Subgroups — #26187002

Order 12032.

590 conjugacy classes of subgroups.

The subgroup lattice has 248 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	#20242240	C₂	1	8	yes	maximal, complemented
4	#20242147	C₂	1	1	yes	maximal, characteristic
5	#20241844	C₂	1	2	yes	maximal
6	C₂×C₄₇×#322	C₂	1	1	yes	maximal, characteristic
7	#15689680	C₂²	1	2	yes	—
8	#15689528	C₄	1	1	yes	characteristic
9	#15689503	C₄	1	1	yes	characteristic
10	#15689474	C₄	1	2	yes	—
11	#15689364	C₄	1	4	yes	—
12	#15689253	—	2	2	no	—
13	#15689253	C₂²	1	4	yes	—
14	C₈×Dic₉₄	—	2	2	no	—
15	C₈×Dic₉₄	C₂²	1	4	yes	—
16	C₄×Dic₁₈₈	C₂²	1	4	yes	—
17	#15689149	—	2	2	no	—
18	#15689147	C₂²	1	2	yes	—
19	#15689078	—	2	2	no	—
20	#15688876	C₂²	1	4	yes	—
21	#15688855	C₂²	1	8	yes	—
22	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₄₇×#322	C₂²	1	4	yes	—
26	C₂×Dic₁₈₈	C₂×C₄	1	4	yes	—
27	C₂×#10625329	—	2	2	no	—
28	#13094024	C₂×C₄	1	2	yes	—
29	#13094024	C₂×C₄	1	2	yes	—
30	#13094016	—	2	2	no	—
31	#13094003	—	2	2	no	—
32	#13094003	C₂×C₄	1	4	yes	—
33	C₄×Dic₉₄	—	2	2	no	—
34	C₄×Dic₉₄	C₂³	1	4	yes	—
35	#13093983	—	2	2	no	—
36	#13093958	C₂×C₄	1	4	yes	—
37	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
38	C₄⋊(C₄₇⋊C₈)	C₂³	1	4	yes	—
39	Dic₄₇⋊C₈	—	2	4	no	—
40	Dic₄₇⋊C₈	—	2	4	no	—
41	#13093949	C₂×C₄	1	8	yes	—
42	#13093934	—	2	4	no	—
43	C₈×Dic₄₇	—	2	4	no	—
44	C₂²×C₈×C₄₇	—	2	1	no	—
45	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
46	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
49	C₄×C₈×C₄₇	—	2	1	no	—
50	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₄₇	D₄	1	2	yes	—
53	C₄×C₈×C₄₇	C₂³	1	2	yes	—
54	C₄×C₈×C₄₇	C₂³	1	2	yes	—
55	C₄×C₈×C₄₇	C₂³	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
59	C₂×Dic₉₄	—	2	2	no	—
60	C₂×Dic₉₄	C₂²×C₄	1	4	yes	—
61	C₂×C₈×C₄₇	—	2	2	no	—
62	C₂×C₈×C₄₇	—	2	2	no	—
63	C₂×C₈×C₄₇	—	4	2	no	—
64	C₂×C₈×C₄₇	—	4	2	no	—
65	C₂×C₈×C₄₇	Q₁₆	1	4	yes	complemented
66	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
70	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
71	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₄₇	#41	1	1	yes	characteristic
73	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
74	C₂²⋊Dic₄₇	—	2	4	no	—
75	C₄⋊Dic₄₇	—	2	2	no	—
76	C₄⋊Dic₄₇	—	2	2	no	—
77	C₄⋊Dic₄₇	—	2	2	no	—
78	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
79	Dic₄₇⋊C₄	—	2	4	no	—
80	Dic₄₇⋊C₄	—	2	4	no	—
81	C₄×Dic₄₇	—	2	4	no	—
82	#10625329	—	4	4	no	—
83	C₂×C₄₇⋊C₈	—	2	4	no	—
84	Dic₁₈₈	—	4	4	no	—
85	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₄₇	—	2	2	no	—
87	C₂×C₄×C₄₇	—	2	2	no	—
88	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
91	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
92	C₂×C₄×C₄₇	#135	1	2	yes	—
93	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
96	C₂×Dic₄₇	—	2	4	no	—
97	C₂×Dic₄₇	—	2	4	no	—
98	Dic₉₄	—	2	8	no	—
99	Dic₉₄	—	4	4	no	—
100	C₈×C₄₇	—	2	2	no	—
101	C₈×C₄₇	—	2	2	no	—
102	C₈×C₄₇	—	2	2	no	—
103	C₈×C₄₇	—	4	2	no	—
104	C₈×C₄₇	C₂×Q₁₆	1	4	yes	—
105	C₄₇⋊C₈	—	4	4	no	—
106	#17977	—	47	1	no	maximal, Sylow
107	C₂²×C₄₇	—	2	1	no	—
108	C₂²×C₄₇	—	2	1	no	—
109	C₂²×C₄₇	C₄×Q₁₆	1	2	yes	—
110	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
111	C₄×C₄₇	—	2	1	no	—
112	C₄×C₄₇	—	2	1	no	—
113	C₄×C₄₇	C₄×Q₁₆	1	2	yes	—
114	C₄×C₄₇	#444	1	2	yes	—
115	C₄×C₄₇	#568	1	2	yes	—
116	C₄×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
117	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
118	Dic₄₇	—	4	4	no	—
119	Dic₄₇	—	4	4	no	—
120	#3170	—	47	1	no	—
121	#2960	—	47	2	no	—
122	#1584	—	47	8	no	—
123	#1567	—	47	1	no	—
124	#1493	—	47	2	no	—
125	C₂×#322	D₄₇	1	1	yes	characteristic
126	C₂×C₄₇	—	2	2	no	—
127	C₂×C₄₇	#2932	1	1	yes	characteristic
128	C₂×C₄₇	C₂×C₄×Q₁₆	1	1	yes	characteristic
129	C₂×C₄₇	C₂×#444	1	1	yes	characteristic
130	#533	—	47	2	no	—
131	#500	—	47	1	no	—
132	#494	—	47	1	no	—
133	#487	—	47	2	no	—
134	#467	—	47	4	no	—
135	#446	—	47	4	no	—
136	#446	—	94	2	no	—
