Subgroups — #26216510

Order 12032.

778 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	#20251951	C₂	1	2	yes	maximal
2	C₂×#15689081	C₂	1	1	yes	maximal, characteristic
3	#20243527	C₂	1	4	yes	maximal, complemented
4	#20243524	C₂	1	4	yes	maximal, complemented
5	#20243115	C₂	1	1	yes	maximal, characteristic
6	C₂×#15688887	C₂	1	2	yes	maximal
7	C₄₇×#1738	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10625329	C₂²	1	2	yes	—
9	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×#13093983	C₂²	1	1	yes	characteristic
11	#15689151	—	2	4	no	—
12	#15689149	—	2	4	no	—
13	#15689146	C₂²	1	8	yes	—
14	#15689081	C₄	1	8	yes	—
15	C₂×#13093934	C₂²	1	2	yes	—
16	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
17	#15688888	C₂²	1	4	yes	—
18	#15688887	C₂²	1	4	yes	—
19	#15688887	C₂²	1	4	yes	—
20	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
21	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
22	C₄₇×#341	C₂²	1	4	yes	—
23	C₂×#10625329	—	2	2	no	—
24	C₂×#10625329	—	2	4	no	—
25	C₂×#10625329	—	2	4	no	—
26	C₂×#10625329	C₂³	1	4	yes	—
27	C₂²×C₄₇⋊C₈	—	2	4	no	—
28	C₂²×C₄₇⋊C₈	—	2	2	no	—
29	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
30	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
31	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
32	#13093983	C₂×C₄	1	4	yes	—
33	#13093983	D₄	1	4	yes	—
34	C₂×C₄⋊Dic₄₇	Q₈	1	2	yes	—
35	C₂×C₂²⋊Dic₄₇	Q₈	1	2	yes	—
36	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
37	Dic₄₇⋊C₈	—	2	16	no	—
38	#13093934	C₂×C₄	1	8	yes	—
39	C₈×Dic₄₇	C₂×C₄	1	8	yes	—
40	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
41	C₂×M₁₆×C₄₇	—	2	4	no	—
42	C₂×M₁₆×C₄₇	—	2	2	no	—
43	C₂×M₁₆×C₄₇	—	2	4	no	—
44	C₂×M₁₆×C₄₇	—	2	4	no	—
45	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
46	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
47	C₂²×C₈×C₄₇	—	2	2	no	—
48	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
49	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	#41	1	4	yes	—
54	C₂²×C₄×C₄₇	#41	1	4	yes	—
55	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
56	C₂²×Dic₄₇	C₂×Q₈	1	4	yes	—
57	M₁₆×C₄₇	—	2	8	no	—
58	M₁₆×C₄₇	—	2	4	no	—
59	M₁₆×C₄₇	—	2	4	no	—
60	M₁₆×C₄₇	—	4	4	no	—
61	C₂×C₈×C₄₇	—	2	4	no	—
62	C₂×C₈×C₄₇	—	2	4	no	—
63	C₂×C₈×C₄₇	—	4	4	no	—
64	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
65	C₂²⋊Dic₄₇	C₄⋊C₄	1	8	yes	—
66	C₄⋊Dic₄₇	C₄⋊C₄	1	8	yes	—
67	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
68	C₄×Dic₄₇	C₂×D₄	1	4	yes	—
69	#10625329	—	2	4	no	—
70	#10625329	—	2	4	no	—
71	#10625329	—	4	4	no	—
72	C₂×C₄₇⋊C₈	—	2	8	no	—
73	C₂×C₄₇⋊C₈	—	2	2	no	—
74	C₂×C₄₇⋊C₈	—	2	2	no	—
75	C₂×C₄₇⋊C₈	—	2	4	no	—
76	C₂×C₄₇⋊C₈	—	4	4	no	—
77	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
78	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
79	C₂³×C₄₇	—	4	2	no	—
80	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
81	C₂³×C₄₇	#125	1	4	yes	—
82	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
83	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
84	C₂×C₄×C₄₇	#117	1	4	yes	—
85	C₂×C₄×C₄₇	#117	1	4	yes	—
86	C₂×C₄×C₄₇	#124	1	4	yes	—
87	C₂×C₄×C₄₇	#124	1	4	yes	—
88	C₂×C₄×C₄₇	#125	1	4	yes	—
89	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
90	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
91	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
92	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
93	C₂×Dic₄₇	—	4	4	no	—
94	C₂×Dic₄₇	C₂×C₄⋊C₄	1	8	yes	—
95	C₈×C₄₇	—	2	4	no	—
96	C₈×C₄₇	—	4	4	no	—
97	C₄₇⋊C₈	—	2	2	no	—
98	C₄₇⋊C₈	—	2	2	no	—
99	C₄₇⋊C₈	—	4	4	no	—
100	#20768	—	47	1	no	maximal, Sylow
101	C₂²×C₄₇	—	2	2	no	—
102	C₂²×C₄₇	—	2	4	no	—
103	C₂²×C₄₇	—	4	4	no	—
104	C₂²×C₄₇	#382	1	4	yes	—
105	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
106	C₂²×C₄₇	C₂×#125	1	2	yes	—
107	C₄×C₄₇	#382	1	4	yes	—
108	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
109	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
110	C₄×C₄₇	C₂×#124	1	2	yes	—
111	Dic₄₇	—	2	8	no	—
112	#2905	—	47	2	no	—
113	C₂×#405	—	47	1	no	—
114	#1838	—	47	4	no	—
