Subgroups — #26226714

Order 12032.

662 conjugacy classes of subgroups.

The subgroup lattice has 236 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	C₂×#15689150	C₂	1	1	yes	maximal, characteristic
3	#20244163	C₂	1	4	yes	maximal, complemented
4	#20244161	C₂	1	4	yes	maximal, complemented
5	#20243428	C₂	1	2	yes	maximal
6	#20243424	C₂	1	2	yes	maximal
7	C₄₇×#2921	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10625329	C₂²	1	1	yes	characteristic
9	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
10	#15689184	—	2	4	no	—
11	C₂×#13093959	C₂²	1	4	yes	—
12	C₂×#13093959	C₂²	1	4	yes	—
13	#15689150	C₄	1	8	yes	—
14	#15689147	C₂²	1	2	yes	—
15	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
16	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
17	#15689087	C₂²	1	2	yes	—
18	C₄×#10625329	C₂²	1	2	yes	—
19	#15688887	C₂²	1	4	yes	—
20	#15688887	C₂²	1	4	yes	—
21	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
22	C₄₇×#433	—	2	4	no	—
23	C₄₇×#423	C₂²	1	2	yes	—
24	C₄₇×#407	C₂²	1	2	yes	—
25	C₄×M₁₆×C₄₇	C₂²	1	2	yes	—
26	C₂×#10625329	—	4	2	no	—
27	C₂×#10625329	C₂×C₄	1	4	yes	—
28	C₂×#10625329	C₂³	1	4	yes	—
29	C₂²×C₄₇⋊C₈	—	4	2	no	—
30	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
31	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
32	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
33	#13093959	—	4	4	no	—
34	#13093959	—	4	4	no	—
35	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
36	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
37	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
38	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
39	#13093935	—	2	2	no	—
40	C₄×C₄₇⋊C₈	—	2	2	no	—
41	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
43	#117×C₄₇	C₂×C₄	1	4	yes	—
44	#117×C₄₇	D₄	1	4	yes	—
45	C₂×C₄⋊C₄×C₄₇	Q₈	1	2	yes	—
46	C₂×C₂²⋊C₄×C₄₇	Q₈	1	2	yes	—
47	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₄₇	—	2	4	no	—
50	C₂²⋊C₈×C₄₇	—	2	4	no	—
51	#97×C₄₇	—	2	2	no	—
52	C₄×C₈×C₄₇	—	2	2	no	—
53	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	—	2	2	no	—
55	C₂²×C₄×C₄₇	—	2	2	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	M₁₆×C₄₇	—	2	4	no	—
63	M₁₆×C₄₇	—	2	4	no	—
64	C₂×C₈×C₄₇	—	2	4	no	—
65	C₄⋊C₄×C₄₇	C₄⋊C₄	1	8	yes	—
66	C₂²⋊C₄×C₄₇	C₄⋊C₄	1	8	yes	—
67	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
68	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
69	C₄²×C₄₇	C₂×D₄	1	2	yes	—
70	C₄²×C₄₇	C₂×D₄	1	2	yes	—
71	#10625329	—	2	4	no	—
72	#10625329	—	2	4	no	—
73	#10625329	—	4	4	no	—
74	C₂×C₄₇⋊C₈	—	2	4	no	—
75	C₂×C₄₇⋊C₈	—	2	4	no	—
76	C₂×C₄₇⋊C₈	—	2	4	no	—
77	C₂×C₄₇⋊C₈	—	4	4	no	—
78	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
79	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
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	2	no	—
86	C₂×C₄×C₄₇	—	2	2	no	—
87	C₂×C₄×C₄₇	—	2	4	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	H(5,𝔽₂)	1	2	yes	—
96	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
97	C₈×C₄₇	—	4	4	no	—
98	C₄₇⋊C₈	—	2	4	no	—
99	C₄₇⋊C₈	—	2	4	no	—
100	C₄₇⋊C₈	—	4	4	no	—
101	#21971	—	47	1	no	maximal, Sylow
102	C₂²×C₄₇	—	2	2	no	—
103	C₂²×C₄₇	—	2	2	no	—
104	C₂²×C₄₇	—	2	4	no	—
105	C₂²×C₄₇	—	4	4	no	—
106	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
107	C₂²×C₄₇	#544	1	2	yes	—
108	C₄×C₄₇	—	2	2	no	—
109	C₄×C₄₇	—	2	4	no	—
110	C₄×C₄₇	—	2	4	no	—
111	C₄×C₄₇	#518	1	1	yes	characteristic
112	C₄×C₄₇	#519	1	1	yes	characteristic
113	C₄×C₄₇	#537	1	2	yes	—
114	#2921	—	47	1	no	—
115	#2921	D₄₇	1	1	yes	characteristic
116	C₂×#424	—	47	1	no	—
117	#1950	—	47	4	no	—
118	#1950	—	47	4	no	—
119	#1811	—	47	2	no	—
120	#1810	—	47	2	no	—
121	C₂×C₄₇	—	2	2	no	—
122	C₂×C₄₇	—	4	2	no	—
123	C₂×C₄₇	#2316	1	1	yes	characteristic
124	C₂×C₄₇	#2916	1	2	yes	—
125	C₂²×M₁₆	—	47	1	no	—
126	C₂³×C₈	—	47	1	no	—
