Subgroups — C₂²×#15688849 #26182067

Order 12032.

1806 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal
2	C₂×#15688849	C₂	1	12	yes	maximal, direct product factor
3	C₂³×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₄₇⋊C₈	—	2	2	no	—
5	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	6	no	—
6	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	12	yes	—
7	#15688849	C₂²	1	16	yes	direct product factor
8	C₂⁴×C₄×C₄₇	C₄	1	1	yes	characteristic
9	C₂⁴×C₄×C₄₇	C₄	1	1	yes	characteristic
10	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
11	C₂²×C₄⋊C₄×C₄₇	—	2	2	no	—
12	C₂²×C₄⋊C₄×C₄₇	—	2	6	no	—
13	C₂²×C₄⋊C₄×C₄₇	C₄	1	6	yes	—
14	C₂²×C₄⋊C₄×C₄₇	C₂²	1	6	yes	—
15	C₂²×C₄₇⋊C₈	—	2	12	no	—
16	C₂²×C₄₇⋊C₈	—	4	8	no	—
17	C₂²⋊(C₄₇⋊C₈)	—	2	12	no	—
18	C₂²⋊(C₄₇⋊C₈)	C₂³	1	8	yes	—
19	C₂⁵×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₄₇	—	2	3	no	—
21	C₂³×C₄×C₄₇	—	2	1	no	—
22	C₂³×C₄×C₄₇	—	2	4	no	—
23	C₂³×C₄×C₄₇	—	2	3	no	—
24	C₂³×C₄×C₄₇	—	2	1	no	—
25	C₂³×C₄×C₄₇	—	4	4	no	—
26	C₂³×C₄×C₄₇	—	4	4	no	—
27	C₂³×C₄×C₄₇	C₈	1	8	yes	complemented
28	C₂³×C₄×C₄₇	C₂×C₄	1	6	yes	—
29	C₂³×C₄×C₄₇	C₂×C₄	1	6	yes	—
30	C₂³×C₄×C₄₇	C₂×C₄	1	3	yes	—
31	C₂³×C₄×C₄₇	D₄	1	2	yes	—
32	C₂³×C₄×C₄₇	D₄	1	6	yes	—
33	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
34	C₂×C₄⋊C₄×C₄₇	—	2	12	no	—
35	C₂×C₄⋊C₄×C₄₇	—	2	12	no	—
36	C₂×C₄⋊C₄×C₄₇	—	2	12	no	—
37	C₂×C₄⋊C₄×C₄₇	—	2	12	no	—
38	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	12	yes	—
39	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
40	C₂⁴×C₄₇	—	2	4	no	—
41	C₂⁴×C₄₇	—	2	4	no	—
42	C₂⁴×C₄₇	C₂²⋊C₄	1	3	yes	—
43	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
44	C₂⁴×C₄₇	C₂×C₈	1	4	yes	—
45	C₂⁴×C₄₇	M₁₆	1	4	yes	—
46	C₂⁴×C₄₇	C₂²×C₄	1	3	yes	—
47	C₂²×C₄×C₄₇	—	2	24	no	—
48	C₂²×C₄×C₄₇	—	2	8	no	—
49	C₂²×C₄×C₄₇	—	2	8	no	—
50	C₂²×C₄×C₄₇	—	2	6	no	—
51	C₂²×C₄×C₄₇	—	2	6	no	—
52	C₂²×C₄×C₄₇	—	2	12	no	—
53	C₂²×C₄×C₄₇	—	2	6	no	—
54	C₂²×C₄×C₄₇	—	2	6	no	—
55	C₂²×C₄×C₄₇	—	4	6	no	—
56	C₂²×C₄×C₄₇	—	4	12	no	—
57	C₂²×C₄×C₄₇	—	4	6	no	—
58	C₂²×C₄×C₄₇	—	4	12	no	—
59	C₂²×C₄×C₄₇	—	4	4	no	—
60	C₂²×C₄×C₄₇	—	4	12	no	—
61	C₂²×C₄×C₄₇	C₂²⋊C₄	1	6	yes	—
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	6	yes	—
63	C₂²×C₄×C₄₇	C₂×C₈	1	24	yes	complemented
64	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₄₇	C₂²×C₄	1	3	yes	—
67	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
