Subgroups — C₂²×#15689177 #27079987

Order 12032.

2830 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×Dic₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689177	C₂	1	24	yes	maximal, direct product factor
3	C₂²×Dic₄₇⋊C₈	C₂	1	2	yes	maximal, complemented
4	C₂²×C₂²⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13093934	C₂	1	2	yes	maximal, complemented
6	C₂²×C₂²⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×Dic₄₇	C₄	1	1	yes	characteristic
8	C₂⁴×Dic₄₇	C₄	1	1	yes	characteristic
9	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
10	C₂²×C₄×Dic₄₇	C₄	1	1	yes	characteristic
11	C₂²×C₄×Dic₄₇	C₄	1	12	yes	—
12	C₂²×C₄×Dic₄₇	C₄	1	1	yes	characteristic
13	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄×Dic₄₇	C₂²	1	12	yes	—
15	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
16	#15689177	C₂²	1	64	yes	direct product factor
17	C₂×Dic₄₇⋊C₈	C₂²	1	24	yes	complemented
18	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	12	yes	—
19	C₂×#13093934	C₂²	1	24	yes	complemented
20	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
21	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₈×C₄₇	C₂²	1	12	yes	—
23	C₂³×Dic₄₇	—	2	8	no	—
24	C₂³×Dic₄₇	—	2	8	no	—
25	C₂³×Dic₄₇	C₂×C₄	1	12	yes	—
26	C₂³×Dic₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂³×Dic₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×Dic₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂³×Dic₄₇	C₂×C₄	1	12	yes	—
30	C₂³×Dic₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄₇⋊C₈	—	2	8	no	—
32	C₂²×C₄₇⋊C₈	C₂³	1	12	yes	—
33	C₂×C₄×Dic₄₇	C₂×C₄	1	12	yes	—
34	C₂×C₄×Dic₄₇	C₂×C₄	1	48	yes	—
35	C₂×C₄×Dic₄₇	C₂×C₄	1	6	yes	—
36	C₂×C₄×Dic₄₇	C₂×C₄	1	12	yes	—
37	C₂×C₄×Dic₄₇	C₂×C₄	1	6	yes	—
38	C₂×C₄×Dic₄₇	C₂³	1	6	yes	—
39	C₂×C₄×Dic₄₇	C₂³	1	6	yes	—
40	C₂×C₄×Dic₄₇	C₂³	1	16	yes	—
41	Dic₄₇⋊C₈	C₂³	1	32	yes	complemented
42	C₂²⋊(C₄₇⋊C₈)	C₂³	1	16	yes	—
43	#13093934	C₂³	1	32	yes	complemented
44	C₂⁵×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₄₇	—	2	8	no	—
46	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₄₇	C₂×C₄	1	6	yes	—
48	C₂³×C₄×C₄₇	C₂³	1	6	yes	—
49	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₄₇	—	2	8	no	—
51	C₂²×C₈×C₄₇	C₂³	1	12	yes	—
52	C₂²⋊C₈×C₄₇	C₂³	1	16	yes	—
53	C₂⁴×C₄₇	—	2	4	no	—
54	C₂⁴×C₄₇	—	2	4	no	—
55	C₂⁴×C₄₇	M₁₆	1	8	yes	—
56	C₂⁴×C₄₇	C₂²×C₄	1	6	yes	—
57	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	—	2	4	no	—
