Subgroups — #26106400

Order 12032.

788 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	C₄×#13093983	C₂	1	2	yes	maximal
2	#20241273	C₂	1	4	yes	maximal
3	C₄₇×#2920	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093983	—	2	2	no	—
5	C₄×C₄⋊Dic₄₇	—	2	2	no	—
6	C₄×C₄⋊Dic₄₇	C₄	1	4	yes	complemented
7	C₄×C₂²⋊Dic₄₇	—	2	2	no	—
8	C₄×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
9	#15688969	C₄	1	4	yes	complemented
10	C₄²×Dic₄₇	—	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	complemented
15	C₄₇×#407	C₂²	1	2	yes	—
16	C₄×M₁₆×C₄₇	C₄	1	2	yes	complemented
17	#13093983	—	2	16	no	—
18	C₂×C₄⋊Dic₄₇	—	2	4	no	—
19	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
20	C₂×C₄×Dic₄₇	—	2	4	no	—
21	C₂×C₄×Dic₄₇	—	2	4	no	—
22	C₂×C₄×Dic₄₇	—	2	4	no	—
23	C₄₇⋊#95	—	2	4	no	—
24	C₄₇⋊#95	C₂×C₄	1	8	yes	—
25	C₂³×C₄×C₄₇	D₄	1	2	yes	—
26	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
27	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₄₇	D₄	1	2	yes	—
33	C₂×C₄²×C₄₇	D₄	1	2	yes	—
34	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
35	C₄⋊C₈×C₄₇	—	2	4	no	—
36	C₂²⋊C₈×C₄₇	—	2	4	no	—
37	#97×C₄₇	—	2	2	no	—
38	C₄×C₈×C₄₇	—	2	2	no	—
39	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	—	2	8	no	—
41	C₂²×C₄×C₄₇	—	4	2	no	—
42	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
50	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
51	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
52	C₂²×Dic₄₇	—	2	8	no	—
53	M₁₆×C₄₇	—	4	4	no	—
54	C₂×C₈×C₄₇	—	2	4	no	—
55	C₄²×C₄₇	C₄²	1	1	yes	characteristic
56	C₄²×C₄₇	C₄²	1	2	yes	—
57	C₄²×C₄₇	C₄²	1	1	yes	characteristic
58	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
59	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
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	C₄⋊C₄	1	2	yes	—
65	C₂²⋊Dic₄₇	—	2	16	no	—
66	C₄⋊Dic₄₇	—	2	16	no	—
67	C₄×Dic₄₇	—	2	16	no	—
68	C₄×Dic₄₇	—	4	8	no	—
69	C₄×Dic₄₇	—	4	8	no	—
70	C₂³×C₄₇	—	2	4	no	—
71	C₂³×C₄₇	—	4	1	no	—
72	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
73	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₄₇	—	2	4	no	—
75	C₂×C₄×C₄₇	—	2	4	no	—
76	C₂×C₄×C₄₇	—	2	4	no	—
77	C₂×C₄×C₄₇	—	2	4	no	—
78	C₂×C₄×C₄₇	—	2	4	no	—
79	C₂×C₄×C₄₇	—	2	4	no	—
80	C₂×C₄×C₄₇	—	2	2	no	—
81	C₂×C₄×C₄₇	—	4	2	no	—
82	C₂×C₄×C₄₇	—	4	1	no	—
83	C₂×C₄×C₄₇	#95	1	2	yes	—
84	C₂×C₄×C₄₇	#95	1	2	yes	—
85	C₂×C₄×C₄₇	#95	1	2	yes	—
86	C₂×C₄×C₄₇	#95	1	2	yes	—
87	C₂×C₄×C₄₇	#104	1	8	yes	—
88	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
89	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
91	C₂×Dic₄₇	—	2	16	no	—
92	C₂×Dic₄₇	—	4	8	no	—
93	C₂×Dic₄₇	—	4	8	no	—
94	C₈×C₄₇	—	4	4	no	—
95	#8048	—	47	1	no	maximal, Sylow
96	C₂²×C₄₇	—	2	2	no	—
97	C₂²×C₄₇	—	2	2	no	—
98	C₂²×C₄₇	—	2	2	no	—
99	C₂²×C₄₇	—	2	1	no	—
100	C₂²×C₄₇	—	4	4	no	—
101	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
102	C₂²×C₄₇	C₂×#104	1	4	yes	—
103	C₄×C₄₇	—	2	2	no	—
104	C₄×C₄₇	—	2	2	no	—
105	C₄×C₄₇	—	2	1	no	—
106	C₄×C₄₇	—	2	2	no	—
107	C₄×C₄₇	#339	1	4	yes	—
108	C₄×C₄₇	#339	1	4	yes	—
109	C₄×C₄₇	#410	1	2	yes	—
110	Dic₄₇	—	4	16	no	—
111	#2920	D₄₇	1	1	yes	characteristic
112	C₄×#117	—	47	2	no	—
113	#1297	—	47	4	no	—
114	C₂×C₄₇	—	2	4	no	—
115	C₂×C₄₇	—	4	1	no	—
116	C₂×C₄₇	C₂×#339	1	2	yes	—
117	C₂×C₄₇	#1740	1	1	yes	characteristic
118	C₂×#117	—	94	2	no	—
119	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
120	#432	—	2	2	no	—
121	#431	—	2	2	no	—
122	#423	Dic₄₇	1	2	yes	complemented
123	#407	C₂×D₄₇	1	2	yes	—
124	C₄×M₁₆	Dic₄₇	1	2	yes	complemented
125	C₄×C₄⋊C₄	—	47	4	no	—
126	C₄×C₄⋊C₄	—	94	2	no	—
