Subgroups — #26245351

Order 12032.

874 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×#15689149	C₂	1	1	yes	maximal, characteristic
2	#20246012	C₂	1	2	yes	maximal
3	#20244887	C₂	1	8	yes	maximal, complemented
4	#20243107	C₂	1	2	yes	maximal
5	#20243105	C₂	1	1	yes	maximal, characteristic
6	C₂²×#106×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10625329	C₂²	1	1	yes	characteristic
8	C₂×#13093983	C₂²	1	2	yes	—
9	C₂×#13093983	C₂²	1	1	yes	characteristic
10	#15689149	—	2	4	no	—
11	C₂×Dic₄₇⋊C₈	C₂²	1	4	yes	—
12	#15689019	C₂²	1	8	yes	—
13	#15688996	—	2	4	no	—
14	#15688992	—	2	2	no	—
15	#15688988	—	2	2	no	—
16	#15688886	C₂²	1	4	yes	—
17	#15688885	C₂²	1	8	yes	—
18	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
20	C₂×#106×C₄₇	—	2	4	no	—
21	C₂×#106×C₄₇	C₂²	1	4	yes	—
22	C₂×#10625329	—	2	2	no	—
23	C₂×#10625329	—	2	4	no	—
24	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
25	#13093983	—	2	2	no	—
26	#13093983	—	2	4	no	—
27	#13093983	—	2	2	no	—
28	#13093983	Q₈	1	8	yes	—
29	C₂×C₄⋊Dic₄₇	D₄	1	2	yes	—
30	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
31	C₂×Dic₄₇⋊C₄	—	2	8	no	—
32	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
33	C₂×C₂²⋊Dic₄₇	D₄	1	2	yes	—
34	C₂×C₄×Dic₄₇	—	2	2	no	—
35	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
36	Dic₄₇⋊C₈	—	2	8	no	—
37	C₄₇⋊#95	—	2	8	no	—
38	C₂³×C₄×C₄₇	—	2	2	no	—
39	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₄₇	—	2	4	no	—
41	C₂²×C₈×C₄₇	D₄	1	4	yes	—
42	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
44	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
45	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
47	#106×C₄₇	—	2	8	no	—
48	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	—	2	4	no	—
50	C₂²×C₄×C₄₇	—	4	4	no	—
51	C₂²×C₄×C₄₇	—	8	4	no	—
52	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
55	C₂²×C₄×C₄₇	#41	1	2	yes	—
56	C₂²×C₄×C₄₇	#41	1	4	yes	—
57	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
58	C₂²×Dic₄₇	—	2	4	no	—
59	C₂²×Dic₄₇	C₂×D₄	1	4	yes	—
60	C₂×C₈×C₄₇	—	2	8	no	—
61	C₂×C₈×C₄₇	—	2	4	no	—
62	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
63	C₄⋊C₄×C₄₇	—	2	4	no	—
64	C₄⋊C₄×C₄₇	—	2	4	no	—
65	C₄⋊C₄×C₄₇	—	4	4	no	—
66	C₂²⋊Dic₄₇	—	2	4	no	—
67	C₂²⋊Dic₄₇	—	2	8	no	—
68	C₄⋊Dic₄₇	—	2	4	no	—
69	C₄⋊Dic₄₇	—	2	4	no	—
70	C₄⋊Dic₄₇	C₂×Q₈	1	8	yes	—
71	Dic₄₇⋊C₄	—	8	8	no	—
72	C₄×Dic₄₇	—	2	8	no	—
73	C₄×Dic₄₇	—	2	2	no	—
74	C₄×Dic₄₇	—	2	2	no	—
75	#10625329	—	4	4	no	—
76	C₂×C₄₇⋊C₈	—	2	2	no	—
77	C₂×C₄₇⋊C₈	—	2	4	no	—
78	C₂³×C₄₇	—	2	4	no	—
79	C₂³×C₄₇	#121	1	2	yes	—
80	C₂³×C₄₇	C₂²⋊Q₈	1	4	yes	—
81	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
82	C₂×C₄×C₄₇	—	4	4	no	—
83	C₂×C₄×C₄₇	—	4	8	no	—
84	C₂×C₄×C₄₇	—	8	8	no	—
85	C₂×C₄×C₄₇	#121	1	2	yes	—
86	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
87	C₂×C₄×C₄₇	#123	1	4	yes	—
88	C₂×C₄×C₄₇	#125	1	8	yes	—
89	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
90	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
91	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
92	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
93	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
94	C₂×Dic₄₇	—	2	4	no	—
95	C₂×Dic₄₇	—	2	8	no	—
96	C₂×Dic₄₇	—	2	8	no	—
97	C₂×Dic₄₇	—	4	4	no	—
98	C₂×Dic₄₇	—	4	4	no	—
99	C₈×C₄₇	—	2	4	no	—
100	C₄₇⋊C₈	—	4	2	no	—
101	#23668	—	47	1	no	maximal, Sylow
102	C₂²×C₄₇	—	2	8	no	—
103	C₂²×C₄₇	—	2	4	no	—
104	C₂²×C₄₇	#398	1	4	yes	—
105	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
106	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
107	C₂²×C₄₇	#577	1	4	yes	—
108	C₄×C₄₇	—	8	4	no	—
109	C₄×C₄₇	#398	1	4	yes	—
110	C₄×C₄₇	C₂×#123	1	1	yes	characteristic
111	C₄×C₄₇	C₂×#125	1	2	yes	—
112	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
113	Dic₄₇	—	4	8	no	—
114	Dic₄₇	—	4	4	no	—
115	C₂²×#106	D₄₇	1	1	yes	characteristic
116	C₂×#424	—	47	1	no	—
117	#2317	—	47	2	no	—
118	#2088	—	47	8	no	—
119	#1736	—	47	2	no	—
