Subgroups — #26105680

Order 12032.

706 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252021	C₂	1	2	yes	maximal
2	#20241217	C₂	1	4	yes	maximal, complemented
3	C₂×C₄₇×#376	C₂	1	1	yes	maximal, characteristic
4	#15689180	—	2	2	no	—
5	#15689180	—	2	2	no	—
6	#15689176	C₄	1	4	yes	complemented
7	#15689176	C₂²	1	4	yes	—
8	#15689147	—	2	2	no	—
9	#15689087	—	2	2	no	—
10	C₄×#10625329	C₄	1	4	yes	complemented
11	C₂²×C₄²×C₄₇	C₄	1	2	yes	—
12	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
13	C₄₇×#376	—	2	2	no	—
14	C₄₇×#376	C₄	1	2	yes	—
15	C₄₇×#376	C₂²	1	2	yes	—
16	C₂×#95×C₄₇	—	2	2	no	—
17	C₂×#10625329	—	2	4	no	—
18	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
19	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
20	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
21	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
22	#13093935	—	2	4	no	—
23	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	complemented
24	C₂³×C₄×C₄₇	—	2	2	no	—
25	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₄₇	D₄	1	2	yes	—
27	C₂×C₄²×C₄₇	—	2	2	no	—
28	C₂×C₄²×C₄₇	—	2	8	no	—
29	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₄₇	D₄	1	2	yes	—
34	C₂×C₄²×C₄₇	D₄	1	2	yes	—
35	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
36	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
37	#95×C₄₇	—	2	4	no	—
38	#95×C₄₇	—	2	4	no	—
39	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	—	2	4	no	—
41	C₂²×C₄×C₄₇	—	2	8	no	—
42	C₂²×C₄×C₄₇	—	2	4	no	—
43	C₂²×C₄×C₄₇	—	4	4	no	—
44	C₂²×C₄×C₄₇	—	4	8	no	—
45	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
46	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
55	C₄²×C₄₇	—	2	8	no	—
56	C₄²×C₄₇	—	2	4	no	—
57	C₄²×C₄₇	C₄²	1	1	yes	characteristic
58	C₄²×C₄₇	C₄²	1	1	yes	characteristic
59	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
60	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
61	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
62	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
63	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
64	C₄²×C₄₇	M₁₆	1	8	yes	complemented
65	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
66	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
68	#10625329	—	4	8	no	—
69	C₂×C₄₇⋊C₈	—	2	8	no	—
70	C₂³×C₄₇	—	2	4	no	—
71	C₂³×C₄₇	—	2	2	no	—
72	C₂³×C₄₇	#95	1	2	yes	—
73	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₄₇	—	2	4	no	—
75	C₂×C₄×C₄₇	—	2	4	no	—
76	C₂×C₄×C₄₇	—	2	2	no	—
77	C₂×C₄×C₄₇	—	2	4	no	—
78	C₂×C₄×C₄₇	—	2	4	no	—
79	C₂×C₄×C₄₇	—	2	8	no	—
80	C₂×C₄×C₄₇	—	4	2	no	—
81	C₂×C₄×C₄₇	—	4	2	no	—
82	C₂×C₄×C₄₇	—	4	2	no	—
83	C₂×C₄×C₄₇	—	4	2	no	—
84	C₂×C₄×C₄₇	—	4	16	no	—
85	C₂×C₄×C₄₇	—	4	8	no	—
86	C₂×C₄×C₄₇	#95	1	2	yes	—
87	C₂×C₄×C₄₇	#95	1	2	yes	—
88	C₂×C₄×C₄₇	#95	1	2	yes	—
89	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
90	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
91	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
92	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
94	C₄₇⋊C₈	—	4	8	no	—
95	#7855	—	47	1	no	maximal, Sylow
96	C₂²×C₄₇	—	2	8	no	—
97	C₂²×C₄₇	—	2	2	no	—
98	C₂²×C₄₇	—	2	2	no	—
99	C₂²×C₄₇	—	2	2	no	—
100	C₂²×C₄₇	H(3,ℤ/4)	1	4	yes	—
101	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
102	C₂²×C₄₇	#407	1	2	yes	—
103	C₄×C₄₇	—	2	2	no	—
104	C₄×C₄₇	—	2	2	no	—
105	C₄×C₄₇	—	2	2	no	—
106	C₄×C₄₇	—	2	2	no	—
107	C₄×C₄₇	—	2	2	no	—
108	C₄×C₄₇	—	4	8	no	—
109	C₄×C₄₇	C₄×M₁₆	1	2	yes	—
110	C₄×C₄₇	#423	1	2	yes	—
111	#2920	—	47	2	no	—
112	C₂×#376	D₄₇	1	1	yes	characteristic
113	#1274	—	47	4	no	—
114	C₂×C₄₇	—	2	2	no	—
115	C₂×C₄₇	—	2	4	no	—
116	C₂×C₄₇	#1728	1	1	yes	characteristic
117	C₂×C₄₇	C₂×H(3,ℤ/4)	1	2	yes	—
118	C₂²×C₄²	Dic₄₇	1	2	yes	—
119	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
120	#432	—	94	2	no	—
121	#432	—	94	2	no	—
122	#431	—	47	4	no	—
123	#431	—	47	4	no	—
124	#423	—	94	2	no	—
125	#407	—	94	2	no	—
126	C₄×M₁₆	—	47	4	no	—
127	#376	—	2	2	no	—
128	#376	Dic₄₇	1	2	yes	—
