Subgroups — #26373016

Order 12032.

696 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689555	C₂	1	2	yes	maximal, complemented
2	#20255148	C₂	1	2	yes	maximal, complemented
3	#20245638	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245474	C₂	1	2	yes	maximal, complemented
5	#20245429	C₂	1	2	yes	maximal, complemented
6	#20245313	C₂	1	4	yes	maximal, complemented
7	C₄×Dic₃₇₆	C₂	1	1	yes	maximal, characteristic, complemented
8	D₄×C₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13094046	C₂²	1	4	yes	—
10	#15689794	—	2	2	no	—
11	#15689555	—	2	8	no	—
12	C₂×Dic₃₇₆	C₂²	1	1	yes	characteristic
13	C₂×#13093969	C₂²	1	4	yes	—
14	C₂×D₇₅₂	C₂²	1	2	yes	—
15	C₂×D₇₅₂	C₂²	1	2	yes	—
16	#15689500	C₂²	1	2	yes	—
17	#15689359	C₂²	1	4	yes	—
18	#15689324	—	2	2	no	—
19	#15689270	—	2	4	no	—
20	C₄×Dic₁₈₈	C₂²	1	2	yes	—
21	#15688949	C₂²	1	1	yes	characteristic
22	#15688928	C₂²	1	2	yes	—
23	#15688925	C₂²	1	4	yes	—
24	C₂²×C₁₆×C₄₇	C₂²	1	2	yes	—
25	C₈×D₄×C₄₇	C₂²	1	1	yes	characteristic
26	C₄₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
27	C₄₇×C₂²⋊C₁₆	C₂²	1	2	yes	—
28	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
29	C₂×#10625350	—	2	4	no	—
30	#13094046	—	2	8	no	—
31	#13094046	—	4	4	no	—
32	C₂×Dic₁₈₈	C₂³	1	2	yes	—
33	C₂×#10625325	—	2	4	no	—
34	C₂×D₃₇₆	—	2	2	no	—
35	C₂×D₃₇₆	C₂³	1	4	yes	—
36	#13094021	—	2	2	no	—
37	#13094020	—	4	2	no	—
38	#13093998	—	2	4	no	—
39	#13093997	—	4	4	no	—
40	C₄×Dic₉₄	—	2	2	no	—
41	#13093986	—	4	2	no	—
42	Dic₃₇₆	D₄	1	4	yes	complemented
43	#13093969	—	2	8	no	—
44	D₇₅₂	—	2	4	no	—
45	D₇₅₂	—	4	2	no	—
46	#13093958	C₂³	1	2	yes	—
47	#13093949	—	2	2	no	—
48	#13093946	—	2	4	no	—
49	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
50	C₄×D₄×C₄₇	D₄	1	1	yes	characteristic
51	C₂×C₁₆×C₄₇	—	2	4	no	—
52	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
53	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
54	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₄₇	D₄	1	1	yes	characteristic
56	C₂²⋊C₈×C₄₇	D₄	1	2	yes	—
57	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
58	C₂×D₄×C₄₇	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
60	C₂×C₂²⋊D₄₇	—	4	4	no	—
61	C₂×C₂²⋊D₄₇	—	4	4	no	—
62	#10625350	—	4	8	no	—
63	#10625350	—	8	4	no	—
64	C₂×D₁₈₈	—	2	4	no	—
65	C₂×D₁₈₈	—	4	2	no	—
66	C₂×D₁₈₈	—	4	2	no	—
67	C₂×C₄×D₄₇	—	4	4	no	—
68	C₂×Dic₉₄	—	2	2	no	—
69	C₂×C₈×C₄₇	—	2	2	no	—
70	C₂×C₈×C₄₇	—	2	2	no	—
71	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
74	C₄⋊C₄×C₄₇	D₈	1	1	yes	characteristic
75	C₂²⋊C₄×C₄₇	D₈	1	2	yes	—
76	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
77	(C₂×D₄₇)⋊C₄	—	4	4	no	—
78	C₄⋊Dic₄₇	—	2	2	no	—
79	Dic₄₇⋊C₄	—	4	2	no	—
80	C₄×Dic₄₇	—	4	2	no	—
81	Dic₁₈₈	—	2	2	no	—
82	Dic₁₈₈	C₂×D₄	1	4	yes	—
83	D₃₇₆	—	2	4	no	—
84	D₃₇₆	—	2	4	no	—
85	D₃₇₆	—	8	2	no	—
86	#10625325	—	4	8	no	—
87	C₁₆×C₄₇	—	2	2	no	—
88	C₁₆×C₄₇	—	2	1	no	—
89	C₁₆×C₄₇	C₂×D₄	1	2	yes	—