68	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
69	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
70	C₄⋊C₄×C₄₇	—	2	24	no	—
71	C₄⋊C₄×C₄₇	—	2	8	no	—
72	C₂×C₄₇⋊C₈	—	2	8	no	—
73	C₂×C₄₇⋊C₈	—	4	24	no	—
74	C₂³×C₄₇	—	2	16	no	—
75	C₂³×C₄₇	—	2	16	no	—
76	C₂³×C₄₇	—	2	6	no	—
77	C₂³×C₄₇	—	2	6	no	—
78	C₂³×C₄₇	—	2	6	no	—
79	C₂³×C₄₇	—	2	6	no	—
80	C₂³×C₄₇	C₂²⋊C₈	1	12	yes	—
81	C₂³×C₄₇	C₂²⋊C₈	1	4	yes	—
82	C₂³×C₄₇	#99	1	4	yes	—
83	C₂³×C₄₇	#101	1	4	yes	—
84	C₂³×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
85	C₂³×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
86	C₂³×C₄₇	C₂²×C₈	1	6	yes	—
87	C₂³×C₄₇	C₂×M₁₆	1	6	yes	—
88	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
89	C₂×C₄×C₄₇	—	2	4	no	—
90	C₂×C₄×C₄₇	—	2	4	no	—
91	C₂×C₄×C₄₇	—	2	24	no	—
92	C₂×C₄×C₄₇	—	2	24	no	—
93	C₂×C₄×C₄₇	—	2	12	no	—
94	C₂×C₄×C₄₇	—	2	12	no	—
95	C₂×C₄×C₄₇	—	2	4	no	—
96	C₂×C₄×C₄₇	—	4	6	no	—
97	C₂×C₄×C₄₇	—	4	3	no	—
98	C₂×C₄×C₄₇	—	4	6	no	—
99	C₂×C₄×C₄₇	—	4	1	no	—
100	C₂×C₄×C₄₇	—	4	12	no	—
101	C₂×C₄×C₄₇	—	4	12	no	—
102	C₂×C₄×C₄₇	—	4	4	no	—
103	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	6	yes	—
104	C₂×C₄×C₄₇	C₂²×C₈	1	8	yes	complemented
105	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
106	C₄₇⋊C₈	—	4	8	no	—
107	C₂²×#324	—	47	1	no	maximal, Sylow
108	C₂²×C₄₇	—	2	3	no	—
109	C₂²×C₄₇	—	2	1	no	—
110	C₂²×C₄₇	—	2	3	no	—
111	C₂²×C₄₇	—	2	1	no	—
112	C₂²×C₄₇	—	2	24	no	—
113	C₂²×C₄₇	—	2	12	no	—
114	C₂²×C₄₇	—	2	12	no	—
115	C₂²×C₄₇	#324	1	16	yes	complemented
116	C₂²×C₄₇	C₂×C₂²⋊C₈	1	6	yes	—
117	C₂²×C₄₇	C₂×C₂²⋊C₈	1	6	yes	—
118	C₂²×C₄₇	C₂×#99	1	6	yes	—
119	C₂²×C₄₇	C₂×#101	1	6	yes	—
120	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
121	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
122	C₂²×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
123	C₄×C₄₇	—	2	8	no	—
124	C₄×C₄₇	—	2	2	no	—
125	C₄×C₄₇	—	2	6	no	—
126	C₄×C₄₇	—	4	4	no	—
127	C₂³×C₄⋊C₄	D₄₇	1	1	yes	characteristic
128	C₂²×C₂²⋊C₈	—	47	2	no	—
129	C₂×#324	—	47	12	no	—
130	C₂×C₄₇	—	2	1	no	—
131	C₂×C₄₇	—	2	1	no	—
132	C₂×C₄₇	—	2	6	no	—
133	C₂×C₄₇	C₂×#324	1	12	yes	complemented
134	C₂×C₄₇	C₂²×C₂²⋊C₈	1	1	yes	characteristic
135	C₂×C₄₇	C₂²×#99	1	1	yes	characteristic
136	C₂×C₄₇	C₂²×#101	1	1	yes	characteristic
137	C₂⁴×C₄	Dic₄₇	1	1	yes	characteristic
138	C₂⁴×C₄	Dic₄₇	1	1	yes	characteristic
139	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
140	C₂³×C₈	—	94	2	no	—
141	C₂²×C₄⋊C₄	—	2	6	no	—
