Subgroups — #26214395

Order 12032.

994 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093959	C₂	1	2	yes	maximal
2	#20251806	C₂	1	2	yes	maximal
3	C₂×#15689081	C₂	1	1	yes	maximal, characteristic
4	#20243293	C₂	1	4	yes	maximal, complemented
5	#20243289	C₂	1	4	yes	maximal, complemented
6	#20243115	C₂	1	1	yes	maximal, characteristic
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	C₂×#13093959	C₄	1	16	yes	complemented
12	C₂×#13093959	C₂²	1	4	yes	—
13	C₂×#13093959	C₂²	1	4	yes	—
14	#15689081	—	2	4	no	—
15	#15689081	C₄	1	8	yes	—
16	#15689078	—	2	4	no	—
17	M₁₆×Dic₄₇	C₂²	1	8	yes	—
18	C₂×#13093934	C₂²	1	2	yes	—
19	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
20	#15688888	C₂²	1	4	yes	—
21	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
22	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
23	C₄₇×#341	C₂²	1	4	yes	—
24	C₂×#10625329	—	2	2	no	—
25	C₂×#10625329	—	2	4	no	—
26	C₂×#10625329	C₂×C₄	1	8	yes	—
27	C₂×#10625329	C₂³	1	4	yes	—
28	C₂²×C₄₇⋊C₈	—	2	2	no	—
29	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
30	C₂²×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
31	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
32	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
33	#13093983	C₂×C₄	1	4	yes	—
34	#13093983	D₄	1	4	yes	—
35	C₂×C₄⋊Dic₄₇	Q₈	1	2	yes	—
36	C₂×C₂²⋊Dic₄₇	Q₈	1	2	yes	—
37	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
38	#13093959	C₂×C₄	1	16	yes	—
39	#13093959	C₂×C₄	1	16	yes	—
40	#13093934	—	2	8	no	—
41	#13093934	C₂×C₄	1	8	yes	—
42	C₈×Dic₄₇	—	2	8	no	—
43	C₈×Dic₄₇	C₂×C₄	1	8	yes	—
44	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
45	C₂×M₁₆×C₄₇	—	2	2	no	—
46	C₂×M₁₆×C₄₇	—	2	4	no	—
47	C₂×M₁₆×C₄₇	—	2	4	no	—
48	C₂×M₁₆×C₄₇	C₂×C₄	1	8	yes	—
49	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
50	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
51	C₂²×C₈×C₄₇	—	2	2	no	—
52	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
53	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
56	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	#41	1	4	yes	—
59	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
60	C₂²×Dic₄₇	C₂×Q₈	1	4	yes	—
61	M₁₆×C₄₇	—	2	8	no	—
62	M₁₆×C₄₇	—	2	8	no	—
63	M₁₆×C₄₇	—	2	4	no	—
64	M₁₆×C₄₇	C₂²×C₄	1	8	yes	—
65	C₂×C₈×C₄₇	—	2	8	no	—
66	C₂×C₈×C₄₇	—	2	4	no	—
67	C₂×C₈×C₄₇	—	2	4	no	—
68	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
69	C₂²⋊Dic₄₇	C₄⋊C₄	1	8	yes	—
70	C₄⋊Dic₄₇	C₄⋊C₄	1	8	yes	—
71	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
72	C₄×Dic₄₇	C₂×D₄	1	4	yes	—
73	#10625329	—	2	8	no	—
74	#10625329	—	2	4	no	—
75	#10625329	C₂²×C₄	1	8	yes	—
76	C₂×C₄₇⋊C₈	—	2	8	no	—
77	C₂×C₄₇⋊C₈	—	2	4	no	—
78	C₂×C₄₇⋊C₈	C₄²	1	16	yes	—
79	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
80	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
81	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
82	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
83	C₂³×C₄₇	—	4	2	no	—
84	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
85	C₂³×C₄₇	C₄×Q₈	1	4	yes	—
86	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
87	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
88	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
89	C₂×C₄×C₄₇	#117	1	4	yes	—
90	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
91	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
92	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
93	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
95	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
97	C₂×Dic₄₇	—	4	4	no	—
98	C₂×Dic₄₇	C₂×C₄⋊C₄	1	8	yes	—
99	C₈×C₄₇	—	2	4	no	—
100	C₈×C₄₇	—	2	8	no	—
101	C₄₇⋊C₈	—	2	8	no	—
102	C₄₇⋊C₈	C₂×C₄²	1	4	yes	—
103	C₄₇⋊C₈	C₂×C₄²	1	4	yes	—
104	#20460	—	47	1	no	maximal, Sylow
105	C₂²×C₄₇	—	2	2	no	—
106	C₂²×C₄₇	—	2	4	no	—
107	C₂²×C₄₇	—	4	4	no	—
108	C₂²×C₄₇	C₄×C₄⋊C₄	1	4	yes	—
109	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
110	C₂²×C₄₇	C₂×C₄×Q₈	1	2	yes	—
111	C₄×C₄₇	C₄×C₄⋊C₄	1	4	yes	—
112	C₄×C₄₇	C₂²×C₄²	1	1	yes	characteristic
113	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
114	C₄×C₄₇	C₂×C₄×D₄	1	2	yes	—
115	Dic₄₇	—	2	8	no	—
