Subgroups — #26105869

Order 12032.

758 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689176	C₂	1	2	yes	maximal
2	#20241231	C₂	1	4	yes	maximal
3	C₂×C₄₇×#376	C₂	1	1	yes	maximal, characteristic
4	#15689176	—	2	4	no	—
5	#15689176	C₄	1	8	yes	complemented
6	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
7	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
8	C₂×C₄×C₄₇⋊C₈	—	2	2	no	—
9	C₂²×C₄²×C₄₇	C₄	1	2	yes	—
10	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
11	C₄₇×#376	—	2	2	no	—
12	C₄₇×#376	—	2	2	no	—
13	C₂×#95×C₄₇	C₄	1	2	yes	—
14	C₂×#95×C₄₇	C₂²	1	2	yes	—
15	C₂²×C₄₇⋊C₈	—	2	4	no	—
16	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
17	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
18	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
19	C₄×C₄₇⋊C₈	—	2	8	no	—
20	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
21	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₄₇	—	2	2	no	—
25	C₂×C₄²×C₄₇	—	2	2	no	—
26	C₂×C₄²×C₄₇	—	2	4	no	—
27	C₂×C₄²×C₄₇	C₈	1	8	yes	complemented
28	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₄₇	D₄	1	2	yes	—
30	C₂×C₄²×C₄₇	D₄	1	2	yes	—
31	C₂×C₄²×C₄₇	D₄	1	2	yes	—
32	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
33	#95×C₄₇	—	2	4	no	—
34	#95×C₄₇	—	2	4	no	—
35	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	—	2	4	no	—
37	C₂²×C₄×C₄₇	—	2	4	no	—
38	C₂²×C₄×C₄₇	—	4	2	no	—
39	C₂²×C₄×C₄₇	—	4	2	no	—
40	C₂²×C₄×C₄₇	—	4	4	no	—
41	C₂²×C₄×C₄₇	—	4	4	no	—
42	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
43	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
44	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₄₇	C₂×C₈	1	8	yes	—
50	C₂²×C₄×C₄₇	C₂×C₈	1	4	yes	—
51	C₂²×C₄×C₄₇	M₁₆	1	4	yes	—
52	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₄₇	—	2	8	no	—
57	C₄²×C₄₇	—	2	8	no	—
58	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
59	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
60	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
61	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
62	C₂×C₄₇⋊C₈	—	2	8	no	—
63	C₂×C₄₇⋊C₈	—	4	8	no	—
64	C₂³×C₄₇	—	2	2	no	—
65	C₂³×C₄₇	—	2	2	no	—
66	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
67	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂³×C₄₇	C₂²×C₈	1	2	yes	—
70	C₂³×C₄₇	C₂×M₁₆	1	2	yes	—
71	C₂×C₄×C₄₇	—	2	4	no	—
72	C₂×C₄×C₄₇	—	2	4	no	—
73	C₂×C₄×C₄₇	—	2	8	no	—
74	C₂×C₄×C₄₇	—	4	2	no	—
75	C₂×C₄×C₄₇	—	4	2	no	—
76	C₂×C₄×C₄₇	—	4	2	no	—
77	C₂×C₄×C₄₇	—	4	1	no	—
78	C₂×C₄×C₄₇	—	4	1	no	—
79	C₂×C₄×C₄₇	—	4	8	no	—
80	C₂×C₄×C₄₇	—	4	4	no	—
81	C₂×C₄×C₄₇	—	4	4	no	—
82	C₂×C₄×C₄₇	—	4	8	no	—
83	C₂×C₄×C₄₇	#95	1	4	yes	—
84	C₂×C₄×C₄₇	#95	1	2	yes	—
85	C₂×C₄×C₄₇	#95	1	2	yes	—
86	C₂×C₄×C₄₇	C₄×C₈	1	4	yes	—
87	C₂×C₄×C₄₇	#97	1	4	yes	—
88	C₂×C₄×C₄₇	C₂²⋊C₈	1	4	yes	—
89	C₂×C₄×C₄₇	C₂²⋊C₈	1	4	yes	—
90	C₂×C₄×C₄₇	C₄⋊C₈	1	8	yes	—
91	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
92	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
93	C₄₇⋊C₈	—	4	8	no	—
94	#7906	—	47	1	no	maximal, Sylow
95	C₂²×C₄₇	—	2	4	no	—
96	C₂²×C₄₇	—	2	4	no	—
97	C₂²×C₄₇	—	2	1	no	—
98	C₂²×C₄₇	—	2	1	no	—
99	C₂²×C₄₇	—	2	1	no	—
100	C₂²×C₄₇	—	2	1	no	—
101	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
102	C₂²×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
103	C₂²×C₄₇	C₂×#97	1	1	yes	characteristic
104	C₂²×C₄₇	C₂×C₂²⋊C₈	1	2	yes	—
105	C₂²×C₄₇	#410	1	2	yes	—
106	C₂²×C₄₇	#413	1	2	yes	—
107	C₂²×C₄₇	C₂×C₄⋊C₈	1	2	yes	—
108	C₄×C₄₇	—	2	2	no	—
109	C₄×C₄₇	—	2	2	no	—
110	C₄×C₄₇	—	2	4	no	—
111	C₄×C₄₇	—	4	8	no	—
112	C₄×C₄₇	#336	1	4	yes	—
113	C₄×C₄₇	#336	1	4	yes	—
114	C₂×#431	—	47	2	no	—
115	C₂×#376	D₄₇	1	1	yes	characteristic
116	#1280	—	47	4	no	—
117	C₂×C₄₇	—	2	1	no	—
118	C₂×C₄₇	—	2	1	no	—
119	C₂×C₄₇	—	2	2	no	—
120	C₂×C₄₇	#1458	1	4	yes	—
121	C₂×C₄₇	C₂×#336	1	1	yes	characteristic
122	C₂×C₄₇	#1740	1	1	yes	characteristic
123	C₂×C₄₇	#1742	1	1	yes	characteristic
124	C₂²×C₄²	Dic₄₇	1	2	yes	—
125	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
126	#431	—	47	8	no	—
127	#431	—	94	4	no	—
128	C₂×C₄⋊C₈	—	94	2	no	—
129	C₂×C₂²⋊C₈	—	47	4	no	—
130	C₂×C₄×C₈	—	94	2	no	—
131	#376	—	2	2	no	—
132	#376	—	2	2	no	—
133	C₂×#95	Dic₄₇	1	2	yes	—
134	C₂×#95	C₂×D₄₇	1	2	yes	—
