Subgroups — #26369472

Order 12032.

674 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689541	C₂	1	2	yes	maximal, complemented
2	#20252101	C₂	1	1	yes	maximal, characteristic
3	#20245212	C₂	1	8	yes	maximal, complemented
4	#20245003	C₂	1	2	yes	maximal, complemented
5	C₂×#15688899	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₇×#2150	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#13094030	C₂²	1	2	yes	—
8	#15689673	C₄	1	1	yes	characteristic
9	#15689541	—	2	8	no	—
10	C₂×#13093963	C₂²	1	4	yes	—
11	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×D₄₇	C₂²	1	4	yes	—
14	#15689202	C₄	1	4	yes	—
15	C₈×D₁₈₈	C₂²	1	4	yes	—
16	#15689147	C₄	1	1	yes	characteristic
17	#15689089	C₄	1	2	yes	—
18	(C₂×D₄₇)⋊C₁₆	C₂²	1	8	yes	—
19	#15688899	C₂²	1	4	yes	—
20	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₄₇×C₄⋊C₁₆	C₂²	1	4	yes	—
23	C₂×#10625350	C₂×C₄	1	2	yes	—
24	#13094030	—	2	4	no	—
25	#13094030	—	2	4	no	—
26	C₂×#10625329	C₂×C₄	1	2	yes	—
27	C₂×#10625324	C₂×C₄	1	4	yes	—
28	C₂×C₈×D₄₇	C₂³	1	4	yes	—
29	#13094024	C₈	1	1	yes	characteristic
30	#13094021	C₈	1	1	yes	characteristic
31	#13094012	C₈	1	2	yes	—
32	#13093998	C₈	1	4	yes	—
33	C₄×D₁₈₈	C₂×C₄	1	4	yes	—
34	#13093963	—	2	8	no	—
35	C₂×C₄₇⋊C₁₆	—	2	2	no	—
36	C₂×C₄₇⋊C₁₆	D₄	1	4	yes	complemented
37	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
38	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
39	C₁₆×D₄₇	—	2	8	no	—
40	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
41	(C₂×D₄₇)⋊C₈	C₂×C₄	1	8	yes	—
42	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
45	M₃₂×C₄₇	—	2	4	no	—
46	C₂×C₁₆×C₄₇	C₂³	1	4	yes	—
47	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₄₇	C₂³	1	2	yes	—
49	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₂²⋊D₄₇	C₂×C₈	1	4	yes	—
52	#10625350	—	2	4	no	—
53	#10625350	—	2	4	no	—
54	C₂×D₁₈₈	C₂×C₈	1	2	yes	—
55	C₂×D₁₈₈	C₂×C₈	1	4	yes	—
56	C₂×C₄×D₄₇	C₂²×C₄	1	4	yes	—
57	C₂×Dic₉₄	C₂×C₈	1	2	yes	—
58	C₂×C₈×C₄₇	—	2	2	no	—
59	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
60	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₄₇	#41	1	2	yes	—
63	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
64	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
65	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
66	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
67	(C₂×D₄₇)⋊C₄	C₂×C₈	1	8	yes	—
68	C₄⋊Dic₄₇	C₂×C₈	1	4	yes	—
69	#10625329	—	2	4	no	—
70	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
71	#10625324	—	2	8	no	—
72	C₈×D₄₇	—	2	4	no	—
73	C₈×D₄₇	—	2	4	no	—
74	C₁₆×C₄₇	—	2	4	no	—
75	C₄₇⋊C₁₆	—	2	2	no	—
76	C₄₇⋊C₁₆	C₂×D₄	1	4	yes	—
77	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
78	C₂²×D₄₇	C₂²×C₈	1	4	yes	—
79	C₂×C₄×C₄₇	—	4	2	no	—
80	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
82	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
83	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₄₇	#131	1	2	yes	—
85	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
86	C₂²⋊D₄₇	—	2	8	no	—
87	C₂×Dic₄₇	C₂²×C₈	1	4	yes	—
88	D₁₈₈	—	2	4	no	—
89	D₁₈₈	—	4	4	no	—
90	C₄×D₄₇	—	2	4	no	—
91	C₄×D₄₇	—	2	4	no	—
92	Dic₉₄	—	2	4	no	—
93	C₈×C₄₇	—	2	2	no	—
94	C₈×C₄₇	C₄×D₄	1	2	yes	—
95	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
96	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
97	C₄₇⋊C₈	—	2	4	no	—
98	#33435	—	47	1	no	maximal, Sylow
99	C₂²×C₄₇	—	2	2	no	—
100	C₂²×C₄₇	C₈×D₄	1	2	yes	—
101	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
102	C₂×D₄₇	—	2	8	no	—
103	C₂×D₄₇	—	2	4	no	—
104	C₄×C₄₇	—	2	2	no	—
105	C₄×C₄₇	—	2	2	no	—
106	C₄×C₄₇	C₈×D₄	1	2	yes	—
107	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
108	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
109	Dic₄₇	—	2	4	no	—
110	C₂×#504	—	47	2	no	—
111	#2931	—	47	1	no	—
112	#2168	—	47	8	no	—
113	#2150	D₄₇	1	1	yes	characteristic, complemented
114	#2113	—	47	2	no	—
