Subgroups — C₄×#15688886 #26121459

Order 12032.

1182 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₄⋊Dic₄₇	C₂	1	2	yes	maximal
2	C₂×#15688886	C₂	1	4	yes	maximal
3	C₂²×C₄×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
5	C₂²×C₄×Dic₄₇	—	2	2	no	—
6	C₄×C₄⋊Dic₄₇	—	2	4	no	—
7	C₄×C₄⋊Dic₄₇	C₄	1	8	yes	complemented
8	C₂×C₄₇⋊#95	—	2	2	no	—
9	#15688886	C₄	1	16	yes	direct product factor
10	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
11	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₄₇	—	2	4	no	—
13	C₂×C₄×C₈×C₄₇	C₄	1	4	yes	complemented
14	C₂³×Dic₄₇	—	2	4	no	—
15	C₂×C₄⋊Dic₄₇	—	2	4	no	—
16	C₂×C₄⋊Dic₄₇	—	2	8	no	—
17	C₂×C₄⋊Dic₄₇	C₂×C₄	1	8	yes	—
18	C₂×C₄⋊Dic₄₇	C₂×C₄	1	16	yes	—
19	C₂×C₄×Dic₄₇	—	2	8	no	—
20	C₂×C₄×Dic₄₇	—	4	4	no	—
21	C₂×C₄×Dic₄₇	—	4	4	no	—
22	C₄₇⋊#95	—	2	8	no	—
23	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₄₇	—	2	4	no	—
27	C₂²×C₈×C₄₇	—	2	4	no	—
28	C₂²×C₈×C₄₇	C₂×C₄	1	8	yes	—
29	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₄₇	D₄	1	4	yes	—
33	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
34	C₄×C₈×C₄₇	—	2	8	no	—
35	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	—	2	4	no	—
37	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
38	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
39	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
40	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
41	C₂²×C₄×C₄₇	D₈	1	2	yes	—
42	C₂²×C₄×C₄₇	SD₁₆	1	4	yes	—
43	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
44	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
46	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
47	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	#41	1	4	yes	—
50	C₂²×Dic₄₇	—	2	8	no	—
51	C₂²×Dic₄₇	—	2	16	no	—
52	C₂²×Dic₄₇	—	4	4	no	—
53	C₂²×Dic₄₇	—	4	4	no	—
54	C₂²×Dic₄₇	—	4	8	no	—
55	C₂×C₈×C₄₇	—	2	4	no	—
56	C₂×C₈×C₄₇	—	2	4	no	—
57	C₂×C₈×C₄₇	—	2	16	no	—
58	C₂×C₈×C₄₇	C₄²	1	8	yes	complemented
59	C₄²×C₄₇	C₄²	1	4	yes	—
60	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
61	C₄²×C₄₇	C₄⋊C₄	1	8	yes	—
62	C₄⋊Dic₄₇	—	2	8	no	—
63	C₄⋊Dic₄₇	—	2	16	no	—
64	C₄⋊Dic₄₇	C₄²	1	16	yes	complemented
65	C₄×Dic₄₇	—	4	16	no	—
66	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
67	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂³×C₄₇	C₄×D₄	1	4	yes	—
69	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
70	C₂³×C₄₇	C₂×SD₁₆	1	2	yes	—
71	C₂³×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
72	C₂³×C₄₇	#135	1	4	yes	—
73	C₂×C₄×C₄₇	—	2	8	no	—
74	C₂×C₄×C₄₇	—	2	4	no	—
75	C₂×C₄×C₄₇	#95	1	4	yes	—
76	C₂×C₄×C₄₇	#95	1	4	yes	—
77	C₂×C₄×C₄₇	#102	1	8	yes	—
78	C₂×C₄×C₄₇	#103	1	8	yes	—
79	C₂×C₄×C₄₇	#106	1	4	yes	—
80	C₂×C₄×C₄₇	#107	1	4	yes	—
81	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
82	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
83	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
84	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
86	C₂×C₄×C₄₇	#117	1	4	yes	—
87	C₂×C₄×C₄₇	#117	1	2	yes	—
88	C₂×C₄×C₄₇	C₄×D₄	1	8	yes	—
