Subgroups — #27085958

Order 12032.

1834 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20257207	C₂	1	2	yes	maximal
2	#20253934	C₂	1	6	yes	maximal, complemented
3	#20253919	C₂	1	12	yes	maximal, complemented
4	#20253902	C₂	1	3	yes	maximal, complemented
5	C₂×#15689521	C₂	1	3	yes	maximal, complemented
6	#20251983	C₂	1	4	yes	maximal
7	C₄×#131×C₄₇	C₂	1	1	yes	maximal, characteristic
8	Q₈×Dic₉₄	—	2	2	no	—
9	#15689852	—	2	2	no	—
10	#15689830	—	2	6	no	—
11	D₄×Dic₉₄	—	2	6	no	—
12	#15689751	C₂²	1	12	yes	—
13	#15689680	C₂²	1	6	yes	—
14	C₂×#13094024	C₂²	1	6	yes	—
15	#15689606	C₂²	1	4	yes	—
16	#15689532	C₂²	1	24	yes	—
17	#15689528	C₂²	1	6	yes	—
18	#15689525	C₂²	1	12	yes	—
19	#15689521	C₂²	1	3	yes	—
20	#15689521	C₂²	1	12	yes	—
21	#15689521	C₂²	1	6	yes	—
22	#15689521	C₂²	1	1	yes	characteristic
23	#15689165	C₂²	1	24	yes	—
24	#15689161	C₂²	1	12	yes	—
25	C₂×#13093956	C₂²	1	12	yes	—
26	C₂×#131×C₄₇	C₂²	1	2	yes	—
27	C₄×#41×C₄₇	C₂²	1	1	yes	characteristic
28	C₄₇×#405	C₂²	1	6	yes	—
29	C₄×M₁₆×C₄₇	C₂²	1	3	yes	—
30	C₂×C₄×C₈×C₄₇	C₂²	1	3	yes	—
31	C₂²×Dic₉₄	—	2	6	no	—
32	#13094024	—	2	6	no	—
33	#13094024	—	2	6	no	—
34	#13094024	C₂³	1	12	yes	—
35	#13094024	C₂³	1	6	yes	—
36	#13094024	C₂³	1	2	yes	—
37	#13094023	—	2	12	no	—
38	#13094016	C₂³	1	12	yes	—
39	#13094015	—	2	12	no	—
40	#13094003	—	2	6	no	—
41	#13094003	—	2	6	no	—
42	#13094001	—	2	24	no	—
43	Q₈×Dic₄₇	—	2	4	no	—
44	C₄×Dic₉₄	—	2	6	no	—
45	C₄×Dic₉₄	—	2	6	no	—
46	D₄×Dic₄₇	—	2	12	no	—
47	#13093983	C₂³	1	12	yes	—
48	C₂×C₄⋊Dic₄₇	—	2	6	no	—
49	C₂×C₄⋊Dic₄₇	C₂³	1	12	yes	—
50	#13093958	C₂³	1	24	yes	—
51	#13093956	C₂³	1	12	yes	—
52	#13093956	C₂³	1	24	yes	—
53	#13093956	C₂³	1	4	yes	—
54	C₂×#41×C₄₇	C₂³	1	1	yes	characteristic
55	#131×C₄₇	Q₈	1	16	yes	complemented
56	C₂×M₁₆×C₄₇	C₂³	1	6	yes	—
57	C₂²×C₈×C₄₇	C₂³	1	6	yes	—
58	C₄×Q₈×C₄₇	D₄	1	2	yes	—
59	C₄×D₄×C₄₇	D₄	1	6	yes	—
60	#117×C₄₇	C₂³	1	3	yes	—
61	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
62	#97×C₄₇	C₂³	1	6	yes	—
63	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₄₇	C₂³	1	6	yes	—
