Subgroups — #26371784

Order 12032.

474 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253924	C₂	1	2	yes	maximal
2	#20245672	C₂	1	4	yes	maximal, complemented
3	#20245665	C₂	1	2	yes	maximal, complemented
4	#20245662	C₂	1	2	yes	maximal, complemented
5	#20245174	C₂	1	2	yes	maximal
6	#20245172	C₂	1	1	yes	maximal, characteristic
7	#20244984	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄₇×#366	C₂	1	1	yes	maximal, characteristic
9	#15689754	—	2	2	no	—
10	C₂×#13093964	C₂²	1	1	yes	characteristic
11	#15689531	—	2	4	no	—
12	#15689527	C₂²	1	4	yes	—
13	#15689524	C₂²	1	4	yes	—
14	#15689163	—	2	2	no	—
15	#15689161	C₂²	1	2	yes	—
16	#15689078	C₂²	1	1	yes	characteristic
17	#15688951	C₂²	1	2	yes	—
18	#15688950	C₂²	1	4	yes	—
19	#15688949	C₂²	1	1	yes	characteristic
20	#15688949	C₂²	1	1	yes	characteristic
21	#15688948	C₂²	1	4	yes	—
22	#15688898	C₂²	1	2	yes	—
23	C₁₆×Dic₄₇	C₂²	1	2	yes	—
24	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×#107×C₄₇	C₂²	1	2	yes	—
26	C₄₇×#366	C₂²	1	2	yes	—
27	C₄₇×#366	C₂²	1	2	yes	—
28	C₂×#10625329	D₄	1	1	yes	characteristic
29	#13094023	—	2	4	no	—
30	#13094023	—	4	2	no	—
31	#13094015	—	2	4	no	—
32	#13094015	—	4	2	no	—
33	#13094003	—	4	4	no	—
34	#13093983	—	2	2	no	—
35	#13093983	D₄	1	1	yes	characteristic
36	#13093964	Q₈	1	4	yes	complemented
37	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
38	#13093958	C₂³	1	2	yes	—
39	#13093958	C₂³	1	2	yes	—
40	#13093957	—	2	4	no	—
41	#13093956	—	2	2	no	—
42	#13093955	—	2	4	no	—
43	#13093934	D₄	1	2	yes	—
44	C₈×Dic₄₇	C₂³	1	2	yes	—
45	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
47	C₂×C₁₆×C₄₇	Q₈	1	4	yes	complemented
48	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
49	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
50	#107×C₄₇	—	2	2	no	—
51	#107×C₄₇	C₂³	1	4	yes	—
52	C₂²×C₄×C₄₇	—	4	2	no	—
53	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂×Dic₉₄	—	4	4	no	—
55	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
57	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
58	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
59	C₄⋊C₄×C₄₇	—	2	4	no	—
60	C₄⋊C₄×C₄₇	—	4	2	no	—
61	C₂²⋊Dic₄₇	—	4	2	no	—
62	C₂²⋊Dic₄₇	D₈	1	2	yes	—
63	C₄⋊Dic₄₇	—	2	2	no	—
64	C₄⋊Dic₄₇	—	2	2	no	—
65	C₄⋊Dic₄₇	—	4	4	no	—
66	C₄⋊Dic₄₇	D₈	1	1	yes	characteristic
67	C₄⋊Dic₄₇	D₈	1	1	yes	characteristic
68	Dic₄₇⋊C₄	—	4	4	no	—
69	Dic₄₇⋊C₄	—	4	4	no	—
70	C₄×Dic₄₇	—	4	2	no	—
71	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
72	#10625329	Q₁₆	1	4	yes	—
73	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
74	C₁₆×C₄₇	C₂×Q₈	1	2	yes	—
75	C₁₆×C₄₇	C₂×Q₈	1	2	yes	—
76	C₄₇⋊C₁₆	C₂×Q₈	1	4	yes	—
77	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₄₇	—	4	4	no	—
79	C₂×C₄×C₄₇	—	16	2	no	—
80	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
81	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
83	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
84	C₂×Dic₄₇	—	4	2	no	—
85	C₂×Dic₄₇	—	4	2	no	—
86	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
87	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
88	Dic₉₄	—	16	4	no	—
89	C₈×C₄₇	C₄⋊Q₈	1	2	yes	—
90	C₈×C₄₇	C₂²×Q₈	1	1	yes	characteristic
91	C₈×C₄₇	C₂²×Q₈	1	1	yes	characteristic
92	C₄₇⋊C₈	C₂×Q₁₆	1	4	yes	—
93	#33729	—	47	1	no	maximal, Sylow
94	C₂²×C₄₇	—	2	1	no	—
95	C₂²×C₄₇	—	2	1	no	—
96	C₂²×C₄₇	#500	1	2	yes	—
97	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
98	C₄×C₄₇	—	8	4	no	—
99	C₄×C₄₇	#500	1	2	yes	—
100	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
101	C₄×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
102	Dic₄₇	—	2	2	no	—
103	Dic₄₇	—	2	2	no	—
104	Dic₄₇	—	8	2	no	—
105	Dic₄₇	—	8	2	no	—
106	#3164	—	47	2	no	—
107	#2255	—	47	4	no	—
108	#2253	—	47	2	no	—
109	#2252	—	47	2	no	—
110	#2159	—	47	2	no	—
111	#2158	—	47	1	no	—
