Subgroups — #26115175

Order 12032.

586 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×C₄×D₃₇₆	C₂	1	1	yes	maximal, characteristic
2	#20244182	C₂	1	2	yes	maximal, complemented
3	C₂×#15688876	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15688852	C₂	1	2	yes	maximal, complemented
5	C₂×C₄₇×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₇₆	C₄	1	1	yes	characteristic
7	C₂×C₄×D₁₈₈	C₂²	1	2	yes	—
8	C₄×D₃₇₆	—	2	2	no	—
9	C₄×D₃₇₆	—	2	2	no	—
10	C₂×#13093958	C₄	1	1	yes	characteristic
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×#13093946	C₄	1	2	yes	—
13	C₂×(C₂×D₄₇)⋊C₈	—	2	4	no	—
14	#15688876	—	2	2	no	—
15	#15688852	—	2	4	no	—
16	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×#322	—	2	2	no	—
20	C₂²×D₁₈₈	C₂×C₄	1	2	yes	—
21	C₂²×C₄×D₄₇	—	2	2	no	—
22	C₂×D₃₇₆	—	2	4	no	—
23	C₂×D₃₇₆	—	2	2	no	—
24	C₂²×C₄₇⋊C₈	—	2	2	no	—
25	C₄×D₁₈₈	—	2	2	no	—
26	C₄×D₁₈₈	—	2	2	no	—
27	C₄×D₁₈₈	—	2	2	no	—
28	C₄×D₁₈₈	—	2	2	no	—
29	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
30	C₂×(C₂×D₄₇)⋊C₄	—	2	2	no	—
31	#13093958	—	2	2	no	—
32	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
33	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
34	#13093946	—	2	4	no	—
35	(C₂×D₄₇)⋊C₈	—	4	8	no	—
36	C₂²×C₈×C₄₇	—	2	1	no	—
37	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₄₇	D₄	1	2	yes	—
40	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
41	C₄×C₈×C₄₇	—	2	1	no	—
42	C₄×C₈×C₄₇	—	2	1	no	—
43	C₄×C₈×C₄₇	—	2	1	no	—
44	C₄×C₈×C₄₇	D₄	1	2	yes	—
45	C₄×C₈×C₄₇	D₄	1	2	yes	—
46	C₄×C₈×C₄₇	D₄	1	2	yes	—
47	C₂³×D₄₇	—	2	2	no	—
48	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
51	C₂²×Dic₄₇	—	2	2	no	—
52	C₂×D₁₈₈	—	2	2	no	—
53	C₂×D₁₈₈	—	2	2	no	—
54	C₂×D₁₈₈	—	2	2	no	—
55	C₂×D₁₈₈	—	2	2	no	—
56	C₂×D₁₈₈	—	2	2	no	—
57	C₂×D₁₈₈	M₁₆	1	4	yes	—
58	C₂×C₄×D₄₇	—	4	2	no	—
59	C₂×C₄×D₄₇	—	4	2	no	—
60	C₂×C₄×D₄₇	—	4	4	no	—
61	C₂×C₄×D₄₇	—	4	4	no	—
62	C₂×C₈×C₄₇	—	2	2	no	—
63	C₂×C₈×C₄₇	—	2	2	no	—
64	C₂×C₈×C₄₇	—	2	2	no	—
65	C₂×C₈×C₄₇	—	2	2	no	—
66	C₂×C₈×C₄₇	—	2	2	no	—
67	C₂×C₈×C₄₇	—	4	2	no	—
68	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
69	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
70	C₂×C₈×C₄₇	D₈	1	4	yes	complemented
71	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₄₇	#41	1	1	yes	characteristic
75	(C₂×D₄₇)⋊C₄	—	4	2	no	—
76	(C₂×D₄₇)⋊C₄	—	4	2	no	—
77	C₄⋊Dic₄₇	—	2	2	no	—
78	C₄⋊Dic₄₇	—	2	2	no	—
79	C₂×C₄₇⋊C₈	—	4	2	no	—
80	C₂×C₄₇⋊C₈	—	4	4	no	—
81	D₃₇₆	—	2	8	no	—
82	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
83	C₂²×D₄₇	—	2	4	no	—
84	C₂²×D₄₇	—	4	2	no	—
85	C₂²×D₄₇	—	4	2	no	—
86	C₂²×D₄₇	—	4	4	no	—
87	C₂²×D₄₇	—	4	4	no	—
88	C₂×C₄×C₄₇	—	2	2	no	—
89	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
91	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
92	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
93	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
94	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