137	C₈×Q₈	—	47	4	no	—
138	C₈×Q₈	—	94	2	no	—
139	C₄×Q₁₆	—	47	4	no	—
140	#424	—	94	2	no	—
141	#423	—	47	2	no	—
142	#405	—	94	2	no	—
143	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
144	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
145	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
146	#335	—	47	4	no	—
147	Q₈⋊C₈	—	47	8	no	—
148	#322	C₂×D₄₇	1	4	yes	—
149	C₄₇	#17977	1	1	yes	characteristic, complemented, Sylow
150	C₂×Q₁₆	—	47	4	no	—
151	C₂×M₁₆	—	94	2	no	—
152	C₂²×C₈	—	2	1	no	—
153	C₂²×C₈	C₄×D₄₇	1	2	yes	—
154	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
155	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
156	C₄⋊Q₈	—	47	2	no	—
157	C₄⋊Q₈	—	47	2	no	—
158	#125	—	94	2	no	—
159	C₂²⋊Q₈	—	47	4	no	—
160	C₂²⋊Q₈	—	94	2	no	—
161	C₄×Q₈	—	47	4	no	—
162	C₄×Q₈	—	94	2	no	—
163	#117	—	94	2	no	—
164	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
165	#107	—	47	4	no	—
166	C₄⋊C₈	—	47	4	no	—
167	C₄⋊C₈	—	94	4	no	—
168	C₄⋊C₈	—	94	4	no	—
169	C₄⋊C₈	—	94	2	no	—
170	#103	—	47	8	no	—
171	#97	—	94	4	no	—
172	C₄×C₈	—	2	1	no	—
173	C₄×C₈	—	94	4	no	—
174	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
175	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
176	C₄×C₈	D₁₈₈	1	2	yes	—
177	C₄×C₈	C₂²×D₄₇	1	2	yes	—
178	C₄×C₈	C₂²×D₄₇	1	2	yes	—
179	C₄×C₈	C₂²×D₄₇	1	2	yes	—
180	C₂×Q₈	—	47	4	no	—
181	C₂×Q₈	—	94	2	no	—
182	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
183	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
184	C₂²×C₄	#10625350	1	1	yes	characteristic
185	Q₁₆	—	188	4	no	—
186	M₁₆	—	188	4	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	—	94	4	no	—
192	C₂×C₈	Dic₁₈₈	1	4	yes	complemented
193	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
194	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
195	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
196	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
197	C₄⋊C₄	—	47	4	no	—
198	C₄⋊C₄	—	94	4	no	—
199	C₄⋊C₄	—	94	2	no	—
200	C₄⋊C₄	—	94	2	no	—
201	C₄⋊C₄	—	94	2	no	—
202	C₄⋊C₄	—	94	4	no	—
203	C₂²⋊C₄	—	94	4	no	—
204	C₄²	—	94	4	no	—
205	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
206	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
207	C₄²	#10625350	1	1	yes	characteristic
208	C₄²	C₂³×D₄₇	1	1	yes	characteristic
209	C₂³	C₄×D₁₈₈	1	1	yes	characteristic
210	Q₈	—	94	8	no	—
211	Q₈	—	188	4	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	94	4	no	—
215	C₂×C₄	—	94	4	no	—
216	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
217	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
218	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
219	C₂×C₄	C₂×Dic₁₈₈	1	2	yes	—
220	C₂×C₄	#13094046	1	2	yes	—
221	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
222	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
223	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	—	2	2	no	—
226	C₈	—	2	2	no	—
227	C₈	—	4	2	no	—
228	C₈	—	188	4	no	—
229	C₈	C₂×Dic₁₈₈	1	4	yes	—
230	C₂²	—	2	1	no	—
231	C₂²	—	2	1	no	—
232	C₂²	C₄×Dic₁₈₈	1	2	yes	—
233	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	188	4	no	—
237	C₄	—	188	4	no	—
238	C₄	C₄×Dic₁₈₈	1	2	yes	—
239	C₄	#15689246	1	2	yes	—
240	C₄	#15689889	1	2	yes	—
241	C₄	C₂²×Dic₁₈₈	1	1	yes	characteristic
242	C₄	C₂×#13094046	1	1	yes	characteristic
243	C₂	—	2	2	no	—
244	C₂	#20252107	1	1	yes	characteristic
245	C₂	C₂×C₄×Dic₁₈₈	1	1	yes	characteristic
246	C₂	C₂×#15689246	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₄₇×#322	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂³×C₄₇	C₄×D₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	C₂×#322	D₄₇
hypocenter	C₄₇	#17977
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
3. #20242240	C₂	semidirect
65. C₂×C₈×C₄₇	Q₁₆	semidirect
149. C₄₇	#17977	semidirect
192. 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
#20242240	2
#20242147	2
#20241844	2
C₂×C₄₇×#322	2
#17977	47