65	C₄×C₈×C₄₇	C₂³	1	3	yes	—
66	#41×C₄₇	C₂×Q₈	1	8	yes	—
67	C₂×Q₈×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂×D₄×C₄₇	C₂×D₄	1	3	yes	—
69	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
70	C₂²×C₄×C₄₇	C₂⁴	1	3	yes	—
71	C₂²×Dic₄₇	—	2	12	no	—
72	C₂×Dic₉₄	—	2	2	no	—
73	C₂×Dic₉₄	—	2	6	no	—
74	C₂×Dic₉₄	—	2	12	no	—
75	C₂×Dic₉₄	—	8	12	no	—
76	M₁₆×C₄₇	C₂×Q₈	1	24	yes	complemented
77	C₂×C₈×C₄₇	C₂×Q₈	1	24	yes	complemented
78	C₂×C₈×C₄₇	C₂⁴	1	2	yes	—
79	C₂×C₈×C₄₇	C₂⁴	1	6	yes	—
80	C₄⋊C₄×C₄₇	C₂×D₄	1	6	yes	—
81	C₂²⋊C₄×C₄₇	C₂×D₄	1	6	yes	—
82	C₄²×C₄₇	C₂×D₄	1	6	yes	—
83	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
84	C₄²×C₄₇	C₂⁴	1	3	yes	—
85	C₂²⋊Dic₄₇	—	2	12	no	—
86	C₄⋊Dic₄₇	—	2	6	no	—
87	C₄⋊Dic₄₇	—	2	12	no	—
88	C₄⋊Dic₄₇	—	2	12	no	—
89	C₄⋊Dic₄₇	—	2	2	no	—
90	C₄⋊Dic₄₇	C₂⁴	1	12	yes	—
91	C₄⋊Dic₄₇	C₂⁴	1	4	yes	—
92	Dic₄₇⋊C₄	—	2	12	no	—
93	Dic₄₇⋊C₄	—	2	12	no	—
94	C₄×Dic₄₇	—	2	12	no	—
95	C₂³×C₄₇	C₂²×D₄	1	3	yes	—
96	Q₈×C₄₇	C₄⋊Q₈	1	4	yes	—
97	D₄×C₄₇	C₄⋊Q₈	1	12	yes	—
98	C₂×C₄×C₄₇	—	2	12	no	—
99	C₂×C₄×C₄₇	C₂²×D₄	1	6	yes	—
100	C₂×C₄×C₄₇	C₂²×D₄	1	3	yes	—
101	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
102	C₂×C₄×C₄₇	C₂²×Q₈	1	12	yes	—
103	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
104	C₂×Dic₄₇	—	2	4	no	—
105	C₂×Dic₄₇	—	2	12	no	—
106	C₂×Dic₄₇	—	8	12	no	—
107	Dic₉₄	—	4	4	no	—
108	Dic₉₄	—	4	12	no	—
109	Dic₉₄	—	8	12	no	—
110	C₈×C₄₇	C₂²×Q₈	1	4	yes	—
111	C₈×C₄₇	C₂²×Q₈	1	12	yes	—
112	#60665	—	47	1	no	maximal, Sylow
113	C₂²×C₄₇	—	2	6	no	—
114	C₂²×C₄₇	C₂×C₄⋊Q₈	1	6	yes	—
115	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
116	C₄×C₄₇	—	2	6	no	—
117	C₄×C₄₇	C₂×C₄⋊Q₈	1	6	yes	—
118	C₄×C₄₇	#577	1	4	yes	—
119	C₄×C₄₇	C₂³×Q₈	1	2	yes	—
120	Dic₄₇	—	4	12	no	—
121	Dic₄₇	—	4	4	no	—
122	#3503	—	47	2	no	—
123	#3165	—	47	6	no	—
124	#3163	—	47	12	no	—
125	#3160	—	47	3	no	—
126	C₂×#498	—	47	3	no	—
127	#2912	—	47	4	no	—
128	C₄×#131	D₄₇	1	1	yes	characteristic
129	C₂×C₄₇	—	2	6	no	—
130	C₂×C₄₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
