Subgroups — C₄×#15689081 #26213620

Order 12032.

1526 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689081	C₂	1	4	yes	maximal
2	C₂²×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄×#13093983	C₂	1	1	yes	maximal, characteristic
4	#20243076	C₂	1	4	yes	maximal, complemented
5	C₄×#13093934	C₂	1	2	yes	maximal, complemented
6	C₄×C₈×Dic₄₇	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
9	C₂×#13093983	C₂²	1	2	yes	—
10	#15689081	C₄	1	32	yes	direct product factor
11	C₂×#13093934	C₂²	1	8	yes	—
12	C₂×C₄×C₄₇⋊C₈	C₄	1	8	yes	complemented
13	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
16	C₂×C₈×Dic₄₇	C₂²	1	8	yes	—
17	C₄×C₄⋊Dic₄₇	C₄	1	4	yes	—
18	C₄×C₂²⋊Dic₄₇	C₄	1	4	yes	—
19	#15688969	C₂²	1	2	yes	—
20	C₄²×Dic₄₇	C₂²	1	2	yes	—
21	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
22	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
23	C₄×M₁₆×C₄₇	C₄	1	8	yes	complemented
24	C₂×#97×C₄₇	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
26	C₂²×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
27	C₂²×C₄₇⋊C₈	C₂×C₄	1	16	yes	—
28	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
29	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
30	#13093983	C₂×C₄	1	16	yes	—
31	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
32	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	4	yes	—
33	C₂×C₄×Dic₄₇	C₂×C₄	1	8	yes	—
34	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
35	#13093934	C₂×C₄	1	32	yes	complemented
36	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
37	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
38	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
39	C₈×Dic₄₇	C₂×C₄	1	32	yes	complemented
40	C₄₇⋊#95	C₂×C₄	1	8	yes	—
41	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
42	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
43	C₂×M₁₆×C₄₇	C₂×C₄	1	16	yes	—
44	C₂×M₁₆×C₄₇	C₂×C₄	1	8	yes	—
45	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
46	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
48	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
51	#97×C₄₇	C₂×C₄	1	8	yes	—
52	C₄×C₈×C₄₇	C₂×C₄	1	8	yes	—
53	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	—	2	8	no	—
55	C₂²×C₄×C₄₇	C₄²	1	8	yes	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
60	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
63	C₂²×Dic₄₇	C₂²×C₄	1	8	yes	—
64	M₁₆×C₄₇	C₄²	1	32	yes	complemented
65	C₂×C₈×C₄₇	C₂²×C₄	1	16	yes	—
66	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
67	C₄²×C₄₇	C₄²	1	8	yes	—
68	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
69	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
70	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
71	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
72	C₂²⋊Dic₄₇	C₄²	1	16	yes	—
73	C₄⋊Dic₄₇	C₄²	1	16	yes	—
74	C₄×Dic₄₇	—	2	16	no	—
75	C₄×Dic₄₇	C₂²×C₄	1	16	yes	—
76	C₂×C₄₇⋊C₈	C₄²	1	32	yes	complemented
77	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
78	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
79	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
