Subgroups — #26215857

Order 12032.

1518 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689161	C₂	1	2	yes	maximal, complemented
2	#20245771	C₂	1	2	yes	maximal
3	#20243421	C₂	1	8	yes	maximal, complemented
4	#20243105	C₂	1	2	yes	maximal, complemented
5	C₂³×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093983	C₂²	1	4	yes	—
7	#15689161	—	2	8	no	—
8	C₂×#13093958	C₂²	1	4	yes	complemented
9	C₂×#13093956	C₂²	1	4	yes	complemented
10	#15688986	—	2	2	no	—
11	#15688983	C₂²	1	8	yes	—
12	#15688979	—	2	2	no	—
13	C₄×C₂²⋊Dic₄₇	—	2	4	no	—
14	#15688886	C₂²	1	8	yes	complemented
15	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
16	C₂²×M₁₆×C₄₇	—	2	8	no	—
17	C₂²×M₁₆×C₄₇	C₄	1	8	yes	complemented
18	C₂²×M₁₆×C₄₇	C₂²	1	4	yes	—
19	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
21	#13093983	—	2	8	no	—
22	#13093983	—	2	8	no	—
23	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
24	C₂×C₂²⋊Dic₄₇	—	2	8	no	—
25	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
26	C₂×C₄×Dic₄₇	—	2	4	no	—
27	#13093958	—	2	8	no	—
28	#13093956	—	2	8	no	—
29	C₄₇⋊#95	—	2	8	no	—
30	C₂⁵×C₄₇	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₄₇	—	2	4	no	—
32	C₂³×C₄×C₄₇	—	2	4	no	—
33	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
34	C₂³×C₄×C₄₇	D₄	1	2	yes	—
35	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₄₇	Q₈	1	4	yes	—
37	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
38	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
39	C₂×M₁₆×C₄₇	—	2	32	no	—
40	C₂×M₁₆×C₄₇	—	2	8	no	—
41	C₂×M₁₆×C₄₇	—	2	8	no	—
42	C₂×M₁₆×C₄₇	C₂×C₄	1	16	yes	—
43	C₂²×C₈×C₄₇	—	2	4	no	—
44	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
45	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
46	C₂²×C₈×C₄₇	D₄	1	8	yes	—
47	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
48	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
49	C₂⁴×C₄₇	—	2	4	no	—
50	C₂⁴×C₄₇	—	4	4	no	—
51	C₂⁴×C₄₇	C₄⋊C₄	1	4	yes	—
52	C₂⁴×C₄₇	C₂×D₄	1	2	yes	—
53	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	—	2	4	no	—
55	C₂²×C₄×C₄₇	—	2	8	no	—
56	C₂²×C₄×C₄₇	—	2	8	no	—
57	C₂²×C₄×C₄₇	—	2	8	no	—
58	C₂²×C₄×C₄₇	—	2	16	no	—
59	C₂²×C₄×C₄₇	—	2	4	no	—
60	C₂²×C₄×C₄₇	—	2	4	no	—
61	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
62	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
67	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
69	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₄₇	#41	1	4	yes	—
71	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
72	C₂²×Dic₄₇	—	2	8	no	—
73	M₁₆×C₄₇	—	2	32	no	—
74	C₂×C₈×C₄₇	—	2	8	no	—
75	C₂×C₈×C₄₇	—	2	4	no	—
76	C₂×C₈×C₄₇	C₂²⋊C₄	1	16	yes	complemented
77	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
78	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
79	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
80	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
81	C₂²⋊Dic₄₇	—	4	16	no	—
82	C₂²⋊Dic₄₇	—	4	8	no	—
83	C₄⋊Dic₄₇	—	2	8	no	—
84	C₄⋊Dic₄₇	—	2	8	no	—
85	C₄×Dic₄₇	—	4	8	no	—
86	C₂³×C₄₇	—	2	4	no	—
87	C₂³×C₄₇	—	2	4	no	—
88	C₂³×C₄₇	—	2	8	no	—
89	C₂³×C₄₇	—	2	8	no	—
90	C₂³×C₄₇	—	4	8	no	—
91	C₂³×C₄₇	—	4	16	no	—
92	C₂³×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
93	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
94	C₂³×C₄₇	#121	1	4	yes	—
95	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₄₇	—	2	4	no	—