131	C₂×C₄₇	C₂×#577	1	2	yes	—
132	C₂×#131	C₂×D₄₇	1	2	yes	—
133	Q₈²	—	94	2	no	—
134	#557	—	94	2	no	—
135	#554	—	94	6	no	—
136	D₄×Q₈	—	94	6	no	—
137	#544	—	47	12	no	—
138	#533	—	47	6	no	—
139	C₂×C₄⋊Q₈	—	47	6	no	—
140	#520	—	47	4	no	—
141	C₄×#41	C₂×D₄₇	1	1	yes	characteristic
142	#501	—	47	24	no	—
143	#500	—	47	6	no	—
144	#499	—	47	12	no	—
145	#498	—	47	3	no	—
146	#498	—	47	12	no	—
147	#498	—	47	6	no	—
148	#498	—	47	1	no	—
149	#428	—	47	24	no	—
150	#427	—	47	12	no	—
151	C₂×#106	—	47	12	no	—
152	#405	C₂×D₄₇	1	6	yes	—
153	C₄×M₁₆	C₂×D₄₇	1	3	yes	—
154	C₂×C₄×C₈	C₂×D₄₇	1	3	yes	—
155	C₄₇	#60665	1	1	yes	characteristic, complemented, Sylow
156	C₂×#41	C₂²×D₄₇	1	1	yes	characteristic
157	C₂²×Q₈	—	94	6	no	—
158	#131	Dic₉₄	1	16	yes	complemented
159	C₂×M₁₆	C₂²×D₄₇	1	6	yes	—
160	C₂²×C₈	C₂²×D₄₇	1	6	yes	—
161	C₄⋊Q₈	—	47	6	no	—
162	C₄⋊Q₈	—	47	12	no	—
163	C₄⋊Q₈	—	47	2	no	—
164	C₄⋊Q₈	—	94	12	no	—
165	C₄⋊Q₈	—	94	6	no	—
166	C₄⋊Q₈	—	94	6	no	—
167	#125	—	47	12	no	—
168	#125	—	94	12	no	—
169	C₂²⋊Q₈	—	94	6	no	—
170	C₂²⋊Q₈	—	94	6	no	—
171	C₂²⋊Q₈	—	94	24	no	—
172	C₄×Q₈	—	94	4	no	—
173	C₄×Q₈	—	94	6	no	—
174	C₄×Q₈	—	94	6	no	—
175	C₄×Q₈	D₁₈₈	1	2	yes	—
176	C₄×D₄	—	94	12	no	—
177	C₄×D₄	D₁₈₈	1	6	yes	—
178	#117	—	47	12	no	—
179	#117	C₂²×D₄₇	1	3	yes	—
180	C₂×C₄⋊C₄	—	47	12	no	—
181	C₂×C₄⋊C₄	—	94	6	no	—
182	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
183	#107	—	47	24	no	—
184	#106	—	47	12	no	—
185	#106	—	47	24	no	—
186	#106	—	47	4	no	—
187	#97	C₂²×D₄₇	1	6	yes	—
188	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
189	C₄×C₈	C₂²×D₄₇	1	3	yes	—
190	C₄×C₈	C₂²×D₄₇	1	6	yes	—
191	#41	C₂×Dic₉₄	1	8	yes	—
192	C₂×Q₈	—	94	12	no	—
193	C₂×Q₈	—	94	2	no	—
194	C₂×Q₈	—	94	6	no	—
195	C₂×Q₈	—	376	12	no	—
196	C₂×Q₈	C₂×D₁₈₈	1	1	yes	characteristic
197	C₂×D₄	C₂×D₁₈₈	1	3	yes	—
198	C₂²×C₄	—	94	12	no	—
199	C₂²×C₄	C₂×D₁₈₈	1	6	yes	—
200	C₂²×C₄	C₂³×D₄₇	1	3	yes	—
201	M₁₆	C₂×Dic₉₄	1	24	yes	complemented
202	C₂×C₈	C₂×Dic₉₄	1	24	yes	complemented
203	C₂×C₈	C₂³×D₄₇	1	2	yes	—