116	C₂²×#108	—	47	2	no	—
117	#2873	—	47	2	no	—
118	C₂×#405	—	47	1	no	—
119	#1778	—	47	4	no	—
120	C₄×#108	—	47	4	no	—
121	#1738	—	47	1	no	—
122	#1738	D₄₇	1	1	yes	characteristic
123	C₂×C₄₇	—	2	2	no	—
124	C₂×C₄₇	—	2	4	no	—
125	C₂×C₄₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄₇	#2916	1	2	yes	—
127	C₂²×M₁₆	—	47	2	no	—
128	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
129	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
130	C₂³×C₈	—	47	1	no	—
131	C₂×#117	—	47	1	no	—
132	C₂×#108	—	47	16	no	—
133	C₂×#108	—	47	4	no	—
134	C₂×#108	—	47	4	no	—
135	#405	—	47	8	no	—
136	#405	—	94	4	no	—
137	#405	—	94	4	no	—
138	C₄×M₁₆	—	47	8	no	—
139	C₂×#97	—	47	2	no	—
140	C₂×C₄×C₈	—	47	2	no	—
141	#341	—	47	4	no	—
142	#341	C₂×D₄₇	1	4	yes	—
143	C₄₇	#20460	1	1	yes	characteristic, complemented, Sylow
144	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
145	C₂×M₁₆	—	2	2	no	—
146	C₂×M₁₆	—	2	4	no	—
147	C₂×M₁₆	—	2	4	no	—
148	C₂×M₁₆	—	47	8	no	—
149	C₂×M₁₆	—	47	4	no	—
150	C₂×M₁₆	—	94	2	no	—
151	C₂×M₁₆	—	94	4	no	—
152	C₂×M₁₆	C₄×D₄₇	1	8	yes	—
153	C₂×M₁₆	C₄×D₄₇	1	4	yes	—
154	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	—	47	4	no	—
157	C₂²×C₈	—	47	8	no	—
158	C₂²×C₈	—	47	1	no	—
159	C₂²×C₈	—	47	1	no	—
160	C₂²×C₈	—	94	2	no	—
161	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
162	#117	—	47	4	no	—
163	#117	—	47	4	no	—
164	C₂×C₄⋊C₄	—	47	2	no	—
165	C₂×C₂²⋊C₄	—	47	2	no	—
166	C₂×C₄²	—	47	2	no	—
167	#108	—	47	16	no	—
168	#108	—	47	16	no	—
169	#97	—	47	8	no	—
170	#97	—	94	8	no	—
171	C₄×C₈	—	47	8	no	—
172	C₄×C₈	—	94	8	no	—
173	C₂⁴	Q₈×D₄₇	1	1	yes	characteristic
174	C₂²×C₄	—	47	4	no	—
175	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
176	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
177	C₂²×C₄	D₄×D₄₇	1	2	yes	—
178	C₂²×C₄	#10625352	1	4	yes	—
179	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
180	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
181	M₁₆	—	2	8	no	—
182	M₁₆	—	2	8	no	—
183	M₁₆	—	2	4	no	—
184	M₁₆	—	47	8	no	—
185	M₁₆	—	94	8	no	—
186	M₁₆	—	94	4	no	—
187	M₁₆	C₂×C₄×D₄₇	1	8	yes	—
188	C₂×C₈	—	2	8	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	47	16	no	—
192	C₂×C₈	—	47	2	no	—
193	C₂×C₈	—	47	2	no	—
194	C₂×C₈	—	47	4	no	—
195	C₂×C₈	—	47	4	no	—
196	C₂×C₈	—	94	8	no	—
197	C₂×C₈	—	94	4	no	—
198	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
199	C₄⋊C₄	—	47	8	no	—
200	C₂²⋊C₄	—	47	8	no	—
201	C₄²	—	47	4	no	—
202	C₄²	—	47	4	no	—
203	C₂³	—	4	2	no	—
204	C₂³	C₄⋊C₄×D₄₇	1	2	yes	—
205	C₂³	#13093989	1	4	yes	—
206	C₂³	C₂×Q₈×D₄₇	1	1	yes	characteristic
207	C₂×C₄	—	47	8	no	—
208	C₂×C₄	—	188	4	no	—
209	C₂×C₄	C₄²×D₄₇	1	4	yes	—
210	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
211	C₂×C₄	#13093980	1	4	yes	—
212	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	4	yes	—
213	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	4	yes	—
214	C₂×C₄	#13093989	1	4	yes	—
215	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
216	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
217	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
218	C₂×C₄	C₂×#10625352	1	2	yes	—
219	C₈	—	2	4	no	—
220	C₈	—	2	8	no	—
221	C₈	—	47	4	no	—
222	C₈	—	47	4	no	—
223	C₈	—	94	8	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	4	no	—
226	C₂²	—	4	4	no	—
227	C₂²	#15688974	1	4	yes	—
228	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
229	C₂²	C₂×#13093989	1	2	yes	—
230	C₄	—	94	8	no	—
231	C₄	#15688974	1	4	yes	—
232	C₄	C₂×C₄²×D₄₇	1	1	yes	characteristic
233	C₄	C₂×#13093980	1	1	yes	characteristic
234	C₄	C₂×(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
235	C₂	—	2	4	no	—
236	C₂	—	2	2	no	—
237	C₂	C₂×#15688974	1	1	yes	characteristic
238	C₂	#20251998	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₄×C₄₇	C₂²×C₄²
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	#20460
LCS	G ▷ 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
4. #20243293	C₂	semidirect
5. #20243289	C₂	semidirect
11. C₂×#13093959	C₄	semidirect
143. C₄₇	#20460	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13093959	2
#20251806	2
C₂×#15689081	2
#20243293	2
#20243289	2
#20243115	2
C₄₇×#1738	2
#20460	47