95	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
96	C₂×C₄×C₄₇	#131	1	2	yes	—
97	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
98	C₂×C₄×C₄₇	#135	1	2	yes	—
99	C₂×Dic₄₇	—	4	2	no	—
100	C₂×Dic₄₇	—	4	4	no	—
101	D₁₈₈	—	2	4	no	—
102	D₁₈₈	—	2	4	no	—
103	D₁₈₈	—	4	2	no	—
104	D₁₈₈	—	4	2	no	—
105	C₄×D₄₇	—	8	4	no	—
106	C₄×D₄₇	—	8	2	no	—
107	C₄×D₄₇	—	8	2	no	—
108	C₈×C₄₇	—	2	2	no	—
109	C₈×C₄₇	—	2	2	no	—
110	C₈×C₄₇	—	2	4	no	—
111	C₈×C₄₇	—	4	2	no	—
112	C₄₇⋊C₈	—	8	2	no	—
113	#9663	—	47	1	no	maximal, Sylow
114	C₂²×C₄₇	#390	1	1	yes	characteristic
115	C₂²×C₄₇	#435	1	2	yes	—
116	C₂²×C₄₇	C₄×D₈	1	2	yes	—
117	C₂²×C₄₇	#444	1	2	yes	—
118	C₂×D₄₇	—	4	8	no	—
119	C₂×D₄₇	—	4	4	no	—
120	C₂×D₄₇	—	8	4	no	—
121	C₂×D₄₇	—	8	2	no	—
122	C₂×D₄₇	—	8	2	no	—
123	C₄×C₄₇	—	2	1	no	—
124	C₄×C₄₇	—	2	1	no	—
125	C₄×C₄₇	#407	1	1	yes	characteristic
126	C₄×C₄₇	#408	1	1	yes	characteristic
127	C₄×C₄₇	#435	1	2	yes	—
128	C₄×C₄₇	#466	1	2	yes	—
129	C₄×C₄₇	#493	1	1	yes	characteristic
130	C₄×C₄₇	#495	1	1	yes	characteristic
131	Dic₄₇	—	8	2	no	—
132	C₂×C₄×D₈	—	47	1	no	—
133	#1954	—	47	2	no	—
134	C₂×#335	—	47	1	no	—
135	C₂×D₄⋊C₈	—	47	2	no	—
136	C₂×#322	D₄₇	1	1	yes	characteristic, complemented
137	C₂×C₄₇	#1581	1	4	yes	—
138	C₂×C₄₇	#1955	1	1	yes	characteristic
139	C₂×C₄₇	#1970	1	1	yes	characteristic
140	C₂×C₄₇	#1975	1	1	yes	characteristic
141	D₄₇	—	8	4	no	—
142	C₂²×D₈	—	47	1	no	—
143	C₂×C₄×D₄	—	47	2	no	—
144	C₄×D₈	—	94	2	no	—
145	C₄×D₈	—	94	2	no	—
146	C₂×#107	—	47	1	no	—
147	C₂×C₄⋊C₈	—	47	2	no	—
148	C₂×#102	—	47	2	no	—
149	C₂×C₂²⋊C₈	—	94	4	no	—
150	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
151	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
152	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
153	#335	—	94	2	no	—
154	D₄⋊C₈	—	94	4	no	—
155	#322	—	2	2	no	—
156	C₄₇	#9663	1	1	yes	characteristic, complemented, Sylow
157	C₂²×D₄	—	47	2	no	—
158	C₂³×C₄	—	94	2	no	—
159	C₂×D₈	—	94	4	no	—
160	C₂×D₈	—	94	2	no	—
161	C₂²×C₈	—	2	1	no	—
162	C₂²×C₈	—	94	2	no	—
163	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
164	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
165	C₂²×C₈	D₁₈₈	1	2	yes	—
166	C₄×D₄	—	94	2	no	—
167	C₄×D₄	—	94	2	no	—
168	C₄×D₄	—	94	2	no	—
169	C₄×D₄	—	94	2	no	—
170	C₂×C₄⋊C₄	—	47	2	no	—
171	C₂×C₂²⋊C₄	—	94	2	no	—
172	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
173	#107	—	94	2	no	—
174	C₄⋊C₈	—	94	2	no	—
175	C₄⋊C₈	—	94	2	no	—
176	#102	—	94	4	no	—
177	C₂²⋊C₈	—	188	8	no	—
178	C₄×C₈	—	2	1	no	—
179	C₄×C₈	—	2	1	no	—
180	C₄×C₈	—	2	1	no	—
181	C₄×C₈	D₁₈₈	1	2	yes	—
182	C₄×C₈	D₁₈₈	1	2	yes	—
183	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
184	C₂⁴	—	94	2	no	—
185	C₂×D₄	—	47	4	no	—
186	C₂×D₄	—	94	2	no	—
187	C₂×D₄	—	94	2	no	—
188	C₂×D₄	—	94	2	no	—
189	C₂×D₄	—	94	2	no	—
190	C₂×D₄	—	94	2	no	—
191	C₂²×C₄	—	94	2	no	—
192	C₂²×C₄	—	188	2	no	—
193	C₂²×C₄	—	188	2	no	—
194	C₂²×C₄	—	188	4	no	—
195	C₂²×C₄	—	188	4	no	—
