Subgroups — #26105875

Order 12032.

802 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689176	C₂	1	2	yes	maximal
2	#20241231	C₂	1	4	yes	maximal
3	C₂×C₄×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
4	#15689176	—	2	4	no	—
5	#15689176	C₄	1	8	yes	complemented
6	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
7	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
8	C₂×C₄×C₄₇⋊C₈	—	2	2	no	—
9	C₂²×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
10	C₂²×C₄²×C₄₇	C₄	1	2	yes	—
11	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₄₇	—	2	2	no	—
13	C₄×C₄⋊C₄×C₄₇	—	2	2	no	—
14	C₂×#95×C₄₇	C₂²	1	2	yes	—
15	C₂²×C₄₇⋊C₈	—	2	4	no	—
16	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
17	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
18	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
19	C₄×C₄₇⋊C₈	—	2	8	no	—
20	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
22	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
25	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
26	C₂×C₄⋊C₄×C₄₇	C₈	1	8	yes	complemented
27	C₂×C₄²×C₄₇	—	2	2	no	—
28	C₂×C₄²×C₄₇	—	2	2	no	—
29	C₂×C₄²×C₄₇	—	4	4	no	—
30	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₄₇	D₄	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	#95×C₄₇	—	2	4	no	—
36	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₄₇	—	2	2	no	—
38	C₂²×C₄×C₄₇	—	2	2	no	—
39	C₂²×C₄×C₄₇	—	2	4	no	—
40	C₂²×C₄×C₄₇	—	2	4	no	—
41	C₂²×C₄×C₄₇	—	4	2	no	—
42	C₂²×C₄×C₄₇	—	4	4	no	—
43	C₂²×C₄×C₄₇	—	4	4	no	—
44	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
45	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
46	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₄₇	C₂×C₈	1	8	yes	—
52	C₂²×C₄×C₄₇	C₂×C₈	1	4	yes	—
53	C₂²×C₄×C₄₇	M₁₆	1	4	yes	—
54	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
58	C₄⋊C₄×C₄₇	—	2	16	no	—
59	C₄²×C₄₇	—	4	4	no	—
60	C₄²×C₄₇	—	4	4	no	—
61	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
62	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
63	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
64	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
65	C₂×C₄₇⋊C₈	—	2	8	no	—
66	C₂×C₄₇⋊C₈	—	4	8	no	—
67	C₂³×C₄₇	—	2	2	no	—
68	C₂³×C₄₇	—	2	1	no	—
69	C₂³×C₄₇	—	2	1	no	—
70	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
71	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂³×C₄₇	C₂²×C₈	1	2	yes	—
74	C₂³×C₄₇	C₂×M₁₆	1	2	yes	—
75	C₂×C₄×C₄₇	—	2	4	no	—
76	C₂×C₄×C₄₇	—	2	1	no	—
77	C₂×C₄×C₄₇	—	2	1	no	—
78	C₂×C₄×C₄₇	—	2	4	no	—
79	C₂×C₄×C₄₇	—	2	4	no	—
80	C₂×C₄×C₄₇	—	2	8	no	—
81	C₂×C₄×C₄₇	—	4	2	no	—
82	C₂×C₄×C₄₇	—	4	1	no	—
83	C₂×C₄×C₄₇	—	4	1	no	—
84	C₂×C₄×C₄₇	—	4	8	no	—
85	C₂×C₄×C₄₇	—	4	4	no	—
86	C₂×C₄×C₄₇	—	4	4	no	—
87	C₂×C₄×C₄₇	—	4	8	no	—
88	C₂×C₄×C₄₇	#95	1	2	yes	—
89	C₂×C₄×C₄₇	#95	1	4	yes	—
90	C₂×C₄×C₄₇	#95	1	2	yes	—
91	C₂×C₄×C₄₇	C₄×C₈	1	4	yes	—
92	C₂×C₄×C₄₇	#97	1	4	yes	—
