Subgroups — #26106846

Order 12032.

518 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13093941	C₂	1	1	yes	maximal, characteristic
2	#20241709	C₂	1	1	yes	maximal, characteristic
3	#20241677	C₂	1	2	yes	maximal, complemented
4	#20241238	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×D₄×C₄₇	C₂	1	1	yes	maximal, characteristic
6	#15689176	C₂²	1	2	yes	—
7	C₂×#13093941	C₄	1	2	yes	—
8	C₄×#10625329	C₂²	1	2	yes	—
9	#15688892	C₄	1	2	yes	—
10	#15688847	C₄	1	4	yes	—
11	#15688845	—	2	2	no	—
12	#15688844	—	2	2	no	—
13	C₂³×C₈×C₄₇	C₄	1	2	yes	—
14	C₂×C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
15	C₈×D₄×C₄₇	—	2	2	no	—
16	C₈×D₄×C₄₇	—	2	2	no	—
17	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₈×C₄₇	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×#10625329	C₂×C₄	1	4	yes	—
21	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
22	#13093941	—	2	4	no	—
23	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
24	#13093935	—	2	2	no	—
25	C₄×C₄₇⋊C₈	—	2	2	no	—
26	C₄×C₄₇⋊C₈	—	2	2	no	—
27	C₂²×D₄×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₄₇	—	2	2	no	—
30	C₂²×C₈×C₄₇	—	2	2	no	—
31	C₂²×C₈×C₄₇	—	4	4	no	—
32	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₄×D₄×C₄₇	—	2	2	no	—
36	C₄×D₄×C₄₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₄₇	—	2	2	no	—
41	C₂²⋊C₈×C₄₇	—	2	4	no	—
42	C₄×C₈×C₄₇	D₄	1	2	yes	—
43	C₄×C₈×C₄₇	D₄	1	2	yes	—
44	C₂⁴×C₄₇	C₄²	1	2	yes	—
45	C₂×D₄×C₄₇	—	2	2	no	—
46	C₂×D₄×C₄₇	—	2	2	no	—
47	C₂×D₄×C₄₇	M₁₆	1	4	yes	—
48	C₂²×C₄×C₄₇	—	2	2	no	—
49	C₂²×C₄×C₄₇	—	2	2	no	—
50	C₂²×C₄×C₄₇	—	4	2	no	—
51	C₂²×C₄×C₄₇	—	4	2	no	—
52	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
53	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₄₇	—	2	2	no	—
57	C₂×C₈×C₄₇	—	2	2	no	—
58	C₂×C₈×C₄₇	—	4	1	no	—
59	C₂×C₈×C₄₇	—	4	4	no	—
60	C₂×C₈×C₄₇	—	4	4	no	—
61	C₂×C₈×C₄₇	—	4	2	no	—
62	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₄⋊C₄×C₄₇	—	2	2	no	—
66	C₂²⋊C₄×C₄₇	—	2	4	no	—
67	C₄²×C₄₇	C₂×D₄	1	2	yes	—
68	C₄²×C₄₇	#41	1	2	yes	—
69	#10625329	—	2	4	no	—
70	#10625329	—	2	4	no	—
71	C₂×C₄₇⋊C₈	—	2	4	no	—
72	C₂×C₄₇⋊C₈	—	2	4	no	—
73	C₂³×C₄₇	—	2	2	no	—
74	C₂³×C₄₇	—	2	2	no	—
75	C₂³×C₄₇	—	4	2	no	—
76	C₂³×C₄₇	—	4	2	no	—
77	C₂³×C₄₇	#97	1	4	yes	—
78	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
79	D₄×C₄₇	—	2	8	no	—
80	C₂×C₄×C₄₇	—	2	1	no	—
81	C₂×C₄×C₄₇	—	2	1	no	—
82	C₂×C₄×C₄₇	—	2	2	no	—
83	C₂×C₄×C₄₇	—	4	1	no	—
84	C₂×C₄×C₄₇	—	4	4	no	—
85	C₂×C₄×C₄₇	—	4	2	no	—
86	C₂×C₄×C₄₇	—	4	2	no	—
87	C₂×C₄×C₄₇	—	4	2	no	—
88	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
90	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
91	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
92	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
93	C₂×C₄×C₄₇	#131	1	2	yes	—
94	C₈×C₄₇	—	2	1	no	—
95	C₈×C₄₇	—	2	2	no	—
96	C₈×C₄₇	—	4	2	no	—
97	C₈×C₄₇	#100	1	2	yes	complemented
98	C₄₇⋊C₈	—	4	4	no	—
99	C₄₇⋊C₈	—	4	4	no	—
100	#8164	—	47	1	no	maximal, Sylow
101	C₂²×C₄₇	—	2	1	no	—
102	C₂²×C₄₇	—	2	1	no	—
103	C₂²×C₄₇	—	2	4	no	—
104	C₂²×C₄₇	—	4	4	no	—
105	C₂²×C₄₇	—	4	4	no	—
106	C₂²×C₄₇	—	4	2	no	—
107	C₂²×C₄₇	—	4	2	no	—
108	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
109	C₂²×C₄₇	C₂×#97	1	1	yes	characteristic
110	C₂²×C₄₇	#405	1	1	yes	characteristic
111	C₄×C₄₇	—	2	1	no	—
112	C₄×C₄₇	—	2	2	no	—
113	C₄×C₄₇	—	4	2	no	—
114	C₄×C₄₇	C₂×#100	1	1	yes	characteristic
115	C₄×C₄₇	#413	1	1	yes	characteristic
116	C₄×C₄₇	#435	1	4	yes	—
117	C₂×C₈×D₄	D₄₇	1	1	yes	characteristic
118	C₄×#100	—	47	1	no	—
119	#1460	—	47	1	no	—
120	#1451	—	47	2	no	—
121	#1283	—	47	2	no	—
122	C₂×C₄₇	—	2	1	no	—
123	C₂×C₄₇	—	2	1	no	—
124	C₂×C₄₇	—	4	4	no	—
125	C₂×C₄₇	#1752	1	1	yes	characteristic
126	C₂×C₄₇	C₄×#100	1	1	yes	characteristic
127	C₂×C₄₇	#1757	1	1	yes	characteristic
128	C₂³×C₈	Dic₄₇	1	2	yes	—
129	C₂×C₄×D₄	C₂×D₄₇	1	1	yes	characteristic
130	C₈×D₄	—	2	2	no	—
