Subgroups — #26186811

Order 12032.

614 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253974	C₂	1	1	yes	maximal, characteristic
2	#20252322	C₂	1	2	yes	maximal
3	#20242286	C₂	1	4	yes	maximal, complemented
4	C₈×Dic₁₈₈	C₂	1	4	yes	maximal, complemented
5	#20242147	C₂	1	1	yes	maximal, characteristic
6	#20241844	C₂	1	2	yes	maximal
7	C₂×C₈²×C₄₇	C₂	1	1	yes	maximal, characteristic
8	#15689680	C₂²	1	2	yes	—
9	#15689528	C₄	1	1	yes	characteristic
10	#15689503	C₄	1	1	yes	characteristic
11	#15689474	C₄	1	2	yes	—
12	#15689364	C₄	1	4	yes	—
13	#15689253	—	2	2	no	—
14	#15689253	C₂²	1	4	yes	—
15	C₈×Dic₉₄	—	2	2	no	—
16	C₈×Dic₉₄	C₂²	1	4	yes	—
17	C₄×Dic₁₈₈	C₂²	1	4	yes	—
18	#15689149	—	2	2	no	—
19	#15689147	C₂²	1	2	yes	—
20	#15689078	—	2	2	no	—
21	#15688876	C₂²	1	4	yes	—
22	#15688855	C₂²	1	8	yes	—
23	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₈²×C₄₇	C₂²	1	2	yes	—
27	C₈²×C₄₇	C₂²	1	2	yes	—
28	C₂×Dic₁₈₈	C₂×C₄	1	4	yes	—
29	C₂×#10625329	—	2	2	no	—
30	#13094024	C₂×C₄	1	2	yes	—
31	#13094024	C₂×C₄	1	2	yes	—
32	#13094016	—	2	2	no	—
33	#13094003	—	2	2	no	—
34	#13094003	C₂×C₄	1	4	yes	—
35	C₄×Dic₉₄	—	2	2	no	—
36	C₄×Dic₉₄	C₂³	1	4	yes	—
37	#13093983	—	2	2	no	—
38	#13093958	C₂×C₄	1	4	yes	—
39	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
40	C₄⋊(C₄₇⋊C₈)	C₂³	1	4	yes	—
41	Dic₄₇⋊C₈	—	2	4	no	—
42	Dic₄₇⋊C₈	—	2	4	no	—
43	#13093949	C₂×C₄	1	8	yes	—
44	#13093934	—	2	4	no	—
45	C₈×Dic₄₇	—	2	4	no	—
46	C₂²×C₈×C₄₇	—	2	1	no	—
47	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
48	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
51	C₄×C₈×C₄₇	—	2	1	no	—
52	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
55	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
56	C₄×C₈×C₄₇	C₂³	1	2	yes	—
57	C₄×C₈×C₄₇	C₂³	1	2	yes	—
58	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
63	C₂×Dic₉₄	—	2	2	no	—
64	C₂×Dic₉₄	C₂²×C₄	1	4	yes	—
65	C₂×C₈×C₄₇	—	2	4	no	—
66	C₂×C₈×C₄₇	—	2	2	no	—
67	C₂×C₈×C₄₇	—	2	2	no	—
68	C₂×C₈×C₄₇	—	2	4	no	—
69	C₂×C₈×C₄₇	Q₁₆	1	4	yes	complemented
70	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₄₇	#41	1	1	yes	characteristic
79	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
80	C₂²⋊Dic₄₇	—	2	4	no	—
81	C₄⋊Dic₄₇	—	2	2	no	—
82	C₄⋊Dic₄₇	—	2	2	no	—
83	C₄⋊Dic₄₇	—	2	2	no	—
84	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
85	Dic₄₇⋊C₄	—	2	4	no	—
86	Dic₄₇⋊C₄	—	2	4	no	—
87	C₄×Dic₄₇	—	2	4	no	—
88	#10625329	—	4	4	no	—
89	C₂×C₄₇⋊C₈	—	2	4	no	—
90	Dic₁₈₈	—	2	8	no	—
91	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₄₇	—	2	2	no	—
93	C₂×C₄×C₄₇	—	2	2	no	—
94	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
98	C₂×C₄×C₄₇	#135	1	2	yes	—
99	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
102	C₂×Dic₄₇	—	2	4	no	—
103	C₂×Dic₄₇	—	2	4	no	—