93	C₂×C₄×C₄₇	C₂²⋊C₈	1	8	yes	—
94	C₂×C₄×C₄₇	#100	1	2	yes	—
95	C₂×C₄×C₄₇	#101	1	2	yes	—
96	C₂×C₄×C₄₇	C₄⋊C₈	1	8	yes	—
97	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
98	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
99	C₄₇⋊C₈	—	4	8	no	—
100	#7908	—	47	1	no	maximal, Sylow
101	C₂²×C₄₇	—	2	2	no	—
102	C₂²×C₄₇	—	2	2	no	—
103	C₂²×C₄₇	—	2	2	no	—
104	C₂²×C₄₇	—	2	2	no	—
105	C₂²×C₄₇	—	2	1	no	—
106	C₂²×C₄₇	—	2	1	no	—
107	C₂²×C₄₇	—	2	1	no	—
108	C₂²×C₄₇	—	2	1	no	—
109	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
110	C₂²×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
111	C₂²×C₄₇	C₂×#97	1	1	yes	characteristic
112	C₂²×C₄₇	C₂×C₂²⋊C₈	1	2	yes	—
113	C₂²×C₄₇	#410	1	2	yes	—
114	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
115	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
116	C₂²×C₄₇	C₂×C₄⋊C₈	1	2	yes	—
117	C₄×C₄₇	—	2	2	no	—
118	C₄×C₄₇	—	2	2	no	—
119	C₄×C₄₇	—	2	2	no	—
120	C₄×C₄₇	—	4	8	no	—
121	C₄×C₄₇	#336	1	8	yes	—
122	C₄×C₄₇	#343	1	4	yes	—
123	C₂×#431	—	47	2	no	—
124	C₂×C₄×C₄⋊C₄	D₄₇	1	1	yes	characteristic
125	#1280	—	47	4	no	—
126	C₂×C₄₇	—	2	1	no	—
127	C₂×C₄₇	—	2	1	no	—
128	C₂×C₄₇	—	2	2	no	—
129	C₂×C₄₇	#1460	1	2	yes	—
130	C₂×C₄₇	#1461	1	2	yes	—
131	C₂×C₄₇	C₂×#336	1	1	yes	characteristic
132	C₂×C₄₇	#1740	1	1	yes	characteristic
133	C₂×C₄₇	C₂×#343	1	1	yes	characteristic
134	C₂²×C₄⋊C₄	Dic₄₇	1	2	yes	—
135	C₂²×C₄²	Dic₄₇	1	2	yes	—
136	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
137	#431	—	47	8	no	—
138	#431	—	94	4	no	—
139	C₂×C₄⋊C₈	—	94	2	no	—
140	C₂×C₂²⋊C₈	—	47	4	no	—
141	C₂×C₄×C₈	—	94	2	no	—
142	C₄×C₄⋊C₄	—	2	2	no	—
143	C₄×C₄⋊C₄	—	2	2	no	—
144	C₂×#95	C₂×D₄₇	1	2	yes	—
145	C₄₇	#7908	1	1	yes	characteristic, complemented, Sylow
146	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
147	C₂³×C₄	C₂×Dic₄₇	1	4	yes	—
148	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
149	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
150	C₂²×C₈	—	94	4	no	—
151	C₂×C₄⋊C₄	—	2	4	no	—
152	C₂×C₄⋊C₄	—	2	4	no	—
153	C₂×C₄⋊C₄	C₄₇⋊C₈	1	8	yes	complemented
154	C₂×C₄²	—	2	2	no	—
155	C₂×C₄²	—	2	2	no	—
156	C₂×C₄²	—	4	4	no	—
157	C₂×C₄²	Dic₉₄	1	2	yes	—
158	C₂×C₄²	C₄×D₄₇	1	4	yes	—
159	C₂×C₄²	D₁₈₈	1	2	yes	—
160	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
161	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
162	C₄⋊C₈	—	94	8	no	—
163	C₂²⋊C₈	—	47	8	no	—
164	C₂²⋊C₈	—	94	4	no	—
165	C₄×C₈	—	94	8	no	—
166	#95	—	2	4	no	—
167	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	2	4	no	—
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	—	4	2	no	—
173	C₂²×C₄	—	4	4	no	—
174	C₂²×C₄	—	4	4	no	—
175	C₂²×C₄	C₂×C₄₇⋊C₈	1	8	yes	—
176	C₂²×C₄	C₂×C₄₇⋊C₈	1	4	yes	—
177	C₂²×C₄	#10625329	1	4	yes	—
178	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
179	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
180	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