90	C₂³×C₄₇	C₂×D₈	1	2	yes	—
91	C₂²×D₄₇	—	4	4	no	—
92	D₄×C₄₇	—	2	2	no	—
93	C₂×C₄×C₄₇	—	2	2	no	—
94	C₂×C₄×C₄₇	—	2	2	no	—
95	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
97	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
99	C₂²⋊D₄₇	—	8	8	no	—
100	C₂²⋊D₄₇	—	16	4	no	—
101	C₂×Dic₄₇	—	4	2	no	—
102	C₂×Dic₄₇	—	4	2	no	—
103	D₁₈₈	—	4	4	no	—
104	D₁₈₈	—	4	4	no	—
105	D₁₈₈	—	8	4	no	—
106	D₁₈₈	—	16	2	no	—
107	C₄×D₄₇	—	8	8	no	—
108	Dic₉₄	—	2	4	no	—
109	Dic₉₄	—	4	2	no	—
110	C₈×C₄₇	—	2	2	no	—
111	C₈×C₄₇	—	2	1	no	—
112	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
113	C₈×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
114	#33853	—	47	1	no	maximal, Sylow
115	C₂²×C₄₇	—	2	2	no	—
116	C₂²×C₄₇	—	2	2	no	—
117	C₂²×C₄₇	—	2	2	no	—
118	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
119	C₂×D₄₇	—	8	4	no	—
120	C₂×D₄₇	—	8	4	no	—
121	C₂×D₄₇	—	8	4	no	—
122	C₄×C₄₇	—	2	1	no	—
123	C₄×C₄₇	—	2	2	no	—
124	C₄×C₄₇	#508	1	2	yes	complemented
125	C₄×C₄₇	D₄²	1	1	yes	characteristic
126	C₄×C₄₇	#576	1	1	yes	characteristic
127	Dic₄₇	—	4	4	no	—
128	Dic₄₇	—	8	2	no	—
129	C₂×#508	—	47	2	no	—
130	#3287	—	47	2	no	—
131	#2246	—	47	1	no	—
132	#2215	—	47	2	no	—
133	#2207	—	47	2	no	—
134	#2186	—	47	4	no	—
135	C₄×Q₃₂	—	47	1	no	—
136	D₄×C₁₆	D₄₇	1	1	yes	characteristic, complemented
137	C₂×C₄₇	—	4	2	no	—
138	C₂×C₄₇	D₄×D₈	1	1	yes	characteristic
139	C₂×C₄₇	C₂×#508	1	1	yes	characteristic
140	C₂×C₄₇	#3415	1	1	yes	characteristic
141	D₄₇	—	16	4	no	—
142	C₂×#135	—	47	4	no	—
143	#550	—	94	2	no	—
144	#508	—	94	8	no	—
145	C₂×Q₃₂	—	47	1	no	—
146	C₂×SD₃₂	—	47	4	no	—
147	C₂×D₁₆	—	47	2	no	—
148	C₂×D₁₆	—	47	2	no	—
149	C₂²×C₁₆	C₂×D₄₇	1	2	yes	—
150	#493	—	47	2	no	—
151	#466	—	47	4	no	—
152	#460	—	94	2	no	—
153	#449	—	94	4	no	—
154	C₄×Q₁₆	—	47	2	no	—
155	C₈×D₄	C₂×D₄₇	1	1	yes	characteristic
156	#366	—	47	1	no	—
157	C₄⋊C₁₆	C₂×D₄₇	1	1	yes	characteristic
158	#358	—	47	2	no	—
159	#357	—	47	4	no	—
160	C₂²⋊C₁₆	C₂×D₄₇	1	2	yes	—
161	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
162	C₄₇	#33853	1	1	yes	characteristic, complemented, Sylow
163	C₂×#41	—	94	4	no	—
164	#135	—	94	8	no	—
165	#135	—	188	4	no	—
166	C₂×Q₁₆	—	47	2	no	—
167	C₂×SD₁₆	—	94	4	no	—
168	C₂×D₈	—	47	4	no	—
169	C₂×D₈	—	94	2	no	—
170	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
171	#127	—	94	2	no	—
172	#127	—	188	2	no	—
173	#121	—	94	4	no	—
174	#121	—	188	4	no	—
175	C₄×Q₈	—	94	2	no	—
176	C₄×D₄	—	188	2	no	—
177	C₄×D₄	D₁₈₈	1	1	yes	characteristic
178	Q₃₂	—	47	4	no	—
179	SD₃₂	—	94	8	no	—
180	D₁₆	—	94	4	no	—
181	D₁₆	—	188	2	no	—
182	C₂×C₁₆	—	2	4	no	—
183	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
184	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
185	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
186	#107	—	47	2	no	—
187	C₄⋊C₈	D₁₈₈	1	1	yes	characteristic
188	#103	—	94	2	no	—
189	#102	—	94	4	no	—
190	C₂²⋊C₈	D₁₈₈	1	2	yes	—