196	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
197	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
198	C₂²×C₄	#10625350	1	1	yes	characteristic
199	D₈	—	94	8	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	188	2	no	—
207	C₂×C₈	—	188	4	no	—
208	C₂×C₈	D₃₇₆	1	4	yes	complemented
209	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
210	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
211	C₄⋊C₄	—	94	2	no	—
212	C₄⋊C₄	—	94	2	no	—
213	C₂²⋊C₄	—	188	2	no	—
214	C₂²⋊C₄	—	188	2	no	—
215	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
216	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
217	C₄²	#10625350	1	1	yes	characteristic
218	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
219	C₂³	—	94	4	no	—
220	C₂³	—	188	4	no	—
221	C₂³	—	188	2	no	—
222	C₂³	—	188	2	no	—
223	C₂³	—	188	4	no	—
224	C₂³	C₄×D₁₈₈	1	1	yes	characteristic
225	D₄	—	94	4	no	—
226	D₄	—	94	4	no	—
227	D₄	—	188	2	no	—
228	D₄	—	188	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	376	2	no	—
231	C₂×C₄	—	376	2	no	—
232	C₂×C₄	—	376	4	no	—
233	C₂×C₄	—	188	2	no	—
234	C₂×C₄	—	188	4	no	—
235	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
236	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
237	C₂×C₄	#13093998	1	1	yes	characteristic
238	C₂×C₄	#13093998	1	1	yes	characteristic
239	C₂×C₄	#13094012	1	1	yes	characteristic
240	C₂×C₄	#13094021	1	1	yes	characteristic
241	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
242	C₂×C₄	#13094032	1	2	yes	—
243	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
244	C₂×C₄	#13094046	1	2	yes	—
245	C₈	—	2	2	no	—
246	C₈	—	2	2	no	—
247	C₈	—	2	4	no	—
248	C₈	—	4	2	no	—
249	C₈	—	376	2	no	—
250	C₂²	—	376	2	no	—
251	C₂²	—	376	4	no	—
252	C₂²	—	376	2	no	—
253	C₂²	—	188	8	no	—
254	C₂²	—	188	4	no	—
255	C₂²	#15689025	1	1	yes	characteristic
256	C₂²	#15689196	1	2	yes	—
257	C₂²	C₄×D₃₇₆	1	2	yes	—
258	C₂²	#15689246	1	2	yes	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	376	2	no	—
262	C₄	#15689086	1	1	yes	characteristic
263	C₄	#15689091	1	1	yes	characteristic
264	C₄	#15689196	1	2	yes	—
265	C₄	#15689359	1	2	yes	—
266	C₄	#15689500	1	1	yes	characteristic
267	C₄	#15689507	1	1	yes	characteristic
268	C₂	—	376	4	no	—
269	C₂	#20242219	1	4	yes	—
270	C₂	#20244186	1	1	yes	characteristic
271	C₂	#20244254	1	1	yes	characteristic
272	C₂	#20244281	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#322	C₂
socle	C₂³×C₄₇	C₄×D₄
prime radical	C₂²×D₃₇₆	C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#322	D₄₇
hypocenter	C₄₇	#9663
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
2. #20244182	C₂	semidirect
3. C₂×#15688876	C₂	semidirect
4. C₂×#15688852	C₂	semidirect
5. C₂×C₄₇×#322	C₂	semidirect
70. C₂×C₈×C₄₇	D₈	semidirect
136. C₂×#322	D₄₇	semidirect
156. C₄₇	#9663	semidirect
208. C₂×C₈	D₃₇₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₃₇₆	2
#20244182	2
C₂×#15688876	2
C₂×#15688852	2
C₂×C₄₇×#322	2
#9663	47