Subgroups — #26216428

Order 12032.

1158 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	C₂×#15689147	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄²×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#20243504	C₂	1	8	yes	maximal, complemented
4	#20243081	C₂	1	4	yes	maximal
5	C₂×C₄₇×#423	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂²×C₄×Dic₄₇	C₄	1	4	yes	—
8	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
9	#15689147	—	2	4	no	—
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
11	C₄²×Dic₄₇	C₄	1	2	yes	—
12	C₄²×Dic₄₇	C₄	1	2	yes	—
13	C₄²×Dic₄₇	C₂²	1	4	yes	—
14	#15688877	C₂²	1	16	yes	—
15	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
16	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#423	—	2	4	no	—
18	C₂×C₄⋊C₈×C₄₇	C₂²	1	4	yes	—
19	C₂³×Dic₄₇	C₂×C₄	1	4	yes	—
20	C₂×#10625329	—	2	4	no	—
21	C₂×#10625329	—	2	8	no	—
22	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
23	C₂×C₄×Dic₄₇	—	2	4	no	—
24	C₂×C₄×Dic₄₇	—	2	4	no	—
25	C₂×C₄×Dic₄₇	—	2	16	no	—
26	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
27	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
28	C₂×C₄×Dic₄₇	C₂×C₄	1	8	yes	—
29	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
30	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
31	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
32	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
33	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
34	C₂×M₁₆×C₄₇	—	2	4	no	—
35	C₂×M₁₆×C₄₇	—	2	8	no	—
36	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
37	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₄₇	D₄	1	4	yes	—
40	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
41	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
42	C₄⋊C₈×C₄₇	—	2	8	no	—
43	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	—	2	8	no	—
45	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
46	C₂²×C₄×C₄₇	C₂²×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×Q₈	1	2	yes	—
51	C₂²×C₄×C₄₇	#41	1	4	yes	—
52	C₂²×C₄×C₄₇	#41	1	4	yes	—
53	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
54	C₂²×Dic₄₇	—	2	8	no	—
55	C₂²×Dic₄₇	—	2	16	no	—
56	C₂²×Dic₄₇	C₂²×C₄	1	8	yes	—
57	M₁₆×C₄₇	—	4	8	no	—
58	C₂×C₈×C₄₇	—	2	4	no	—
59	C₂×C₈×C₄₇	—	2	8	no	—
60	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
61	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
62	C₄²×C₄₇	C₂×D₄	1	4	yes	—
63	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
64	C₄×Dic₄₇	—	2	8	no	—
65	C₄×Dic₄₇	—	2	8	no	—
66	C₄×Dic₄₇	—	2	4	no	—
67	C₄×Dic₄₇	—	2	4	no	—
68	C₄×Dic₄₇	—	2	16	no	—
69	C₄×Dic₄₇	M₁₆	1	16	yes	complemented
70	#10625329	—	4	8	no	—
71	C₂×C₄₇⋊C₈	—	2	4	no	—
72	C₂×C₄₇⋊C₈	—	2	8	no	—
73	C₂³×C₄₇	—	2	4	no	—
74	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
75	C₂³×C₄₇	#117	1	4	yes	—
76	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
77	C₂×C₄×C₄₇	—	2	4	no	—
78	C₂×C₄×C₄₇	—	2	16	no	—
79	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
80	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
81	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
82	C₂×C₄×C₄₇	#117	1	4	yes	—
83	C₂×C₄×C₄₇	#124	1	4	yes	—
84	C₂×C₄×C₄₇	#124	1	4	yes	—
85	C₂×C₄×C₄₇	#125	1	8	yes	—
86	C₂×C₄×C₄₇	C₂×M₁₆	1	8	yes	—
87	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
88	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
89	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
90	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
91	C₂×Dic₄₇	—	2	32	no	—
92	C₂×Dic₄₇	—	2	8	no	—
93	C₂×Dic₄₇	—	2	8	no	—
94	C₂×Dic₄₇	C₂×M₁₆	1	16	yes	—
95	C₈×C₄₇	—	4	4	no	—
96	C₄₇⋊C₈	—	4	4	no	—
97	#20752	—	47	1	no	maximal, Sylow
98	C₂²×C₄₇	—	2	4	no	—
99	C₂²×C₄₇	—	2	8	no	—
100	C₂²×C₄₇	#382	1	4	yes	—
101	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
102	C₂²×C₄₇	C₂×#117	1	2	yes	—
103	C₂²×C₄₇	C₂²×M₁₆	1	4	yes	—
104	C₄×C₄₇	—	2	4	no	—
105	C₄×C₄₇	#382	1	4	yes	—
106	C₄×C₄₇	#423	1	8	yes	—
107	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
108	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
109	C₄×C₄₇	C₂×#125	1	2	yes	—
110	Dic₄₇	—	2	16	no	—
111	C₂×#423	—	47	1	no	—
