Subgroups — #26105744

Order 12032.

758 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689147	C₂	1	2	yes	maximal
2	#20241223	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#10625329	—	2	2	no	—
5	#15689147	—	2	4	no	—
6	#15689147	C₄	1	8	yes	—
7	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
8	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
9	C₂³×C₈×C₄₇	C₄	1	2	yes	—
10	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₄₇	—	2	4	no	—
12	C₂×C₄×C₈×C₄₇	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
14	C₂×#10625329	—	2	8	no	—
15	C₂×#10625329	—	4	4	no	—
16	C₂×#10625329	—	4	4	no	—
17	C₂²×C₄₇⋊C₈	—	2	4	no	—
18	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
19	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
20	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
21	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
24	C₂²×C₈×C₄₇	—	2	4	no	—
25	C₂²×C₈×C₄₇	—	2	8	no	—
26	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₄₇	D₄	1	4	yes	—
31	C₂×C₄²×C₄₇	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
33	C₄×C₈×C₄₇	—	2	4	no	—
34	C₄×C₈×C₄₇	—	2	4	no	—
35	C₂⁴×C₄₇	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	—	2	4	no	—
37	C₂²×C₄×C₄₇	—	2	4	no	—
38	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
39	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
41	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
43	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₄₇	D₈	1	2	yes	—
46	C₂²×C₄×C₄₇	SD₁₆	1	4	yes	—
47	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
48	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₄₇	—	2	16	no	—
52	C₂×C₈×C₄₇	—	2	4	no	—
53	C₂×C₈×C₄₇	—	2	4	no	—
54	C₂×C₈×C₄₇	M₁₆	1	8	yes	complemented
55	C₄²×C₄₇	C₄²	1	2	yes	—
56	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
57	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
58	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
59	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
60	C₄²×C₄₇	C₂×D₄	1	2	yes	—
61	#10625329	—	4	16	no	—
62	C₂×C₄₇⋊C₈	—	2	8	no	—
63	C₂×C₄₇⋊C₈	—	4	4	no	—
64	C₂×C₄₇⋊C₈	—	4	4	no	—
65	C₂³×C₄₇	—	2	2	no	—
66	C₂³×C₄₇	—	2	2	no	—
67	C₂³×C₄₇	#95	1	2	yes	—
68	C₂³×C₄₇	#106	1	2	yes	—
69	C₂³×C₄₇	#107	1	2	yes	—
70	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₄₇	—	2	2	no	—
72	C₂×C₄×C₄₇	—	2	2	no	—
73	C₂×C₄×C₄₇	—	2	8	no	—
74	C₂×C₄×C₄₇	—	2	4	no	—
75	C₂×C₄×C₄₇	—	2	4	no	—
76	C₂×C₄×C₄₇	#95	1	2	yes	—
77	C₂×C₄×C₄₇	#95	1	2	yes	—
78	C₂×C₄×C₄₇	#95	1	2	yes	—
79	C₂×C₄×C₄₇	#102	1	8	yes	—
80	C₂×C₄×C₄₇	#103	1	8	yes	—
81	C₂×C₄×C₄₇	#106	1	2	yes	—
82	C₂×C₄×C₄₇	#107	1	2	yes	—
83	C₂×C₄×C₄₇	#108	1	4	yes	—
84	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
85	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
86	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
88	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
89	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
91	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
92	C₈×C₄₇	—	2	8	no	—
93	C₄₇⋊C₈	—	4	8	no	—
94	#7874	—	47	1	no	maximal, Sylow
95	C₂²×C₄₇	—	2	4	no	—
96	C₂²×C₄₇	—	2	4	no	—
97	C₂²×C₄₇	—	2	2	no	—
98	C₂²×C₄₇	—	2	2	no	—
99	C₂²×C₄₇	#340	1	4	yes	—
100	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
101	C₂²×C₄₇	#423	1	2	yes	—
102	C₂²×C₄₇	C₂×#106	1	1	yes	characteristic
103	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
104	C₂²×C₄₇	C₂×#108	1	2	yes	—
105	C₄×C₄₇	—	2	2	no	—
106	C₄×C₄₇	—	2	2	no	—
107	C₄×C₄₇	#340	1	4	yes	—
108	C₄×C₄₇	#341	1	4	yes	—
109	C₄×C₄₇	C₄×M₁₆	1	2	yes	—
110	C₄×C₄₇	#407	1	2	yes	—
111	C₄×C₄₇	C₂×#102	1	2	yes	—
112	C₄×C₄₇	C₂×#103	1	2	yes	—
113	C₂×#423	—	47	2	no	—
114	C₂²×C₄×C₈	D₄₇	1	1	yes	characteristic
115	#1277	—	47	4	no	—
116	C₂×C₄₇	—	2	4	no	—
117	C₂×C₄₇	#1566	1	2	yes	—
118	C₂×C₄₇	#1567	1	2	yes	—
119	C₂×C₄₇	#1728	1	1	yes	characteristic
120	C₂×C₄₇	C₂×#340	1	1	yes	characteristic
121	C₂×C₄₇	C₂×#341	1	1	yes	characteristic
122	C₂²×M₁₆	—	94	2	no	—
123	C₂³×C₈	Dic₄₇	1	2	yes	—
124	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
125	#423	—	47	8	no	—
126	#423	—	94	4	no	—
127	C₂×C₄⋊C₈	—	47	4	no	—
128	C₂×C₄⋊C₈	—	94	2	no	—
129	C₂×C₄×C₈	—	2	4	no	—
130	C₂×C₄×C₈	Dic₄₇	1	2	yes	—
