Subgroups — #26181990

Order 12032.

900 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#10625329	C₂	1	2	yes	maximal
2	#20241691	C₂	1	4	yes	maximal, complemented
3	#20241678	C₂	1	4	yes	maximal, complemented
4	#20241669	C₂	1	4	yes	maximal, complemented
5	C₄₇×#2920	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₄	1	4	yes	—
7	C₄×#10625329	C₄	1	4	yes	—
8	C₄×#10625329	C₄	1	4	yes	—
9	C₄×#10625329	C₂²	1	2	yes	—
10	C₄×#10625329	C₂²	1	2	yes	—
11	C₄×#10625329	C₂²	1	4	yes	—
12	C₂×#13093935	C₄	1	4	yes	—
13	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
14	#15688845	C₂²	1	8	yes	complemented
15	#15688844	C₂²	1	4	yes	complemented
16	C₈×C₄₇⋊C₈	C₂²	1	4	yes	complemented
17	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#432	C₄	1	4	yes	—
19	C₄₇×#431	C₂²	1	4	yes	—
20	C₄₇×#423	C₄	1	2	yes	—
21	C₄₇×#407	C₄	1	2	yes	—
22	C₄×M₁₆×C₄₇	C₂²	1	2	yes	—
23	C₂×#10625329	—	2	16	no	—
24	C₂×#10625329	C₂×C₄	1	4	yes	—
25	C₂×#10625329	C₂×C₄	1	8	yes	—
26	C₂×#10625329	C₂×C₄	1	4	yes	—
27	C₂²×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
28	#13093935	C₂×C₄	1	4	yes	—
29	#13093935	C₂×C₄	1	8	yes	—
30	#13093935	C₂×C₄	1	4	yes	—
31	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
32	C₄×C₄₇⋊C₈	C₂³	1	4	yes	—
33	C₄×C₄₇⋊C₈	C₂³	1	4	yes	—
34	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
35	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
43	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
45	C₂²⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
46	#97×C₄₇	C₂×C₄	1	4	yes	—
47	C₄×C₈×C₄₇	C₂³	1	4	yes	—
48	C₂⁴×C₄₇	C₄²	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	—	2	8	no	—
50	C₂²×C₄×C₄₇	—	4	2	no	—
51	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
52	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
53	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
54	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
61	M₁₆×C₄₇	—	4	4	no	—
62	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
63	C₄²×C₄₇	C₄²	1	1	yes	characteristic
64	C₄²×C₄₇	C₄²	1	2	yes	—
65	C₄²×C₄₇	C₄²	1	2	yes	—
66	C₄²×C₄₇	C₄²	1	1	yes	characteristic
67	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
68	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
69	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
70	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
71	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
73	#10625329	—	2	8	no	—
74	#10625329	—	2	16	no	—
75	#10625329	M₁₆	1	16	yes	complemented
76	C₂×C₄₇⋊C₈	—	2	16	no	—
77	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
78	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
79	C₂³×C₄₇	—	2	4	no	—
80	C₂³×C₄₇	—	4	1	no	—
81	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
82	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
83	C₂×C₄×C₄₇	—	2	4	no	—
84	C₂×C₄×C₄₇	—	2	4	no	—
85	C₂×C₄×C₄₇	—	2	4	no	—
86	C₂×C₄×C₄₇	—	2	2	no	—
87	C₂×C₄×C₄₇	—	2	4	no	—
88	C₂×C₄×C₄₇	—	2	4	no	—
89	C₂×C₄×C₄₇	—	2	4	no	—
90	C₂×C₄×C₄₇	—	4	2	no	—
91	C₂×C₄×C₄₇	—	4	1	no	—
92	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
93	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
94	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
95	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
96	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
97	C₂×C₄×C₄₇	C₂×M₁₆	1	8	yes	—
98	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
99	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
100	C₈×C₄₇	—	2	8	no	—
101	C₄₇⋊C₈	—	2	8	no	—
102	C₄₇⋊C₈	C₂×M₁₆	1	16	yes	complemented
103	M₁₆²	—	47	1	no	maximal, Sylow
104	C₂²×C₄₇	—	2	2	no	—
105	C₂²×C₄₇	—	2	2	no	—
106	C₂²×C₄₇	—	2	2	no	—
107	C₂²×C₄₇	—	2	1	no	—
108	C₂²×C₄₇	—	4	4	no	—
109	C₂²×C₄₇	C₄×M₁₆	1	4	yes	—
110	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
111	C₄×C₄₇	—	2	2	no	—
112	C₄×C₄₇	—	2	2	no	—
113	C₄×C₄₇	—	2	1	no	—
114	C₄×C₄₇	—	2	2	no	—
115	C₄×C₄₇	C₄×M₁₆	1	4	yes	—
116	C₄×C₄₇	C₂²×M₁₆	1	4	yes	—
117	C₄×C₄₇	#568	1	2	yes	—
118	#2920	D₄₇	1	1	yes	characteristic
119	C₂×C₄×M₁₆	—	47	2	no	—
120	#1455	—	47	4	no	—
121	#1451	—	47	4	no	—
122	C₈×M₁₆	—	47	4	no	—
123	C₂×C₄₇	—	2	4	no	—
124	C₂×C₄₇	—	4	1	no	—
125	C₂×C₄₇	C₂×C₄×M₁₆	1	2	yes	—
126	C₂×C₄₇	#2873	1	1	yes	characteristic
127	C₂²×M₁₆	—	47	4	no	—
128	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