112	C₂×#366	D₄₇	1	1	yes	characteristic
113	#2108	—	47	1	no	—
114	C₂×C₄₇	—	2	2	no	—
115	C₂×C₄₇	C₂×#500	1	1	yes	characteristic
116	C₂×C₄₇	#3415	1	1	yes	characteristic
117	C₂×C₄₇	#3417	1	1	yes	characteristic
118	#544	—	94	2	no	—
119	C₂×M₃₂	—	47	1	no	—
120	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
121	#501	—	94	4	no	—
122	#500	—	47	4	no	—
123	#499	—	47	4	no	—
124	#428	—	94	2	no	—
125	#427	—	47	2	no	—
126	C₂×#107	C₂×D₄₇	1	2	yes	—
127	#405	—	47	1	no	—
128	#367	—	47	2	no	—
129	#367	—	47	4	no	—
130	#366	—	47	1	no	—
131	#366	—	47	1	no	—
132	#366	—	47	4	no	—
133	#366	C₂×D₄₇	1	2	yes	—
134	#366	C₂×D₄₇	1	2	yes	—
135	#346	—	47	2	no	—
136	C₄×C₁₆	—	47	2	no	—
137	C₄₇	#33729	1	1	yes	characteristic, complemented, Sylow
138	C₂×M₁₆	—	47	1	no	—
139	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
140	C₄⋊Q₈	—	94	4	no	—
141	C₄⋊Q₈	—	188	2	no	—
142	#125	—	94	4	no	—
143	#125	—	188	2	no	—
144	C₂²⋊Q₈	—	188	4	no	—
145	#117	—	47	1	no	—
146	#117	—	94	2	no	—
147	C₂×C₄⋊C₄	—	2	2	no	—
148	M₃₂	—	47	4	no	—
149	C₂×C₁₆	—	47	2	no	—
150	C₂×C₁₆	Dic₉₄	1	4	yes	complemented
151	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
152	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
153	#107	—	2	2	no	—
154	#107	—	47	2	no	—
155	#107	—	47	2	no	—
156	#107	—	94	4	no	—
157	#107	C₂²×D₄₇	1	4	yes	—
158	#106	—	94	4	no	—
159	#106	—	94	2	no	—
160	#97	—	47	2	no	—
161	C₄×C₈	—	47	2	no	—
162	C₂×Q₈	—	188	4	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
165	M₁₆	—	47	4	no	—
166	C₂×C₈	—	47	2	no	—
167	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
168	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
169	C₂×C₈	Q₈×D₄₇	1	2	yes	—
170	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
171	C₄⋊C₄	—	2	4	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄⋊C₄	—	47	1	no	—
174	C₄⋊C₄	—	47	1	no	—
175	C₄⋊C₄	—	94	2	no	—
176	C₄⋊C₄	—	94	2	no	—
177	C₄⋊C₄	—	188	4	no	—
178	C₄⋊C₄	—	188	4	no	—
179	C₄⋊C₄	—	188	4	no	—
180	C₂²⋊C₄	—	47	2	no	—
181	C₂²⋊C₄	—	188	2	no	—
182	C₄²	—	47	2	no	—
183	C₄²	—	188	2	no	—
184	C₁₆	—	47	4	no	—
185	C₁₆	C₂×Dic₉₄	1	2	yes	—
186	C₁₆	C₂×Dic₉₄	1	2	yes	—
187	C₂³	D₈×D₄₇	1	1	yes	characteristic
188	Q₈	—	752	4	no	—
189	C₂×C₄	—	4	4	no	—
190	C₂×C₄	—	16	2	no	—
191	C₂×C₄	—	47	2	no	—
192	C₂×C₄	—	47	2	no	—
193	C₂×C₄	—	188	2	no	—
194	C₂×C₄	—	188	2	no	—
195	C₂×C₄	#13094023	1	2	yes	—
196	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
197	C₂×C₄	Q₁₆×D₄₇	1	2	yes	—
198	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
199	C₈	—	47	4	no	—
200	C₈	#13094023	1	2	yes	—
201	C₈	C₂×Q₈×D₄₇	1	1	yes	characteristic
202	C₈	C₂²×Dic₉₄	1	1	yes	characteristic
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	#15689527	1	2	yes	—
206	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
207	C₄	—	8	4	no	—
208	C₄	—	94	2	no	—
209	C₄	—	94	2	no	—
210	C₄	—	376	2	no	—
211	C₄	—	376	2	no	—
212	C₄	#15689527	1	2	yes	—
213	C₄	C₂×#13094023	1	1	yes	characteristic
214	C₄	C₂×Q₁₆×D₄₇	1	1	yes	characteristic
215	C₂	—	2	2	no	—
216	C₂	C₂×#15689527	1	1	yes	characteristic
217	C₂	#20256525	1	1	yes	characteristic
218	C₂	#20256537	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₇
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄₇×#366	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂³×C₄₇	C₂×D₈
hypercenter	C₂×#366	D₄₇
hypocenter	C₄₇	#33729
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
2. #20245672	C₂	semidirect
3. #20245665	C₂	semidirect
4. #20245662	C₂	semidirect
36. #13093964	Q₈	semidirect
47. C₂×C₁₆×C₄₇	Q₈	semidirect
137. C₄₇	#33729	semidirect
150. C₂×C₁₆	Dic₉₄	semidirect

Maximal subgroups

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

subgroup	index
#20253924	2
#20245672	2
#20245665	2
#20245662	2
#20245174	2
#20245172	2
#20244984	2
C₂×C₄₇×#366	2
#33729	47