142	C₂²×C₄⋊C₄	—	2	2	no	—
143	C₂²×C₄⋊C₄	Dic₄₇	1	6	yes	—
144	C₂²×C₄⋊C₄	C₂×D₄₇	1	6	yes	—
145	C₂×C₂²⋊C₈	—	47	12	no	—
146	C₂×C₂²⋊C₈	—	94	6	no	—
147	#324	—	47	16	no	—
148	C₄₇	C₂²×#324	1	1	yes	characteristic, complemented, Sylow
149	C₂⁵	C₂×Dic₄₇	1	1	yes	characteristic
150	C₂³×C₄	—	2	4	no	—
151	C₂³×C₄	—	2	3	no	—
152	C₂³×C₄	—	2	1	no	—
153	C₂³×C₄	—	2	3	no	—
154	C₂³×C₄	—	2	1	no	—
155	C₂³×C₄	—	4	4	no	—
156	C₂³×C₄	—	4	4	no	—
157	C₂³×C₄	C₄₇⋊C₈	1	8	yes	complemented
158	C₂³×C₄	C₂×Dic₄₇	1	6	yes	—
159	C₂³×C₄	C₂×Dic₄₇	1	6	yes	—
160	C₂³×C₄	C₂×Dic₄₇	1	3	yes	—
161	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
162	C₂³×C₄	C₂²⋊D₄₇	1	6	yes	—
163	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
164	C₂²×C₈	—	94	12	no	—
165	C₂²×C₈	—	188	8	no	—
166	C₂×C₄⋊C₄	—	2	12	no	—
167	C₂×C₄⋊C₄	—	2	12	no	—
168	C₂×C₄⋊C₄	—	2	12	no	—
169	C₂×C₄⋊C₄	—	2	12	no	—
170	C₂×C₄⋊C₄	C₂×Dic₄₇	1	12	yes	—
171	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
172	C₂²⋊C₈	—	47	8	no	—
173	C₂²⋊C₈	—	94	12	no	—
174	C₂⁴	—	2	4	no	—
175	C₂⁴	—	2	4	no	—
176	C₂⁴	C₂×C₄₇⋊C₈	1	4	yes	—
177	C₂⁴	#10625329	1	4	yes	—
178	C₂⁴	C₂²⋊Dic₄₇	1	3	yes	—
179	C₂⁴	C₂²⋊Dic₄₇	1	1	yes	characteristic
180	C₂⁴	C₂²×Dic₄₇	1	3	yes	—
181	C₂²×C₄	—	2	24	no	—
182	C₂²×C₄	—	2	8	no	—
183	C₂²×C₄	—	2	8	no	—
184	C₂²×C₄	—	2	6	no	—
185	C₂²×C₄	—	2	6	no	—
186	C₂²×C₄	—	2	12	no	—
187	C₂²×C₄	—	2	6	no	—
188	C₂²×C₄	—	2	6	no	—
189	C₂²×C₄	—	4	6	no	—
190	C₂²×C₄	—	4	12	no	—
191	C₂²×C₄	—	4	6	no	—
192	C₂²×C₄	—	4	12	no	—
193	C₂²×C₄	—	4	4	no	—
194	C₂²×C₄	—	4	12	no	—
195	C₂²×C₄	C₂×C₄₇⋊C₈	1	24	yes	complemented
196	C₂²×C₄	C₂²⋊Dic₄₇	1	6	yes	—
197	C₂²×C₄	C₂²⋊Dic₄₇	1	6	yes	—
198	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
199	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
200	C₂²×C₄	C₂²×Dic₄₇	1	3	yes	—
201	C₂²×C₄	C₂×C₂²⋊D₄₇	1	6	yes	—
202	C₂²×C₄	C₂×C₂²⋊D₄₇	1	6	yes	—
203	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
204	C₂×C₈	—	94	8	no	—
205	C₂×C₈	—	188	24	no	—
206	C₄⋊C₄	—	2	24	no	—
207	C₄⋊C₄	—	2	8	no	—
208	C₂³	—	2	6	no	—
209	C₂³	—	2	6	no	—
210	C₂³	—	2	6	no	—
211	C₂³	—	2	6	no	—
212	C₂³	—	2	16	no	—
213	C₂³	—	2	16	no	—
214	C₂³	C₂²⋊(C₄₇⋊C₈)	1	12	yes	—
215	C₂³	C₂²⋊(C₄₇⋊C₈)	1	4	yes	—
216	C₂³	#13093939	1	4	yes	—
217	C₂³	#13093943	1	4	yes	—
218	C₂³	C₂×C₂²⋊Dic₄₇	1	3	yes	—
219	C₂³	C₂×C₂²⋊Dic₄₇	1	3	yes	—
220	C₂³	C₂²×C₄₇⋊C₈	1	6	yes	—
221	C₂³	C₂×#10625329	1	6	yes	—