131	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
132	C₄₇	#7874	1	1	yes	characteristic, complemented, Sylow
133	C₂³×C₄	Dic₉₄	1	1	yes	characteristic
134	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
135	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
136	C₂×M₁₆	—	94	8	no	—
137	C₂×M₁₆	—	188	4	no	—
138	C₂×M₁₆	—	188	4	no	—
139	C₂²×C₈	—	2	4	no	—
140	C₂²×C₈	—	2	8	no	—
141	C₂²×C₈	—	94	4	no	—
142	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
143	C₂×C₄²	Dic₉₄	1	1	yes	characteristic
144	C₂×C₄²	C₄×D₄₇	1	4	yes	—
145	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
146	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
147	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
148	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
149	C₄⋊C₈	—	47	8	no	—
150	C₄⋊C₈	—	94	8	no	—
151	C₄⋊C₈	—	94	4	no	—
152	C₄×C₈	—	2	4	no	—
153	C₄×C₈	—	2	4	no	—
154	C₂⁴	C₄⋊Dic₄₇	1	1	yes	characteristic
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	—	2	4	no	—
157	C₂²×C₄	#10625325	1	4	yes	—
158	C₂²×C₄	D₃₇₆	1	2	yes	—
159	C₂²×C₄	Dic₁₈₈	1	2	yes	—
160	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
161	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
162	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
163	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
164	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
165	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
166	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
167	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
168	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
169	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
170	M₁₆	—	188	16	no	—
171	C₂×C₈	—	2	16	no	—
172	C₂×C₈	—	2	4	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	94	8	no	—
175	C₂×C₈	—	188	4	no	—
176	C₂×C₈	—	188	4	no	—
177	C₂×C₈	#10625329	1	8	yes	complemented
178	C₄²	C₄×Dic₄₇	1	2	yes	—
179	C₄²	Dic₄₇⋊C₄	1	4	yes	—
180	C₄²	(C₂×D₄₇)⋊C₄	1	4	yes	—
181	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
182	C₄²	C₂×C₄×D₄₇	1	2	yes	—
183	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
184	C₂³	—	2	2	no	—
185	C₂³	—	2	2	no	—
186	C₂³	C₄₇⋊#95	1	2	yes	—
187	C₂³	#13093956	1	2	yes	—
188	C₂³	#13093958	1	2	yes	—
189	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	8	no	—
195	C₂×C₄	C₄₇⋊#95	1	2	yes	—
196	C₂×C₄	C₄₇⋊#95	1	2	yes	—
197	C₂×C₄	C₄₇⋊#95	1	2	yes	—
198	C₂×C₄	#13093946	1	8	yes	—
199	C₂×C₄	#13093949	1	8	yes	—
200	C₂×C₄	#13093956	1	2	yes	—
201	C₂×C₄	#13093958	1	2	yes	—
202	C₂×C₄	#13093960	1	4	yes	—
203	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
204	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
205	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
206	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
207	C₂×C₄	C₂×#10625329	1	4	yes	—
208	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
209	C₂×C₄	C₂×#10625325	1	2	yes	—
210	C₂×C₄	C₂×Dic₁₈₈	1	1	yes	characteristic
211	C₈	—	2	8	no	—
212	C₈	—	188	8	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	2	4	no	—
216	C₂²	—	2	4	no	—
217	C₂²	#15688886	1	4	yes	—
218	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
219	C₂²	#15689147	1	2	yes	—
220	C₂²	C₂×#13093956	1	1	yes	characteristic
221	C₂²	C₂×#13093958	1	1	yes	characteristic
222	C₂²	C₂×#13093960	1	2	yes	—
223	C₄	—	2	2	no	—
224	C₄	—	2	2	no	—
225	C₄	#15688886	1	4	yes	—
226	C₄	#15688888	1	4	yes	—
227	C₄	C₄×#10625329	1	2	yes	—
228	C₄	#15689087	1	2	yes	—
229	C₄	C₂×#13093946	1	2	yes	—
230	C₄	C₂×#13093949	1	2	yes	—
231	C₂	—	2	4	no	—
232	C₂	#20242144	1	2	yes	—
233	C₂	#20242147	1	2	yes	—
234	C₂	#20243082	1	1	yes	characteristic
235	C₂	C₂×#15688886	1	1	yes	characteristic
236	C₂	C₂×#15688888	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂²×C₄×C₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₄₇	C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂²×C₄×C₈	D₄₇
hypocenter	C₄₇	#7874
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #20241223	C₂	semidirect
54. C₂×C₈×C₄₇	M₁₆	semidirect
132. C₄₇	#7874	semidirect
177. C₂×C₈	#10625329	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689147	2
#20241223	2
C₂²×C₄×C₈×C₄₇	2
#7874	47