120	#1735	—	47	1	no	—
121	C₂×C₄₇	—	2	4	no	—
122	C₂×C₄₇	C₂×#398	1	1	yes	characteristic
123	C₂×C₄₇	#3057	1	2	yes	—
124	C₂×C₄₇	#3084	1	4	yes	—
125	C₂²×M₁₆	—	47	1	no	—
126	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
127	C₂×#117	—	47	1	no	—
128	C₂×#117	—	47	2	no	—
129	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
130	C₂×#106	—	2	4	no	—
131	C₂×#106	C₂×D₄₇	1	4	yes	—
132	#424	—	94	4	no	—
133	C₂×C₄⋊C₈	—	47	4	no	—
134	#389	—	47	8	no	—
135	#385	—	94	4	no	—
136	#384	—	94	2	no	—
137	#382	—	94	2	no	—
138	#340	—	47	4	no	—
139	#340	—	47	8	no	—
140	C₄₇	#23668	1	1	yes	characteristic, complemented, Sylow
141	C₂³×C₄	—	2	2	no	—
142	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
143	C₂×M₁₆	—	94	4	no	—
144	C₂×M₁₆	—	94	2	no	—
145	C₂²×C₈	—	2	4	no	—
146	C₂²×C₈	—	47	2	no	—
147	C₂²×C₈	C₂²⋊D₄₇	1	4	yes	—
148	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
149	#117	—	47	8	no	—
150	#117	—	94	4	no	—
151	#117	—	94	2	no	—
152	#117	—	94	2	no	—
153	C₂×C₄⋊C₄	—	2	4	no	—
154	C₂×C₄⋊C₄	—	2	4	no	—
155	C₂×C₄⋊C₄	—	2	2	no	—
156	C₂×C₄⋊C₄	—	47	4	no	—
157	C₂×C₄⋊C₄	—	47	2	no	—
158	C₂×C₄⋊C₄	—	94	8	no	—
159	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
160	C₂×C₂²⋊C₄	—	47	2	no	—
161	C₂×C₂²⋊C₄	—	94	2	no	—
162	C₂×C₄²	—	47	2	no	—
163	C₂×C₄²	—	94	2	no	—
164	#106	—	2	8	no	—
165	C₄⋊C₈	—	94	8	no	—
166	#95	—	94	8	no	—
167	C₂⁴	D₄×D₄₇	1	1	yes	characteristic
168	C₂²×C₄	—	2	4	no	—
169	C₂²×C₄	—	4	4	no	—
170	C₂²×C₄	—	8	4	no	—
171	C₂²×C₄	—	47	4	no	—
172	C₂²×C₄	—	94	4	no	—
173	C₂²×C₄	#10625350	1	4	yes	—
174	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
175	C₂²×C₄	#10625352	1	2	yes	—
176	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
177	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
178	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
179	M₁₆	—	188	4	no	—
180	C₂×C₈	—	2	8	no	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	94	2	no	—
183	C₂×C₈	—	94	4	no	—
184	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
185	C₄⋊C₄	—	2	4	no	—
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	4	4	no	—
188	C₄⋊C₄	—	47	8	no	—
189	C₄⋊C₄	—	94	4	no	—
190	C₄⋊C₄	—	94	4	no	—
191	C₄⋊C₄	—	376	8	no	—
192	C₂²⋊C₄	—	94	4	no	—
193	C₂²⋊C₄	—	94	8	no	—
194	C₄²	—	94	8	no	—
195	C₄²	—	94	2	no	—
196	C₄²	—	94	2	no	—
197	C₂³	—	2	4	no	—
198	C₂³	#13093995	1	2	yes	—
199	C₂³	#13094002	1	4	yes	—
200	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	4	8	no	—
203	C₂×C₄	—	8	8	no	—
204	C₂×C₄	—	94	4	no	—
205	C₂×C₄	—	94	8	no	—
206	C₂×C₄	—	94	8	no	—
207	C₂×C₄	—	188	4	no	—
208	C₂×C₄	—	188	4	no	—
209	C₂×C₄	#13093995	1	2	yes	—
210	C₂×C₄	#13094002	1	4	yes	—
211	C₂×C₄	#13094007	1	4	yes	—
212	C₂×C₄	#13094014	1	8	yes	—
213	C₂×C₄	#13094022	1	4	yes	—
214	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
215	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
216	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
217	C₂×C₄	C₂×#10625350	1	2	yes	—
218	C₈	—	2	4	no	—
219	C₈	—	188	2	no	—
220	C₂²	—	2	8	no	—
221	C₂²	—	2	4	no	—
222	C₂²	#15689054	1	4	yes	—
223	C₂²	C₂×#13093995	1	1	yes	characteristic
224	C₂²	C₂×#13094002	1	2	yes	—
225	C₂²	#15689929	1	4	yes	—
226	C₄	—	8	4	no	—
227	C₄	—	188	8	no	—
228	C₄	—	188	4	no	—
229	C₄	#15689054	1	4	yes	—
230	C₄	C₂×#13094007	1	1	yes	characteristic
231	C₄	C₂×#13094014	1	2	yes	—
232	C₄	C₂×#13094022	1	1	yes	characteristic
233	C₂	—	2	4	no	—
234	C₂	C₂×#15689054	1	1	yes	characteristic
235	C₂	#20253023	1	2	yes	—
236	C₂	#20253255	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×#106×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
hypercenter	C₂²×#106	D₄₇
hypocenter	C₄₇	#23668
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#106

Complements

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

N	H ≅ G/N	type
3. #20244887	C₂	semidirect
140. C₄₇	#23668	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689149	2
#20246012	2
#20244887	2
#20243107	2
#20243105	2
C₂²×#106×C₄₇	2
#23668	47