129	#432	Dic₄₇	1	4	yes	—
130	#431	C₂×D₄₇	1	4	yes	—
131	#423	Dic₄₇	1	2	yes	—
132	#407	Dic₄₇	1	2	yes	—
133	C₄×M₁₆	—	47	4	no	—
134	C₄×M₁₆	—	47	4	no	—
135	C₄×M₁₆	—	47	2	no	—
136	C₄×M₁₆	—	47	2	no	—
137	C₄×M₁₆	—	47	4	no	—
138	C₄×M₁₆	C₂×D₄₇	1	2	yes	—
139	C₂×#97	—	47	4	no	—
140	C₂×C₄×C₈	—	47	4	no	—
141	#322	—	47	4	no	—
142	#322	—	47	8	no	—
143	C₈²	—	47	4	no	—
144	C₄₇	M₁₆²	1	1	yes	characteristic, complemented, Sylow
145	C₂³×C₄	C₄×D₄₇	1	2	yes	—
146	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
147	C₂×M₁₆	—	47	4	no	—
148	C₂×M₁₆	—	47	8	no	—
149	C₂×M₁₆	—	47	4	no	—
150	C₂×M₁₆	—	94	16	no	—
151	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
152	C₂²×C₈	—	47	8	no	—
153	C₂×C₄²	C₄×D₄₇	1	2	yes	—
154	C₂×C₄²	C₄×D₄₇	1	2	yes	—
155	C₂×C₄²	C₄×D₄₇	1	2	yes	—
156	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
157	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
158	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
159	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
160	C₄⋊C₈	C₂×Dic₄₇	1	8	yes	—
161	C₂²⋊C₈	C₂×Dic₄₇	1	8	yes	—
162	#97	—	47	4	no	—
163	#97	—	47	8	no	—
164	#97	—	47	4	no	—
165	#97	C₂×Dic₄₇	1	4	yes	—
166	C₄×C₈	—	47	8	no	—
167	C₄×C₈	—	47	4	no	—
168	C₄×C₈	—	47	4	no	—
169	C₄×C₈	C₂²×D₄₇	1	4	yes	—
170	C₂⁴	C₄×Dic₄₇	1	1	yes	characteristic
171	C₂²×C₄	—	2	8	no	—
172	C₂²×C₄	—	4	2	no	—
173	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
174	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
175	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
176	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
177	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
178	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
179	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
180	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
181	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
182	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
183	M₁₆	—	4	4	no	—
184	M₁₆	—	47	16	no	—
185	M₁₆	—	94	16	no	—
186	M₁₆	—	94	8	no	—
187	C₂×C₈	—	47	8	no	—
188	C₂×C₈	—	47	8	no	—
189	C₂×C₈	—	94	16	no	—
190	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
191	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
192	C₄²	C₄×Dic₄₇	1	2	yes	—
193	C₄²	C₄×Dic₄₇	1	2	yes	—
194	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
195	C₄²	C₂×C₄×D₄₇	1	2	yes	—
196	C₄²	C₂×C₄×D₄₇	1	2	yes	—
197	C₄²	C₂×C₄×D₄₇	1	2	yes	—
198	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
199	C₄²	C₂²×Dic₄₇	1	2	yes	—
200	C₄²	C₂³×D₄₇	1	1	yes	characteristic
201	C₂³	—	2	4	no	—
202	C₂³	—	4	1	no	—
203	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
204	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	1	no	—
214	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
215	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
216	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
217	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
218	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
219	C₂×C₄	M₁₆×D₄₇	1	8	yes	—
220	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
221	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
222	C₈	—	2	8	no	—
223	C₈	—	47	16	no	—
224	C₈	—	94	8	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	1	no	—
229	C₂²	—	4	4	no	—
230	C₂²	M₁₆×Dic₄₇	1	4	yes	—
231	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	1	no	—
235	C₄	—	2	2	no	—
236	C₄	M₁₆×Dic₄₇	1	4	yes	—
237	C₄	C₂×M₁₆×D₄₇	1	4	yes	—
238	C₄	#15689890	1	2	yes	—
239	C₂	—	2	4	no	—
240	C₂	—	4	1	no	—
241	C₂	C₂×M₁₆×Dic₄₇	1	2	yes	—
242	C₂	#20251807	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂²×C₄²
Fitting	C₄₇×#2920	C₂
prime radical	C₂⁴×C₄₇	C₄²
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#2920	D₄₇
hypocenter	C₄₇	M₁₆²
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ #2920

Complements

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

N	H ≅ G/N	type
2. #20241691	C₂	semidirect
3. #20241678	C₂	semidirect
4. #20241669	C₂	semidirect
14. #15688845	C₂²	semidirect
15. #15688844	C₂²	semidirect
16. C₈×C₄₇⋊C₈	C₂²	semidirect
75. #10625329	M₁₆	semidirect
102. C₄₇⋊C₈	C₂×M₁₆	semidirect
144. C₄₇	M₁₆²	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#10625329	2
#20241691	2
#20241678	2
#20241669	2
C₄₇×#2920	2
M₁₆²	47