104	Dic₉₄	—	2	8	no	—
105	Dic₉₄	—	4	4	no	—
106	C₈×C₄₇	—	2	4	no	—
107	C₈×C₄₇	—	2	2	no	—
108	C₈×C₄₇	—	2	1	no	—
109	C₈×C₄₇	—	2	1	no	—
110	C₈×C₄₇	#131	1	4	yes	—
111	C₈×C₄₇	C₂×Q₁₆	1	2	yes	—
112	C₈×C₄₇	C₂×Q₁₆	1	2	yes	—
113	C₄₇⋊C₈	—	4	4	no	—
114	#17958	—	47	1	no	maximal, Sylow
115	C₂²×C₄₇	—	2	1	no	—
116	C₂²×C₄₇	—	2	1	no	—
117	C₂²×C₄₇	C₄×Q₁₆	1	2	yes	—
118	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
119	C₄×C₄₇	—	2	1	no	—
120	C₄×C₄₇	—	2	1	no	—
121	C₄×C₄₇	C₄×Q₁₆	1	2	yes	—
122	C₄×C₄₇	#443	1	2	yes	—
123	C₄×C₄₇	C₂×#131	1	2	yes	—
124	C₄×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
125	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
126	Dic₄₇	—	4	4	no	—
127	Dic₄₇	—	4	4	no	—
128	#3170	—	47	1	no	—
129	#2960	—	47	2	no	—
130	#1591	—	47	4	no	—
131	C₈×Q₁₆	—	47	4	no	—
132	#1567	—	47	1	no	—
133	#1493	—	47	2	no	—
134	C₂×C₈²	D₄₇	1	1	yes	characteristic
135	C₂×C₄₇	—	2	2	no	—
136	C₂×C₄₇	#2931	1	1	yes	characteristic
137	C₂×C₄₇	C₂×C₄×Q₁₆	1	1	yes	characteristic
138	C₂×C₄₇	C₂×#443	1	1	yes	characteristic
139	#533	—	47	2	no	—
140	#500	—	47	1	no	—
141	#494	—	47	1	no	—
142	#487	—	47	2	no	—
143	#467	—	47	4	no	—
144	#446	—	47	4	no	—
145	#446	—	94	2	no	—
146	C₈×Q₈	—	47	4	no	—
147	C₈×Q₈	—	94	2	no	—
148	C₄×Q₁₆	—	47	4	no	—
149	#424	—	94	2	no	—
150	#423	—	47	2	no	—
151	#405	—	94	2	no	—
152	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
153	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
154	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
155	#335	—	47	4	no	—
156	Q₈⋊C₈	—	47	8	no	—
157	C₈²	C₂×D₄₇	1	2	yes	—
158	C₈²	C₂×D₄₇	1	2	yes	—
159	C₄₇	#17958	1	1	yes	characteristic, complemented, Sylow
160	C₂×Q₁₆	—	47	4	no	—
161	C₂×M₁₆	—	94	2	no	—
162	C₂²×C₈	—	2	1	no	—
163	C₂²×C₈	C₄×D₄₇	1	2	yes	—
164	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
165	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
166	C₄⋊Q₈	—	47	2	no	—
167	C₄⋊Q₈	—	47	2	no	—
168	#125	—	94	2	no	—
169	C₂²⋊Q₈	—	47	4	no	—
170	C₂²⋊Q₈	—	94	2	no	—
171	C₄×Q₈	—	47	4	no	—
172	C₄×Q₈	—	94	2	no	—
173	#117	—	94	2	no	—
174	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
175	#107	—	47	4	no	—
176	C₄⋊C₈	—	47	4	no	—
177	C₄⋊C₈	—	94	4	no	—
178	C₄⋊C₈	—	94	4	no	—
179	C₄⋊C₈	—	94	2	no	—
180	#103	—	47	8	no	—
181	#97	—	94	4	no	—
182	C₄×C₈	—	2	1	no	—
183	C₄×C₈	—	94	4	no	—
184	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
185	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
186	C₄×C₈	D₁₈₈	1	1	yes	characteristic
187	C₄×C₈	D₁₈₈	1	1	yes	characteristic
188	C₄×C₈	C₂²×D₄₇	1	2	yes	—
189	C₄×C₈	C₂²×D₄₇	1	2	yes	—
190	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
191	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
192	C₂×Q₈	—	47	4	no	—
193	C₂×Q₈	—	94	2	no	—
194	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
