Subgroups — #26221229

Order 12032.

714 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252026	C₂	1	1	yes	maximal, characteristic
2	#20251807	C₂	1	1	yes	maximal, characteristic
3	#20244164	C₂	1	4	yes	maximal, complemented
4	#20243595	C₂	1	4	yes	maximal
5	#20243294	C₂	1	2	yes	maximal, complemented
6	C₄×#13093960	C₂	1	2	yes	maximal, complemented
7	C₂×C₄₇×#433	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10625329	C₂²	1	2	yes	—
9	#15689184	—	2	4	no	—
10	C₂×#13093960	C₂²	1	4	yes	—
11	C₂×#13093960	C₂²	1	4	yes	—
12	#15689147	C₂²	1	2	yes	—
13	#15689087	C₂²	1	2	yes	—
14	#15689079	—	2	4	no	—
15	C₄×#10625329	C₂²	1	2	yes	—
16	C₄×#10625329	C₂²	1	2	yes	—
17	C₄×#10625329	C₂²	1	2	yes	—
18	#15688888	C₂²	1	4	yes	—
19	#15688888	C₂²	1	4	yes	—
20	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
21	C₄₇×#433	C₄	1	8	yes	—
22	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
24	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×#10625329	—	2	2	no	—
27	C₂×#10625329	—	2	8	no	—
28	C₂×#10625329	C₂³	1	4	yes	—
29	C₂×#10625329	C₂³	1	4	yes	—
30	C₂²×C₄₇⋊C₈	—	2	2	no	—
31	#13093960	—	2	8	no	—
32	#13093960	—	2	8	no	—
33	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
34	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
35	#13093935	—	2	2	no	—
36	#13093935	—	2	2	no	—
37	#13093935	—	2	2	no	—
38	C₄×C₄₇⋊C₈	—	2	2	no	—
39	C₄×C₄₇⋊C₈	—	2	2	no	—
40	C₄×C₄₇⋊C₈	—	2	2	no	—
41	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
43	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
44	#117×C₄₇	C₂×C₄	1	4	yes	—
45	#117×C₄₇	D₄	1	4	yes	—
46	C₂×C₄⋊C₄×C₄₇	Q₈	1	2	yes	—
47	C₂×C₂²⋊C₄×C₄₇	Q₈	1	2	yes	—
48	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
50	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
51	C₂²⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
52	#97×C₄₇	C₂×C₄	1	4	yes	—
53	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
54	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	—	2	4	no	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
62	C₂×C₈×C₄₇	—	2	4	no	—
63	C₂×C₈×C₄₇	—	2	4	no	—
64	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
65	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
66	C₄⋊C₄×C₄₇	C₄⋊C₄	1	8	yes	—
67	C₂²⋊C₄×C₄₇	C₄⋊C₄	1	8	yes	—
68	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
69	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
70	C₄²×C₄₇	C₂×D₄	1	2	yes	—
71	C₄²×C₄₇	C₂×D₄	1	2	yes	—
72	#10625329	—	2	8	no	—
73	#10625329	—	2	4	no	—
74	#10625329	—	2	4	no	—
75	#10625329	—	2	4	no	—
76	#10625329	#41	1	8	yes	—
77	C₂×C₄₇⋊C₈	—	2	8	no	—
78	C₂×C₄₇⋊C₈	—	2	4	no	—
79	C₂×C₄₇⋊C₈	—	2	4	no	—
80	C₂³×C₄₇	—	2	2	no	—
81	C₂³×C₄₇	—	4	2	no	—
82	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
83	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
84	C₂×C₄×C₄₇	—	2	2	no	—
85	C₂×C₄×C₄₇	—	2	4	no	—
86	C₂×C₄×C₄₇	—	2	2	no	—
87	C₂×C₄×C₄₇	—	2	2	no	—
88	C₂×C₄×C₄₇	—	4	2	no	—
89	C₂×C₄×C₄₇	—	4	2	no	—
90	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
91	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
92	C₂×C₄×C₄₇	C₂×C₄⋊C₄	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	4	yes	—
96	C₈×C₄₇	—	2	4	no	—
97	C₈×C₄₇	—	2	4	no	—
98	C₄₇⋊C₈	—	2	8	no	—
99	C₄₇⋊C₈	—	4	4	no	—
100	#21372	—	47	1	no	maximal, Sylow
101	C₂²×C₄₇	—	2	2	no	—
102	C₂²×C₄₇	—	2	4	no	—
103	C₂²×C₄₇	—	2	2	no	—
104	C₂²×C₄₇	—	8	2	no	—
105	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
106	C₂²×C₄₇	#533	1	2	yes	—
107	C₄×C₄₇	—	2	4	no	—
108	C₄×C₄₇	—	2	4	no	—
109	C₄×C₄₇	—	2	2	no	—
110	C₄×C₄₇	C₄×#41	1	2	yes	—
111	C₄×C₄₇	#525	1	2	yes	—
112	#2921	—	47	1	no	—
113	C₂×#433	D₄₇	1	1	yes	characteristic
114	#2873	—	47	1	no	—
115	#1950	—	47	4	no	—
116	#1855	—	47	4	no	—
117	#1778	—	47	2	no	—
118	C₄×#108	—	47	2	no	—
119	C₂×C₄₇	—	2	2	no	—
120	C₂×C₄₇	—	4	2	no	—
121	C₂×C₄₇	#2296	1	1	yes	characteristic
122	C₂×C₄₇	#2916	1	2	yes	—
123	C₂²×M₁₆	—	47	2	no	—
124	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
125	#433	—	94	4	no	—
