Subgroups — C₂²×C₄×#10625324 #27079663

Order 12032.

3950 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#10625324	C₂	1	2	yes	maximal
2	C₂²×C₄²×D₄₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#10625324	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#13093934	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₄×D₄₇	C₄	1	2	yes	—
8	C₂³×C₄×D₄₇	C₂²	1	1	yes	characteristic
9	C₂²×#10625324	C₄	1	32	yes	direct product factor
10	C₂²×#10625324	C₂²	1	24	yes	—
11	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
12	C₂²×C₄×Dic₄₇	C₄	1	2	yes	—
13	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
14	C₂×C₄²×D₄₇	C₄	1	12	yes	—
15	C₂×C₄²×D₄₇	C₂²	1	12	yes	—
16	C₄×#10625324	C₂²	1	64	yes	direct product factor
17	C₂×#13093934	C₂²	1	24	yes	complemented
18	C₂×C₄×C₄₇⋊C₈	C₂²	1	12	yes	complemented
19	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
20	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₄₇	C₂²	1	12	yes	complemented
22	C₂⁴×D₄₇	C₂×C₄	1	1	yes	characteristic
23	C₂³×Dic₄₇	C₂×C₄	1	2	yes	—
24	C₂³×Dic₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×D₄₇	—	2	16	no	—
26	C₂²×C₄×D₄₇	C₂×C₄	1	24	yes	—
27	C₂²×C₄×D₄₇	C₂×C₄	1	6	yes	—
28	C₂²×C₄×D₄₇	C₂×C₄	1	16	yes	—
29	C₂²×C₄×D₄₇	C₂³	1	6	yes	—
30	C₂×#10625324	C₂×C₄	1	192	yes	direct product factor
31	C₂×#10625324	C₂³	1	32	yes	—
32	C₂²×C₄₇⋊C₈	C₂×C₄	1	16	yes	complemented
33	C₂²×C₄₇⋊C₈	C₂³	1	12	yes	—
34	C₂×C₄×Dic₄₇	C₂×C₄	1	6	yes	—
35	C₂×C₄×Dic₄₇	C₂×C₄	1	24	yes	—
36	C₂×C₄×Dic₄₇	C₂³	1	6	yes	—
37	C₄²×D₄₇	C₂×C₄	1	48	yes	—
38	C₄²×D₄₇	C₂³	1	16	yes	—
39	#13093934	C₂³	1	32	yes	complemented
40	C₄×C₄₇⋊C₈	C₂³	1	16	yes	complemented
41	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
42	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₄₇	C₂×C₄	1	16	yes	complemented
44	C₂²×C₈×C₄₇	C₂³	1	12	yes	—
45	C₂×C₄²×C₄₇	C₂×C₄	1	6	yes	—
46	C₂×C₄²×C₄₇	C₂³	1	6	yes	—
47	C₄×C₈×C₄₇	C₂³	1	16	yes	complemented
48	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
49	C₂³×D₄₇	—	2	8	no	—
50	C₂³×D₄₇	C₄²	1	8	yes	—
51	C₂³×D₄₇	C₂²×C₄	1	6	yes	—
52	C₂²×C₄×C₄₇	M₁₆	1	16	yes	complemented
53	C₂²×C₄×C₄₇	C₂²×C₄	1	3	yes	—
54	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
55	C₂²×C₄×C₄₇	C₂²×C₄	1	12	yes	—
56	C₂²×C₄×C₄₇	C₂⁴	1	3	yes	—
57	C₂²×Dic₄₇	—	2	8	no	—
58	C₂²×Dic₄₇	C₄²	1	8	yes	—
59	C₂²×Dic₄₇	C₂²×C₄	1	12	yes	—
60	C₂²×Dic₄₇	C₂²×C₄	1	6	yes	—