129	#376	C₂×D₄₇	1	2	yes	—
130	C₂×#95	—	2	2	no	—
131	C₄₇	#7855	1	1	yes	characteristic, complemented, Sylow
132	C₂³×C₄	—	2	2	no	—
133	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
134	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
135	C₂×M₁₆	—	94	4	no	—
136	C₂×C₄²	—	2	2	no	—
137	C₂×C₄²	—	2	8	no	—
138	C₂×C₄²	Dic₉₄	1	2	yes	—
139	C₂×C₄²	C₄×D₄₇	1	2	yes	—
140	C₂×C₄²	C₄×D₄₇	1	2	yes	—
141	C₂×C₄²	D₁₈₈	1	2	yes	—
142	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
143	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
144	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
145	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
146	C₄⋊C₈	—	94	4	no	—
147	C₄⋊C₈	—	94	4	no	—
148	C₂²⋊C₈	—	94	4	no	—
149	C₂²⋊C₈	—	94	4	no	—
150	#97	—	94	4	no	—
151	C₄×C₈	—	47	8	no	—
152	#95	—	2	4	no	—
153	#95	—	2	4	no	—
154	C₂⁴	C₂²⋊Dic₄₇	1	1	yes	characteristic
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	—	2	8	no	—
157	C₂²×C₄	—	2	4	no	—
158	C₂²×C₄	—	4	4	no	—
159	C₂²×C₄	—	4	8	no	—
160	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
161	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
162	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
163	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
164	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
165	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
166	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
167	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
168	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
169	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
170	M₁₆	—	188	8	no	—
171	C₂×C₈	—	94	8	no	—
172	C₄²	—	2	8	no	—
173	C₄²	—	2	4	no	—
174	C₄²	#10625329	1	8	yes	complemented
175	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
176	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
177	C₄²	Dic₄₇⋊C₄	1	2	yes	—
178	C₄²	Dic₄₇⋊C₄	1	2	yes	—
179	C₄²	C₄⋊Dic₄₇	1	2	yes	—
180	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
181	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
182	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
183	C₄²	C₂×C₄×D₄₇	1	2	yes	—
184	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
185	C₂³	—	2	2	no	—
186	C₂³	—	2	4	no	—
187	C₂³	C₄₇⋊#95	1	2	yes	—
188	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	4	no	—
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	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	16	no	—
200	C₂×C₄	—	4	8	no	—
201	C₂×C₄	C₄₇⋊#95	1	2	yes	—
202	C₂×C₄	C₄₇⋊#95	1	2	yes	—
203	C₂×C₄	C₄₇⋊#95	1	2	yes	—
204	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
205	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
206	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
207	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
208	C₂×C₄	C₂×#10625329	1	4	yes	—
209	C₈	—	188	8	no	—
210	C₂²	—	2	8	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	#15688879	1	4	yes	—
215	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
216	C₂²	#15689087	1	2	yes	—
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	2	2	no	—
220	C₄	—	2	2	no	—
221	C₄	—	2	2	no	—
222	C₄	—	4	8	no	—
223	C₄	C₄×#10625329	1	2	yes	—
224	C₄	#15689147	1	2	yes	—
225	C₂	—	2	2	no	—
226	C₂	—	2	4	no	—
227	C₂	#20243082	1	1	yes	characteristic
228	C₂	C₂×#15688879	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#376	C₂
socle	C₂²×C₄₇	C₂×#95
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#376	D₄₇
hypocenter	C₄₇	#7855
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#376

Complements

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

N	H ≅ G/N	type
2. #20241217	C₂	semidirect
6. #15689176	C₄	semidirect
10. C₄×#10625329	C₄	semidirect
23. C₄×C₄₇⋊C₈	C₂×C₄	semidirect
64. C₄²×C₄₇	M₁₆	semidirect
131. C₄₇	#7855	semidirect
174. C₄²	#10625329	semidirect

Maximal subgroups

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

subgroup	index
#20252021	2
#20241217	2
C₂×C₄₇×#376	2
#7855	47