97	C₂×C₄×C₄₇	—	2	8	no	—
98	C₂×C₄×C₄₇	—	2	4	no	—
99	C₂×C₄×C₄₇	—	2	4	no	—
100	C₂×C₄×C₄₇	—	2	8	no	—
101	C₂×C₄×C₄₇	—	2	16	no	—
102	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
103	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
104	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₄₇	#117	1	4	yes	—
106	C₂×C₄×C₄₇	#121	1	4	yes	—
107	C₂×C₄×C₄₇	C₂²⋊Q₈	1	8	yes	—
108	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
110	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
112	C₂×Dic₄₇	—	4	8	no	—
113	C₂×Dic₄₇	—	4	16	no	—
114	C₈×C₄₇	—	2	4	no	—
115	C₈×C₄₇	C₂×C₂²⋊C₄	1	8	yes	—
116	#20660	—	47	1	no	maximal, Sylow
117	C₂²×C₄₇	—	2	4	no	—
118	C₂²×C₄₇	—	2	8	no	—
119	C₂²×C₄₇	—	2	8	no	—
120	C₂²×C₄₇	—	2	4	no	—
121	C₂²×C₄₇	—	4	16	no	—
122	C₂²×C₄₇	—	4	8	no	—
123	C₂²×C₄₇	#380	1	4	yes	—
124	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
125	C₂²×C₄₇	C₂×#121	1	2	yes	—
126	C₂²×C₄₇	#575	1	4	yes	—
127	C₄×C₄₇	—	2	4	no	—
128	C₄×C₄₇	#380	1	4	yes	—
129	C₄×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
130	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
131	C₄×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
132	Dic₄₇	—	8	8	no	—
133	C₂³×M₁₆	D₄₇	1	1	yes	characteristic
134	C₂×#427	—	47	2	no	—
135	#2280	—	47	2	no	—
136	#1809	—	47	8	no	—
137	#1735	—	47	2	no	—
138	C₂×C₄₇	—	2	4	no	—
139	C₂×C₄₇	—	4	4	no	—
140	C₂×C₄₇	C₂×#380	1	1	yes	characteristic
141	C₂×C₄₇	#2911	1	2	yes	—
142	C₂×C₄₇	#3054	1	4	yes	—
143	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
144	C₂²×M₁₆	—	2	8	no	—
145	C₂²×M₁₆	Dic₄₇	1	8	yes	complemented
146	C₂²×M₁₆	C₂×D₄₇	1	4	yes	—
147	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
148	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
149	C₂×#117	—	47	4	no	—
150	#427	—	94	8	no	—
151	C₂×#107	—	47	4	no	—
152	C₂×#106	—	47	4	no	—
153	#381	—	94	2	no	—
154	#380	—	47	8	no	—
155	#379	—	94	2	no	—
156	C₄×C₂²⋊C₄	—	94	4	no	—
157	#340	—	47	8	no	—
158	C₄₇	#20660	1	1	yes	characteristic, complemented, Sylow
159	C₂⁵	D₁₈₈	1	1	yes	characteristic
160	C₂³×C₄	—	2	4	no	—
161	C₂³×C₄	—	2	4	no	—
162	C₂³×C₄	Dic₉₄	1	4	yes	—
163	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
164	C₂³×C₄	D₁₈₈	1	2	yes	—
165	C₂³×C₄	C₂×Dic₄₇	1	4	yes	—
166	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
167	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
168	C₂×M₁₆	—	2	32	no	—
169	C₂×M₁₆	—	2	8	no	—
170	C₂×M₁₆	—	2	8	no	—
171	C₂×M₁₆	C₂×Dic₄₇	1	16	yes	—
172	C₂²×C₈	—	2	4	no	—
173	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
174	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
175	C₂²×C₈	C₂²⋊D₄₇	1	8	yes	—
176	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
177	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
178	#117	—	94	8	no	—
179	#117	—	94	8	no	—
180	C₂×C₄⋊C₄	—	47	8	no	—
181	C₂×C₂²⋊C₄	—	94	4	no	—
182	C₂×C₂²⋊C₄	—	94	8	no	—
183	C₂×C₄²	—	94	4	no	—
184	#107	—	94	8	no	—
185	#106	—	94	8	no	—
186	#95	—	94	8	no	—
187	C₂⁴	—	2	4	no	—
188	C₂⁴	—	4	4	no	—
189	C₂⁴	C₄⋊Dic₄₇	1	4	yes	—
190	C₂⁴	C₂×D₁₈₈	1	2	yes	—
191	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	2	8	no	—
194	C₂²×C₄	—	2	8	no	—
195	C₂²×C₄	—	2	8	no	—
196	C₂²×C₄	—	2	16	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	94	8	no	—
200	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
201	C₂²×C₄	C₄⋊Dic₄₇	1	8	yes	—