112	C₂×#423	D₄₇	1	1	yes	characteristic
113	C₂×C₄³	—	47	1	no	—
114	#1831	—	47	8	no	—
115	#1728	—	47	4	no	—
116	C₂×C₄₇	—	2	4	no	—
117	C₂×C₄₇	C₂×#382	1	1	yes	characteristic
118	C₂×C₄₇	C₂×#423	1	2	yes	—
119	C₂×C₄₇	#2920	1	4	yes	—
120	C₂²×M₁₆	—	47	2	no	—
121	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
122	C₂²×C₄²	—	47	4	no	—
123	C₂²×C₄²	—	47	2	no	—
124	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
125	#423	—	2	4	no	—
126	#423	—	94	4	no	—
127	C₂×C₄⋊C₈	—	47	4	no	—
128	C₂×C₄⋊C₈	C₂×D₄₇	1	4	yes	—
129	C₄³	—	47	2	no	—
130	C₄³	—	47	2	no	—
131	C₄³	—	47	4	no	—
132	#336	—	47	16	no	—
133	C₄₇	#20752	1	1	yes	characteristic, complemented, Sylow
134	C₂³×C₄	—	47	4	no	—
135	C₂³×C₄	C₄×D₄₇	1	2	yes	—
136	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
137	C₂×M₁₆	—	2	8	no	—
138	C₂×M₁₆	—	2	4	no	—
139	C₂×M₁₆	—	94	8	no	—
140	C₂×M₁₆	—	94	4	no	—
141	C₂²×C₈	—	47	4	no	—
142	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
143	C₂×C₄²	—	47	4	no	—
144	C₂×C₄²	—	47	8	no	—
145	C₂×C₄²	—	47	4	no	—
146	C₂×C₄²	—	47	4	no	—
147	C₂×C₄²	—	47	4	no	—
148	C₂×C₄²	—	94	16	no	—
149	C₂×C₄²	—	94	4	no	—
150	C₂×C₄²	—	94	4	no	—
151	C₂×C₄²	Dic₉₄	1	4	yes	—
152	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
153	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
154	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
155	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
156	C₄⋊C₈	—	2	8	no	—
157	C₄⋊C₈	—	94	8	no	—
158	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
159	C₂²×C₄	—	2	8	no	—
160	C₂²×C₄	—	47	8	no	—
161	C₂²×C₄	—	94	16	no	—
162	C₂²×C₄	—	94	8	no	—
163	C₂²×C₄	Dic₄₇⋊C₄	1	8	yes	—
164	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
165	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
166	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
168	C₂²×C₄	#10625352	1	4	yes	—
169	C₂²×C₄	#10625354	1	4	yes	—
170	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
171	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
172	M₁₆	—	4	8	no	—
173	M₁₆	—	188	8	no	—
174	C₂×C₈	—	2	4	no	—
175	C₂×C₈	—	2	8	no	—
176	C₂×C₈	—	94	4	no	—
177	C₂×C₈	—	94	8	no	—
178	C₄²	—	47	16	no	—
179	C₄²	—	94	16	no	—
180	C₄²	—	94	8	no	—
181	C₄²	—	94	8	no	—
182	C₄²	—	94	4	no	—
183	C₄²	—	94	4	no	—
184	C₄²	Dic₄₇⋊C₄	1	4	yes	—
185	C₄²	Dic₄₇⋊C₄	1	4	yes	—
186	C₄²	C₂×Dic₉₄	1	4	yes	—
187	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
188	C₂³	—	2	4	no	—
189	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
190	C₂³	#13093982	1	4	yes	—
191	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	16	no	—
194	C₂×C₄	—	47	16	no	—
195	C₂×C₄	—	94	32	no	—
196	C₂×C₄	—	94	8	no	—
197	C₂×C₄	—	94	8	no	—
198	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
199	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
200	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
201	C₂×C₄	#13093982	1	4	yes	—
202	C₂×C₄	#13094011	1	4	yes	—
203	C₂×C₄	#13094013	1	4	yes	—
204	C₂×C₄	#13094015	1	8	yes	—
205	C₂×C₄	M₁₆×D₄₇	1	8	yes	—
206	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
207	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
208	C₂×C₄	C₂×#10625352	1	2	yes	—
209	C₂×C₄	C₂×#10625354	1	2	yes	—
210	C₈	—	4	4	no	—
211	C₈	—	188	4	no	—
212	C₂²	—	2	4	no	—
213	C₂²	—	2	8	no	—
214	C₂²	#15688988	1	4	yes	—
215	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
216	C₂²	C₂×#13093982	1	2	yes	—
217	C₂²	C₂×M₁₆×D₄₇	1	4	yes	—
218	C₄	—	2	4	no	—
219	C₄	—	94	16	no	—
220	C₄	#15688988	1	4	yes	—
221	C₄	#15689146	1	8	yes	—
222	C₄	C₂×#13094011	1	1	yes	characteristic
223	C₄	C₂×#13094013	1	1	yes	characteristic
224	C₄	C₂×#13094015	1	2	yes	—
225	C₂	—	2	4	no	—
226	C₂	C₂×#15688988	1	1	yes	characteristic
227	C₂	C₂×#15689146	1	2	yes	—
228	C₂	#20252020	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₂×C₄₇×#423	C₂
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#423	D₄₇
hypocenter	C₄₇	#20752
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
3. #20243504	C₂	semidirect
69. C₄×Dic₄₇	M₁₆	semidirect
133. C₄₇	#20752	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689147	2
C₂×C₄²×Dic₄₇	2
#20243504	2
#20243081	2
C₂×C₄₇×#423	2
#20752	47