115	C₂×#346	—	47	1	no	—
116	C₂×C₄₇	—	2	2	no	—
117	C₂×C₄₇	C₂×C₈×D₄	1	1	yes	characteristic
118	C₂×C₄₇	#3407	1	2	yes	—
119	D₄₇	—	4	4	no	—
120	C₂×#131	—	47	2	no	—
121	#532	—	47	1	no	—
122	#504	—	94	8	no	—
123	C₂×M₃₂	—	47	4	no	—
124	C₂×M₃₂	C₂×D₄₇	1	2	yes	—
125	C₂²×C₁₆	—	47	1	no	—
126	C₂²×C₁₆	—	47	1	no	—
127	C₂²×C₁₆	—	47	4	no	—
128	#436	—	47	4	no	—
129	C₈×D₄	—	47	4	no	—
130	#423	—	47	1	no	—
131	#408	—	47	2	no	—
132	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
133	C₄⋊C₁₆	C₂×D₄₇	1	4	yes	—
134	C₂²⋊C₁₆	—	47	8	no	—
135	#346	—	47	4	no	—
136	C₄₇	#33435	1	1	yes	characteristic, complemented, Sylow
137	C₂×#41	—	47	2	no	—
138	#131	—	94	4	no	—
139	#131	—	94	4	no	—
140	C₂×M₁₆	—	47	2	no	—
141	C₂×M₁₆	—	47	4	no	—
142	C₂²×C₈	—	47	4	no	—
143	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
144	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
145	C₄⋊Q₈	—	47	1	no	—
146	#127	—	47	1	no	—
147	#124	—	47	2	no	—
148	#121	—	47	4	no	—
149	C₄×D₄	—	47	4	no	—
150	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
151	M₃₂	—	2	4	no	—
152	M₃₂	—	94	8	no	—
153	C₂×C₁₆	—	47	4	no	—
154	C₂×C₁₆	—	47	2	no	—
155	C₂×C₁₆	—	47	2	no	—
156	C₂×C₁₆	—	94	8	no	—
157	C₂×C₁₆	—	94	2	no	—
158	C₂×C₁₆	C₂²×D₄₇	1	4	yes	—
159	C₄⋊C₈	—	47	4	no	—
160	C₂²⋊C₈	—	47	8	no	—
161	C₄×C₈	C₄×D₄₇	1	2	yes	—
162	C₄×C₈	C₂²×D₄₇	1	2	yes	—
163	#41	—	94	4	no	—
164	#41	—	94	4	no	—
165	C₂×Q₈	—	47	2	no	—
166	C₂×D₄	—	47	4	no	—
167	C₂×D₄	—	47	2	no	—
168	C₂×D₄	—	47	4	no	—
169	C₂²×C₄	—	47	4	no	—
170	C₂²×C₄	C₈×D₄₇	1	2	yes	—
171	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
172	M₁₆	—	94	4	no	—
173	M₁₆	—	94	8	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	47	4	no	—
176	C₂×C₈	—	94	4	no	—
177	C₂×C₈	—	94	4	no	—
178	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
179	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
180	C₂×C₈	D₄×D₄₇	1	2	yes	—
181	C₂×C₈	#10625354	1	2	yes	—
182	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
183	C₄⋊C₄	—	47	4	no	—
184	C₂²⋊C₄	—	47	8	no	—
185	C₄²	C₈×D₄₇	1	1	yes	characteristic
186	C₄²	C₈×D₄₇	1	1	yes	characteristic
187	C₄²	C₂×C₄×D₄₇	1	2	yes	—
188	C₁₆	—	2	4	no	—
189	C₁₆	—	47	4	no	—
190	C₁₆	—	94	2	no	—
191	C₂³	—	47	4	no	—
192	C₂³	C₂×C₈×D₄₇	1	1	yes	characteristic
193	Q₈	—	94	4	no	—
194	D₄	—	94	8	no	—
195	D₄	—	94	4	no	—
196	D₄	—	188	4	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	47	4	no	—
199	C₂×C₄	—	94	4	no	—
200	C₂×C₄	—	94	4	no	—
201	C₂×C₄	#13093986	1	2	yes	—
202	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
203	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
204	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
205	C₂×C₄	#13094032	1	2	yes	—
206	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
207	C₈	—	2	2	no	—
208	C₈	—	94	4	no	—
209	C₈	#13093986	1	2	yes	—
210	C₈	C₂×D₄×D₄₇	1	1	yes	characteristic
211	C₈	C₂×#10625354	1	1	yes	characteristic
212	C₂²	—	2	2	no	—
213	C₂²	—	94	4	no	—
214	C₂²	—	94	8	no	—
215	C₂²	#15689188	1	2	yes	—
216	C₂²	C₂²×C₈×D₄₇	1	1	yes	characteristic
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	94	4	no	—
220	C₄	#15689188	1	2	yes	—
221	C₄	C₂×#13093986	1	1	yes	characteristic
222	C₄	C₂×#13094032	1	1	yes	characteristic
223	C₂	—	2	2	no	—
224	C₂	—	188	4	no	—
225	C₂	C₂×#15689188	1	1	yes	characteristic
226	C₂	#20256490	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₈
Fitting	C₄₇×#2150	C₂
prime radical	#15689673	C₄
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	#2150	D₄₇
hypocenter	C₄₇	#33435
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
1. C₂×#15689541	C₂	semidirect
3. #20245212	C₂	semidirect
4. #20245003	C₂	semidirect
5. C₂×#15688899	C₂	semidirect
6. C₄₇×#2150	C₂	semidirect
36. C₂×C₄₇⋊C₁₆	D₄	semidirect
113. #2150	D₄₇	semidirect
136. C₄₇	#33435	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689541	2
#20252101	2
#20245212	2
#20245003	2
C₂×#15688899	2
C₄₇×#2150	2
#33435	47