Subgroups — #26105665

Order 12032.

674 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689174	C₂	1	2	yes	maximal
2	C₂×C₄×#10625324	C₂	1	1	yes	maximal, characteristic
3	C₄₇⋊#1273	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇⋊#1273	C₂	1	2	yes	maximal, complemented
5	C₄₇×#1273	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#10625324	C₄	1	2	yes	—
7	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
8	#15689174	—	2	4	no	—
9	#15689174	—	2	4	no	—
10	C₂×Dic₄₇⋊C₈	C₄	1	4	yes	—
11	C₂×(C₂×D₄₇)⋊C₈	C₄	1	4	yes	—
12	C₄×#10625324	—	2	4	no	—
13	C₂×#13093934	C₄	1	2	yes	—
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
15	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄×D₄₇	C₂×C₄	1	2	yes	—
19	C₂²×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
20	C₂×#10625324	—	2	4	no	—
21	C₂×#10625324	—	2	4	no	—
22	C₂×#10625324	—	2	4	no	—
23	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
24	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
25	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
27	C₄²×D₄₇	D₄	1	4	yes	—
28	C₄²×D₄₇	D₄	1	2	yes	—
29	C₄²×D₄₇	Q₈	1	2	yes	—
30	Dic₄₇⋊C₈	—	2	8	no	—
31	(C₂×D₄₇)⋊C₈	—	2	8	no	—
32	#13093934	—	2	4	no	—
33	C₄×C₄₇⋊C₈	—	2	2	no	—
34	C₄×C₄₇⋊C₈	—	2	2	no	—
35	C₄×C₄₇⋊C₈	—	2	2	no	—
36	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
37	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
39	C₄×C₈×C₄₇	—	2	2	no	—
40	C₄×C₈×C₄₇	—	2	2	no	—
41	C₄×C₈×C₄₇	—	2	2	no	—
42	C₂³×D₄₇	C₄²	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
45	C₂²×Dic₄₇	C₄²	1	1	yes	characteristic
46	C₂²×Dic₄₇	C₄²	1	2	yes	—
47	C₂×C₄×D₄₇	—	2	2	no	—
48	C₂×C₄×D₄₇	—	2	2	no	—
49	C₂×C₄×D₄₇	—	2	4	no	—
50	C₂×C₄×D₄₇	—	2	4	no	—
51	C₂×C₄×D₄₇	C₂²⋊C₄	1	4	yes	—
52	C₂×C₄×D₄₇	C₂²⋊C₄	1	2	yes	—
53	C₂×C₄×D₄₇	C₄⋊C₄	1	2	yes	—
54	C₂×C₄×D₄₇	C₄⋊C₄	1	4	yes	—
55	C₂×C₈×C₄₇	—	2	2	no	—
56	C₂×C₈×C₄₇	—	2	2	no	—
57	C₂×C₈×C₄₇	—	2	4	no	—
58	C₂×C₈×C₄₇	—	2	4	no	—
59	C₂×C₈×C₄₇	—	2	4	no	—
60	C₂×C₈×C₄₇	M₁₆	1	4	yes	—
61	C₄²×C₄₇	C₂×D₄	1	2	yes	—
62	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
64	C₄×Dic₄₇	C₂²⋊C₄	1	2	yes	—
65	C₄×Dic₄₇	C₂²⋊C₄	1	4	yes	—
66	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
67	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
68	C₂×C₄₇⋊C₈	—	2	2	no	—
69	C₂×C₄₇⋊C₈	—	2	2	no	—
70	C₂×C₄₇⋊C₈	—	2	4	no	—
71	C₂×C₄₇⋊C₈	—	2	4	no	—
72	C₂×C₄₇⋊C₈	—	2	4	no	—
73	C₂×C₄₇⋊C₈	M₁₆	1	4	yes	—
74	#10625324	—	2	16	no	—
75	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
76	C₂²×D₄₇	—	2	1	no	—
77	C₂²×D₄₇	—	2	1	no	—
78	C₂²×D₄₇	—	2	2	no	—
79	C₂²×D₄₇	—	2	2	no	—
80	C₂²×D₄₇	#95	1	2	yes	—
