Subgroups — #27085974

Order 12032.

1834 conjugacy classes of subgroups.

The subgroup lattice has 274 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	#20253927	C₂	1	12	yes	maximal, complemented
4	#20253902	C₂	1	3	yes	maximal, complemented
5	C₂×#15689528	C₂	1	3	yes	maximal, complemented
6	#20251988	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	3	yes	—
18	#15689528	C₂²	1	12	yes	—
19	#15689528	C₂²	1	6	yes	—
20	#15689528	C₂²	1	1	yes	characteristic
21	#15689525	C₂²	1	12	yes	—
22	#15689521	C₂²	1	6	yes	—
23	#15689165	C₂²	1	24	yes	—
24	#15689161	C₂²	1	12	yes	—
25	C₂×#13093958	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	2	yes	—
36	#13094024	C₂³	1	6	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	4	yes	—
51	#13093958	C₂³	1	12	yes	—
52	#13093958	C₂³	1	24	yes	—
53	#13093956	C₂³	1	24	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	1	yes	characteristic
59	C₄×Q₈×C₄₇	D₄	1	1	yes	characteristic
60	C₄×D₄×C₄₇	D₄	1	3	yes	—
61	C₄×D₄×C₄₇	D₄	1	3	yes	—
62	#117×C₄₇	C₂³	1	3	yes	—
63	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
64	#97×C₄₇	C₂³	1	6	yes	—
65	C₄×C₈×C₄₇	C₂³	1	3	yes	—
66	C₄×C₈×C₄₇	C₂³	1	6	yes	—
67	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
68	#41×C₄₇	C₂×Q₈	1	8	yes	—
69	C₂×Q₈×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂×D₄×C₄₇	C₂×D₄	1	3	yes	—
71	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
72	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
73	C₂²×C₄×C₄₇	C₂⁴	1	3	yes	—
74	C₂²×Dic₄₇	—	2	12	no	—
75	C₂×Dic₉₄	—	2	2	no	—
76	C₂×Dic₉₄	—	2	6	no	—
77	C₂×Dic₉₄	—	2	12	no	—
78	C₂×Dic₉₄	—	8	12	no	—
79	M₁₆×C₄₇	C₂×Q₈	1	24	yes	complemented
80	C₂×C₈×C₄₇	C₂×Q₈	1	24	yes	complemented
81	C₂×C₈×C₄₇	C₂⁴	1	2	yes	—
82	C₂×C₈×C₄₇	C₂⁴	1	6	yes	—
83	C₄⋊C₄×C₄₇	C₂×D₄	1	3	yes	—
84	C₄⋊C₄×C₄₇	C₂×D₄	1	3	yes	—
85	C₂²⋊C₄×C₄₇	C₂×D₄	1	6	yes	—
86	C₄²×C₄₇	C₂×D₄	1	3	yes	—
87	C₄²×C₄₇	C₂×D₄	1	3	yes	—
88	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
89	C₄²×C₄₇	C₂⁴	1	3	yes	—
90	C₂²⋊Dic₄₇	—	2	12	no	—
91	C₄⋊Dic₄₇	—	2	6	no	—
92	C₄⋊Dic₄₇	—	2	12	no	—
93	C₄⋊Dic₄₇	—	2	12	no	—
94	C₄⋊Dic₄₇	—	2	2	no	—
95	C₄⋊Dic₄₇	C₂⁴	1	4	yes	—
96	C₄⋊Dic₄₇	C₂⁴	1	12	yes	—
97	Dic₄₇⋊C₄	—	2	12	no	—
98	Dic₄₇⋊C₄	—	2	12	no	—
99	C₄×Dic₄₇	—	2	12	no	—
100	C₂³×C₄₇	C₂²×D₄	1	3	yes	—
101	Q₈×C₄₇	C₄⋊Q₈	1	4	yes	—
102	D₄×C₄₇	C₄⋊Q₈	1	12	yes	—
103	C₂×C₄×C₄₇	—	2	6	no	—
104	C₂×C₄×C₄₇	—	2	6	no	—
105	C₂×C₄×C₄₇	C₂²×D₄	1	3	yes	—
106	C₂×C₄×C₄₇	C₂²×D₄	1	3	yes	—
107	C₂×C₄×C₄₇	C₂²×D₄	1	3	yes	—
108	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂²×Q₈	1	12	yes	—
