Subgroups — #27080687

Order 12032.

1810 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20256793	C₂	1	1	yes	maximal, characteristic
2	#20252332	C₂	1	16	yes	maximal, complemented
3	#20252328	C₂	1	4	yes	maximal, complemented
4	#20252322	C₂	1	4	yes	maximal, complemented
5	#20251952	C₂	1	1	yes	maximal, characteristic
6	C₂×#15689149	C₂	1	2	yes	maximal
7	#20251806	C₂	1	2	yes	maximal
8	C₂×C₄₇×#405	C₂	1	1	yes	maximal, characteristic
9	C₂²×#10625329	C₂²	1	2	yes	—
10	#15689754	C₄	1	4	yes	—
11	#15689751	C₄	1	8	yes	—
12	#15689749	C₄	1	4	yes	—
13	#15689680	C₂²	1	4	yes	—
14	#15689676	C₂²	1	8	yes	—
15	C₂×#13093983	C₂²	1	2	yes	—
16	#15689253	C₂²	1	16	yes	complemented
17	#15689251	C₂²	1	32	yes	complemented
18	C₈×Dic₉₄	C₂²	1	16	yes	complemented
19	#15689150	C₂²	1	8	yes	—
20	#15689149	C₂²	1	4	yes	—
21	#15689149	C₂²	1	4	yes	—
22	#15689149	C₂²	1	8	yes	—
23	#15689147	C₂²	1	4	yes	—
24	C₂×Dic₄₇⋊C₈	C₂²	1	8	yes	—
25	#15689081	C₂²	1	8	yes	—
26	#15689078	C₂²	1	8	yes	—
27	M₁₆×Dic₄₇	C₂²	1	8	yes	—
28	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
29	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
31	C₄₇×#405	C₂²	1	8	yes	—
32	C₂×#97×C₄₇	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
34	C₂×#10625329	—	2	8	no	—
35	C₂×#10625329	C₂³	1	4	yes	—
36	C₂×#10625329	C₂³	1	4	yes	—
37	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
38	#13094024	C₂×C₄	1	8	yes	—
39	#13094023	C₂×C₄	1	16	yes	—
40	#13094016	C₂×C₄	1	8	yes	—
41	#13094015	C₂×C₄	1	16	yes	—
42	#13094003	C₂×C₄	1	16	yes	—
43	#13094001	C₂×C₄	1	32	yes	—
44	C₄×Dic₉₄	C₂³	1	16	yes	—
45	#13093983	C₂×C₄	1	4	yes	—
46	#13093983	C₂×C₄	1	4	yes	—
47	#13093983	C₂³	1	4	yes	—
48	#13093983	C₂³	1	4	yes	—
49	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
50	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	4	yes	—
51	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
52	C₄⋊(C₄₇⋊C₈)	C₂³	1	16	yes	—
53	Dic₄₇⋊C₈	C₂³	1	16	yes	—
54	Dic₄₇⋊C₈	C₂³	1	16	yes	—
55	#13093934	C₂³	1	16	yes	—
56	C₈×Dic₄₇	C₂³	1	16	yes	—
57	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
58	C₂×M₁₆×C₄₇	Q₈	1	8	yes	complemented
59	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
60	C₂²×C₈×C₄₇	—	2	4	no	—
61	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
62	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
65	#117×C₄₇	C₂×C₄	1	4	yes	—
66	#117×C₄₇	C₂³	1	4	yes	—
67	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
68	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
69	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
70	#97×C₄₇	C₂³	1	8	yes	—
71	C₄×C₈×C₄₇	C₂³	1	8	yes	—
72	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
77	C₂²×C₄×C₄₇	#41	1	4	yes	—
78	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
80	C₂²×Dic₄₇	C₂²×C₄	1	8	yes	—
81	C₂×Dic₉₄	C₂²×C₄	1	16	yes	—
82	M₁₆×C₄₇	C₂×Q₈	1	16	yes	—
83	C₂×C₈×C₄₇	—	2	8	no	—
84	C₂×C₈×C₄₇	—	2	2	no	—
85	C₂×C₈×C₄₇	—	2	2	no	—
86	C₂×C₈×C₄₇	C₂×Q₈	1	8	yes	—
87	C₂×C₈×C₄₇	C₂⁴	1	2	yes	—
88	C₂×C₈×C₄₇	C₂⁴	1	2	yes	—
89	C₂×C₈×C₄₇	C₂⁴	1	4	yes	—
90	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
91	C₂²⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
92	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
93	C₄²×C₄₇	C₂⁴	1	4	yes	—
94	C₂²⋊Dic₄₇	C₂²×C₄	1	8	yes	—
95	C₂²⋊Dic₄₇	C₂²×C₄	1	8	yes	—
96	C₄⋊Dic₄₇	C₂²×C₄	1	8	yes	—
97	C₄⋊Dic₄₇	C₂²×C₄	1	8	yes	—
98	C₄⋊Dic₄₇	C₂²×C₄	1	16	yes	—