59	C₂²×C₄×C₄₇	—	2	24	no	—
60	C₂²×C₄×C₄₇	—	2	24	no	—
61	C₂²×C₄×C₄₇	C₂²×C₄	1	6	yes	—
62	C₂²×C₄×C₄₇	C₂²×C₄	1	12	yes	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	3	yes	—
64	C₂²×C₄×C₄₇	#41	1	8	yes	—
65	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
66	C₂²×C₄×C₄₇	C₂⁴	1	3	yes	—
67	C₂²×Dic₄₇	—	2	4	no	—
68	C₂²×Dic₄₇	—	2	48	no	—
69	C₂²×Dic₄₇	—	2	4	no	—
70	C₂²×Dic₄₇	—	2	4	no	—
71	C₂²×Dic₄₇	—	2	48	no	—
72	C₂²×Dic₄₇	M₁₆	1	8	yes	—
73	C₂²×Dic₄₇	C₂²×C₄	1	16	yes	—
74	C₂²×Dic₄₇	C₂²×C₄	1	6	yes	—
75	C₂²×Dic₄₇	C₂²×C₄	1	6	yes	—
76	C₂²×Dic₄₇	C₂²×C₄	1	6	yes	—
77	C₂²×Dic₄₇	C₂²×C₄	1	16	yes	—
78	C₂²×Dic₄₇	C₂²×C₄	1	6	yes	—
79	C₂×C₈×C₄₇	—	2	24	no	—
80	C₂×C₈×C₄₇	C₂⁴	1	8	yes	—
81	C₄×Dic₄₇	C₂²×C₄	1	16	yes	—
82	C₄×Dic₄₇	C₂²×C₄	1	12	yes	—
83	C₄×Dic₄₇	C₂²×C₄	1	16	yes	—
84	C₄×Dic₄₇	C₂²×C₄	1	12	yes	—
85	C₄×Dic₄₇	C₂⁴	1	4	yes	—
86	C₄×Dic₄₇	C₂⁴	1	4	yes	—
87	C₂×C₄₇⋊C₈	—	2	24	no	—
88	C₂×C₄₇⋊C₈	C₂⁴	1	8	yes	—
89	C₂³×C₄₇	—	2	12	no	—
90	C₂³×C₄₇	—	2	12	no	—
91	C₂³×C₄₇	—	2	32	no	—
92	C₂³×C₄₇	#117	1	4	yes	—
93	C₂³×C₄₇	C₂×M₁₆	1	4	yes	—
94	C₂³×C₄₇	C₂×M₁₆	1	24	yes	—
95	C₂³×C₄₇	C₂×M₁₆	1	4	yes	—
96	C₂³×C₄₇	C₂³×C₄	1	4	yes	—
97	C₂³×C₄₇	C₂³×C₄	1	3	yes	—
98	C₂×C₄×C₄₇	—	2	12	no	—
99	C₂×C₄×C₄₇	—	2	24	no	—
100	C₂×C₄×C₄₇	—	2	8	no	—
101	C₂×C₄×C₄₇	#117	1	12	yes	—
102	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
103	C₂×C₄×C₄₇	C₂³×C₄	1	3	yes	—
104	C₂×C₄×C₄₇	C₂×#41	1	12	yes	—
105	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
106	C₂×Dic₄₇	—	2	12	no	—
107	C₂×Dic₄₇	—	2	32	no	—
108	C₂×Dic₄₇	—	2	12	no	—
109	C₂×Dic₄₇	—	2	12	no	—
110	C₂×Dic₄₇	—	2	32	no	—
111	C₂×Dic₄₇	C₂×M₁₆	1	24	yes	—
112	C₂×Dic₄₇	C₂³×C₄	1	4	yes	—
113	C₂×Dic₄₇	C₂³×C₄	1	4	yes	—
114	C₂×Dic₄₇	C₂³×C₄	1	4	yes	—
115	C₂×Dic₄₇	C₂³×C₄	1	4	yes	—
116	C₈×C₄₇	—	2	8	no	—
117	C₄₇⋊C₈	—	2	8	no	—
118	C₂²×#432	—	47	1	no	maximal, Sylow
119	C₂²×C₄₇	—	2	4	no	—
120	C₂²×C₄₇	—	2	48	no	—
121	C₂²×C₄₇	—	2	4	no	—
122	C₂²×C₄₇	#432	1	16	yes	complemented
123	C₂²×C₄₇	C₂×#117	1	6	yes	—
124	C₂²×C₄₇	C₂²×M₁₆	1	6	yes	—
125	C₂²×C₄₇	C₂²×M₁₆	1	8	yes	—
126	C₂²×C₄₇	C₂²×M₁₆	1	6	yes	—
127	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
128	C₄×C₄₇	—	2	4	no	—
129	C₄×C₄₇	C₂×#117	1	6	yes	—
130	C₄×C₄₇	C₂²×#41	1	2	yes	—
131	Dic₄₇	—	2	4	no	—
132	Dic₄₇	—	2	4	no	—
133	Dic₄₇	—	2	4	no	—
134	Dic₄₇	C₂²×M₁₆	1	8	yes	—
135	C₂³×C₄²	—	47	1	no	—
136	C₂×#432	—	47	24	no	—
137	C₂²×C₄⋊C₈	—	47	2	no	—
138	C₂²×C₂²⋊C₈	—	47	1	no	—
139	C₂²×C₂²⋊C₈	D₄₇	1	1	yes	characteristic
