Subgroups — #26225157

Order 12032.

670 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253912	C₂	1	1	yes	maximal, characteristic
2	#20252328	C₂	1	2	yes	maximal
3	#20244365	C₂	1	2	yes	maximal
4	#20244141	C₂	1	2	yes	maximal
5	#20244136	C₂	1	2	yes	maximal
6	#20243563	C₂	1	1	yes	maximal, characteristic
7	#20243272	C₂	1	2	yes	maximal
8	#20243267	C₂	1	2	yes	maximal
9	C₄×#97×C₄₇	C₂	1	1	yes	maximal, characteristic
10	#15689680	C₂²	1	2	yes	—
11	#15689532	C₄	1	8	yes	complemented
12	#15689251	—	2	8	no	—
13	C₂×#13093960	C₂²	1	2	yes	—
14	C₂×#13093960	C₂²	1	2	yes	—
15	#15689161	C₂²	1	2	yes	—
16	#15689161	C₂²	1	2	yes	—
17	#15689149	C₂²	1	4	yes	—
18	#15689147	C₂²	1	2	yes	—
19	#15689078	C₂²	1	4	yes	—
20	#15688883	C₂²	1	4	yes	—
21	#15688883	C₂²	1	4	yes	—
22	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₄³×C₄₇	C₂²	1	1	yes	characteristic
29	C₂×#10625329	C₂³	1	4	yes	—
30	#13094024	C₂×C₄	1	4	yes	—
31	#13094016	C₂×C₄	1	4	yes	—
32	#13094003	—	2	4	no	—
33	C₄×Dic₉₄	—	2	4	no	—
34	#13093983	C₂³	1	4	yes	—
35	#13093960	—	2	4	no	—
36	#13093960	—	2	4	no	—
37	#13093958	C₂×C₄	1	4	yes	—
38	#13093958	C₂×C₄	1	4	yes	—
39	#13093956	C₂×C₄	1	4	yes	—
40	#13093956	C₂×C₄	1	4	yes	—
41	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
42	Dic₄₇⋊C₈	—	2	8	no	—
43	#13093934	—	2	4	no	—
44	C₈×Dic₄₇	—	2	4	no	—
45	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	—	2	2	no	—
50	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
53	#97×C₄₇	—	2	2	no	—
54	#97×C₄₇	C₂×C₄	1	4	yes	—
55	#97×C₄₇	D₄	1	4	yes	—
56	#97×C₄₇	Q₈	1	4	yes	complemented
57	C₄×C₈×C₄₇	—	2	2	no	—
58	C₄×C₈×C₄₇	Q₈	1	4	yes	complemented
59	C₂²×C₄×C₄₇	—	2	1	no	—
60	C₂²×C₄×C₄₇	—	2	1	no	—
61	C₂²×C₄×C₄₇	#41	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
63	C₂×Dic₉₄	—	2	4	no	—
64	C₂×C₈×C₄₇	—	2	1	no	—
65	C₂×C₈×C₄₇	—	2	1	no	—
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	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
71	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
72	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
73	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
74	C₂×C₈×C₄₇	#41	1	2	yes	—
75	C₂×C₈×C₄₇	#41	1	2	yes	—
76	C₄²×C₄₇	—	2	1	no	—
77	C₄²×C₄₇	—	2	1	no	—
78	C₄²×C₄₇	—	4	4	no	—
79	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
80	C₄²×C₄₇	C₂×D₄	1	2	yes	—
81	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
82	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
83	C₂²⋊Dic₄₇	—	2	4	no	—
84	C₄⋊Dic₄₇	—	2	4	no	—
85	C₄⋊Dic₄₇	C₂²×C₄	1	8	yes	—
86	Dic₄₇⋊C₄	—	2	8	no	—
87	C₄×Dic₄₇	—	2	4	no	—
88	#10625329	—	2	4	no	—
89	#10625329	—	2	4	no	—
90	C₂×C₄₇⋊C₈	—	2	4	no	—
91	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
92	C₂×C₄×C₄₇	—	2	4	no	—
93	C₂×C₄×C₄₇	—	2	2	no	—
94	C₂×C₄×C₄₇	—	2	2	no	—
95	C₂×C₄×C₄₇	—	4	2	no	—
96	C₂×C₄×C₄₇	—	4	2	no	—
97	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
98	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
99	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
102	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
103	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
104	C₂×Dic₄₇	—	2	8	no	—
105	Dic₉₄	—	8	4	no	—
106	C₈×C₄₇	—	2	2	no	—
107	C₈×C₄₇	C₄×Q₈	1	4	yes	—
108	C₈×C₄₇	C₄×Q₈	1	4	yes	—
109	C₈×C₄₇	C₄⋊Q₈	1	4	yes	—
110	C₄₇⋊C₈	—	4	4	no	—
111	#21736	—	47	1	no	maximal, Sylow
112	C₂²×C₄₇	—	2	2	no	—
113	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
114	C₂²×C₄₇	#533	1	1	yes	characteristic
115	C₂²×C₄₇	#533	1	1	yes	characteristic
116	C₄×C₄₇	—	2	4	no	—
117	C₄×C₄₇	—	2	4	no	—
118	C₄×C₄₇	—	4	2	no	—
119	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
120	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
121	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
122	C₄×C₄₇	#532	1	1	yes	characteristic
123	Dic₄₇	—	4	8	no	—
124	#3162	—	47	1	no	—
125	#2961	—	47	2	no	—
126	#1994	—	47	2	no	—
127	#1946	—	47	2	no	—
128	#1945	—	47	2	no	—
129	#1849	—	47	1	no	—
130	#1773	—	47	2	no	—
131	#1772	—	47	2	no	—
132	C₄×#97	D₄₇	1	1	yes	characteristic
133	C₂×C₄₇	—	2	1	no	—
134	C₂×C₄₇	—	2	1	no	—
135	C₂×C₄₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
136	C₂×C₄₇	#2956	1	2	yes	—
137	#533	—	47	2	no	—
138	#501	—	47	8	no	—
139	#446	—	94	8	no	—
140	C₂×#108	—	47	2	no	—
141	C₂×#108	—	47	2	no	—
142	#427	—	47	2	no	—
143	#427	—	47	2	no	—
144	#424	—	47	4	no	—
145	#423	—	47	2	no	—