181	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
182	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
183	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
184	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
185	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
187	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
188	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
189	C₂×C₈	—	94	8	no	—
190	C₂×C₈	—	188	8	no	—
191	C₄⋊C₄	—	2	16	no	—
192	C₄²	—	4	4	no	—
193	C₄²	—	4	4	no	—
194	C₄²	Dic₄₇⋊C₄	1	4	yes	—
195	C₄²	Dic₄₇⋊C₄	1	4	yes	—
196	C₄²	(C₂×D₄₇)⋊C₄	1	4	yes	—
197	C₄²	(C₂×D₄₇)⋊C₄	1	4	yes	—
198	C₂³	—	2	2	no	—
199	C₂³	—	2	1	no	—
200	C₂³	—	2	1	no	—
201	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
202	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
203	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
204	C₂³	C₂²×C₄₇⋊C₈	1	2	yes	—
205	C₂³	C₂×#10625329	1	2	yes	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	8	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	1	no	—
215	C₂×C₄	—	4	8	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	—	4	8	no	—
219	C₂×C₄	C₄₇⋊#95	1	2	yes	—
220	C₂×C₄	C₄₇⋊#95	1	4	yes	—
221	C₂×C₄	C₄₇⋊#95	1	2	yes	—
222	C₂×C₄	C₄×C₄₇⋊C₈	1	4	yes	—
223	C₂×C₄	#13093935	1	4	yes	—
224	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	8	yes	—
225	C₂×C₄	#13093941	1	2	yes	—
226	C₂×C₄	#13093943	1	2	yes	—
227	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	8	yes	—
228	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
229	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
230	C₈	—	188	8	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	1	no	—
239	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
240	C₂²	C₂×C₄×C₄₇⋊C₈	1	1	yes	characteristic
241	C₂²	C₂×#13093935	1	1	yes	characteristic
242	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
243	C₂²	#15689097	1	2	yes	—
244	C₂²	C₂×#13093941	1	1	yes	characteristic
245	C₂²	C₂×#13093943	1	1	yes	characteristic
246	C₂²	C₂×C₄⋊(C₄₇⋊C₈)	1	2	yes	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	4	8	no	—
251	C₄	#15688878	1	8	yes	—
252	C₄	#15688892	1	4	yes	—
253	C₂	—	2	1	no	—
254	C₂	—	2	1	no	—
255	C₂	—	2	2	no	—
256	C₂	#20241709	1	2	yes	—
257	C₂	#20241712	1	2	yes	—
258	C₂	C₂×#15688878	1	1	yes	characteristic
259	C₂	#20243121	1	1	yes	characteristic
260	C₂	C₂×#15688892	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₄₇
commutator	C₂²×C₄₇	C₂×C₄×C₈
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄×C₄⋊C₄×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×C₄×C₄⋊C₄	D₄₇
hypocenter	C₄₇	#7908
LCS	G ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
5. #15689176	C₄	semidirect
26. C₂×C₄⋊C₄×C₄₇	C₈	semidirect
145. C₄₇	#7908	semidirect
153. C₂×C₄⋊C₄	C₄₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689176	2
#20241231	2
C₂×C₄×C₄⋊C₄×C₄₇	2
#7908	47