80	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
81	C₂³×C₄₇	—	2	4	no	—
82	C₂³×C₄₇	C₂×C₄²	1	4	yes	—
83	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
84	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
85	C₂×C₄×C₄₇	—	2	4	no	—
86	C₂×C₄×C₄₇	—	2	16	no	—
87	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
88	C₂×C₄×C₄₇	C₂×C₄²	1	8	yes	—
89	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
90	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
91	C₂×C₄×C₄₇	C₂×C₄²	1	8	yes	—
92	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
93	C₂×C₄×C₄₇	#131	1	8	yes	—
94	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
96	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
97	C₂×Dic₄₇	—	2	16	no	—
98	C₂×Dic₄₇	C₂×C₄²	1	16	yes	—
99	C₈×C₄₇	C₂×C₄²	1	16	yes	—
100	C₄₇⋊C₈	C₂×C₄²	1	8	yes	—
101	C₄₇⋊C₈	C₂×C₄²	1	8	yes	—
102	C₄×#405	—	47	1	no	maximal, Sylow
103	C₂²×C₄₇	—	2	4	no	—
104	C₂²×C₄₇	—	2	8	no	—
105	C₂²×C₄₇	C₄³	1	4	yes	—
106	C₂²×C₄₇	C₂²×C₄²	1	2	yes	—
107	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
108	C₂²×C₄₇	C₂×#131	1	4	yes	—
109	C₄×C₄₇	—	2	4	no	—
110	C₄×C₄₇	C₄³	1	4	yes	—
111	C₄×C₄₇	#405	1	8	yes	complemented
112	C₄×C₄₇	C₂²×C₄²	1	2	yes	—
113	C₄×C₄₇	C₂²×C₄²	1	2	yes	—
114	Dic₄₇	—	2	16	no	—
115	C₂×#405	—	47	4	no	—
116	C₂×C₄×M₁₆	D₄₇	1	1	yes	characteristic
117	C₂²×C₄×C₈	—	47	1	no	—
118	C₄×#117	—	47	1	no	—
119	#1726	—	47	4	no	—
120	C₄×#97	—	47	2	no	—
121	C₄²×C₈	—	47	2	no	—
122	C₂×C₄₇	—	2	4	no	—
123	C₂×C₄₇	C₂×C₄³	1	1	yes	characteristic
124	C₂×C₄₇	C₂×#405	1	2	yes	—
125	C₂×C₄₇	C₄×#131	1	4	yes	—
126	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
127	C₂³×C₈	—	47	2	no	—
128	C₂×#117	—	47	2	no	—
129	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
130	#405	—	47	32	no	—
131	C₄×M₁₆	Dic₄₇	1	8	yes	complemented
132	C₂×#97	—	47	8	no	—
133	C₂×#97	C₂×D₄₇	1	2	yes	—
134	C₂×C₄×C₈	—	47	8	no	—
135	C₂×C₄×C₈	—	47	8	no	—
136	C₂×C₄×C₈	—	47	2	no	—
137	C₂×C₄×C₈	—	47	1	no	—
138	C₂×C₄×C₈	—	47	1	no	—
139	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
140	C₄×C₄⋊C₄	—	47	4	no	—
141	C₄×C₂²⋊C₄	—	47	4	no	—
142	#376	—	47	2	no	—
143	C₄³	—	47	2	no	—
144	C₄₇	C₄×#405	1	1	yes	characteristic, complemented, Sylow
145	C₂³×C₄	C₄×D₄₇	1	2	yes	—
146	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
147	C₂×M₁₆	C₄×D₄₇	1	16	yes	—
148	C₂×M₁₆	C₂×Dic₄₇	1	8	yes	—
149	C₂²×C₈	—	47	8	no	—
150	C₂²×C₈	—	47	16	no	—
151	C₂²×C₈	—	47	2	no	—
152	C₂²×C₈	—	47	2	no	—
153	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
154	#117	—	47	16	no	—
155	C₂×C₄⋊C₄	—	47	4	no	—
156	C₂×C₂²⋊C₄	—	47	4	no	—
157	C₂×C₄²	—	47	4	no	—
158	C₂×C₄²	—	47	8	no	—
159	C₂×C₄²	C₄×D₄₇	1	2	yes	—
160	C₂×C₄²	C₄×D₄₇	1	4	yes	—
161	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
162	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
163	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
164	#97	—	47	32	no	—
165	#97	C₂×Dic₄₇	1	8	yes	—
166	C₄×C₈	—	47	32	no	—
167	C₄×C₈	—	47	8	no	—
168	C₄×C₈	—	47	4	no	—
169	C₄×C₈	—	47	4	no	—
170	C₄×C₈	C₂×Dic₄₇	1	8	yes	—
171	#95	—	47	8	no	—
172	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
173	C₂²×C₄	—	2	8	no	—
174	C₂²×C₄	—	47	8	no	—
