Subgroups — #26122083

Order 12032.

842 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245795	C₂	1	2	yes	maximal
2	C₂×#15688880	C₂	1	4	yes	maximal
3	C₂×C₄×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093983	C₂²	1	4	yes	—
5	#15688993	—	2	2	no	—
6	#15688989	—	2	2	no	—
7	#15688983	—	2	4	no	—
8	C₄×C₄⋊Dic₄₇	C₄	1	8	yes	complemented
9	#15688880	—	2	8	no	—
10	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
11	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
12	C₄×M₁₆×C₄₇	—	2	2	no	—
13	C₄×M₁₆×C₄₇	—	2	2	no	—
14	C₂×#97×C₄₇	C₄	1	2	yes	complemented
15	C₂×C₄×C₈×C₄₇	C₄	1	2	yes	complemented
16	#13093983	—	2	8	no	—
17	#13093983	—	2	8	no	—
18	C₂×C₄⋊Dic₄₇	—	2	8	no	—
19	C₂×C₄⋊Dic₄₇	—	2	4	no	—
20	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
21	C₂×C₄×Dic₄₇	C₂×C₄	1	8	yes	—
22	C₄₇⋊#95	—	2	8	no	—
23	C₂³×C₄×C₄₇	D₄	1	2	yes	—
24	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
25	C₂×M₁₆×C₄₇	—	2	4	no	—
26	C₂×M₁₆×C₄₇	—	2	4	no	—
27	C₂×M₁₆×C₄₇	D₄	1	4	yes	—
28	C₂×M₁₆×C₄₇	Q₈	1	4	yes	—
29	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₄₇	D₄	1	2	yes	—
34	C₂×C₄²×C₄₇	D₄	1	2	yes	—
35	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
36	#97×C₄₇	—	2	4	no	—
37	C₄×C₈×C₄₇	—	2	2	no	—
38	C₄×C₈×C₄₇	—	2	2	no	—
39	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	—	4	4	no	—
41	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
42	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
43	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
44	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
51	C₂²×C₄×C₄₇	#41	1	4	yes	—
52	C₂²×Dic₄₇	—	2	8	no	—
53	M₁₆×C₄₇	—	2	8	no	—
54	M₁₆×C₄₇	—	2	8	no	—
55	C₂×C₈×C₄₇	—	2	4	no	—
56	C₂×C₈×C₄₇	—	2	4	no	—
57	C₂×C₈×C₄₇	C₄⋊C₄	1	8	yes	complemented
58	C₄²×C₄₇	C₄²	1	2	yes	—
59	C₄²×C₄₇	C₄²	1	2	yes	—
60	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
63	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
64	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
65	C₂²⋊Dic₄₇	—	4	4	no	—
66	C₂²⋊Dic₄₇	—	4	4	no	—
67	C₄⋊Dic₄₇	—	4	16	no	—
68	C₄⋊Dic₄₇	—	4	8	no	—
69	C₄×Dic₄₇	—	2	16	no	—
70	C₂³×C₄₇	—	4	2	no	—
71	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
72	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂³×C₄₇	#124	1	2	yes	—
74	C₂³×C₄₇	#127	1	1	yes	characteristic
75	C₂³×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₄₇	—	2	8	no	—
77	C₂×C₄×C₄₇	—	2	4	no	—
78	C₂×C₄×C₄₇	—	4	4	no	—
79	C₂×C₄×C₄₇	—	4	2	no	—
80	C₂×C₄×C₄₇	#95	1	2	yes	—
81	C₂×C₄×C₄₇	#95	1	4	yes	—
82	C₂×C₄×C₄₇	#95	1	2	yes	—
83	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
84	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
86	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
87	C₂×C₄×C₄₇	#117	1	2	yes	—
88	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
89	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
90	C₂×C₄×C₄₇	#121	1	4	yes	—
91	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
92	C₂×C₄×C₄₇	#125	1	2	yes	—
93	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
94	C₂×Dic₄₇	—	4	16	no	—
95	C₂×Dic₄₇	—	4	8	no	—
96	C₈×C₄₇	—	2	8	no	—
97	#10837	—	47	1	no	maximal, Sylow
98	C₂²×C₄₇	—	2	4	no	—
99	C₂²×C₄₇	—	2	2	no	—
100	C₂²×C₄₇	—	4	2	no	—
101	C₂²×C₄₇	—	4	1	no	—
102	C₂²×C₄₇	—	4	1	no	—
103	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
104	C₂²×C₄₇	#382	1	1	yes	characteristic
105	C₂²×C₄₇	#384	1	1	yes	characteristic
106	C₂²×C₄₇	#390	1	2	yes	—
107	C₂²×C₄₇	#391	1	2	yes	—
108	C₄×C₄₇	—	2	4	no	—
109	C₄×C₄₇	—	2	2	no	—
110	C₄×C₄₇	H(3,ℤ/4)	1	4	yes	complemented
111	C₄×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
112	C₄×C₄₇	#380	1	2	yes	—
113	C₄×C₄₇	#389	1	4	yes	—
114	Dic₄₇	—	8	8	no	—
115	C₂×C₄×M₁₆	D₄₇	1	1	yes	characteristic
116	#2285	—	47	2	no	—
117	C₂×H(3,ℤ/4)	—	47	4	no	—
118	C₂×C₄₇	—	2	2	no	—
119	C₂×C₄₇	—	2	1	no	—
120	C₂×C₄₇	—	2	1	no	—
121	C₂×C₄₇	—	2	2	no	—
122	C₂×C₄₇	C₄⋊#95	1	1	yes	characteristic
123	C₂×C₄₇	C₂×H(3,ℤ/4)	1	2	yes	—
124	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
125	C₂×#117	—	47	4	no	—
126	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
127	C₄×M₁₆	—	2	2	no	—
128	C₄×M₁₆	—	2	2	no	—
129	C₂×#97	Dic₄₇	1	2	yes	complemented
130	C₂×C₄×C₈	Dic₄₇	1	2	yes	complemented
131	#384	—	94	2	no	—