140	C₂²×#97	—	47	2	no	—
141	C₂×C₄₇	—	2	8	no	—
142	C₂×C₄₇	C₂×#432	1	12	yes	complemented
143	C₂×C₄₇	C₂²×#117	1	1	yes	characteristic
144	C₂×C₄₇	C₂³×M₁₆	1	1	yes	characteristic
145	C₂×C₄₇	C₂³×M₁₆	1	1	yes	characteristic
146	C₂⁴×C₄	—	47	1	no	—
147	C₂⁴×C₄	—	47	1	no	—
148	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
149	C₂³×C₈	—	47	2	no	—
150	C₂³×C₈	C₂×D₄₇	1	2	yes	—
151	C₂²×C₄²	—	47	1	no	—
152	C₂²×C₄²	—	47	12	no	—
153	C₂²×C₄²	—	47	1	no	—
154	C₂²×C₄²	—	47	12	no	—
155	C₂²×C₄²	—	47	1	no	—
156	C₂²×C₄²	—	47	1	no	—
157	#432	—	47	64	no	—
158	C₂×C₄⋊C₈	—	47	24	no	—
159	C₂×C₂²⋊C₈	—	47	12	no	—
160	C₂×C₂²⋊C₈	C₂×D₄₇	1	12	yes	—
161	C₂×#97	—	47	24	no	—
162	C₄₇	C₂²×#432	1	1	yes	characteristic, complemented, Sylow
163	C₂⁵	C₄×D₄₇	1	1	yes	characteristic
164	C₂³×C₄	—	2	8	no	—
165	C₂³×C₄	—	47	1	no	—
166	C₂³×C₄	—	47	12	no	—
167	C₂³×C₄	—	47	1	no	—
168	C₂³×C₄	—	47	12	no	—
169	C₂³×C₄	—	47	1	no	—
170	C₂³×C₄	—	47	1	no	—
171	C₂³×C₄	—	94	8	no	—
172	C₂³×C₄	—	94	8	no	—
173	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
174	C₂³×C₄	C₄×D₄₇	1	6	yes	—
175	C₂³×C₄	C₂²×D₄₇	1	6	yes	—
176	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
177	C₂²×C₈	—	2	8	no	—
178	C₂²×C₈	—	47	12	no	—
179	C₂²×C₈	—	94	8	no	—
180	C₂²×C₈	C₂²×D₄₇	1	12	yes	—
181	C₂×C₄²	—	47	12	no	—
182	C₂×C₄²	—	47	6	no	—
183	C₂×C₄²	—	47	48	no	—
184	C₂×C₄²	—	47	12	no	—
185	C₂×C₄²	—	47	6	no	—
186	C₂×C₄²	—	47	6	no	—
187	C₂×C₄²	—	47	6	no	—
188	C₂×C₄²	—	47	16	no	—
189	C₄⋊C₈	—	47	32	no	—
190	C₂²⋊C₈	—	47	16	no	—
191	C₂²⋊C₈	C₂²×D₄₇	1	16	yes	—
192	#97	—	47	32	no	—
193	C₂⁴	—	2	4	no	—
194	C₂⁴	—	2	4	no	—
195	C₂⁴	#10625324	1	8	yes	—
196	C₂⁴	C₂×C₄×D₄₇	1	6	yes	—
197	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	2	24	no	—
200	C₂²×C₄	—	2	24	no	—
201	C₂²×C₄	—	47	8	no	—
202	C₂²×C₄	—	47	6	no	—
203	C₂²×C₄	—	47	16	no	—
204	C₂²×C₄	—	47	6	no	—
205	C₂²×C₄	—	47	16	no	—
206	C₂²×C₄	—	47	6	no	—
207	C₂²×C₄	—	47	6	no	—
208	C₂²×C₄	—	94	48	no	—
209	C₂²×C₄	—	94	4	no	—
210	C₂²×C₄	—	94	48	no	—
211	C₂²×C₄	—	94	4	no	—
212	C₂²×C₄	—	94	4	no	—
213	C₂²×C₄	C₂×C₄×D₄₇	1	6	yes	—
214	C₂²×C₄	C₂×C₄×D₄₇	1	12	yes	—
215	C₂²×C₄	C₂×C₄×D₄₇	1	3	yes	—
216	C₂²×C₄	#10625352	1	8	yes	—
217	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
218	C₂²×C₄	C₂³×D₄₇	1	3	yes	—
219	C₂×C₈	—	2	24	no	—
220	C₂×C₈	—	47	8	no	—
221	C₂×C₈	—	94	24	no	—
222	C₂×C₈	C₂³×D₄₇	1	8	yes	—
223	C₄²	—	47	16	no	—
224	C₄²	—	47	12	no	—
225	C₄²	—	47	16	no	—