61	C₂×C₄×D₄₇	—	2	96	no	—
62	C₂×C₄×D₄₇	C₄²	1	48	yes	—
63	C₂×C₄×D₄₇	C₂²×C₄	1	32	yes	—
64	C₂×C₄×D₄₇	C₂²×C₄	1	12	yes	—
65	C₂×C₄×D₄₇	C₂²×C₄	1	48	yes	—
66	C₂×C₄×D₄₇	C₂⁴	1	4	yes	—
67	C₂×C₈×C₄₇	C₂²×C₄	1	48	yes	complemented
68	C₂×C₈×C₄₇	C₂⁴	1	8	yes	—
69	C₄²×C₄₇	C₂²×C₄	1	12	yes	—
70	C₄²×C₄₇	C₂⁴	1	4	yes	—
71	C₄×Dic₄₇	C₂²×C₄	1	12	yes	—
72	C₄×Dic₄₇	C₂²×C₄	1	32	yes	—
73	C₄×Dic₄₇	C₂⁴	1	4	yes	—
74	C₂×C₄₇⋊C₈	C₂²×C₄	1	48	yes	complemented
75	C₂×C₄₇⋊C₈	C₂⁴	1	8	yes	—
76	#10625324	C₂²×C₄	1	128	yes	direct product factor
77	C₂³×C₄₇	C₂×C₄²	1	4	yes	—
78	C₂³×C₄₇	C₂×M₁₆	1	8	yes	—
79	C₂³×C₄₇	C₂³×C₄	1	3	yes	—
80	C₂²×D₄₇	—	2	24	no	—
81	C₂²×D₄₇	—	2	32	no	—
82	C₂²×D₄₇	C₂×C₄²	1	24	yes	—
83	C₂²×D₄₇	C₂³×C₄	1	4	yes	—
84	C₂×C₄×C₄₇	C₂×C₄²	1	12	yes	—
85	C₂×C₄×C₄₇	C₂×M₁₆	1	48	yes	complemented
86	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
87	C₂×C₄×C₄₇	C₂³×C₄	1	12	yes	—
88	C₂×C₄×C₄₇	C₂³×C₄	1	3	yes	—
89	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
90	C₂×Dic₄₇	—	2	24	no	—
91	C₂×Dic₄₇	C₂×C₄²	1	24	yes	—
92	C₂×Dic₄₇	C₂³×C₄	1	8	yes	—
93	C₂×Dic₄₇	C₂³×C₄	1	4	yes	—
94	C₄×D₄₇	—	2	64	no	—
95	C₄×D₄₇	C₂×C₄²	1	48	yes	—
96	C₄×D₄₇	C₂³×C₄	1	16	yes	—
97	C₈×C₄₇	C₂³×C₄	1	16	yes	complemented
98	C₄₇⋊C₈	C₂³×C₄	1	16	yes	complemented
99	C₂²×C₄×M₁₆	—	47	1	no	maximal, Sylow
100	C₂²×C₄₇	C₄×M₁₆	1	16	yes	complemented
101	C₂²×C₄₇	C₂²×C₄²	1	6	yes	—
102	C₂²×C₄₇	C₂²×M₁₆	1	12	yes	—
103	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
104	C₂×D₄₇	—	2	48	no	—
105	C₂×D₄₇	—	2	8	no	—
106	C₂×D₄₇	C₂²×C₄²	1	8	yes	—
107	C₄×C₄₇	C₂²×C₄²	1	6	yes	—
108	C₄×C₄₇	C₂²×M₁₆	1	16	yes	complemented
109	C₄×C₄₇	C₂⁴×C₄	1	2	yes	—
110	Dic₄₇	—	2	8	no	—
111	Dic₄₇	C₂²×C₄²	1	8	yes	—
112	C₂³×M₁₆	—	47	2	no	—
113	C₂³×C₄²	—	47	1	no	—
114	C₂×C₄×M₁₆	—	47	24	no	—
115	C₂²×#97	—	47	2	no	—
116	C₂²×C₄×C₈	—	47	1	no	—
117	C₂²×C₄×C₈	D₄₇	1	1	yes	characteristic, complemented
118	C₂×C₄₇	C₂×C₄×M₁₆	1	12	yes	complemented
119	C₂×C₄₇	C₂³×C₄²	1	1	yes	characteristic
120	C₂×C₄₇	C₂³×M₁₆	1	2	yes	—
121	D₄₇	—	2	8	no	—
122	C₂⁴×C₄	—	47	1	no	—
123	C₂⁴×C₄	—	47	2	no	—
124	C₂²×M₁₆	—	47	32	no	—
125	C₂²×M₁₆	—	47	24	no	—
126	C₂³×C₈	—	47	2	no	—
127	C₂³×C₈	C₂×D₄₇	1	2	yes	—
128	C₂²×C₄²	—	47	2	no	—
129	C₂²×C₄²	—	47	1	no	—
130	C₂²×C₄²	—	47	12	no	—
131	C₂²×C₄²	—	47	12	no	—
132	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
133	C₄×M₁₆	—	47	64	no	—
134	C₂×#97	—	47	24	no	—
