Subgroups — #26368960

Order 12032.

1172 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689540	C₂	1	2	yes	maximal, complemented
2	#20245002	C₂	1	6	yes	maximal, complemented
3	#20244999	C₂	1	3	yes	maximal, complemented
4	C₂×C₂²⋊(C₄₇⋊C₁₆)	C₂	1	3	yes	maximal, complemented
5	C₂²×#131×C₄₇	C₂	1	1	yes	maximal, characteristic
6	#15689540	—	2	8	no	—
7	C₂×#13093964	C₂²	1	6	yes	complemented
8	C₂²×C₄₇⋊C₁₆	C₂²	1	6	yes	complemented
9	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	1	yes	characteristic
10	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	3	yes	—
11	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	6	yes	complemented
12	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	6	yes	complemented
13	C₂²×#41×C₄₇	C₄	1	1	yes	characteristic
14	C₂×#131×C₄₇	—	2	8	no	—
15	C₂×#131×C₄₇	C₄	1	1	yes	characteristic
16	C₂×#131×C₄₇	C₄	1	3	yes	—
17	C₂×#131×C₄₇	C₂²	1	3	yes	—
18	C₂×#131×C₄₇	C₂²	1	1	yes	characteristic
19	C₂²×M₁₆×C₄₇	C₄	1	3	yes	—
20	C₂³×C₈×C₄₇	C₂²	1	3	yes	—
21	#13093964	—	2	12	no	—
22	C₂×C₄₇⋊C₁₆	—	2	12	no	—
23	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
24	C₂×C₄₇⋊C₁₆	C₂³	1	6	yes	—
25	C₂×#41×C₄₇	—	2	4	no	—
26	C₂×#41×C₄₇	—	2	4	no	—
27	C₂×#41×C₄₇	C₈	1	1	yes	characteristic
28	C₂×#41×C₄₇	C₈	1	3	yes	—
29	C₂×#41×C₄₇	C₂×C₄	1	3	yes	—
30	C₂×#41×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×Q₈×C₄₇	C₈	1	1	yes	characteristic
32	C₂²×D₄×C₄₇	C₈	1	3	yes	—
33	C₂³×C₄×C₄₇	C₂×C₄	1	3	yes	—
34	#131×C₄₇	—	2	4	no	—
35	#131×C₄₇	—	2	12	no	—
36	#131×C₄₇	—	2	12	no	—
37	#131×C₄₇	D₄	1	8	yes	—
38	C₂×M₁₆×C₄₇	—	2	12	no	—
39	C₂×M₁₆×C₄₇	C₂×C₄	1	3	yes	—
40	C₂×M₁₆×C₄₇	C₂×C₄	1	6	yes	—
41	C₂×M₁₆×C₄₇	C₂×C₄	1	3	yes	—
42	C₂²×C₈×C₄₇	—	2	12	no	—
43	C₂²×C₈×C₄₇	C₂×C₄	1	3	yes	—
44	C₂²×C₈×C₄₇	C₂×C₄	1	3	yes	—
45	C₂²×C₈×C₄₇	C₂×C₄	1	3	yes	—
46	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₄₇	C₂³	1	3	yes	—
48	C₂²×C₈×C₄₇	C₂³	1	3	yes	—
49	C₂⁴×C₄₇	C₂×C₈	1	3	yes	—
50	#41×C₄₇	—	2	4	no	—
51	#41×C₄₇	—	2	12	no	—
52	#41×C₄₇	—	2	6	no	—
53	#41×C₄₇	—	2	6	no	—
54	#41×C₄₇	C₂²⋊C₄	1	4	yes	—
55	#41×C₄₇	M₁₆	1	4	yes	—
56	C₂×Q₈×C₄₇	—	2	4	no	—
57	C₂×Q₈×C₄₇	C₂×C₈	1	3	yes	—
58	C₂×Q₈×C₄₇	C₂×C₈	1	1	yes	characteristic
59	C₂×D₄×C₄₇	—	2	12	no	—
60	C₂×D₄×C₄₇	C₂×C₈	1	6	yes	—
61	C₂×D₄×C₄₇	C₂×C₈	1	3	yes	—
62	C₂×D₄×C₄₇	C₂×C₈	1	3	yes	—
63	C₂²×C₄×C₄₇	—	2	6	no	—
64	C₂²×C₄×C₄₇	—	2	6	no	—
65	C₂²×C₄×C₄₇	C₂×C₈	1	3	yes	—
