Subgroups — #26141553

Order 12032.

1342 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688969	C₂	1	2	yes	maximal
2	#20241641	C₂	1	4	yes	maximal
3	#20241622	C₂	1	8	yes	maximal
4	C₂×C₄₇×#382	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
6	C₂²×C₄×Dic₄₇	C₂²	1	4	yes	—
7	#15688969	—	2	8	no	—
8	C₂×C₄₇⋊#95	C₂²	1	4	yes	—
9	C₂×C₄₇⋊#95	C₂²	1	4	yes	—
10	C₂×C₄₇⋊#95	C₂²	1	8	yes	—
11	C₂×C₄₇⋊#95	C₂²	1	4	yes	—
12	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
13	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×#382	C₄	1	8	yes	complemented
15	C₂×#95×C₄₇	C₂²	1	4	yes	—
16	C₂³×Dic₄₇	C₂³	1	8	yes	—
17	C₂×C₄⋊Dic₄₇	—	2	16	no	—
18	C₂×C₄⋊Dic₄₇	—	4	4	no	—
19	C₂×C₄×Dic₄₇	—	2	8	no	—
20	C₂×C₄×Dic₄₇	—	2	16	no	—
21	C₄₇⋊#95	—	2	8	no	—
22	C₄₇⋊#95	—	2	8	no	—
23	C₄₇⋊#95	—	2	16	no	—
24	C₄₇⋊#95	—	4	4	no	—
25	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
26	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
27	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
28	C₂×C₄⋊C₄×C₄₇	—	4	2	no	—
29	C₂×C₄⋊C₄×C₄₇	—	4	2	no	—
30	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
31	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₄₇	D₄	1	4	yes	—
33	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
34	#95×C₄₇	C₂×C₄	1	16	yes	—
35	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	—	2	8	no	—
37	C₂²×C₄×C₄₇	—	2	2	no	—
38	C₂²×C₄×C₄₇	—	2	2	no	—
39	C₂²×C₄×C₄₇	—	2	8	no	—
40	C₂²×C₄×C₄₇	—	2	8	no	—
41	C₂²×C₄×C₄₇	—	2	4	no	—
42	C₂²×C₄×C₄₇	—	2	4	no	—
43	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
44	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
45	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
46	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
48	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
49	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
50	C₂²×Dic₄₇	—	2	8	no	—
51	C₂²×Dic₄₇	—	2	8	no	—
52	C₂²×Dic₄₇	—	2	8	no	—
53	C₂²×Dic₄₇	—	2	8	no	—
54	C₂²×Dic₄₇	—	2	8	no	—
55	C₂²×Dic₄₇	—	2	8	no	—
56	C₂²×Dic₄₇	#41	1	16	yes	—
57	C₄⋊C₄×C₄₇	—	4	8	no	—
58	C₄²×C₄₇	C₄⋊C₄	1	16	yes	complemented
59	C₄⋊Dic₄₇	—	4	16	no	—
60	C₄×Dic₄₇	—	2	32	no	—
61	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
62	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
63	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
64	C₂³×C₄₇	C₂×#41	1	8	yes	—
65	C₂³×C₄₇	H(5,𝔽₂)	1	2	yes	—
66	C₂³×C₄₇	2₋¹⁺⁴	1	2	yes	—
67	C₂×C₄×C₄₇	—	2	8	no	—
68	C₂×C₄×C₄₇	—	2	2	no	—
69	C₂×C₄×C₄₇	—	2	2	no	—
70	C₂×C₄×C₄₇	—	2	8	no	—
71	C₂×C₄×C₄₇	—	2	8	no	—
72	C₂×C₄×C₄₇	—	2	4	no	—
73	C₂×C₄×C₄₇	—	2	4	no	—
74	C₂×C₄×C₄₇	—	2	16	no	—
75	C₂×C₄×C₄₇	—	2	8	no	—
76	C₂×C₄×C₄₇	—	4	4	no	—
77	C₂×C₄×C₄₇	—	4	4	no	—
78	C₂×C₄×C₄₇	—	4	8	no	—
79	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
80	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
81	C₂×Dic₄₇	—	2	16	no	—
82	C₂×Dic₄₇	—	2	16	no	—
83	C₂×Dic₄₇	—	2	16	no	—
84	C₂×Dic₄₇	—	4	8	no	—
85	C₂×Dic₄₇	—	4	8	no	—
86	C₂×Dic₄₇	—	4	8	no	—
87	C₂×Dic₄₇	—	4	8	no	—
88	#14168	—	47	1	no	maximal, Sylow
89	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₄₇	C₄×#41	1	4	yes	—
91	C₂²×C₄₇	#518	1	1	yes	characteristic
92	C₂²×C₄₇	#519	1	1	yes	characteristic
93	C₂²×C₄₇	#525	1	4	yes	—
94	C₂²×C₄₇	#533	1	4	yes	—
95	C₂²×C₄₇	#537	1	2	yes	—
96	C₂²×C₄₇	#544	1	2	yes	—
97	C₂²×C₄₇	#551	1	4	yes	—
98	C₂²×C₄₇	#552	1	8	yes	—
99	C₂²×C₄₇	#555	1	4	yes	—
100	C₄×C₄₇	—	2	4	no	—
101	C₄×C₄₇	—	2	8	no	—
102	C₄×C₄₇	—	4	8	no	—
103	Dic₄₇	—	4	16	no	—
104	C₂×#382	D₄₇	1	1	yes	characteristic
105	C₂×#376	—	47	2	no	—
106	#1441	—	47	4	no	—
107	#1436	—	47	8	no	—
108	C₂×C₄₇	#2296	1	2	yes	—
109	C₂×C₄₇	#2316	1	1	yes	characteristic
110	C₂×C₄₇	#2371	1	2	yes	—
111	C₂×C₄₇	#2372	1	2	yes	—
112	C₂×C₄₇	#2609	1	4	yes	—
113	C₂×C₄₇	#2612	1	4	yes	—
114	C₂²×C₄⋊C₄	—	47	4	no	—
115	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
116	C₂²×C₄²	—	47	4	no	—