175	C₂²×C₄	C₄×Dic₄₇	1	8	yes	—
176	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
177	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
178	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
179	C₂²×C₄	C₂×C₄×D₄₇	1	8	yes	—
180	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
181	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
182	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
183	M₁₆	C₄×Dic₄₇	1	32	yes	complemented
184	C₂×C₈	—	47	32	no	—
185	C₂×C₈	—	47	4	no	—
186	C₂×C₈	—	47	4	no	—
187	C₂×C₈	—	47	8	no	—
188	C₂×C₈	—	47	8	no	—
189	C₂×C₈	C₂×C₄×D₄₇	1	16	yes	—
190	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
191	C₄⋊C₄	—	47	16	no	—
192	C₂²⋊C₄	—	47	16	no	—
193	C₄²	—	47	16	no	—
194	C₄²	—	94	16	no	—
195	C₄²	C₄×Dic₄₇	1	8	yes	—
196	C₄²	C₂×C₄×D₄₇	1	2	yes	—
197	C₄²	C₂×C₄×D₄₇	1	2	yes	—
198	C₄²	C₂²×Dic₄₇	1	2	yes	—
199	C₄²	C₂²×Dic₄₇	1	2	yes	—
200	C₂³	—	2	4	no	—
201	C₂³	C₄²×D₄₇	1	4	yes	—
202	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
203	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	16	no	—
206	C₂×C₄	—	47	16	no	—
207	C₂×C₄	—	94	16	no	—
208	C₂×C₄	C₄²×D₄₇	1	4	yes	—
209	C₂×C₄	C₄²×D₄₇	1	8	yes	—
210	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
211	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
212	C₂×C₄	C₂×C₄×Dic₄₇	1	8	yes	—
213	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
214	C₂×C₄	#13094032	1	8	yes	—
215	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
216	C₂×C₄	C₂²×C₄×D₄₇	1	4	yes	—
217	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
218	C₈	—	47	8	no	—
219	C₈	—	47	8	no	—
220	C₈	C₂×C₄×Dic₄₇	1	16	yes	—
221	C₂²	—	2	4	no	—
222	C₂²	—	2	8	no	—
223	C₂²	C₄²×Dic₄₇	1	4	yes	—
224	C₂²	C₂×C₄²×D₄₇	1	2	yes	—
225	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
226	C₂²	C₂×#13094032	1	4	yes	—
227	C₄	—	2	4	no	—
228	C₄	—	94	16	no	—
229	C₄	C₄²×Dic₄₇	1	4	yes	—
230	C₄	#15689081	1	8	yes	direct product factor
231	C₄	C₂×C₄²×D₄₇	1	2	yes	—
232	C₄	C₂²×C₄×Dic₄₇	1	2	yes	—
233	C₂	—	2	4	no	—
234	C₂	C₂×C₄²×Dic₄₇	1	1	yes	characteristic
235	C₂	C₂×#15689081	1	2	yes	—
236	C₂	C₄×#13094032	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₄₇
commutator	C₂×C₄₇	C₂×C₄³
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄×M₁₆×C₄₇	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₄₇
hypocenter	C₄₇	C₄×#405
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
4. #20243076	C₂	semidirect
5. C₄×#13093934	C₂	semidirect
6. C₄×C₈×Dic₄₇	C₂	semidirect
10. #15689081	C₄	direct (N × H)
12. C₂×C₄×C₄₇⋊C₈	C₄	semidirect
23. C₄×M₁₆×C₄₇	C₄	semidirect
35. #13093934	C₂×C₄	semidirect
39. C₈×Dic₄₇	C₂×C₄	semidirect
64. M₁₆×C₄₇	C₄²	semidirect
76. C₂×C₄₇⋊C₈	C₄²	semidirect
111. C₄×C₄₇	#405	semidirect
131. C₄×M₁₆	Dic₄₇	semidirect
144. C₄₇	C₄×#405	semidirect
183. M₁₆	C₄×Dic₄₇	semidirect
230. C₄	#15689081	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×#15689081	2
C₂²×C₄×C₄₇⋊C₈	2
C₄×#13093983	2
#20243076	2
C₄×#13093934	2
C₄×C₈×Dic₄₇	2
C₂×C₄×M₁₆×C₄₇	2
C₄×#405	47

Subgroups — C4×#15689081 #26213620

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15689081 #26213620