66	C₂²×C₄×C₄₇	C₂×C₈	1	3	yes	—
67	C₂²×C₄×C₄₇	C₂×C₈	1	3	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂²×C₄	1	3	yes	—
70	C₂²×C₄×C₄₇	C₂²×C₄	1	3	yes	—
71	M₁₆×C₄₇	—	2	6	no	—
72	M₁₆×C₄₇	—	2	12	no	—
73	M₁₆×C₄₇	C₂²⋊C₄	1	12	yes	—
74	C₂×C₈×C₄₇	—	2	6	no	—
75	C₂×C₈×C₄₇	—	2	6	no	—
76	C₂×C₈×C₄₇	—	2	6	no	—
77	C₂×C₈×C₄₇	—	2	2	no	—
78	C₂×C₈×C₄₇	—	2	6	no	—
79	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₄₇	C₂²×C₄	1	3	yes	—
81	C₂×C₈×C₄₇	C₂²×C₄	1	3	yes	—
82	C₂×C₈×C₄₇	C₂×D₄	1	12	yes	—
83	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
84	C₄₇⋊C₁₆	—	2	2	no	—
85	C₄₇⋊C₁₆	—	2	6	no	—
86	C₂³×C₄₇	—	2	6	no	—
87	C₂³×C₄₇	—	2	6	no	—
88	C₂³×C₄₇	—	4	3	no	—
89	C₂³×C₄₇	C₂²×C₈	1	3	yes	—
90	C₂³×C₄₇	C₂²×C₈	1	3	yes	—
91	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
92	Q₈×C₄₇	—	2	6	no	—
93	Q₈×C₄₇	C₂²⋊C₈	1	4	yes	—
94	D₄×C₄₇	—	2	12	no	—
95	D₄×C₄₇	—	2	6	no	—
96	D₄×C₄₇	C₂²⋊C₈	1	12	yes	—
97	C₂×C₄×C₄₇	—	2	6	no	—
98	C₂×C₄×C₄₇	—	2	1	no	—
99	C₂×C₄×C₄₇	—	2	3	no	—
100	C₂×C₄×C₄₇	—	2	1	no	—
101	C₂×C₄×C₄₇	—	2	3	no	—
102	C₂×C₄×C₄₇	—	2	6	no	—
103	C₂×C₄×C₄₇	—	2	6	no	—
104	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	6	yes	—
105	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂²×C₈	1	3	yes	—
107	C₂×C₄×C₄₇	C₂²×C₈	1	3	yes	—
108	C₂×C₄×C₄₇	C₂×M₁₆	1	6	yes	—
109	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
110	C₈×C₄₇	—	2	1	no	—
111	C₈×C₄₇	—	2	3	no	—
112	C₈×C₄₇	C₂×C₂²⋊C₄	1	6	yes	—
113	C₈×C₄₇	C₂²×D₄	1	2	yes	—
114	#33334	—	47	1	no	maximal, Sylow
115	C₂²×C₄₇	—	2	1	no	—
116	C₂²×C₄₇	—	2	3	no	—
117	C₂²×C₄₇	—	2	1	no	—
118	C₂²×C₄₇	—	2	3	no	—
119	C₂²×C₄₇	—	2	3	no	—
120	C₂²×C₄₇	—	2	3	no	—
121	C₂²×C₄₇	—	2	3	no	—
122	C₂²×C₄₇	—	4	12	no	—
123	C₂²×C₄₇	C₂×C₂²⋊C₈	1	6	yes	—
124	C₂²×C₄₇	#504	1	2	yes	complemented
125	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
126	C₄×C₄₇	—	2	1	no	—
127	C₄×C₄₇	—	2	3	no	—
128	C₄×C₄₇	C₂×C₂²⋊C₈	1	6	yes	—
129	C₄×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
130	C₄×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
131	C₂²×#131	D₄₇	1	1	yes	characteristic
132	C₂×#504	—	47	2	no	—
133	#2113	—	47	6	no	—
134	#2112	—	47	3	no	—
135	C₂×C₂²⋊C₁₆	—	47	3	no	—
136	C₂×C₄₇	—	2	2	no	—
137	C₂×C₄₇	—	2	6	no	—
138	C₂×C₄₇	—	4	3	no	—
139	C₂×C₄₇	C₂²×C₂²⋊C₈	1	1	yes	characteristic
140	C₂×C₄₇	C₂×#504	1	1	yes	characteristic
141	C₂×C₄₇	#3407	1	1	yes	characteristic
142	C₂²×#41	Dic₄₇	1	1	yes	characteristic