132	#382	—	94	2	no	—
133	#380	—	94	4	no	—
134	C₄×C₄⋊C₄	—	47	8	no	—
135	H(3,ℤ/4)	—	94	8	no	—
136	C₄₇	#10837	1	1	yes	characteristic, complemented, Sylow
137	C₂³×C₄	D₁₈₈	1	2	yes	—
138	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
139	C₂×M₁₆	—	2	4	no	—
140	C₂×M₁₆	—	2	4	no	—
141	C₂×M₁₆	Dic₉₄	1	4	yes	—
142	C₂×M₁₆	D₁₈₈	1	4	yes	—
143	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
144	#117	—	94	8	no	—
145	#117	—	94	8	no	—
146	C₂×C₄⋊C₄	—	94	8	no	—
147	C₂×C₄⋊C₄	—	94	4	no	—
148	C₂×C₂²⋊C₄	—	94	4	no	—
149	C₂×C₄²	—	47	8	no	—
150	C₂×C₄²	Dic₉₄	1	2	yes	—
151	C₂×C₄²	C₄×D₄₇	1	4	yes	—
152	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
153	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
154	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
155	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
156	#97	—	2	4	no	—
157	C₄×C₈	—	2	2	no	—
158	C₄×C₈	—	2	2	no	—
159	#95	—	94	8	no	—
160	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
161	C₂²×C₄	—	4	4	no	—
162	C₂²×C₄	—	94	8	no	—
163	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
164	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
165	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
166	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
167	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
168	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
169	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
170	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
171	C₂²×C₄	#10625350	1	4	yes	—
172	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
174	M₁₆	—	2	8	no	—
175	M₁₆	—	2	8	no	—
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	C₄⋊Dic₄₇	1	8	yes	complemented
179	C₄⋊C₄	—	188	16	no	—
180	C₄⋊C₄	—	188	8	no	—
181	C₂²⋊C₄	—	188	4	no	—
182	C₂²⋊C₄	—	188	4	no	—
183	C₄²	—	94	16	no	—
184	C₄²	C₄×Dic₄₇	1	2	yes	—
185	C₄²	C₄×Dic₄₇	1	2	yes	—
186	C₄²	Dic₄₇⋊C₄	1	4	yes	—
187	C₄²	Dic₄₇⋊C₄	1	4	yes	—
188	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
189	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
190	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
191	C₂³	—	4	2	no	—
192	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
193	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
194	C₂³	#13094012	1	2	yes	—
195	C₂³	#13094021	1	1	yes	characteristic
196	C₂³	#13094024	1	1	yes	characteristic
197	C₂×C₄	—	2	8	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	188	16	no	—
202	C₂×C₄	—	188	8	no	—
203	C₂×C₄	C₄₇⋊#95	1	2	yes	—
204	C₂×C₄	C₄₇⋊#95	1	4	yes	—
205	C₂×C₄	C₄₇⋊#95	1	2	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	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
210	C₂×C₄	#13093983	1	2	yes	—
211	C₂×C₄	C₄×D₁₈₈	1	4	yes	—
212	C₂×C₄	C₄×Dic₉₄	1	4	yes	—
213	C₂×C₄	#13093998	1	4	yes	—
214	C₂×C₄	#13094003	1	4	yes	—
215	C₂×C₄	#13094016	1	2	yes	—
216	C₂×C₄	#13094024	1	2	yes	—
217	C₈	—	2	8	no	—
218	C₂²	—	2	4	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	4	2	no	—
221	C₂²	—	4	1	no	—
222	C₂²	—	4	1	no	—
223	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
224	C₂²	#15688989	1	1	yes	characteristic
225	C₂²	#15688993	1	1	yes	characteristic
226	C₂²	#15689025	1	2	yes	—
227	C₂²	#15689029	1	2	yes	—
228	C₄	—	2	4	no	—
229	C₄	—	2	2	no	—
230	C₄	—	376	8	no	—
231	C₄	#15688880	1	4	yes	complemented
232	C₄	C₄×C₄⋊Dic₄₇	1	2	yes	—
233	C₄	#15688983	1	2	yes	—
234	C₄	#15689021	1	4	yes	—
235	C₂	—	2	2	no	—
236	C₂	—	2	1	no	—
237	C₂	—	2	1	no	—
238	C₂	—	2	2	no	—
239	C₂	#20241618	1	1	yes	characteristic
240	C₂	C₂×#15688880	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄×M₁₆×C₄₇	C₂
socle	C₂²×C₄₇	C₂×#95
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×C₄×M₁₆	D₄₇
hypocenter	C₄₇	#10837
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
8. C₄×C₄⋊Dic₄₇	C₄	semidirect
14. C₂×#97×C₄₇	C₄	semidirect
15. C₂×C₄×C₈×C₄₇	C₄	semidirect
57. C₂×C₈×C₄₇	C₄⋊C₄	semidirect
110. C₄×C₄₇	H(3,ℤ/4)	semidirect
129. C₂×#97	Dic₄₇	semidirect
130. C₂×C₄×C₈	Dic₄₇	semidirect
136. C₄₇	#10837	semidirect
178. C₂×C₈	C₄⋊Dic₄₇	semidirect
231. C₄	#15688880	semidirect

Maximal subgroups

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

subgroup	index
#20245795	2
C₂×#15688880	2
C₂×C₄×M₁₆×C₄₇	2
#10837	47