111	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
112	C₂×Dic₄₇	—	2	4	no	—
113	C₂×Dic₄₇	—	2	12	no	—
114	C₂×Dic₄₇	—	8	12	no	—
115	Dic₉₄	—	4	12	no	—
116	Dic₉₄	—	4	4	no	—
117	Dic₉₄	—	8	12	no	—
118	C₈×C₄₇	C₂²×Q₈	1	4	yes	—
119	C₈×C₄₇	C₂²×Q₈	1	12	yes	—
120	#60667	—	47	1	no	maximal, Sylow
121	C₂²×C₄₇	—	2	3	no	—
122	C₂²×C₄₇	—	2	3	no	—
123	C₂²×C₄₇	C₂×C₄⋊Q₈	1	6	yes	—
124	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
125	C₄×C₄₇	—	2	3	no	—
126	C₄×C₄₇	—	2	3	no	—
127	C₄×C₄₇	C₂×C₄⋊Q₈	1	6	yes	—
128	C₄×C₄₇	#576	1	2	yes	—
129	C₄×C₄₇	#578	1	2	yes	—
130	C₄×C₄₇	C₂³×Q₈	1	2	yes	—
131	Dic₄₇	—	4	12	no	—
132	Dic₄₇	—	4	4	no	—
133	#3503	—	47	2	no	—
134	#3165	—	47	6	no	—
135	#3164	—	47	12	no	—
136	#3160	—	47	3	no	—
137	C₂×#500	—	47	3	no	—
138	#2913	—	47	4	no	—
139	C₄×#131	D₄₇	1	1	yes	characteristic
140	C₂×C₄₇	—	2	6	no	—
141	C₂×C₄₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
142	C₂×C₄₇	C₂×#576	1	1	yes	characteristic
143	C₂×C₄₇	C₂×#578	1	1	yes	characteristic
144	C₂×#131	C₂×D₄₇	1	2	yes	—
145	Q₈²	—	94	2	no	—
146	#557	—	94	2	no	—
147	#554	—	94	6	no	—
148	D₄×Q₈	—	94	6	no	—
149	#544	—	47	12	no	—
150	#533	—	47	6	no	—
151	C₂×C₄⋊Q₈	—	47	6	no	—
152	#520	—	47	4	no	—
153	C₄×#41	C₂×D₄₇	1	1	yes	characteristic
154	#501	—	47	24	no	—
155	#500	—	47	3	no	—
156	#500	—	47	12	no	—
157	#500	—	47	6	no	—
158	#500	—	47	1	no	—
159	#499	—	47	12	no	—
160	#498	—	47	6	no	—
161	#428	—	47	24	no	—
162	#427	—	47	12	no	—
163	C₂×#107	—	47	12	no	—
164	#405	C₂×D₄₇	1	6	yes	—
165	C₄×M₁₆	C₂×D₄₇	1	3	yes	—
166	C₂×C₄×C₈	C₂×D₄₇	1	3	yes	—
167	C₄₇	#60667	1	1	yes	characteristic, complemented, Sylow
168	C₂×#41	C₂²×D₄₇	1	1	yes	characteristic
169	C₂²×Q₈	—	94	6	no	—
170	#131	Dic₉₄	1	16	yes	complemented
171	C₂×M₁₆	C₂²×D₄₇	1	6	yes	—
172	C₂²×C₈	C₂²×D₄₇	1	6	yes	—
173	C₄⋊Q₈	—	47	6	no	—
174	C₄⋊Q₈	—	47	12	no	—
175	C₄⋊Q₈	—	47	2	no	—
176	C₄⋊Q₈	—	94	12	no	—
177	C₄⋊Q₈	—	94	6	no	—
178	C₄⋊Q₈	—	94	6	no	—
179	#125	—	47	12	no	—
180	#125	—	94	12	no	—
181	C₂²⋊Q₈	—	94	6	no	—
182	C₂²⋊Q₈	—	94	6	no	—
183	C₂²⋊Q₈	—	94	24	no	—
184	C₄×Q₈	—	94	4	no	—
185	C₄×Q₈	—	94	6	no	—
186	C₄×Q₈	—	94	6	no	—
187	C₄×Q₈	D₁₈₈	1	1	yes	characteristic
188	C₄×Q₈	D₁₈₈	1	1	yes	characteristic
189	C₄×D₄	—	94	12	no	—
190	C₄×D₄	D₁₈₈	1	3	yes	—
191	C₄×D₄	D₁₈₈	1	3	yes	—
192	#117	—	47	12	no	—
193	#117	C₂²×D₄₇	1	3	yes	—
194	C₂×C₄⋊C₄	—	47	12	no	—
195	C₂×C₄⋊C₄	—	94	6	no	—
196	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
197	#107	—	47	12	no	—
198	#107	—	47	24	no	—
199	#107	—	47	4	no	—
200	#106	—	47	24	no	—
201	#97	C₂²×D₄₇	1	6	yes	—
202	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
