Subgroups — #26106420

Order 12032.

564 conjugacy classes of subgroups.

The subgroup lattice has 214 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	#20241275	C₂	1	4	yes	maximal
3	C₄₇×#2920	C₂	1	1	yes	maximal, characteristic
4	#15689176	—	2	2	no	—
5	#15689176	—	2	2	no	—
6	#15689176	C₄	1	4	yes	—
7	#15689176	C₄	1	4	yes	—
8	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
9	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
10	C₂×C₄×C₄₇⋊C₈	—	2	2	no	—
11	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
12	C₄₇×#432	—	2	2	no	—
13	C₄₇×#431	—	2	2	no	—
14	C₄₇×#423	C₄	1	2	yes	—
15	C₄₇×#407	C₂²	1	2	yes	—
16	C₄×M₁₆×C₄₇	C₄	1	2	yes	—
17	C₂²×C₄₇⋊C₈	—	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)	C₂×C₄	1	8	yes	—
22	C₄×C₄₇⋊C₈	—	2	4	no	—
23	C₄×C₄₇⋊C₈	—	2	4	no	—
24	C₂³×C₄×C₄₇	Q₈	1	2	yes	—
25	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
26	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₄₇	D₄	1	2	yes	—
32	C₂×C₄²×C₄₇	D₄	1	2	yes	—
33	C₂×C₄²×C₄₇	D₄	1	2	yes	—
34	C₄⋊C₈×C₄₇	—	2	4	no	—
35	C₂²⋊C₈×C₄₇	—	2	4	no	—
36	#97×C₄₇	—	2	2	no	—
37	C₄×C₈×C₄₇	—	2	2	no	—
38	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₄×C₄₇	—	2	8	no	—
40	C₂²×C₄×C₄₇	—	4	2	no	—
41	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
42	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
44	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
46	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
51	M₁₆×C₄₇	—	4	4	no	—
52	C₂×C₈×C₄₇	—	2	4	no	—
53	C₄²×C₄₇	C₄²	1	1	yes	characteristic
54	C₄²×C₄₇	C₄²	1	2	yes	—
55	C₄²×C₄₇	C₄²	1	1	yes	characteristic
56	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
57	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
58	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
59	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
60	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
61	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄₇⋊C₈	—	2	8	no	—
64	C₂×C₄₇⋊C₈	—	4	8	no	—
65	C₂³×C₄₇	—	2	4	no	—
66	C₂³×C₄₇	—	4	1	no	—
67	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
68	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₄₇	—	2	4	no	—
70	C₂×C₄×C₄₇	—	2	4	no	—
71	C₂×C₄×C₄₇	—	2	4	no	—
72	C₂×C₄×C₄₇	—	2	4	no	—
73	C₂×C₄×C₄₇	—	2	4	no	—
74	C₂×C₄×C₄₇	—	2	4	no	—
75	C₂×C₄×C₄₇	—	2	2	no	—
76	C₂×C₄×C₄₇	—	4	2	no	—
77	C₂×C₄×C₄₇	—	4	1	no	—
78	C₂×C₄×C₄₇	#95	1	2	yes	—
79	C₂×C₄×C₄₇	#95	1	2	yes	—
80	C₂×C₄×C₄₇	#95	1	2	yes	—
81	C₂×C₄×C₄₇	#95	1	2	yes	—
82	C₂×C₄×C₄₇	#108	1	8	yes	—
83	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
84	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
85	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₈×C₄₇	—	4	4	no	—
87	C₄₇⋊C₈	—	4	8	no	—
88	#8054	—	47	1	no	maximal, Sylow
89	C₂²×C₄₇	—	2	2	no	—
90	C₂²×C₄₇	—	2	2	no	—
91	C₂²×C₄₇	—	2	2	no	—
92	C₂²×C₄₇	—	2	1	no	—
93	C₂²×C₄₇	—	4	4	no	—
94	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
95	C₂²×C₄₇	C₂×#108	1	4	yes	—
96	C₄×C₄₇	—	2	2	no	—
97	C₄×C₄₇	—	2	2	no	—
98	C₄×C₄₇	—	2	2	no	—
99	C₄×C₄₇	—	2	1	no	—
100	C₄×C₄₇	#341	1	4	yes	—
101	C₄×C₄₇	#341	1	4	yes	—
102	C₄×C₄₇	#410	1	2	yes	—
103	#2920	D₄₇	1	1	yes	characteristic
104	C₂×#431	—	47	2	no	—
105	#1298	—	47	4	no	—
106	C₂×C₄₇	—	2	4	no	—
107	C₂×C₄₇	—	4	1	no	—
108	C₂×C₄₇	C₂×#341	1	2	yes	—
109	C₂×C₄₇	#1740	1	1	yes	characteristic
110	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
111	#432	—	2	2	no	—
112	#431	—	2	2	no	—
113	#431	—	47	4	no	—
114	#431	—	47	4	no	—
115	#431	—	94	2	no	—
116	#431	—	94	2	no	—
117	#423	Dic₄₇	1	2	yes	—
118	C₂×C₄⋊C₈	—	94	2	no	—
119	#407	C₂×D₄₇	1	2	yes	—
120	C₂×C₂²⋊C₈	—	47	4	no	—