226	C₄²	—	47	12	no	—
227	C₄²	—	47	4	no	—
228	C₄²	—	47	4	no	—
229	C₂³	—	2	12	no	—
230	C₂³	—	2	12	no	—
231	C₂³	—	2	32	no	—
232	C₂³	#13093980	1	4	yes	—
233	C₂³	M₁₆×D₄₇	1	4	yes	—
234	C₂³	C₂×#10625324	1	24	yes	—
235	C₂³	C₂×#10625324	1	4	yes	—
236	C₂³	C₂²×C₄×D₄₇	1	4	yes	—
237	C₂³	C₂²×C₄×D₄₇	1	3	yes	—
238	C₂×C₄	—	2	12	no	—
239	C₂×C₄	—	2	24	no	—
240	C₂×C₄	—	2	8	no	—
241	C₂×C₄	—	47	24	no	—
242	C₂×C₄	—	47	4	no	—
243	C₂×C₄	—	47	4	no	—
244	C₂×C₄	—	47	4	no	—
245	C₂×C₄	—	47	4	no	—
246	C₂×C₄	—	94	32	no	—
247	C₂×C₄	—	94	12	no	—
248	C₂×C₄	—	94	32	no	—
249	C₂×C₄	—	94	12	no	—
250	C₂×C₄	—	94	12	no	—
251	C₂×C₄	#13093980	1	12	yes	—
252	C₂×C₄	C₂²×C₄×D₄₇	1	4	yes	—
253	C₂×C₄	C₂²×C₄×D₄₇	1	3	yes	—
254	C₂×C₄	C₂×#10625352	1	12	yes	—
255	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
256	C₈	—	2	8	no	—
257	C₈	—	94	8	no	—
258	C₂²	—	2	4	no	—
259	C₂²	—	2	4	no	—
260	C₂²	—	2	48	no	—
261	C₂²	#15689177	1	16	yes	direct product factor
262	C₂²	C₂×#13093980	1	6	yes	—
263	C₂²	C₂×M₁₆×D₄₇	1	6	yes	—
264	C₂²	C₂²×#10625324	1	8	yes	—
265	C₂²	C₂²×#10625324	1	6	yes	—
266	C₂²	C₂³×C₄×D₄₇	1	1	yes	characteristic
267	C₄	—	2	4	no	—
268	C₄	—	47	8	no	—
269	C₄	—	94	4	no	—
270	C₄	—	94	4	no	—
271	C₄	—	94	4	no	—
272	C₄	C₂×#13093980	1	6	yes	—
273	C₄	C₂²×#10625352	1	2	yes	—
274	C₂	—	2	8	no	—
275	C₂	C₂×#15689177	1	12	yes	direct product factor
276	C₂	C₂²×#13093980	1	1	yes	characteristic
277	C₂	C₂²×M₁₆×D₄₇	1	1	yes	characteristic
278	C₂	C₂³×#10625324	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂²×C₂²⋊C₈×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂²×C₄
prime radical	C₂⁵×C₄₇	C₂×C₄
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	C₂²×C₂²⋊C₈	D₄₇
hypocenter	C₄₇	C₂²×#432
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#15689177	C₂	direct (N × H)
3. C₂²×Dic₄₇⋊C₈	C₂	semidirect
5. C₂²×#13093934	C₂	semidirect
16. #15689177	C₂²	direct (N × H)
17. C₂×Dic₄₇⋊C₈	C₂²	semidirect
19. C₂×#13093934	C₂²	semidirect
41. Dic₄₇⋊C₈	C₂³	semidirect
43. #13093934	C₂³	semidirect
122. C₂²×C₄₇	#432	semidirect
142. C₂×C₄₇	C₂×#432	semidirect
162. C₄₇	C₂²×#432	semidirect
261. C₂²	#15689177	direct (N × H)
275. C₂	C₂×#15689177	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×C₄×Dic₄₇	2
C₂×#15689177	2
C₂²×Dic₄₇⋊C₈	2
C₂²×C₂²⋊(C₄₇⋊C₈)	2
C₂²×#13093934	2
C₂²×C₂²⋊C₈×C₄₇	2
C₂²×#432	47

Subgroups — C22×#15689177 #27079987

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15689177 #27079987