117	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
118	#382	Dic₄₇	1	8	yes	complemented
119	#376	—	94	8	no	—
120	C₂×#95	—	47	8	no	—
121	C₂×#95	—	47	4	no	—
122	C₂×#95	—	47	4	no	—
123	C₂×#95	—	47	4	no	—
124	C₂×#95	C₂×D₄₇	1	4	yes	—
125	C₄₇	#14168	1	1	yes	characteristic, complemented, Sylow
126	C₂³×C₄	—	47	8	no	—
127	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
128	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
129	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
130	C₂×C₄⋊C₄	—	4	2	no	—
131	C₂×C₄⋊C₄	—	4	2	no	—
132	C₂×C₄⋊C₄	—	94	16	no	—
133	C₂×C₄⋊C₄	—	188	4	no	—
134	C₂×C₄⋊C₄	C₂×Dic₄₇	1	8	yes	—
135	C₂×C₄²	—	94	16	no	—
136	C₂×C₄²	—	94	8	no	—
137	C₂×C₄²	Dic₉₄	1	4	yes	—
138	C₂×C₄²	D₁₈₈	1	4	yes	—
139	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
140	#95	—	94	16	no	—
141	#95	—	94	8	no	—
142	#95	—	94	8	no	—
143	#95	—	188	4	no	—
144	#95	C₂×Dic₄₇	1	16	yes	—
145	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
146	C₂²×C₄	—	2	8	no	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	2	8	no	—
150	C₂²×C₄	—	2	8	no	—
151	C₂²×C₄	—	2	4	no	—
152	C₂²×C₄	—	2	4	no	—
153	C₂²×C₄	—	47	16	no	—
154	C₂²×C₄	—	94	8	no	—
155	C₂²×C₄	—	94	8	no	—
156	C₂²×C₄	—	94	8	no	—
157	C₂²×C₄	—	94	8	no	—
158	C₂²×C₄	—	94	8	no	—
159	C₂²×C₄	—	94	8	no	—
160	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
161	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
162	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
163	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
164	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
165	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
166	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
167	C₄⋊C₄	—	4	8	no	—
168	C₄⋊C₄	—	188	16	no	—
169	C₄²	—	94	32	no	—
170	C₄²	C₄⋊Dic₄₇	1	16	yes	complemented
171	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
172	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
173	C₂³	C₂²×Dic₉₄	1	1	yes	characteristic
174	C₂³	#41×D₄₇	1	8	yes	—
175	C₂³	#13094069	1	2	yes	—
176	C₂³	#13094072	1	2	yes	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	8	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	8	no	—
182	C₂×C₄	—	2	8	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	16	no	—
185	C₂×C₄	—	2	8	no	—
186	C₂×C₄	—	4	4	no	—
187	C₂×C₄	—	4	4	no	—
188	C₂×C₄	—	4	8	no	—
189	C₂×C₄	—	94	16	no	—
190	C₂×C₄	—	94	16	no	—
191	C₂×C₄	—	94	16	no	—
192	C₂×C₄	—	188	8	no	—
193	C₂×C₄	—	188	8	no	—
194	C₂×C₄	—	188	8	no	—
195	C₂×C₄	—	188	8	no	—
196	C₂×C₄	C₂×C₄⋊Dic₄₇	1	8	yes	—
197	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
198	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
199	C₂²	#41×Dic₄₇	1	4	yes	—
200	C₂²	#15689598	1	1	yes	characteristic
201	C₂²	#15689601	1	1	yes	characteristic
202	C₂²	#15689635	1	4	yes	—
203	C₂²	#15689676	1	4	yes	—
204	C₂²	#15689704	1	2	yes	—
205	C₂²	#15689754	1	2	yes	—
206	C₂²	#15689803	1	4	yes	—
207	C₂²	#15689805	1	8	yes	—
208	C₂²	#15689835	1	4	yes	—
209	C₄	—	2	8	no	—
210	C₄	—	2	4	no	—
211	C₄	—	4	8	no	—
212	C₄	—	188	16	no	—
213	C₂	#20245872	1	2	yes	—
214	C₂	#20246006	1	1	yes	characteristic
215	C₂	#20246494	1	2	yes	—
216	C₂	#20246500	1	2	yes	—
217	C₂	#20249065	1	4	yes	—
218	C₂	#20249096	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₄₇
commutator	C₂³×C₄₇	C₂³×C₄
Fitting	C₂×C₄₇×#382	C₂
socle, prime radical, cosocle	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂×#382	D₄₇
hypocenter	C₄₇	#14168
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂×#382

Complements

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

N	H ≅ G/N	type
14. C₄₇×#382	C₄	semidirect
58. C₄²×C₄₇	C₄⋊C₄	semidirect
118. #382	Dic₄₇	semidirect
125. C₄₇	#14168	semidirect
170. C₄²	C₄⋊Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688969	2
#20241641	2
#20241622	2
C₂×C₄₇×#382	2
#14168	47