115	#1837	—	47	4	no	—
116	#1738	—	47	1	no	—
117	#1738	D₄₇	1	1	yes	characteristic
118	C₂×#341	—	47	2	no	—
119	C₂×C₄₇	—	2	2	no	—
120	C₂×C₄₇	—	2	4	no	—
121	C₂×C₄₇	C₂×#382	1	1	yes	characteristic
122	C₂×C₄₇	#2916	1	2	yes	—
123	C₂²×M₁₆	—	47	2	no	—
124	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
125	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
126	C₂³×C₈	—	47	1	no	—
127	C₂×#117	—	47	1	no	—
128	#424	—	94	4	no	—
129	#424	—	94	4	no	—
130	#423	—	47	8	no	—
131	#405	—	47	8	no	—
132	C₂×#97	—	47	2	no	—
133	C₂×C₄×C₈	—	47	2	no	—
134	#341	—	47	4	no	—
135	#341	—	47	4	no	—
136	#341	—	47	4	no	—
137	#341	C₂×D₄₇	1	4	yes	—
138	C₄₇	#20768	1	1	yes	characteristic, complemented, Sylow
139	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
140	C₂×M₁₆	—	2	4	no	—
141	C₂×M₁₆	—	2	2	no	—
142	C₂×M₁₆	—	2	4	no	—
143	C₂×M₁₆	—	2	4	no	—
144	C₂×M₁₆	—	47	4	no	—
145	C₂×M₁₆	—	94	2	no	—
146	C₂×M₁₆	—	94	4	no	—
147	C₂×M₁₆	—	94	4	no	—
148	C₂×M₁₆	C₄×D₄₇	1	4	yes	—
149	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
150	C₂²×C₈	—	2	2	no	—
151	C₂²×C₈	—	47	4	no	—
152	C₂²×C₈	—	47	1	no	—
153	C₂²×C₈	—	47	1	no	—
154	C₂²×C₈	—	94	4	no	—
155	C₂²×C₈	—	94	2	no	—
156	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
157	#117	—	47	4	no	—
158	#117	—	47	4	no	—
159	C₂×C₄⋊C₄	—	47	2	no	—
160	C₂×C₂²⋊C₄	—	47	2	no	—
161	C₂×C₄²	—	47	2	no	—
162	C₄⋊C₈	—	94	16	no	—
163	#97	—	47	8	no	—
164	C₄×C₈	—	47	8	no	—
165	C₂⁴	Q₈×D₄₇	1	1	yes	characteristic
166	C₂²×C₄	—	47	4	no	—
167	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
168	C₂²×C₄	#10625350	1	4	yes	—
169	C₂²×C₄	D₄×D₄₇	1	2	yes	—
170	C₂²×C₄	#10625352	1	4	yes	—
171	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
172	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
173	M₁₆	—	2	8	no	—
174	M₁₆	—	2	4	no	—
175	M₁₆	—	2	4	no	—
176	M₁₆	—	4	4	no	—
177	M₁₆	—	94	4	no	—
178	M₁₆	—	94	4	no	—
179	M₁₆	—	188	4	no	—
180	C₂×C₈	—	2	4	no	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	4	4	no	—
183	C₂×C₈	—	47	2	no	—
184	C₂×C₈	—	47	2	no	—
185	C₂×C₈	—	94	8	no	—
186	C₂×C₈	—	94	2	no	—
187	C₂×C₈	—	94	2	no	—
188	C₂×C₈	—	94	4	no	—
189	C₂×C₈	—	188	4	no	—
190	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
191	C₄⋊C₄	—	47	8	no	—
192	C₂²⋊C₄	—	47	8	no	—
193	C₄²	—	47	4	no	—
194	C₄²	—	47	4	no	—
195	C₂³	—	4	2	no	—
196	C₂³	C₄⋊C₄×D₄₇	1	2	yes	—
197	C₂³	#13094014	1	4	yes	—
198	C₂³	C₂×Q₈×D₄₇	1	1	yes	characteristic
199	C₂×C₄	—	47	8	no	—
200	C₂×C₄	—	188	4	no	—
201	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
202	C₂×C₄	#13093980	1	4	yes	—
203	C₂×C₄	#13093981	1	4	yes	—
204	C₂×C₄	#13094010	1	4	yes	—
205	C₂×C₄	#13094010	1	4	yes	—
206	C₂×C₄	#13094014	1	4	yes	—
207	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
208	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
209	C₂×C₄	C₂×#10625352	1	2	yes	—
210	C₂×C₄	C₂×#10625350	1	2	yes	—
211	C₈	—	2	4	no	—
212	C₈	—	4	4	no	—
213	C₈	—	94	2	no	—
214	C₈	—	94	2	no	—
215	C₈	—	188	4	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	2	4	no	—
218	C₂²	—	4	4	no	—
219	C₂²	#15688987	1	4	yes	—
220	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
221	C₂²	C₂×#13094014	1	2	yes	—
222	C₄	—	94	8	no	—
223	C₄	#15688987	1	4	yes	—
224	C₄	C₂×#13093980	1	1	yes	characteristic
225	C₄	C₂×#13093981	1	1	yes	characteristic
226	C₄	C₂×#13094010	1	2	yes	—
227	C₂	—	2	4	no	—
228	C₂	—	2	2	no	—
229	C₂	C₂×#15688987	1	1	yes	characteristic
230	C₂	#20251998	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1738	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₄₇	C₂×Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1738	D₄₇
hypocenter	C₄₇	#20768
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1738

Complements

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

N	H ≅ G/N	type
3. #20243527	C₂	semidirect
4. #20243524	C₂	semidirect
138. C₄₇	#20768	semidirect

Maximal subgroups

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

subgroup	index
#20251951	2
C₂×#15689081	2
#20243527	2
#20243524	2
#20243115	2
C₂×#15688887	2
C₄₇×#1738	2
#20768	47