81	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
83	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
85	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
86	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
87	C₂×C₄×C₄₇	#131	1	4	yes	—
88	C₂×C₄×C₄₇	#131	1	4	yes	—
89	C₂×Dic₄₇	—	2	1	no	—
90	C₂×Dic₄₇	—	2	1	no	—
91	C₂×Dic₄₇	—	2	2	no	—
92	C₂×Dic₄₇	—	2	2	no	—
93	C₂×Dic₄₇	#95	1	2	yes	—
94	C₂×Dic₄₇	#95	1	4	yes	—
95	C₄×D₄₇	—	2	4	no	—
96	C₄×D₄₇	—	2	8	no	—
97	C₄×D₄₇	—	2	4	no	—
98	C₄×D₄₇	—	2	8	no	—
99	C₈×C₄₇	—	2	4	no	—
100	C₈×C₄₇	—	4	4	no	—
101	C₄₇⋊C₈	—	2	4	no	—
102	C₄₇⋊C₈	—	4	4	no	—
103	#7850	—	47	1	no	maximal, Sylow
104	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
105	C₂²×C₄₇	C₄×M₁₆	1	1	yes	characteristic
106	C₂²×C₄₇	C₄×M₁₆	1	1	yes	characteristic
107	C₂²×C₄₇	#405	1	2	yes	—
108	C₂²×C₄₇	#405	1	2	yes	—
109	C₂×D₄₇	—	2	2	no	—
110	C₂×D₄₇	—	2	4	no	—
111	C₂×D₄₇	—	4	1	no	—
112	C₂×D₄₇	—	4	2	no	—
113	C₂×D₄₇	—	4	1	no	—
114	C₄×C₄₇	#407	1	2	yes	—
115	C₄×C₄₇	#408	1	4	yes	—
116	C₄×C₄₇	#423	1	2	yes	—
117	C₄×C₄₇	#424	1	4	yes	—
118	Dic₄₇	—	2	2	no	—
119	Dic₄₇	—	2	4	no	—
120	C₂×#431	—	47	2	no	—
121	C₂×C₄×M₁₆	—	47	1	no	—
122	#1273	—	47	2	no	—
123	#1273	—	47	1	no	—
124	#1273	D₄₇	1	1	yes	characteristic, complemented
125	C₂×C₄₇	#1453	1	1	yes	characteristic
126	C₂×C₄₇	#1453	1	2	yes	—
127	C₂×C₄₇	#1453	1	1	yes	characteristic
128	C₂×C₄₇	#1728	1	1	yes	characteristic
129	C₂×C₄₇	#1729	1	2	yes	—
130	D₄₇	—	4	2	no	—
131	C₂²×M₁₆	—	47	2	no	—
132	C₂²×C₄²	—	47	1	no	—
133	#431	—	94	4	no	—
134	#431	—	94	4	no	—
135	C₂×C₄⋊C₈	—	47	4	no	—
136	C₂×C₂²⋊C₈	—	47	4	no	—
137	C₄×M₁₆	—	94	4	no	—
138	C₂×#97	—	47	2	no	—
139	C₂×C₄×C₈	—	47	1	no	—
140	C₂×C₄×C₈	—	47	2	no	—
141	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
142	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
143	C₄₇	#7850	1	1	yes	characteristic, complemented, Sylow
144	C₂³×C₄	—	47	1	no	—
145	C₂³×C₄	—	47	2	no	—
146	C₂×M₁₆	—	94	4	no	—
147	C₂×M₁₆	—	94	4	no	—
148	C₂×M₁₆	—	94	4	no	—
149	C₂²×C₈	—	47	2	no	—
150	C₂²×C₈	—	47	4	no	—
151	C₂²×C₈	C₄×D₄₇	1	4	yes	—
152	C₂²×C₈	C₄×D₄₇	1	2	yes	—
153	C₂×C₄²	—	47	1	no	—
154	C₂×C₄²	—	47	2	no	—
155	C₂×C₄²	—	47	4	no	—
156	C₂×C₄²	—	47	2	no	—
157	C₂×C₄²	—	47	2	no	—
158	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
159	C₄⋊C₈	—	94	8	no	—
160	C₂²⋊C₈	—	94	8	no	—
161	#97	—	94	4	no	—
162	C₄×C₈	—	2	2	no	—
163	C₄×C₈	—	2	2	no	—
164	C₄×C₈	—	2	2	no	—
165	C₄×C₈	—	94	2	no	—
166	C₄×C₈	—	94	2	no	—
167	C₄×C₈	—	94	2	no	—
168	C₂⁴	—	47	1	no	—
169	C₂²×C₄	—	47	1	no	—
170	C₂²×C₄	—	47	2	no	—
171	C₂²×C₄	—	47	4	no	—
172	C₂²×C₄	—	47	2	no	—
173	C₂²×C₄	—	47	2	no	—
174	C₂²×C₄	—	47	4	no	—