127	C₄×C₂²⋊C₄	—	47	4	no	—
128	C₄×C₂²⋊C₄	—	94	2	no	—
129	#376	—	47	4	no	—
130	C₄³	—	94	2	no	—
131	C₄₇	#8048	1	1	yes	characteristic, complemented, Sylow
132	C₂³×C₄	D₁₈₈	1	2	yes	—
133	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
134	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
135	#117	—	94	16	no	—
136	C₂×C₄⋊C₄	—	94	4	no	—
137	C₂×C₂²⋊C₄	—	94	4	no	—
138	C₂×C₄²	—	94	4	no	—
139	C₂×C₄²	—	94	4	no	—
140	C₂×C₄²	—	94	4	no	—
141	C₂×C₄²	Dic₉₄	1	2	yes	—
142	C₂×C₄²	C₄×D₄₇	1	2	yes	—
143	C₂×C₄²	C₄×D₄₇	1	2	yes	—
144	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
145	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
146	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
147	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
148	C₄⋊C₈	—	2	4	no	—
149	C₂²⋊C₈	—	2	4	no	—
150	#97	—	2	2	no	—
151	C₄×C₈	—	2	2	no	—
152	#95	—	47	8	no	—
153	#95	—	94	4	no	—
154	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
155	C₂²×C₄	—	2	8	no	—
156	C₂²×C₄	—	4	2	no	—
157	C₂²×C₄	—	94	8	no	—
158	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
159	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
160	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
161	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
162	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
163	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
164	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
165	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
166	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
168	M₁₆	—	4	4	no	—
169	C₂×C₈	—	2	4	no	—
170	C₄⋊C₄	—	94	16	no	—
171	C₂²⋊C₄	—	94	16	no	—
172	C₄²	—	94	16	no	—
173	C₄²	—	188	8	no	—
174	C₄²	—	188	8	no	—
175	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
176	C₄²	C₄×Dic₄₇	1	2	yes	—
177	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
178	C₄²	Dic₄₇⋊C₄	1	2	yes	—
179	C₄²	Dic₄₇⋊C₄	1	2	yes	—
180	C₄²	Dic₄₇⋊C₄	1	2	yes	—
181	C₄²	Dic₄₇⋊C₄	1	2	yes	—
182	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
183	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
184	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
185	C₂³	—	2	4	no	—
186	C₂³	—	4	1	no	—
187	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
188	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	94	16	no	—
199	C₂×C₄	—	188	8	no	—
200	C₂×C₄	—	188	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	C₄₇⋊#95	1	2	yes	—
205	C₂×C₄	#13093950	1	8	yes	—
206	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
207	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
208	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
209	C₈	—	4	4	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	2	1	no	—
214	C₂²	—	4	4	no	—
215	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
216	C₂²	C₂×#13093950	1	4	yes	—
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	2	no	—
221	C₄	—	188	16	no	—
222	C₄	#15688884	1	4	yes	—
223	C₄	#15688884	1	4	yes	—
224	C₄	#15689097	1	2	yes	—
225	C₂	—	2	4	no	—
226	C₂	—	4	1	no	—
227	C₂	C₂×#15688884	1	2	yes	—
228	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×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#2920	D₄₇
hypocenter	C₄₇	#8048
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
6. C₄×C₄⋊Dic₄₇	C₄	semidirect
9. #15688969	C₄	semidirect
14. C₄₇×#423	C₄	semidirect
16. C₄×M₁₆×C₄₇	C₄	semidirect
122. #423	Dic₄₇	semidirect
124. C₄×M₁₆	Dic₄₇	semidirect
131. C₄₇	#8048	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13093983	2
#20241273	2
C₄₇×#2920	2
#8048	47