146	#405	—	47	4	no	—
147	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
148	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
149	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
150	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
151	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
152	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
153	C₄³	C₂×D₄₇	1	1	yes	characteristic
154	#339	—	47	4	no	—
155	#339	—	47	4	no	—
156	C₄₇	#21736	1	1	yes	characteristic, complemented, Sylow
157	C₂×M₁₆	—	47	4	no	—
158	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
159	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
160	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
161	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
162	C₄⋊Q₈	—	47	4	no	—
163	#125	—	47	4	no	—
164	C₂²⋊Q₈	—	94	4	no	—
165	C₄×Q₈	—	94	4	no	—
166	#117	—	47	4	no	—
167	C₂×C₄²	—	2	2	no	—
168	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
169	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
170	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
171	#108	—	94	4	no	—
172	#108	—	94	4	no	—
173	#107	—	47	4	no	—
174	#107	—	47	4	no	—
175	#106	—	47	4	no	—
176	#106	—	47	4	no	—
177	C₄⋊C₈	—	94	8	no	—
178	C₄⋊C₈	—	94	4	no	—
179	#97	—	2	2	no	—
180	#97	—	94	4	no	—
181	#97	Dic₉₄	1	4	yes	complemented
182	#97	C₄×D₄₇	1	4	yes	—
183	#97	D₁₈₈	1	4	yes	—
184	C₄×C₈	—	2	2	no	—
185	C₄×C₈	—	94	4	no	—
186	C₄×C₈	Dic₉₄	1	4	yes	complemented
187	C₂×Q₈	—	94	4	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	#10625350	1	2	yes	—
191	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
192	M₁₆	—	94	4	no	—
193	M₁₆	—	94	4	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	94	4	no	—
197	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
198	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
199	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
200	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
201	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
202	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
203	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
204	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
205	C₂×C₈	#10625350	1	2	yes	—
206	C₂×C₈	#10625350	1	2	yes	—
207	C₄⋊C₄	—	47	8	no	—
208	C₄⋊C₄	—	94	8	no	—
209	C₄⋊C₄	—	94	4	no	—
210	C₂²⋊C₄	—	94	4	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	2	1	no	—
213	C₄²	—	4	4	no	—
214	C₄²	—	94	4	no	—
215	C₄²	C₂×Dic₉₄	1	2	yes	—
216	C₄²	C₂×Dic₉₄	1	2	yes	—
217	C₄²	C₂×C₄×D₄₇	1	2	yes	—
218	C₄²	C₂×D₁₈₈	1	2	yes	—
219	C₂³	C₂×#10625350	1	1	yes	characteristic
220	Q₈	—	376	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	94	8	no	—
227	C₂×C₄	C₄×D₁₈₈	1	4	yes	—
228	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
229	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
230	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
231	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
232	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
233	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
234	C₈	—	2	2	no	—
235	C₈	—	188	4	no	—
236	C₈	C₄×Dic₉₄	1	4	yes	—
237	C₈	C₄×Dic₉₄	1	4	yes	—
238	C₈	#13094024	1	4	yes	—
239	C₂²	—	2	2	no	—
240	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
241	C₂²	#15689680	1	1	yes	characteristic
242	C₂²	#15689680	1	1	yes	characteristic
243	C₄	—	2	4	no	—
244	C₄	—	2	4	no	—
245	C₄	—	4	2	no	—
246	C₄	—	188	8	no	—
247	C₄	C₂×C₄×Dic₉₄	1	1	yes	characteristic
248	C₄	C₂×C₄×Dic₉₄	1	1	yes	characteristic
249	C₄	C₂×#13094024	1	1	yes	characteristic
250	C₄	#15689673	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	2	1	no	—
253	C₂	C₄×#13094024	1	1	yes	characteristic
254	C₂	#20252299	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄×#97×C₄₇	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂³×C₄₇	C₂×#41
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#97	D₄₇
hypocenter	C₄₇	#21736
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
11. #15689532	C₄	semidirect
56. #97×C₄₇	Q₈	semidirect
58. C₄×C₈×C₄₇	Q₈	semidirect
156. C₄₇	#21736	semidirect
181. #97	Dic₉₄	semidirect
186. C₄×C₈	Dic₉₄	semidirect

Maximal subgroups

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

subgroup	index
#20253912	2
#20252328	2
#20244365	2
#20244141	2
#20244136	2
#20243563	2
#20243272	2
#20243267	2
C₄×#97×C₄₇	2
#21736	47