99	Dic₄₇⋊C₄	C₂²×C₄	1	32	yes	—
100	C₄×Dic₄₇	C₂²×C₄	1	8	yes	—
101	C₄×Dic₄₇	C₂⁴	1	8	yes	—
102	#10625329	—	2	8	no	—
103	#10625329	—	2	8	no	—
104	C₂×C₄₇⋊C₈	—	2	8	no	—
105	C₂×C₄₇⋊C₈	C₂⁴	1	8	yes	—
106	C₂³×C₄₇	—	4	2	no	—
107	C₂³×C₄₇	C₄×Q₈	1	4	yes	—
108	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
109	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
110	C₂×C₄×C₄₇	—	4	4	no	—
111	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
112	C₂×C₄×C₄₇	C₄×Q₈	1	8	yes	—
113	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
114	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
115	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
117	C₂×C₄×C₄₇	C₂²×Q₈	1	4	yes	—
118	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
119	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
120	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
121	C₂×Dic₄₇	—	4	8	no	—
122	C₂×Dic₄₇	C₂³×C₄	1	16	yes	—
123	Dic₉₄	—	4	16	no	—
124	C₈×C₄₇	—	2	2	no	—
125	C₈×C₄₇	—	2	2	no	—
126	C₈×C₄₇	C₂²×Q₈	1	8	yes	—
127	C₄₇⋊C₈	—	2	8	no	—
128	#60193	—	47	1	no	maximal, Sylow
129	C₂²×C₄₇	—	2	4	no	—
130	C₂²×C₄₇	—	2	2	no	—
131	C₂²×C₄₇	—	4	4	no	—
132	C₂²×C₄₇	C₂×C₄×Q₈	1	2	yes	—
133	C₂²×C₄₇	C₂×C₄×Q₈	1	4	yes	—
134	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
135	C₄×C₄₇	—	2	8	no	—
136	C₄×C₄₇	C₂×C₄×Q₈	1	4	yes	—
137	C₄×C₄₇	C₂×C₄×Q₈	1	2	yes	—
138	C₄×C₄₇	C₂³×Q₈	1	1	yes	characteristic
139	C₄×C₄₇	C₂²×#41	1	1	yes	characteristic
140	Dic₄₇	—	2	16	no	—
141	#3458	—	47	1	no	—
142	#2962	—	47	16	no	—
143	#2961	—	47	4	no	—
144	#2960	—	47	4	no	—
145	#2905	—	47	1	no	—
146	C₂×#424	—	47	2	no	—
147	#2873	—	47	2	no	—
148	C₂×#405	D₄₇	1	1	yes	characteristic
149	C₂×C₄₇	—	2	4	no	—
150	C₂×C₄₇	—	2	2	no	—
151	C₂×C₄₇	C₂²×C₄×Q₈	1	1	yes	characteristic
152	C₂×C₄₇	#3573	1	2	yes	—
153	C₂²×M₁₆	—	47	2	no	—
154	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
155	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
156	#544	—	47	8	no	—
157	#544	—	47	4	no	—
158	#543	—	47	4	no	—
159	#533	—	47	8	no	—
160	#533	—	47	4	no	—
161	C₂×#117	—	47	2	no	—
162	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
163	#446	—	47	32	no	—
164	#446	—	47	16	no	—
165	C₈×Q₈	—	47	16	no	—
166	#424	—	47	4	no	—
167	#424	—	47	4	no	—
168	#424	—	47	8	no	—
169	#424	—	47	8	no	—
170	#423	—	47	4	no	—
171	C₂×C₄⋊C₈	—	47	8	no	—
172	#405	—	47	8	no	—
173	#405	—	47	8	no	—
174	#405	C₂×D₄₇	1	8	yes	—
175	C₄×M₁₆	—	47	8	no	—
176	C₂×#97	C₂×D₄₇	1	2	yes	—
177	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
178	C₄₇	#60193	1	1	yes	characteristic, complemented, Sylow
179	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
180	C₂×M₁₆	—	47	4	no	—
181	C₂×M₁₆	—	47	4	no	—
182	C₂×M₁₆	—	94	8	no	—
183	C₂×M₁₆	Dic₉₄	1	8	yes	complemented
184	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
185	C₂²×C₈	—	2	4	no	—
186	C₂²×C₈	—	47	4	no	—
187	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
188	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
189	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
190	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
191	C₄⋊Q₈	—	47	16	no	—
192	C₄⋊Q₈	—	47	8	no	—
193	#125	—	47	16	no	—
194	#125	—	47	8	no	—
195	C₂²⋊Q₈	—	47	16	no	—
196	C₂²⋊Q₈	—	47	32	no	—
197	C₄×Q₈	—	47	16	no	—
198	#117	—	47	4	no	—
199	#117	—	47	4	no	—
200	#117	—	47	4	no	—
201	#117	—	47	4	no	—
202	#117	C₄×D₄₇	1	4	yes	—