191	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
192	#41	—	376	4	no	—
193	#41	—	188	8	no	—
194	C₂×Q₈	—	94	2	no	—
195	C₂×D₄	—	94	4	no	—
196	C₂×D₄	—	188	4	no	—
197	C₂×D₄	—	188	4	no	—
198	C₂×D₄	—	188	2	no	—
199	C₂×D₄	—	188	2	no	—
200	C₂×D₄	D₃₇₆	1	1	yes	characteristic
201	C₂²×C₄	—	188	4	no	—
202	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
203	Q₁₆	—	47	4	no	—
204	Q₁₆	—	94	2	no	—
205	SD₁₆	—	188	8	no	—
206	D₈	—	94	4	no	—
207	D₈	—	94	4	no	—
208	D₈	—	376	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
212	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
213	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
214	C₄⋊C₄	—	94	2	no	—
215	C₄⋊C₄	—	188	2	no	—
216	C₄⋊C₄	D₃₇₆	1	1	yes	characteristic
217	C₂²⋊C₄	—	188	4	no	—
218	C₂²⋊C₄	D₃₇₆	1	2	yes	—
219	C₄²	—	188	2	no	—
220	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
221	C₁₆	—	2	2	no	—
222	C₁₆	—	2	1	no	—
223	C₁₆	D₄×D₄₇	1	2	yes	—
224	C₂³	—	188	4	no	—
225	C₂³	C₂×D₃₇₆	1	2	yes	—
226	Q₈	—	94	4	no	—
227	Q₈	—	188	2	no	—
228	D₄	—	2	2	no	—
229	D₄	—	376	8	no	—
230	D₄	—	376	4	no	—
231	D₄	—	188	4	no	—
232	D₄	—	188	4	no	—
233	D₄	—	752	2	no	—
234	D₄	—	752	4	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	376	8	no	—
238	C₂×C₄	—	188	2	no	—
239	C₂×C₄	—	188	2	no	—
240	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
241	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
242	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
243	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
244	C₈	—	2	2	no	—
245	C₈	—	2	1	no	—
246	C₈	C₂×D₄×D₄₇	1	1	yes	characteristic
247	C₈	#13094070	1	1	yes	characteristic
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	376	4	no	—
252	C₂²	—	376	4	no	—
253	C₂²	—	376	4	no	—
254	C₂²	C₂²×D₃₇₆	1	1	yes	characteristic
255	C₄	—	2	1	no	—
256	C₄	—	2	2	no	—
257	C₄	—	376	2	no	—
258	C₄	—	188	4	no	—
259	C₄	#15689555	1	2	yes	complemented
260	C₄	D₄×D₁₈₈	1	1	yes	characteristic
261	C₄	#15689928	1	1	yes	characteristic
262	C₂	—	4	2	no	—
263	C₂	—	752	4	no	—
264	C₂	D₄×D₃₇₆	1	1	yes	characteristic
265	C₂	C₂×#15689555	1	1	yes	characteristic
266	C₂	#20256529	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₇₆
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	D₄×C₁₆×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×D₈
hypercenter	D₄×C₁₆	D₄₇
hypocenter	C₄₇	#33853
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#15689555	C₂	semidirect
2. #20255148	C₂	semidirect
3. #20245638	C₂	semidirect
4. #20245474	C₂	semidirect
5. #20245429	C₂	semidirect
6. #20245313	C₂	semidirect
7. C₄×Dic₃₇₆	C₂	semidirect
8. D₄×C₁₆×C₄₇	C₂	semidirect
42. Dic₃₇₆	D₄	semidirect
124. C₄×C₄₇	#508	semidirect
136. D₄×C₁₆	D₄₇	semidirect
162. C₄₇	#33853	semidirect
259. C₄	#15689555	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689555	2
#20255148	2
#20245638	2
#20245474	2
#20245429	2
#20245313	2
C₄×Dic₃₇₆	2
D₄×C₁₆×C₄₇	2
#33853	47