89	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
90	C₂×Dic₄₇	—	2	16	no	—
91	C₂×Dic₄₇	—	4	8	no	—
92	C₂×Dic₄₇	—	4	8	no	—
93	C₂×Dic₄₇	—	4	16	no	—
94	C₂×Dic₄₇	—	4	16	no	—
95	C₈×C₄₇	—	2	8	no	—
96	C₄×#340	—	47	1	no	maximal, Sylow
97	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
98	C₂²×C₄₇	C₄×C₂²⋊C₄	1	4	yes	—
99	C₂²×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
100	C₂²×C₄₇	C₂×#102	1	2	yes	—
101	C₂²×C₄₇	C₂×#103	1	2	yes	—
102	C₂²×C₄₇	#416	1	4	yes	—
103	C₂²×C₄₇	C₂×#106	1	1	yes	characteristic
104	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
105	C₂²×C₄₇	#427	1	2	yes	—
106	C₂²×C₄₇	C₄×D₈	1	4	yes	—
107	C₂²×C₄₇	C₄×SD₁₆	1	8	yes	—
108	C₂²×C₄₇	C₄×Q₁₆	1	4	yes	—
109	C₄×C₄₇	—	2	4	no	—
110	C₄×C₄₇	#340	1	8	yes	complemented
111	C₄×C₄₇	C₄³	1	1	yes	characteristic
112	C₄×C₄₇	#376	1	1	yes	characteristic
113	C₄×C₄₇	C₄×C₂²⋊C₄	1	2	yes	—
114	C₄×C₄₇	C₄×C₄⋊C₄	1	4	yes	—
115	Dic₄₇	—	4	16	no	—
116	C₂²×C₄×C₈	D₄₇	1	1	yes	characteristic
117	C₂×C₄×C₄⋊C₄	—	47	2	no	—
118	C₂×#340	—	47	4	no	—
119	C₂×C₄₇	C₄×#95	1	1	yes	characteristic
120	C₂×C₄₇	C₂×#340	1	1	yes	characteristic
121	C₂×C₄₇	#1735	1	1	yes	characteristic
122	C₂×C₄₇	C₄×#102	1	4	yes	—
123	C₂×C₄₇	C₄×#103	1	4	yes	—
124	C₂×C₄₇	C₄×#106	1	2	yes	—
125	C₂×C₄₇	C₄×#107	1	2	yes	—
126	C₂³×C₈	C₂×D₄₇	1	2	yes	—
127	C₂²×C₄⋊C₄	—	47	4	no	—
128	C₂²×C₄²	—	94	2	no	—
129	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
130	C₂×C₄×C₈	—	2	4	no	—
131	C₂×C₄×C₈	Dic₄₇	1	4	yes	complemented
132	C₄×C₄⋊C₄	—	47	8	no	—
133	C₄×C₄⋊C₄	—	94	4	no	—
134	C₂×#95	—	94	2	no	—
135	#340	—	47	16	no	—
136	C₄₇	C₄×#340	1	1	yes	characteristic, complemented, Sylow
137	C₂³×C₄	—	94	4	no	—
138	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
139	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
140	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
141	C₂²×C₈	—	2	4	no	—
142	C₂²×C₈	—	2	4	no	—
143	C₂²×C₈	C₄×D₄₇	1	8	yes	—
144	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
145	C₂×C₄⋊C₄	—	47	16	no	—
146	C₂×C₄⋊C₄	—	47	8	no	—
147	C₂×C₄⋊C₄	—	94	8	no	—
148	C₂×C₄⋊C₄	—	94	4	no	—
149	C₂×C₄²	—	94	8	no	—
150	C₂×C₄²	—	188	4	no	—
151	C₂×C₄²	—	188	4	no	—
152	C₂×C₄²	Dic₉₄	1	2	yes	—
153	C₂×C₄²	C₄×D₄₇	1	4	yes	—
154	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
155	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
156	C₄×C₈	—	2	8	no	—
157	#95	—	94	8	no	—
158	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
159	C₂²×C₄	—	2	4	no	—
160	C₂²×C₄	—	94	8	no	—
161	C₂²×C₄	—	94	16	no	—
162	C₂²×C₄	—	188	4	no	—
163	C₂²×C₄	—	188	4	no	—
164	C₂²×C₄	—	188	8	no	—
165	C₂²×C₄	#10625325	1	4	yes	—
166	C₂²×C₄	D₃₇₆	1	2	yes	—
167	C₂²×C₄	Dic₁₈₈	1	2	yes	—
168	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
169	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
170	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
171	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
172	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
174	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
175	C₂²×C₄	#10625350	1	4	yes	—