203	#117	C₂²×D₄₇	1	4	yes	—
204	C₂×C₄⋊C₄	—	47	4	no	—
205	C₂×C₄⋊C₄	C₄×D₄₇	1	2	yes	—
206	C₂×C₂²⋊C₄	—	47	4	no	—
207	C₂×C₂²⋊C₄	C₄×D₄₇	1	2	yes	—
208	C₂×C₄²	—	47	4	no	—
209	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
210	C₄⋊C₈	—	47	16	no	—
211	C₄⋊C₈	—	47	16	no	—
212	C₄⋊C₈	—	47	16	no	—
213	#97	—	47	16	no	—
214	#97	C₂²×D₄₇	1	8	yes	—
215	C₄×C₈	—	47	16	no	—
216	C₄×C₈	C₂²×D₄₇	1	8	yes	—
217	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
218	C₂×Q₈	—	47	16	no	—
219	C₂²×C₄	—	47	8	no	—
220	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
221	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
222	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
223	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
224	C₂²×C₄	#10625350	1	4	yes	—
225	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
226	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
227	M₁₆	—	94	8	no	—
228	M₁₆	—	94	8	no	—
229	M₁₆	C₂×Dic₉₄	1	16	yes	—
230	C₂×C₈	—	2	8	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	47	8	no	—
234	C₂×C₈	—	94	8	no	—
235	C₂×C₈	C₂×Dic₉₄	1	8	yes	—
236	C₂×C₈	C₂³×D₄₇	1	2	yes	—
237	C₂×C₈	C₂³×D₄₇	1	2	yes	—
238	C₂×C₈	C₂³×D₄₇	1	4	yes	—
239	C₄⋊C₄	—	47	32	no	—
240	C₄⋊C₄	—	47	8	no	—
241	C₄⋊C₄	—	47	8	no	—
242	C₄⋊C₄	—	47	16	no	—
243	C₄⋊C₄	C₂×C₄×D₄₇	1	8	yes	—
244	C₂²⋊C₄	—	47	8	no	—
245	C₂²⋊C₄	—	47	8	no	—
246	C₂²⋊C₄	C₂×C₄×D₄₇	1	8	yes	—
247	C₄²	—	47	8	no	—
248	C₄²	—	47	8	no	—
249	C₄²	C₂×C₄×D₄₇	1	4	yes	—
250	C₄²	C₂³×D₄₇	1	4	yes	—
251	C₂³	—	4	2	no	—
252	C₂³	C₄×Dic₉₄	1	4	yes	—
253	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
254	C₂³	C₂²×C₄×D₄₇	1	2	yes	—
255	Q₈	—	188	16	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	47	16	no	—
258	C₂×C₄	—	188	8	no	—
259	C₂×C₄	C₄×Dic₉₄	1	8	yes	—
260	C₂×C₄	C₄×Dic₉₄	1	4	yes	—
261	C₂×C₄	C₂²×C₄×D₄₇	1	8	yes	—
262	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
263	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
264	C₂×C₄	C₂²×Dic₉₄	1	2	yes	—
265	C₂×C₄	C₂²×Dic₉₄	1	4	yes	—
266	C₂×C₄	C₂×#10625350	1	4	yes	—
267	C₂×C₄	C₂×#10625350	1	2	yes	—
268	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
269	C₈	—	2	2	no	—
270	C₈	—	2	2	no	—
271	C₈	—	94	8	no	—
272	C₈	C₂²×Dic₉₄	1	8	yes	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	4	4	no	—
276	C₂²	C₂×C₄×Dic₉₄	1	2	yes	—
277	C₂²	C₂×C₄×Dic₉₄	1	4	yes	—
278	C₂²	C₂³×C₄×D₄₇	1	1	yes	characteristic
279	C₄	—	2	8	no	—
280	C₄	—	94	16	no	—
281	C₄	C₂×C₄×Dic₉₄	1	4	yes	—
282	C₄	C₂×C₄×Dic₉₄	1	2	yes	—
283	C₄	C₂³×Dic₉₄	1	1	yes	characteristic
284	C₄	C₂²×#10625350	1	1	yes	characteristic
285	C₂	—	2	2	no	—
286	C₂	—	2	4	no	—
287	C₂	C₂²×C₄×Dic₉₄	1	1	yes	characteristic
288	C₂	#20258099	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂⁴×D₄₇
commutator, socle	C₂²×C₄₇	C₂⁴×C₄
Fitting	C₂×C₄₇×#405	C₂
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	C₂×#405	D₄₇
hypocenter	C₄₇	#60193
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #20252332	C₂	semidirect
3. #20252328	C₂	semidirect
4. #20252322	C₂	semidirect
16. #15689253	C₂²	semidirect
17. #15689251	C₂²	semidirect
18. C₈×Dic₉₄	C₂²	semidirect
58. C₂×M₁₆×C₄₇	Q₈	semidirect
178. C₄₇	#60193	semidirect
183. C₂×M₁₆	Dic₉₄	semidirect

Maximal subgroups

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

subgroup	index
#20256793	2
#20252332	2
#20252328	2
#20252322	2
#20251952	2
C₂×#15689149	2
#20251806	2
C₂×C₄₇×#405	2
#60193	47