196	C₂²×C₄	#10625350	1	1	yes	characteristic
197	Q₁₆	—	94	8	no	—
198	M₁₆	—	188	4	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	94	4	no	—
204	C₂×C₈	Dic₁₈₈	1	4	yes	complemented
205	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
206	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
207	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
208	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
209	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
210	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
211	C₄⋊C₄	—	47	4	no	—
212	C₄⋊C₄	—	94	4	no	—
213	C₄⋊C₄	—	94	2	no	—
214	C₄⋊C₄	—	94	2	no	—
215	C₄⋊C₄	—	94	2	no	—
216	C₄⋊C₄	—	94	4	no	—
217	C₂²⋊C₄	—	94	4	no	—
218	C₄²	—	94	4	no	—
219	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
220	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
221	C₄²	#10625350	1	1	yes	characteristic
222	C₄²	C₂³×D₄₇	1	1	yes	characteristic
223	C₂³	C₄×D₁₈₈	1	1	yes	characteristic
224	Q₈	—	94	8	no	—
225	Q₈	—	188	4	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	94	4	no	—
229	C₂×C₄	—	94	4	no	—
230	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
231	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
232	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
233	C₂×C₄	C₂×Dic₁₈₈	1	2	yes	—
234	C₂×C₄	#13094046	1	2	yes	—
235	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
236	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
237	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
238	C₈	—	2	4	no	—
239	C₈	—	2	2	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	188	4	no	—
243	C₈	#13094030	1	4	yes	—
244	C₈	C₂×Dic₁₈₈	1	2	yes	—
245	C₈	C₂×Dic₁₈₈	1	2	yes	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	C₄×Dic₁₈₈	1	2	yes	—
249	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	188	4	no	—
253	C₄	—	188	4	no	—
254	C₄	C₄×Dic₁₈₈	1	2	yes	—
255	C₄	#15689241	1	2	yes	—
256	C₄	C₂×#13094030	1	2	yes	—
257	C₄	C₂²×Dic₁₈₈	1	1	yes	characteristic
258	C₄	C₂×#13094046	1	1	yes	characteristic
259	C₂	—	2	2	no	—
260	C₂	#20252101	1	1	yes	characteristic
261	C₂	C₂×C₄×Dic₁₈₈	1	1	yes	characteristic
262	C₂	C₂×#15689241	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₈²×C₄₇	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂³×C₄₇	C₄×D₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	C₂×C₈²	D₄₇
hypocenter	C₄₇	#17958
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
3. #20242286	C₂	semidirect
4. C₈×Dic₁₈₈	C₂	semidirect
69. C₂×C₈×C₄₇	Q₁₆	semidirect
159. C₄₇	#17958	semidirect
204. C₂×C₈	Dic₁₈₈	semidirect

Maximal subgroups

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

subgroup	index
#20253974	2
#20252322	2
#20242286	2
C₈×Dic₁₈₈	2
#20242147	2
#20241844	2
C₂×C₈²×C₄₇	2
#17958	47