131	C₈×D₄	—	2	2	no	—
132	#431	—	47	2	no	—
133	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
134	C₂×#100	—	47	2	no	—
135	C₂×C₂²⋊C₈	Dic₄₇	1	2	yes	—
136	C₄×M₁₆	—	47	2	no	—
137	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
138	#343	—	47	2	no	—
139	#323	—	47	4	no	—
140	#322	—	94	2	no	—
141	#322	—	94	2	no	—
142	C₄₇	#8164	1	1	yes	characteristic, complemented, Sylow
143	C₂²×D₄	C₄×D₄₇	1	1	yes	characteristic
144	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
145	C₂×M₁₆	—	47	4	no	—
146	C₂²×C₈	—	2	2	no	—
147	C₂²×C₈	—	2	2	no	—
148	C₂²×C₈	—	4	4	no	—
149	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
150	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
151	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
152	C₄×D₄	—	2	2	no	—
153	C₄×D₄	—	2	2	no	—
154	C₂×C₄⋊C₄	C₄×D₄₇	1	1	yes	characteristic
155	C₂×C₂²⋊C₄	C₄×D₄₇	1	2	yes	—
156	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	94	2	no	—
159	#100	—	94	4	no	—
160	C₂²⋊C₈	—	2	4	no	—
161	C₂²⋊C₈	—	47	4	no	—
162	#97	—	94	2	no	—
163	C₄×C₈	—	94	2	no	—
164	C₄×C₈	—	94	2	no	—
165	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
166	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
167	C₂⁴	C₄×Dic₄₇	1	2	yes	—
168	C₂×D₄	—	2	2	no	—
169	C₂×D₄	—	2	2	no	—
170	C₂×D₄	#10625324	1	4	yes	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	4	2	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
176	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
178	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
179	M₁₆	—	94	4	no	—
180	M₁₆	—	94	4	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	4	1	no	—
184	C₂×C₈	—	4	4	no	—
185	C₂×C₈	—	4	4	no	—
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	94	4	no	—
188	C₂×C₈	—	94	4	no	—
189	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
190	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
191	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
192	C₄⋊C₄	—	2	2	no	—
193	C₂²⋊C₄	—	2	4	no	—
194	C₄²	#10625350	1	2	yes	—
195	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
196	C₂³	—	2	2	no	—
197	C₂³	—	2	2	no	—
198	C₂³	—	4	2	no	—
199	C₂³	—	4	2	no	—
200	C₂³	#13093934	1	4	yes	—
201	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
202	D₄	—	2	8	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
212	C₂×C₄	#13093983	1	1	yes	characteristic
213	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
214	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
215	C₂×C₄	C₂×#10625324	1	2	yes	—
216	C₂×C₄	#13094030	1	2	yes	—
217	C₈	—	2	1	no	—
218	C₈	—	2	2	no	—
219	C₈	—	4	2	no	—
220	C₈	—	188	4	no	—
221	C₈	—	188	4	no	—
222	C₈	#13093941	1	2	yes	complemented
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	4	no	—
226	C₂²	—	4	4	no	—
227	C₂²	—	4	4	no	—
228	C₂²	—	4	2	no	—
229	C₂²	—	4	2	no	—
230	C₂²	C₄×C₂²⋊Dic₄₇	1	1	yes	characteristic
231	C₂²	C₂×#13093934	1	1	yes	characteristic
232	C₂²	#15689078	1	1	yes	characteristic
233	C₄	—	2	1	no	—
234	C₄	—	2	2	no	—
235	C₄	—	4	2	no	—
236	C₄	C₂×#13093941	1	1	yes	characteristic
237	C₄	#15689107	1	1	yes	characteristic
238	C₄	#15689194	1	4	yes	—
239	C₂	—	2	1	no	—
240	C₂	—	2	1	no	—
241	C₂	—	4	4	no	—
242	C₂	#20243164	1	1	yes	characteristic
243	C₂	C₄×#13093941	1	1	yes	characteristic
244	C₂	#20243188	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₄₇
commutator	C₂³×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₈×D₄×C₄₇	C₂
socle	C₂²×C₄₇	C₄×C₂²⋊C₄
prime radical	C₂²×D₄×C₄₇	C₂×C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×C₈×D₄	D₄₇
hypocenter	C₄₇	#8164
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
3. #20241677	C₂	semidirect
4. #20241238	C₂	semidirect
97. C₈×C₄₇	#100	semidirect
142. C₄₇	#8164	semidirect
222. C₈	#13093941	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13093941	2
#20241709	2
#20241677	2
#20241238	2
C₂×C₈×D₄×C₄₇	2
#8164	47