143	C₂×#131	—	2	8	no	—
144	C₂×#131	Dic₄₇	1	1	yes	characteristic
145	C₂×#131	Dic₄₇	1	3	yes	—
146	C₂×#131	C₂×D₄₇	1	3	yes	—
147	C₂×#131	C₂×D₄₇	1	1	yes	characteristic
148	C₂²×M₁₆	Dic₄₇	1	3	yes	—
149	C₂³×C₈	C₂×D₄₇	1	3	yes	—
150	#504	—	94	8	no	—
151	C₂×M₃₂	—	47	6	no	—
152	C₂²×C₁₆	—	47	6	no	—
153	C₂²⋊C₁₆	—	47	1	no	—
154	C₂²⋊C₁₆	—	47	3	no	—
155	C₂²⋊C₁₆	—	47	6	no	—
156	C₂²⋊C₁₆	—	47	6	no	—
157	C₄₇	#33334	1	1	yes	characteristic, complemented, Sylow
158	C₂×#41	—	2	4	no	—
159	C₂×#41	—	2	4	no	—
160	C₂×#41	C₄₇⋊C₈	1	1	yes	characteristic
161	C₂×#41	C₄₇⋊C₈	1	3	yes	—
162	C₂×#41	C₂×Dic₄₇	1	3	yes	—
163	C₂×#41	C₂×Dic₄₇	1	1	yes	characteristic
164	C₂²×Q₈	C₄₇⋊C₈	1	1	yes	characteristic
165	C₂²×D₄	C₄₇⋊C₈	1	3	yes	—
166	C₂³×C₄	C₂×Dic₄₇	1	3	yes	—
167	#131	—	2	4	no	—
168	#131	—	2	12	no	—
169	#131	—	2	12	no	—
170	#131	C₂²⋊D₄₇	1	8	yes	—
171	C₂×M₁₆	—	2	12	no	—
172	C₂×M₁₆	C₂×Dic₄₇	1	3	yes	—
173	C₂×M₁₆	C₂×Dic₄₇	1	6	yes	—
174	C₂×M₁₆	C₂×Dic₄₇	1	3	yes	—
175	C₂²×C₈	—	2	12	no	—
176	C₂²×C₈	C₂×Dic₄₇	1	3	yes	—
177	C₂²×C₈	C₂×Dic₄₇	1	3	yes	—
178	C₂²×C₈	C₂×Dic₄₇	1	3	yes	—
179	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
180	C₂²×C₈	C₂²×D₄₇	1	3	yes	—
181	C₂²×C₈	C₂²×D₄₇	1	3	yes	—
182	M₃₂	—	94	12	no	—
183	C₂×C₁₆	—	47	2	no	—
184	C₂×C₁₆	—	47	6	no	—
185	C₂×C₁₆	—	94	12	no	—
186	C₂⁴	C₂×C₄₇⋊C₈	1	3	yes	—
187	#41	—	2	4	no	—
188	#41	—	2	12	no	—
189	#41	—	2	6	no	—
190	#41	—	2	6	no	—
191	#41	#10625329	1	4	yes	—
192	#41	C₂²⋊Dic₄₇	1	4	yes	—
193	C₂×Q₈	—	2	4	no	—
194	C₂×Q₈	C₂×C₄₇⋊C₈	1	3	yes	—
195	C₂×Q₈	C₂×C₄₇⋊C₈	1	1	yes	characteristic
196	C₂×D₄	—	2	12	no	—
197	C₂×D₄	C₂×C₄₇⋊C₈	1	6	yes	—
198	C₂×D₄	C₂×C₄₇⋊C₈	1	3	yes	—
199	C₂×D₄	C₂×C₄₇⋊C₈	1	3	yes	—
200	C₂²×C₄	—	2	6	no	—
201	C₂²×C₄	—	2	6	no	—
202	C₂²×C₄	C₂×C₄₇⋊C₈	1	3	yes	—
203	C₂²×C₄	C₂×C₄₇⋊C₈	1	3	yes	—
204	C₂²×C₄	C₂×C₄₇⋊C₈	1	3	yes	—
205	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
206	C₂²×C₄	C₂²×Dic₄₇	1	3	yes	—
207	C₂²×C₄	C₂²×Dic₄₇	1	3	yes	—
208	M₁₆	—	2	6	no	—
209	M₁₆	—	2	12	no	—
210	M₁₆	C₂²⋊Dic₄₇	1	12	yes	—
211	C₂×C₈	—	2	6	no	—
212	C₂×C₈	—	2	6	no	—
213	C₂×C₈	—	2	6	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	6	no	—
216	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
217	C₂×C₈	C₂²×Dic₄₇	1	3	yes	—
218	C₂×C₈	C₂²×Dic₄₇	1	3	yes	—
219	C₂×C₈	C₂×C₂²⋊D₄₇	1	12	yes	—
220	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
221	C₁₆	—	94	2	no	—
222	C₁₆	—	94	6	no	—