127	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
128	#433	—	2	4	no	—
129	#433	—	94	4	no	—
130	C₂×#108	—	47	4	no	—
131	C₂×#108	—	47	4	no	—
132	#424	—	47	8	no	—
133	#423	—	47	2	no	—
134	#423	C₂×D₄₇	1	2	yes	—
135	C₂×C₄⋊C₈	—	47	2	no	—
136	C₂×C₄⋊C₈	—	47	2	no	—
137	#407	—	47	2	no	—
138	#407	C₂×D₄₇	1	2	yes	—
139	C₄×M₁₆	—	47	2	no	—
140	C₄×M₁₆	C₂×D₄₇	1	2	yes	—
141	#341	—	47	4	no	—
142	#341	—	47	4	no	—
143	C₄₇	#21971	1	1	yes	characteristic, complemented, Sylow
144	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
145	C₂×M₁₆	—	47	4	no	—
146	C₂×M₁₆	—	47	4	no	—
147	C₂×M₁₆	—	188	2	no	—
148	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
149	C₂²×C₈	—	47	2	no	—
150	C₂²×C₈	—	47	2	no	—
151	C₂²×C₈	—	47	4	no	—
152	C₂²×C₈	—	188	2	no	—
153	#117	C₄×D₄₇	1	4	yes	—
154	#117	C₂²⋊D₄₇	1	4	yes	—
155	C₂×C₄⋊C₄	Dic₉₄	1	2	yes	—
156	C₂×C₂²⋊C₄	Dic₉₄	1	2	yes	—
157	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
158	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
159	#108	—	188	4	no	—
160	#108	—	188	4	no	—
161	C₄⋊C₈	—	2	4	no	—
162	C₄⋊C₈	—	47	8	no	—
163	C₄⋊C₈	—	47	8	no	—
164	C₄⋊C₈	—	94	4	no	—
165	C₂²⋊C₈	—	2	4	no	—
166	C₂²⋊C₈	—	94	4	no	—
167	#97	—	2	2	no	—
168	#97	—	94	2	no	—
169	C₄×C₈	—	2	2	no	—
170	C₄×C₈	—	94	2	no	—
171	C₂⁴	C₂×Dic₉₄	1	1	yes	characteristic
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
175	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
176	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
177	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
178	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
179	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
180	M₁₆	—	2	4	no	—
181	M₁₆	—	2	4	no	—
182	M₁₆	—	94	4	no	—
183	M₁₆	—	94	4	no	—
184	M₁₆	—	188	4	no	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	—	47	4	no	—
187	C₂×C₈	—	47	4	no	—
188	C₂×C₈	—	94	4	no	—
189	C₂×C₈	—	94	4	no	—
190	C₂×C₈	—	94	4	no	—
191	C₂×C₈	—	188	4	no	—
192	C₄⋊C₄	Dic₄₇⋊C₄	1	8	yes	—
193	C₂²⋊C₄	Dic₄₇⋊C₄	1	8	yes	—
194	C₄²	C₂×C₄×D₄₇	1	2	yes	—
195	C₄²	C₂×C₄×D₄₇	1	2	yes	—
196	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
197	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
198	C₂³	—	2	2	no	—
199	C₂³	—	4	2	no	—
200	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
201	C₂³	C₂²×Dic₉₄	1	1	yes	characteristic
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
209	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
210	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
211	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
212	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
213	C₂×C₄	#13094070	1	2	yes	—
214	C₂×C₄	#13094071	1	2	yes	—
215	C₈	—	4	4	no	—
216	C₈	—	94	4	no	—
217	C₈	—	94	4	no	—
218	C₈	—	188	4	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	2	4	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	4	4	no	—
223	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
224	C₂²	#15689751	1	2	yes	—
225	C₄	—	2	4	no	—
226	C₄	—	2	4	no	—
227	C₄	—	2	2	no	—
228	C₄	#15689597	1	1	yes	characteristic
229	C₄	#15689600	1	1	yes	characteristic
230	C₄	#15689701	1	2	yes	—
231	C₂	—	2	2	no	—
232	C₂	—	4	2	no	—
233	C₂	#20246004	1	1	yes	characteristic
234	C₂	#20251997	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#2921	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₄₇	C₂×Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#2921	D₄₇
hypocenter	C₄₇	#21971
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ #2921

Complements

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

N	H ≅ G/N	type
3. #20244163	C₂	semidirect
4. #20244161	C₂	semidirect
143. C₄₇	#21971	semidirect

Maximal subgroups

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

subgroup	index
#20252026	2
C₂×#15689150	2
#20244163	2
#20244161	2
#20243428	2
#20243424	2
C₄₇×#2921	2
#21971	47