈	—	94	4	no	—
154	C₄×Q₈	—	94	16	no	—
155	C₂×C₄⋊C₄	—	47	4	no	—
156	C₂×C₄⋊C₄	—	94	4	no	—
157	C₂×C₄⋊C₄	—	94	4	no	—
158	C₂×C₄⋊C₄	—	94	4	no	—
159	C₂×C₄²	—	94	4	no	—
160	C₂×C₄²	D₁₈₈	1	2	yes	—
161	C₂×C₄²	D₁₈₈	1	2	yes	—
162	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
163	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
164	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
165	#107	—	94	8	no	—
166	#103	—	94	16	no	—
167	C₄×C₈	—	2	4	no	—
168	C₄×C₈	—	2	2	no	—
169	C₄×C₈	C₄×D₄₇	1	4	yes	—
170	C₄×C₈	D₁₈₈	1	8	yes	—
171	#95	—	94	8	no	—
172	C₂×Q₈	—	47	8	no	—
173	C₂×Q₈	—	94	4	no	—
174	C₂×Q₈	—	94	8	no	—
175	C₂×Q₈	—	94	8	no	—
176	C₂²×C₄	—	94	4	no	—
177	C₂²×C₄	—	94	4	no	—
178	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
180	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
181	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
182	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
183	Q₁₆	—	94	32	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
187	C₂×C₈	C₂×D₁₈₈	1	4	yes	—
188	C₂×C₈	C₂×D₁₈₈	1	4	yes	—
189	C₂×C₈	#10625350	1	4	yes	—
190	C₄⋊C₄	—	94	16	no	—
191	C₄⋊C₄	—	94	8	no	—
192	C₄⋊C₄	—	188	8	no	—
193	C₄⋊C₄	—	188	8	no	—
194	C₄²	—	2	4	no	—
195	C₄²	—	2	2	no	—
196	C₄²	—	188	8	no	—
197	C₄²	Dic₁₈₈	1	8	yes	complemented
198	C₄²	C₂×C₄×D₄₇	1	2	yes	—
199	C₄²	C₂×D₁₈₈	1	4	yes	—
200	C₄²	#10625350	1	2	yes	—
201	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
202	Q₈	—	94	16	no	—
203	Q₈	—	188	8	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	94	8	no	—
208	C₂×C₄	—	188	8	no	—
209	C₂×C₄	—	188	4	no	—
210	C₂×C₄	—	188	4	no	—
211	C₂×C₄	—	188	4	no	—
212	C₂×C₄	C₄×D₁₈₈	1	4	yes	—
213	C₂×C₄	#13094021	1	2	yes	—
214	C₂×C₄	#13094021	1	2	yes	—
215	C₂×C₄	C₂×Dic₁₈₈	1	4	yes	—
216	C₂×C₄	C₂×Dic₁₈₈	1	4	yes	—
217	C₂×C₄	C₂×Dic₁₈₈	1	4	yes	—
218	C₂×C₄	#13094046	1	4	yes	—
219	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
220	C₂×C₄	C₂²×D₁₈₈	1	2	yes	—
221	C₂×C₄	C₂×#10625350	1	2	yes	—
222	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
223	C₈	—	2	4	no	—
224	C₈	C₄×D₁₈₈	1	8	yes	—
225	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
226	C₂²	C₂×#13094021	1	1	yes	characteristic
227	C₂²	#15689673	1	1	yes	characteristic
228	C₂²	C₂²×Dic₁₈₈	1	2	yes	—
229	C₂²	C₂×#13094046	1	2	yes	—
230	C₄	—	2	4	no	—
231	C₄	—	2	2	no	—
232	C₄	—	376	4	no	—
233	C₄	—	188	8	no	—
234	C₄	C₄×Dic₁₈₈	1	8	yes	complemented
235	C₄	#15689503	1	4	yes	direct product factor
236	C₄	C₂×C₄×D₁₈₈	1	2	yes	—
237	C₄	#15689673	1	2	yes	—
238	C₂	C₄×#13094021	1	1	yes	characteristic
239	C₂	C₂×C₄×Dic₁₈₈	1	2	yes	—
240	C₂	C₂×#15689503	1	1	yes	characteristic
241	C₂	#20253816	1	1	yes	characteristic
242	C₂	#20253974	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄²×C₈×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄²×C₈	D₄₇
hypocenter	C₄₇	C₄×#494
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
11. #15689503	C₄	direct (N × H)
69. C₄²×C₄₇	Q₁₆	semidirect
111. C₄×C₄₇	C₄×Q₁₆	semidirect
112. C₄×C₄₇	#494	semidirect
145. C₄₇	C₄×#494	semidirect
197. C₄²	Dic₁₈₈	semidirect
234. C₄	C₄×Dic₁₈₈	semidirect
235. C₄	#15689503	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×#15689503	2
C₂×C₄×Dic₁₈₈	2
C₄×#13094024	2
#20244249	2
#20244190	2
#20243548	2
C₄²×C₈×C₄₇	2
C₄×#494	47

Subgroups — C4×#15689503 #26224715

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15689503 #26224715