203	C₄×C₈	C₂²×D₄₇	1	3	yes	—
204	C₄×C₈	C₂²×D₄₇	1	6	yes	—
205	#41	C₂×Dic₉₄	1	8	yes	—
206	C₂×Q₈	—	94	12	no	—
207	C₂×Q₈	—	94	2	no	—
208	C₂×Q₈	—	94	6	no	—
209	C₂×Q₈	—	376	12	no	—
210	C₂×Q₈	C₂×D₁₈₈	1	1	yes	characteristic
211	C₂×D₄	C₂×D₁₈₈	1	3	yes	—
212	C₂²×C₄	—	94	12	no	—
213	C₂²×C₄	C₂×D₁₈₈	1	3	yes	—
214	C₂²×C₄	C₂×D₁₈₈	1	3	yes	—
215	C₂²×C₄	C₂³×D₄₇	1	3	yes	—
216	M₁₆	C₂×Dic₉₄	1	24	yes	complemented
217	C₂×C₈	C₂×Dic₉₄	1	24	yes	complemented
218	C₂×C₈	C₂³×D₄₇	1	2	yes	—
219	C₂×C₈	C₂³×D₄₇	1	6	yes	—
220	C₄⋊C₄	—	47	4	no	—
221	C₄⋊C₄	—	47	12	no	—
222	C₄⋊C₄	—	94	12	no	—
223	C₄⋊C₄	—	94	12	no	—
224	C₄⋊C₄	—	94	12	no	—
225	C₄⋊C₄	—	94	6	no	—
226	C₄⋊C₄	—	94	12	no	—
227	C₄⋊C₄	—	94	2	no	—
228	C₄⋊C₄	C₂×D₁₈₈	1	3	yes	—
229	C₄⋊C₄	C₂×D₁₈₈	1	3	yes	—
230	C₂²⋊C₄	—	94	12	no	—
231	C₂²⋊C₄	C₂×D₁₈₈	1	6	yes	—
232	C₄²	—	94	12	no	—
233	C₄²	C₂×D₁₈₈	1	3	yes	—
234	C₄²	C₂×D₁₈₈	1	3	yes	—
235	C₄²	C₂³×D₄₇	1	1	yes	characteristic
236	C₄²	C₂³×D₄₇	1	3	yes	—
237	C₂³	C₂²×D₁₈₈	1	3	yes	—
238	Q₈	—	376	12	no	—
239	Q₈	—	188	12	no	—
240	Q₈	—	188	4	no	—
241	Q₈	#13094024	1	4	yes	—
242	D₄	#13094024	1	12	yes	—
243	C₂×C₄	—	2	6	no	—
244	C₂×C₄	—	2	6	no	—
245	C₂×C₄	—	94	4	no	—
246	C₂×C₄	—	94	12	no	—
247	C₂×C₄	—	376	12	no	—
248	C₂×C₄	C₂²×D₁₈₈	1	3	yes	—
249	C₂×C₄	C₂²×D₁₈₈	1	3	yes	—
250	C₂×C₄	C₂²×D₁₈₈	1	3	yes	—
251	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
252	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
253	C₂×C₄	C₂²×Dic₉₄	1	12	yes	—
254	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
255	C₈	C₂²×Dic₉₄	1	4	yes	—
256	C₈	C₂²×Dic₉₄	1	12	yes	—
257	C₂²	—	2	3	no	—
258	C₂²	—	2	3	no	—
259	C₂²	C₂×#13094024	1	6	yes	—
260	C₂²	C₂³×D₁₈₈	1	1	yes	characteristic
261	C₄	—	2	3	no	—
262	C₄	—	2	3	no	—
263	C₄	—	188	12	no	—
264	C₄	—	188	4	no	—
265	C₄	C₂×#13094024	1	6	yes	—
266	C₄	#15689928	1	2	yes	—
267	C₄	#15689940	1	2	yes	—
268	C₄	C₂³×Dic₉₄	1	2	yes	—
269	C₂	—	2	6	no	—
270	C₂	C₂²×#13094024	1	1	yes	characteristic
271	C₂	C₂×#15689928	1	1	yes	characteristic
272	C₂	C₂×#15689940	1	1	yes	characteristic

	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	#60667
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. #20253927	C₂	semidirect
4. #20253902	C₂	semidirect
5. C₂×#15689528	C₂	semidirect
55. #131×C₄₇	Q₈	semidirect
79. M₁₆×C₄₇	C₂×Q₈	semidirect
80. C₂×C₈×C₄₇	C₂×Q₈	semidirect
167. C₄₇	#60667	semidirect
170. #131	Dic₉₄	semidirect
216. M₁₆	C₂×Dic₉₄	semidirect
217. 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
#20253927	2
#20253902	2
C₂×#15689528	2
#20251988	2
C₄×#131×C₄₇	2
#60667	47