121	C₄×M₁₆	Dic₄₇	1	2	yes	—
122	C₂×C₄×C₈	—	94	2	no	—
123	C₄₇	#8054	1	1	yes	characteristic, complemented, Sylow
124	C₂³×C₄	Dic₉₄	1	2	yes	—
125	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
126	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
127	C₂²×C₈	—	94	4	no	—
128	C₂×C₄²	C₄×D₄₇	1	2	yes	—
129	C₂×C₄²	C₄×D₄₇	1	2	yes	—
130	C₂×C₄²	D₁₈₈	1	2	yes	—
131	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
132	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
133	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
134	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
135	C₄⋊C₈	—	2	4	no	—
136	C₄⋊C₈	—	94	4	no	—
137	C₄⋊C₈	—	94	4	no	—
138	C₂²⋊C₈	—	2	4	no	—
139	C₂²⋊C₈	—	47	8	no	—
140	C₂²⋊C₈	—	94	4	no	—
141	#97	—	2	2	no	—
142	C₄×C₈	—	2	2	no	—
143	C₄×C₈	—	94	4	no	—
144	C₄×C₈	—	94	4	no	—
145	C₂⁴	C₂×Dic₉₄	1	1	yes	characteristic
146	C₂²×C₄	—	2	8	no	—
147	C₂²×C₄	—	4	2	no	—
148	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
149	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
150	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
151	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
152	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
153	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
154	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
155	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
156	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
157	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
158	M₁₆	—	4	4	no	—
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	—	94	8	no	—
161	C₂×C₈	—	188	8	no	—
162	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
163	C₄²	C₄×Dic₄₇	1	2	yes	—
164	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
165	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
166	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
167	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
168	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
169	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
170	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
171	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
172	C₂³	—	2	4	no	—
173	C₂³	—	4	1	no	—
174	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
175	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	4	1	no	—
185	C₂×C₄	C₄₇⋊#95	1	2	yes	—
186	C₂×C₄	C₄₇⋊#95	1	2	yes	—
187	C₂×C₄	C₄₇⋊#95	1	2	yes	—
188	C₂×C₄	C₄₇⋊#95	1	2	yes	—
189	C₂×C₄	#13093959	1	8	yes	—
190	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
191	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
192	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
193	C₈	—	4	4	no	—
194	C₈	—	188	8	no	—
195	C₂²	—	2	2	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	2	2	no	—
198	C₂²	—	2	1	no	—
199	C₂²	—	4	4	no	—
200	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
201	C₂²	C₂×#13093959	1	4	yes	—
202	C₄	—	2	2	no	—
203	C₄	—	2	2	no	—
204	C₄	—	2	2	no	—
205	C₄	—	2	1	no	—
206	C₄	#15688887	1	4	yes	—
207	C₄	#15688887	1	4	yes	—
208	C₄	#15689097	1	2	yes	—
209	C₂	—	2	4	no	—
210	C₂	—	4	1	no	—
211	C₂	C₂×#15688887	1	2	yes	—
212	C₂	#20243121	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#2920	C₂
socle	C₂²×C₄₇	C₂×#95
prime radical	C₂⁴×C₄₇	C₂×Q₈
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#2920	D₄₇
hypocenter	C₄₇	#8054
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ #2920

Complements

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

N	H ≅ G/N	type
123. C₄₇	#8054	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
#20241275	2
C₄₇×#2920	2
#8054	47