126	#433	Dic₄₇	1	8	yes	—
127	C₂×#108	—	47	4	no	—
128	C₂×#108	—	47	4	no	—
129	#423	—	47	2	no	—
130	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
131	#407	—	47	2	no	—
132	C₂×C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
133	#405	—	94	4	no	—
134	C₄×M₁₆	—	47	2	no	—
135	C₄×M₁₆	—	47	2	no	—
136	C₄×M₁₆	—	47	2	no	—
137	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
138	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
139	#341	—	47	4	no	—
140	#341	—	47	4	no	—
141	C₄₇	#21372	1	1	yes	characteristic, complemented, Sylow
142	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
143	C₂×M₁₆	—	47	4	no	—
144	C₂×M₁₆	—	47	4	no	—
145	C₂×M₁₆	—	94	8	no	—
146	C₂×M₁₆	—	94	2	no	—
147	C₂²×C₈	—	94	2	no	—
148	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
149	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
150	#117	D₁₈₈	1	4	yes	—
151	#117	C₂×Dic₄₇	1	4	yes	—
152	C₂×C₄⋊C₄	Dic₉₄	1	2	yes	—
153	C₂×C₂²⋊C₄	Dic₉₄	1	2	yes	—
154	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
155	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
156	#108	—	94	8	no	—
157	#108	—	94	8	no	—
158	C₄⋊C₈	—	94	4	no	—
159	C₄⋊C₈	C₂×Dic₄₇	1	8	yes	—
160	C₂²⋊C₈	—	94	4	no	—
161	C₂²⋊C₈	C₂×Dic₄₇	1	8	yes	—
162	#97	—	94	2	no	—
163	#97	—	94	2	no	—
164	#97	—	94	2	no	—
165	#97	C₂×Dic₄₇	1	4	yes	—
166	C₄×C₈	—	94	2	no	—
167	C₄×C₈	—	94	2	no	—
168	C₄×C₈	—	94	2	no	—
169	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
170	C₂⁴	C₂×Dic₉₄	1	1	yes	characteristic
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
173	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
174	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
175	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
176	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
177	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
178	M₁₆	—	47	8	no	—
179	M₁₆	—	94	8	no	—
180	M₁₆	—	94	4	no	—
181	M₁₆	—	94	4	no	—
182	M₁₆	—	94	4	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	94	8	no	—
186	C₂×C₈	—	94	4	no	—
187	C₂×C₈	—	94	4	no	—
188	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
189	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
190	C₄⋊C₄	C₄⋊Dic₄₇	1	8	yes	—
191	C₂²⋊C₄	C₄⋊Dic₄₇	1	8	yes	—
192	C₄²	C₂×D₁₈₈	1	2	yes	—
193	C₄²	C₂×D₁₈₈	1	2	yes	—
194	C₄²	C₂²×Dic₄₇	1	2	yes	—
195	C₄²	C₂²×Dic₄₇	1	2	yes	—
196	C₂³	—	2	2	no	—
197	C₂³	—	4	2	no	—
198	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
199	C₂³	C₂²×Dic₉₄	1	1	yes	characteristic
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
207	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
208	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
209	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
210	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
211	C₂×C₄	#41×D₄₇	1	4	yes	—
212	C₈	—	2	4	no	—
213	C₈	—	2	4	no	—
214	C₈	—	94	8	no	—
215	C₈	—	188	4	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	2	4	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	8	2	no	—
220	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
221	C₂²	#15689676	1	2	yes	—
222	C₄	—	2	4	no	—
223	C₄	—	2	4	no	—
224	C₄	—	2	2	no	—
225	C₄	#41×Dic₄₇	1	2	yes	—
226	C₄	#15689635	1	2	yes	—
227	C₂	—	2	2	no	—
228	C₂	—	4	2	no	—
229	C₂	#20245872	1	1	yes	characteristic
230	C₂	#20251999	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#433	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₄₇	C₂×Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#433	D₄₇
hypocenter	C₄₇	#21372
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#433

Complements

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

N	H ≅ G/N	type
3. #20244164	C₂	semidirect
5. #20243294	C₂	semidirect
6. C₄×#13093960	C₂	semidirect
141. C₄₇	#21372	semidirect

Maximal subgroups

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

subgroup	index
#20252026	2
#20251807	2
#20244164	2
#20243595	2
#20243294	2
C₄×#13093960	2
C₂×C₄₇×#433	2
#21372	47