135	C₂×C₄×C₈	—	47	12	no	—
136	C₂×C₄×C₈	C₂×D₄₇	1	12	yes	complemented
137	C₄₇	C₂²×C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
138	C₂⁵	—	47	1	no	—
139	C₂³×C₄	—	47	1	no	—
140	C₂³×C₄	—	47	16	no	—
141	C₂³×C₄	—	47	6	no	—
142	C₂³×C₄	—	47	6	no	—
143	C₂³×C₄	—	47	2	no	—
144	C₂³×C₄	—	47	24	no	—
145	C₂³×C₄	—	94	16	no	—
146	C₂³×C₄	C₄×D₄₇	1	2	yes	—
147	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
148	C₂×M₁₆	—	47	192	no	—
149	C₂×M₁₆	—	47	32	no	—
150	C₂²×C₈	—	47	16	no	—
151	C₂²×C₈	—	47	12	no	—
152	C₂²×C₈	C₄×D₄₇	1	16	yes	complemented
153	C₂²×C₈	C₂²×D₄₇	1	12	yes	—
154	C₂×C₄²	—	47	24	no	—
155	C₂×C₄²	—	47	6	no	—
156	C₂×C₄²	—	47	6	no	—
157	C₂×C₄²	—	47	48	no	—
158	C₂×C₄²	—	47	16	no	—
159	C₂×C₄²	C₄×D₄₇	1	6	yes	—
160	C₂×C₄²	C₂²×D₄₇	1	6	yes	—
161	#97	—	47	32	no	—
162	C₄×C₈	—	47	16	no	—
163	C₄×C₈	C₂²×D₄₇	1	16	yes	complemented
164	C₂⁴	—	47	8	no	—
165	C₂⁴	—	47	6	no	—
166	C₂⁴	—	94	8	no	—
167	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
168	C₂²×C₄	—	47	8	no	—
169	C₂²×C₄	—	47	6	no	—
170	C₂²×C₄	—	47	48	no	—
171	C₂²×C₄	—	47	12	no	—
172	C₂²×C₄	—	47	48	no	—
173	C₂²×C₄	—	47	4	no	—
174	C₂²×C₄	—	47	12	no	—
175	C₂²×C₄	—	47	32	no	—
176	C₂²×C₄	—	94	8	no	—
177	C₂²×C₄	—	94	96	no	—
178	C₂²×C₄	#10625324	1	16	yes	direct product factor
179	C₂²×C₄	C₂×C₄×D₄₇	1	3	yes	—
180	C₂²×C₄	C₂×C₄×D₄₇	1	8	yes	—
181	C₂²×C₄	C₂×C₄×D₄₇	1	12	yes	—
182	C₂²×C₄	C₂³×D₄₇	1	3	yes	—
183	M₁₆	—	47	128	no	—
184	C₂×C₈	—	47	48	no	—
185	C₂×C₈	—	47	8	no	—
186	C₂×C₈	C₂×C₄×D₄₇	1	48	yes	complemented
187	C₂×C₈	C₂³×D₄₇	1	8	yes	—
188	C₄²	—	47	32	no	—
189	C₄²	—	47	12	no	—
190	C₄²	—	47	4	no	—
191	C₄²	C₂×C₄×D₄₇	1	12	yes	—
192	C₄²	C₂³×D₄₇	1	4	yes	—
193	C₂³	—	47	24	no	—
194	C₂³	—	47	4	no	—
195	C₂³	—	94	32	no	—
196	C₂³	—	94	24	no	—
197	C₂³	C₄²×D₄₇	1	4	yes	—
198	C₂³	C₂×#10625324	1	8	yes	—
199	C₂³	C₂²×C₄×D₄₇	1	3	yes	—
200	C₂×C₄	—	47	24	no	—
201	C₂×C₄	—	47	4	no	—
202	C₂×C₄	—	47	48	no	—
203	C₂×C₄	—	47	16	no	—
204	C₂×C₄	—	47	8	no	—
205	C₂×C₄	—	94	24	no	—
206	C₂×C₄	—	94	64	no	—
207	C₂×C₄	C₄²×D₄₇	1	12	yes	—
208	C₂×C₄	C₂×#10625324	1	48	yes	direct product factor
209	C₂×C₄	C₂²×C₄×D₄₇	1	3	yes	—
210	C₂×C₄	C₂²×C₄×D₄₇	1	12	yes	—
211	C₂×C₄	C₂²×C₄×D₄₇	1	8	yes	—
212	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
213	C₈	—	47	16	no	—
214	C₈	C₂²×C₄×D₄₇	1	16	yes	complemented
215	C₂²	—	47	8	no	—
216	C₂²	—	94	8	no	—