202	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
203	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
204	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
205	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
206	C₂²×C₄	#10625350	1	4	yes	—
207	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
208	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
209	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
210	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
211	M₁₆	—	2	32	no	—
212	C₂×C₈	—	2	8	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	C₂²⋊Dic₄₇	1	16	yes	complemented
215	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
216	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
217	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
218	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
219	C₄⋊C₄	—	94	8	no	—
220	C₄⋊C₄	—	94	8	no	—
221	C₂²⋊C₄	—	188	16	no	—
222	C₂²⋊C₄	—	188	8	no	—
223	C₄²	—	188	8	no	—
224	C₂³	—	2	4	no	—
225	C₂³	—	2	8	no	—
226	C₂³	—	2	4	no	—
227	C₂³	—	2	8	no	—
228	C₂³	—	4	8	no	—
229	C₂³	—	4	16	no	—
230	C₂³	C₂×C₄⋊Dic₄₇	1	4	yes	—
231	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
232	C₂³	#13093998	1	4	yes	—
233	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
234	C₂×C₄	—	2	8	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	16	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	188	8	no	—
241	C₂×C₄	—	188	16	no	—
242	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
243	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
244	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
245	C₂×C₄	#13093983	1	4	yes	—
246	C₂×C₄	#13093998	1	4	yes	—
247	C₂×C₄	#13094003	1	8	yes	—
248	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
249	C₂×C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
250	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
251	C₂×C₄	C₂×#10625350	1	2	yes	—
252	C₈	—	2	4	no	—
253	C₈	C₂×C₂²⋊Dic₄₇	1	8	yes	—
254	C₂²	—	2	8	no	—
255	C₂²	—	2	4	no	—
256	C₂²	—	2	4	no	—
257	C₂²	—	2	8	no	—
258	C₂²	—	4	16	no	—
259	C₂²	—	4	8	no	—
260	C₂²	#15688983	1	4	yes	—
261	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
262	C₂²	C₂×#13093998	1	2	yes	—
263	C₂²	#15689923	1	4	yes	—
264	C₄	—	2	4	no	—
265	C₄	—	376	8	no	—
266	C₄	#15688983	1	4	yes	—
267	C₄	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
268	C₄	C₂×#13093983	1	1	yes	characteristic
269	C₄	C₂×#13094003	1	2	yes	—
270	C₂	—	2	4	no	—
271	C₂	—	4	4	no	—
272	C₂	C₂×#15688983	1	1	yes	characteristic
273	C₂	#20251978	1	2	yes	—
274	C₂	#20252997	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂³×M₁₆×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁵×C₄₇	D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂³×M₁₆	D₄₇
hypocenter	C₄₇	#20660
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ C₂³×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#15689161	C₂	semidirect
3. #20243421	C₂	semidirect
4. #20243105	C₂	semidirect
8. C₂×#13093958	C₂²	semidirect
9. C₂×#13093956	C₂²	semidirect
14. #15688886	C₂²	semidirect
17. C₂²×M₁₆×C₄₇	C₄	semidirect
76. C₂×C₈×C₄₇	C₂²⋊C₄	semidirect
145. C₂²×M₁₆	Dic₄₇	semidirect
158. C₄₇	#20660	semidirect
214. C₂×C₈	C₂²⋊Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689161	2
#20245771	2
#20243421	2
#20243105	2
C₂³×M₁₆×C₄₇	2
#20660	47