175	C₂²×C₄	—	94	2	no	—
176	C₂²×C₄	—	94	2	no	—
177	C₂²×C₄	—	94	4	no	—
178	C₂²×C₄	—	94	4	no	—
179	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
181	M₁₆	—	94	16	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	47	4	no	—
188	C₂×C₈	—	94	2	no	—
189	C₂×C₈	—	94	2	no	—
190	C₂×C₈	—	94	4	no	—
191	C₂×C₈	—	94	4	no	—
192	C₂×C₈	—	94	4	no	—
193	C₂×C₈	#10625324	1	4	yes	—
194	C₄²	—	47	2	no	—
195	C₄²	—	47	4	no	—
196	C₄²	—	47	2	no	—
197	C₄²	—	47	4	no	—
198	C₄²	D₄×D₄₇	1	2	yes	—
199	C₄²	D₄×D₄₇	1	1	yes	characteristic
200	C₄²	Q₈×D₄₇	1	1	yes	characteristic
201	C₂³	—	47	2	no	—
202	C₂³	—	94	1	no	—
203	C₂³	—	94	1	no	—
204	C₂³	—	94	2	no	—
205	C₂³	—	94	2	no	—
206	C₂³	C₄²×D₄₇	1	1	yes	characteristic
207	C₂×C₄	—	47	2	no	—
208	C₂×C₄	—	47	4	no	—
209	C₂×C₄	—	94	1	no	—
210	C₂×C₄	—	94	1	no	—
211	C₂×C₄	—	94	2	no	—
212	C₂×C₄	—	94	2	no	—
213	C₂×C₄	—	94	4	no	—
214	C₂×C₄	—	94	8	no	—
215	C₂×C₄	—	94	4	no	—
216	C₂×C₄	—	94	8	no	—
217	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
218	C₂×C₄	C₂²⋊C₄×D₄₇	1	2	yes	—
219	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
220	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
221	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
222	C₂×C₄	C₂×#10625324	1	2	yes	—
223	C₂×C₄	#13094030	1	4	yes	—
224	C₂×C₄	#13094032	1	4	yes	—
225	C₈	—	2	4	no	—
226	C₈	—	4	4	no	—
227	C₈	—	94	4	no	—
228	C₈	—	188	4	no	—
229	C₂²	—	94	2	no	—
230	C₂²	—	94	4	no	—
231	C₂²	—	188	1	no	—
232	C₂²	—	188	2	no	—
233	C₂²	—	188	1	no	—
234	C₂²	#95×D₄₇	1	1	yes	characteristic
235	C₂²	C₄×#10625324	1	1	yes	characteristic
236	C₂²	#15689075	1	1	yes	characteristic
237	C₂²	#15689077	1	2	yes	—
238	C₂²	#15689080	1	2	yes	—
239	C₄	—	94	2	no	—
240	C₄	—	94	4	no	—
241	C₄	#15689085	1	2	yes	—
242	C₄	#15689088	1	4	yes	—
243	C₄	#15689145	1	2	yes	—
244	C₄	#15689148	1	4	yes	—
245	C₂	—	188	2	no	—
246	C₂	#20241682	1	1	yes	characteristic
247	C₂	#20241684	1	2	yes	—
248	C₂	#20241686	1	1	yes	characteristic
249	C₂	#20243080	1	1	yes	characteristic
250	C₂	#20243083	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₄₇
commutator, socle	C₂³×C₄₇	C₂×C₄²
Fitting	C₄₇×#1273	C₂
prime radical	C₂³×D₄₇	C₄²
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1273	D₄₇
hypocenter	C₄₇	#7850
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ #1273

Complements

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

N	H ≅ G/N	type
3. C₄₇⋊#1273	C₂	semidirect
4. C₄₇⋊#1273	C₂	semidirect
5. C₄₇×#1273	C₂	semidirect
124. #1273	D₄₇	semidirect
143. C₄₇	#7850	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689174	2
C₂×C₄×#10625324	2
C₄₇⋊#1273	2
C₄₇×#1273	2
#7850	47