176	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	2	16	no	—
181	C₂×C₈	C₄×Dic₄₇	1	8	yes	complemented
182	C₄⋊C₄	—	47	16	no	—
183	C₄⋊C₄	—	94	8	no	—
184	C₄⋊C₄	—	94	16	no	—
185	C₄²	—	188	16	no	—
186	C₄²	C₄×Dic₄₇	1	4	yes	—
187	C₄²	Dic₄₇⋊C₄	1	8	yes	—
188	C₄²	C₂²⋊Dic₄₇	1	4	yes	—
189	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
190	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
191	C₂³	C₄×D₁₈₈	1	4	yes	—
192	C₂³	C₂×D₃₇₆	1	1	yes	characteristic
193	C₂³	C₂×#10625325	1	2	yes	—
194	C₂³	C₂×Dic₁₈₈	1	1	yes	characteristic
195	C₂³	#13094046	1	4	yes	—
196	C₂×C₄	—	2	8	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	94	16	no	—
199	C₂×C₄	—	188	8	no	—
200	C₂×C₄	—	188	8	no	—
201	C₂×C₄	—	188	16	no	—
202	C₂×C₄	—	188	16	no	—
203	C₂×C₄	C₄₇⋊#95	1	4	yes	—
204	C₂×C₄	C₄₇⋊#95	1	4	yes	—
205	C₂×C₄	#13093946	1	8	yes	—
206	C₂×C₄	#13093949	1	8	yes	—
207	C₂×C₄	#13093956	1	4	yes	—
208	C₂×C₄	#13093958	1	4	yes	—
209	C₂×C₄	C₄²×D₄₇	1	4	yes	—
210	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
211	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
212	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
213	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
214	C₂×C₄	#13093981	1	4	yes	—
215	C₂×C₄	#13093983	1	2	yes	—
216	C₂×C₄	C₄×C₂²⋊D₄₇	1	8	yes	—
217	C₂×C₄	C₄×Dic₉₄	1	4	yes	—
218	C₈	—	2	8	no	—
219	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
220	C₂²	C₄×(C₂×D₄₇)⋊C₄	1	4	yes	—
221	C₂²	C₄×C₄⋊Dic₄₇	1	2	yes	—
222	C₂²	C₂×#13093946	1	2	yes	—
223	C₂²	C₂×#13093949	1	2	yes	—
224	C₂²	#15689120	1	4	yes	—
225	C₂²	C₂×#13093956	1	1	yes	characteristic
226	C₂²	C₂×#13093958	1	1	yes	characteristic
227	C₂²	#15689161	1	2	yes	—
228	C₂²	C₄×D₃₇₆	1	4	yes	—
229	C₂²	C₄×#10625325	1	8	yes	—
230	C₂²	C₄×Dic₁₈₈	1	4	yes	—
231	C₄	—	2	4	no	—
232	C₄	—	188	16	no	—
233	C₄	#15688886	1	8	yes	direct product factor
234	C₄	C₄²×Dic₄₇	1	1	yes	characteristic
235	C₄	#15688969	1	1	yes	characteristic
236	C₄	C₄×C₂²⋊Dic₄₇	1	2	yes	—
237	C₄	C₄×Dic₄₇⋊C₄	1	4	yes	—
238	C₂	C₄×C₄₇⋊#95	1	1	yes	characteristic
239	C₂	C₂×#15688886	1	1	yes	characteristic
240	C₂	#20243105	1	1	yes	characteristic
241	C₂	C₄×#13093946	1	4	yes	—
242	C₂	C₄×#13093949	1	4	yes	—
243	C₂	C₄×#13093956	1	2	yes	—
244	C₂	C₄×#13093958	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	D₁₈₈
commutator	C₄×C₄₇	C₄³
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂²×C₄×C₈×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂²×C₄×C₈	D₄₇
hypocenter	C₄₇	C₄×#340
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
7. C₄×C₄⋊Dic₄₇	C₄	semidirect
9. #15688886	C₄	direct (N × H)
13. C₂×C₄×C₈×C₄₇	C₄	semidirect
58. C₂×C₈×C₄₇	C₄²	semidirect
64. C₄⋊Dic₄₇	C₄²	semidirect
110. C₄×C₄₇	#340	semidirect
131. C₂×C₄×C₈	Dic₄₇	semidirect
136. C₄₇	C₄×#340	semidirect
181. C₂×C₈	C₄×Dic₄₇	semidirect
233. C₄	#15688886	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×C₄⋊Dic₄₇	2
C₂×#15688886	2
C₂²×C₄×C₈×C₄₇	2
C₄×#340	47

Subgroups — C4×#15688886 #26121459

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15688886 #26121459