222	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	24	no	—
225	C₂×C₄	—	2	24	no	—
226	C₂×C₄	—	2	12	no	—
227	C₂×C₄	—	2	12	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	4	6	no	—
231	C₂×C₄	—	4	3	no	—
232	C₂×C₄	—	4	6	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	12	no	—
235	C₂×C₄	—	4	12	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	6	yes	—
238	C₂×C₄	C₂²×C₄₇⋊C₈	1	8	yes	complemented
239	C₂×C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
240	C₈	—	188	8	no	—
241	C₂²	—	2	24	no	—
242	C₂²	—	2	12	no	—
243	C₂²	—	2	12	no	—
244	C₂²	—	2	3	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	3	no	—
247	C₂²	—	2	1	no	—
248	C₂²	#15688849	1	16	yes	direct product factor
249	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	6	yes	—
250	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	6	yes	—
251	C₂²	C₂×#13093939	1	6	yes	—
252	C₂²	C₂×#13093943	1	6	yes	—
253	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
254	C₂²	C₂³×C₄₇⋊C₈	1	1	yes	characteristic
255	C₂²	C₂²×#10625329	1	1	yes	characteristic
256	C₄	—	2	8	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	6	no	—
259	C₄	—	4	4	no	—
260	C₂	—	2	6	no	—
261	C₂	—	2	1	no	—
262	C₂	—	2	1	no	—
263	C₂	C₂×#15688849	1	12	yes	direct product factor
264	C₂	C₂²×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
265	C₂	C₂²×#13093939	1	1	yes	characteristic
266	C₂	C₂²×#13093943	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²⋊Dic₄₇
commutator	C₂²×C₄₇	C₂³×C₈
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂³×C₄⋊C₄×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂²⋊C₄
prime radical	C₂⁵×C₄₇	C₂×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂³×C₄⋊C₄	D₄₇
hypocenter	C₄₇	C₂²×#324
LCS	G ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×#15688849	C₂	direct (N × H)
7. #15688849	C₂²	direct (N × H)
27. C₂³×C₄×C₄₇	C₈	semidirect
63. C₂²×C₄×C₄₇	C₂×C₈	semidirect
104. C₂×C₄×C₄₇	C₂²×C₈	semidirect
115. C₂²×C₄₇	#324	semidirect
133. C₂×C₄₇	C₂×#324	semidirect
148. C₄₇	C₂²×#324	semidirect
157. C₂³×C₄	C₄₇⋊C₈	semidirect
195. C₂²×C₄	C₂×C₄₇⋊C₈	semidirect
238. C₂×C₄	C₂²×C₄₇⋊C₈	semidirect
248. C₂²	#15688849	direct (N × H)
263. C₂	C₂×#15688849	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₂²⋊(C₄₇⋊C₈)	2
C₂×#15688849	2
C₂³×C₄⋊C₄×C₄₇	2
C₂²×#324	47

Subgroups — C22×#15688849 #26182067

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15688849 #26182067