204	C₂×C₈	C₂³×D₄₇	1	6	yes	—
205	C₄⋊C₄	—	47	12	no	—
206	C₄⋊C₄	—	47	4	no	—
207	C₄⋊C₄	—	94	12	no	—
208	C₄⋊C₄	—	94	12	no	—
209	C₄⋊C₄	—	94	12	no	—
210	C₄⋊C₄	—	94	6	no	—
211	C₄⋊C₄	—	94	12	no	—
212	C₄⋊C₄	—	94	2	no	—
213	C₄⋊C₄	C₂×D₁₈₈	1	6	yes	—
214	C₂²⋊C₄	—	94	12	no	—
215	C₂²⋊C₄	C₂×D₁₈₈	1	6	yes	—
216	C₄²	—	94	12	no	—
217	C₄²	C₂×D₁₈₈	1	6	yes	—
218	C₄²	C₂³×D₄₇	1	1	yes	characteristic
219	C₄²	C₂³×D₄₇	1	3	yes	—
220	C₂³	C₂²×D₁₈₈	1	3	yes	—
221	Q₈	—	376	12	no	—
222	Q₈	—	188	4	no	—
223	Q₈	—	188	12	no	—
224	Q₈	#13094024	1	4	yes	—
225	D₄	#13094024	1	12	yes	—
226	C₂×C₄	—	2	12	no	—
227	C₂×C₄	—	94	4	no	—
228	C₂×C₄	—	94	12	no	—
229	C₂×C₄	—	376	12	no	—
230	C₂×C₄	C₂²×D₁₈₈	1	6	yes	—
231	C₂×C₄	C₂²×D₁₈₈	1	3	yes	—
232	C₂×C₄	C₂²×D₁₈₈	1	2	yes	—
233	C₂×C₄	C₂²×Dic₉₄	1	12	yes	—
234	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
235	C₈	C₂²×Dic₉₄	1	4	yes	—
236	C₈	C₂²×Dic₉₄	1	12	yes	—
237	C₂²	—	2	6	no	—
238	C₂²	C₂×#13094024	1	6	yes	—
239	C₂²	C₂³×D₁₈₈	1	1	yes	characteristic
240	C₄	—	2	6	no	—
241	C₄	—	188	12	no	—
242	C₄	—	188	4	no	—
243	C₄	C₂×#13094024	1	6	yes	—
244	C₄	#15689935	1	4	yes	—
245	C₄	C₂³×Dic₉₄	1	2	yes	—
246	C₂	—	2	6	no	—
247	C₂	C₂²×#13094024	1	1	yes	characteristic
248	C₂	C₂×#15689935	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂³×D₁₈₈
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄×#131×C₄₇	C₂
socle	C₂²×C₄₇	C₂³×D₄
prime radical	C₂×#41×C₄₇	C₂³
hypercenter	C₄×#131	D₄₇
hypocenter	C₄₇	#60665
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
2. #20253934	C₂	semidirect
3. #20253919	C₂	semidirect
4. #20253902	C₂	semidirect
5. C₂×#15689521	C₂	semidirect
55. #131×C₄₇	Q₈	semidirect
76. M₁₆×C₄₇	C₂×Q₈	semidirect
77. C₂×C₈×C₄₇	C₂×Q₈	semidirect
155. C₄₇	#60665	semidirect
158. #131	Dic₉₄	semidirect
201. M₁₆	C₂×Dic₉₄	semidirect
202. C₂×C₈	C₂×Dic₉₄	semidirect

Maximal subgroups

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

subgroup	index
#20257207	2
#20253934	2
#20253919	2
#20253902	2
C₂×#15689521	2
#20251983	2
C₄×#131×C₄₇	2
#60665	47