135	C₄₇	#7906	1	1	yes	characteristic, complemented, Sylow
136	C₂³×C₄	C₂×Dic₄₇	1	4	yes	—
137	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
138	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
139	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
140	C₂²×C₈	—	94	4	no	—
141	C₂×C₄²	—	2	4	no	—
142	C₂×C₄²	—	2	2	no	—
143	C₂×C₄²	—	2	2	no	—
144	C₂×C₄²	C₄₇⋊C₈	1	8	yes	complemented
145	C₂×C₄²	Dic₉₄	1	2	yes	—
146	C₂×C₄²	C₄×D₄₇	1	4	yes	—
147	C₂×C₄²	D₁₈₈	1	2	yes	—
148	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
149	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
150	C₄⋊C₈	—	94	8	no	—
151	C₂²⋊C₈	—	47	8	no	—
152	C₂²⋊C₈	—	94	4	no	—
153	C₄×C₈	—	94	8	no	—
154	#95	—	2	4	no	—
155	#95	—	2	4	no	—
156	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
157	C₂²×C₄	—	2	4	no	—
158	C₂²×C₄	—	2	4	no	—
159	C₂²×C₄	—	4	2	no	—
160	C₂²×C₄	—	4	2	no	—
161	C₂²×C₄	—	4	4	no	—
162	C₂²×C₄	—	4	4	no	—
163	C₂²×C₄	C₂×C₄₇⋊C₈	1	8	yes	—
164	C₂²×C₄	C₂×C₄₇⋊C₈	1	4	yes	—
165	C₂²×C₄	#10625329	1	4	yes	—
166	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
167	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
168	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
169	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
170	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
171	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
172	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
173	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
174	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
175	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
176	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
177	C₂×C₈	—	94	8	no	—
178	C₂×C₈	—	188	8	no	—
179	C₄²	—	2	8	no	—
180	C₄²	—	2	8	no	—
181	C₄²	Dic₄₇⋊C₄	1	4	yes	—
182	C₄²	Dic₄₇⋊C₄	1	4	yes	—
183	C₄²	(C₂×D₄₇)⋊C₄	1	4	yes	—
184	C₄²	(C₂×D₄₇)⋊C₄	1	4	yes	—
185	C₂³	—	2	2	no	—
186	C₂³	—	2	2	no	—
187	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
188	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
189	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
190	C₂³	C₂²×C₄₇⋊C₈	1	2	yes	—
191	C₂³	C₂×#10625329	1	2	yes	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	8	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	1	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	1	no	—
200	C₂×C₄	—	4	8	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	—	4	8	no	—
204	C₂×C₄	C₄₇⋊#95	1	4	yes	—
205	C₂×C₄	C₄₇⋊#95	1	2	yes	—
206	C₂×C₄	C₄₇⋊#95	1	2	yes	—
207	C₂×C₄	C₄×C₄₇⋊C₈	1	4	yes	—
208	C₂×C₄	#13093935	1	4	yes	—
209	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	4	yes	—
210	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	4	yes	—
211	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	8	yes	—
212	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
213	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
214	C₈	—	188	8	no	—
215	C₂²	—	2	4	no	—
216	C₂²	—	2	4	no	—
217	C₂²	—	2	1	no	—
218	C₂²	—	2	1	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
222	C₂²	C₂×C₄×C₄₇⋊C₈	1	1	yes	characteristic
223	C₂²	C₂×#13093935	1	1	yes	characteristic
224	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
225	C₂²	#15689097	1	2	yes	—
226	C₂²	#15689107	1	2	yes	—
227	C₂²	C₂×C₄⋊(C₄₇⋊C₈)	1	2	yes	—
228	C₄	—	2	2	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	4	no	—
231	C₄	—	4	8	no	—
232	C₄	#15688878	1	4	yes	—
233	C₄	#15688878	1	4	yes	—
234	C₂	—	2	1	no	—
235	C₂	—	2	1	no	—
236	C₂	—	2	2	no	—
237	C₂	#20241702	1	4	yes	—
238	C₂	C₂×#15688878	1	1	yes	characteristic
239	C₂	#20243121	1	1	yes	characteristic
240	C₂	#20243127	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₄₇
commutator	C₂²×C₄₇	C₂×C₄×C₈
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#376	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#376	D₄₇
hypocenter	C₄₇	#7906
LCS	G ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#376

Complements

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

N	H ≅ G/N	type
5. #15689176	C₄	semidirect
27. C₂×C₄²×C₄₇	C₈	semidirect
135. C₄₇	#7906	semidirect
144. C₂×C₄²	C₄₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689176	2
#20241231	2
C₂×C₄₇×#376	2
#7906	47