223	C₂³	—	2	6	no	—
224	C₂³	—	2	6	no	—
225	C₂³	—	4	3	no	—
226	C₂³	C₂²×C₄₇⋊C₈	1	3	yes	—
227	C₂³	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
228	C₂³	C₂²×C₄₇⋊C₈	1	3	yes	—
229	Q₈	—	2	6	no	—
230	Q₈	C₂²⋊(C₄₇⋊C₈)	1	4	yes	—
231	D₄	—	2	12	no	—
232	D₄	—	2	6	no	—
233	D₄	C₂²⋊(C₄₇⋊C₈)	1	12	yes	—
234	C₂×C₄	—	2	6	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	3	no	—
237	C₂×C₄	—	2	6	no	—
238	C₂×C₄	—	2	6	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	3	no	—
241	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	6	yes	—
242	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
243	C₂×C₄	C₂²×C₄₇⋊C₈	1	3	yes	—
244	C₂×C₄	C₂²×C₄₇⋊C₈	1	3	yes	—
245	C₂×C₄	C₂×#10625329	1	6	yes	—
246	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
247	C₈	—	2	1	no	—
248	C₈	—	2	3	no	—
249	C₈	C₂×C₂²⋊Dic₄₇	1	6	yes	—
250	C₈	C₂²×C₂²⋊D₄₇	1	2	yes	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	3	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	3	no	—
255	C₂²	—	2	3	no	—
256	C₂²	—	2	3	no	—
257	C₂²	—	2	3	no	—
258	C₂²	—	4	12	no	—
259	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	6	yes	—
260	C₂²	#15689540	1	2	yes	complemented
261	C₂²	C₂³×C₄₇⋊C₈	1	1	yes	characteristic
262	C₄	—	2	1	no	—
263	C₄	—	2	3	no	—
264	C₄	C₂×C₂²⋊(C₄₇⋊C₈)	1	6	yes	—
265	C₄	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
266	C₄	C₂²×#10625329	1	1	yes	characteristic
267	C₂	—	2	2	no	—
268	C₂	—	2	6	no	—
269	C₂	—	4	3	no	—
270	C₂	C₂²×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
271	C₂	C₂×#15689540	1	1	yes	characteristic
272	C₂	#20256491	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₈
Fitting	C₂²×#131×C₄₇	C₂
prime radical	C₂²×#41×C₄₇	C₄
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₂²×#131	D₄₇
hypocenter	C₄₇	#33334
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×#131

Complements

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

N	H ≅ G/N	type
1. C₂×#15689540	C₂	semidirect
2. #20245002	C₂	semidirect
3. #20244999	C₂	semidirect
4. C₂×C₂²⋊(C₄₇⋊C₁₆)	C₂	semidirect
7. C₂×#13093964	C₂²	semidirect
8. C₂²×C₄₇⋊C₁₆	C₂²	semidirect
11. C₂²⋊(C₄₇⋊C₁₆)	C₂²	semidirect
12. C₂²⋊(C₄₇⋊C₁₆)	C₂²	semidirect
124. C₂²×C₄₇	#504	semidirect
157. C₄₇	#33334	semidirect
260. C₂²	#15689540	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689540	2
#20245002	2
#20244999	2
C₂×C₂²⋊(C₄₇⋊C₁₆)	2
C₂²×#131×C₄₇	2
#33334	47