217	C₂²	—	94	48	no	—
218	C₂²	C₄×#10625324	1	16	yes	direct product factor
219	C₂²	C₂×C₄²×D₄₇	1	6	yes	—
220	C₂²	C₂²×#10625324	1	12	yes	—
221	C₂²	C₂³×C₄×D₄₇	1	1	yes	characteristic
222	C₄	—	47	8	no	—
223	C₄	—	94	8	no	—
224	C₄	C₂×C₄²×D₄₇	1	6	yes	—
225	C₄	C₂²×#10625324	1	16	yes	direct product factor
226	C₄	C₂³×C₄×D₄₇	1	2	yes	—
227	C₂	—	94	8	no	—
228	C₂	C₂×C₄×#10625324	1	12	yes	direct product factor
229	C₂	C₂²×C₄²×D₄₇	1	1	yes	characteristic
230	C₂	C₂³×#10625324	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄²	C₂×D₄₇
commutator	C₂×C₄₇	C₂³×C₄²
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂²×C₄×C₈×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂²×C₄
prime radical	C₂⁴×D₄₇	C₂×C₄
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	C₂²×C₄×C₈	D₄₇
hypocenter	C₄₇	C₂²×C₄×M₁₆
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×#10625324	C₂	direct (N × H)
4. C₂²×#13093934	C₂	semidirect
5. C₂²×C₄×C₄₇⋊C₈	C₂	semidirect
6. C₂²×C₄×C₈×C₄₇	C₂	semidirect
9. C₂²×#10625324	C₄	direct (N × H)
16. C₄×#10625324	C₂²	direct (N × H)
17. C₂×#13093934	C₂²	semidirect
18. C₂×C₄×C₄₇⋊C₈	C₂²	semidirect
21. C₂×C₄×C₈×C₄₇	C₂²	semidirect
30. C₂×#10625324	C₂×C₄	direct (N × H)
32. C₂²×C₄₇⋊C₈	C₂×C₄	semidirect
39. #13093934	C₂³	semidirect
40. C₄×C₄₇⋊C₈	C₂³	semidirect
43. C₂²×C₈×C₄₇	C₂×C₄	semidirect
47. C₄×C₈×C₄₇	C₂³	semidirect
52. C₂²×C₄×C₄₇	M₁₆	semidirect
67. C₂×C₈×C₄₇	C₂²×C₄	semidirect
74. C₂×C₄₇⋊C₈	C₂²×C₄	semidirect
76. #10625324	C₂²×C₄	direct (N × H)
85. C₂×C₄×C₄₇	C₂×M₁₆	semidirect
97. C₈×C₄₇	C₂³×C₄	semidirect
98. C₄₇⋊C₈	C₂³×C₄	semidirect
100. C₂²×C₄₇	C₄×M₁₆	semidirect
108. C₄×C₄₇	C₂²×M₁₆	semidirect
117. C₂²×C₄×C₈	D₄₇	semidirect
118. C₂×C₄₇	C₂×C₄×M₁₆	semidirect
136. C₂×C₄×C₈	C₂×D₄₇	semidirect
137. C₄₇	C₂²×C₄×M₁₆	semidirect
152. C₂²×C₈	C₄×D₄₇	semidirect
163. C₄×C₈	C₂²×D₄₇	semidirect
178. C₂²×C₄	#10625324	direct (N × H)
186. C₂×C₈	C₂×C₄×D₄₇	semidirect
208. C₂×C₄	C₂×#10625324	direct (N × H)
214. C₈	C₂²×C₄×D₄₇	semidirect
218. C₂²	C₄×#10625324	direct (N × H)
225. C₄	C₂²×#10625324	direct (N × H)
228. C₂	C₂×C₄×#10625324	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×#10625324	2
C₂²×C₄²×D₄₇	2
C₂×C₄×#10625324	2
C₂²×#13093934	2
C₂²×C₄×C₄₇⋊C₈	2
C₂²×C₄×C₈×C₄₇	2
C₂²×C₄×M₁₆	47

Subgroups — C22×C4×#10625324 #27079663

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×#10625324 #27079663