Subgroups — #26134868

Order 12032.

366 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245144	C₂	1	1	yes	maximal, characteristic
2	C₄×#13093960	C₂	1	1	yes	maximal, characteristic
3	#20241496	C₂	1	4	yes	maximal
4	C₂×C₄₇×#500	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093964	C₂²	1	2	yes	—
6	C₂×#13093960	C₂²	1	2	yes	—
7	C₄×#10625329	—	2	1	no	—
8	C₄⋊(C₄₇⋊C₁₆)	—	2	1	no	—
9	C₄⋊(C₄₇⋊C₁₆)	—	2	1	no	—
10	#15688888	—	2	1	no	—
11	C₂×C₄⋊Q₈×C₄₇	—	2	1	no	—
12	C₄₇×#500	—	2	1	no	—
13	C₄₇×#500	—	2	1	no	—
14	C₄₇×#500	C₄	1	2	yes	—
15	C₄₇×#500	C₄	1	2	yes	—
16	C₂×#107×C₄₇	—	2	1	no	—
17	C₂×#107×C₄₇	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×#10625329	—	2	2	no	—
20	#13093964	—	2	4	no	—
21	C₂×C₄₇⋊C₁₆	—	2	2	no	—
22	#13093960	—	2	2	no	—
23	#13093960	Q₈	1	4	yes	complemented
24	#13093935	—	4	1	no	—
25	C₄×C₄₇⋊C₈	—	4	1	no	—
26	C₂²×Q₈×C₄₇	—	4	1	no	—
27	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
29	C₄⋊Q₈×C₄₇	—	2	2	no	—
30	C₄⋊Q₈×C₄₇	—	2	2	no	—
31	C₄⋊Q₈×C₄₇	—	4	1	no	—
32	C₄⋊Q₈×C₄₇	—	4	1	no	—
33	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
34	C₂×C₄⋊C₄×C₄₇	—	4	1	no	—
35	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
36	#107×C₄₇	—	2	2	no	—
37	#107×C₄₇	—	2	2	no	—
38	#107×C₄₇	—	2	2	no	—
39	#107×C₄₇	C₂×C₄	1	4	yes	—
40	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
43	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
44	C₂×Q₈×C₄₇	—	4	2	no	—
45	C₂×Q₈×C₄₇	—	4	2	no	—
46	C₂×Q₈×C₄₇	—	8	4	no	—
47	C₂²×C₄×C₄₇	—	4	2	no	—
48	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₄₇	—	2	2	no	—
52	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
57	C₂×C₈×C₄₇	#41	1	2	yes	—
58	C₄⋊C₄×C₄₇	—	2	4	no	—
59	C₄⋊C₄×C₄₇	—	4	2	no	—
60	C₄⋊C₄×C₄₇	—	4	2	no	—
61	C₄⋊C₄×C₄₇	—	4	2	no	—
62	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₄²×C₄₇	D₈	1	1	yes	characteristic
65	C₄²×C₄₇	D₈	1	1	yes	characteristic
66	#10625329	—	2	4	no	—
67	#10625329	—	4	2	no	—
68	C₂×C₄₇⋊C₈	—	4	2	no	—
69	C₄₇⋊C₁₆	—	4	2	no	—
70	C₂³×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
71	Q₈×C₄₇	—	8	4	no	—
72	Q₈×C₄₇	—	8	4	no	—
73	C₂×C₄×C₄₇	—	2	2	no	—
74	C₂×C₄×C₄₇	—	2	2	no	—
75	C₂×C₄×C₄₇	—	4	4	no	—
76	C₂×C₄×C₄₇	—	8	4	no	—
77	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
78	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
79	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
80	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
81	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
83	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₄₇	#137	1	2	yes	—
85	C₈×C₄₇	—	2	1	no	—
86	C₈×C₄₇	—	2	1	no	—
87	C₈×C₄₇	C₄×Q₈	1	2	yes	—
88	C₈×C₄₇	#124	1	1	yes	characteristic
89	C₈×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
90	C₄₇⋊C₈	—	4	4	no	—
91	#13279	—	47	1	no	maximal, Sylow
92	C₂²×C₄₇	—	2	1	no	—
93	C₂²×C₄₇	—	2	1	no	—
94	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
95	C₂²×C₄₇	#475	1	2	yes	—
96	C₄×C₄₇	—	2	1	no	—
97	C₄×C₄₇	—	2	1	no	—
98	C₄×C₄₇	—	8	4	no	—
99	C₄×C₄₇	#361	1	2	yes	—
100	C₄×C₄₇	#362	1	2	yes	—
101	C₄×C₄₇	#391	1	1	yes	characteristic
102	C₄×C₄₇	#419	1	1	yes	characteristic
103	C₄×C₄₇	#474	1	2	yes	—
104	C₂×#500	D₄₇	1	1	yes	characteristic
105	#2150	—	47	1	no	—
106	C₄×#108	—	47	1	no	—
107	#1387	—	47	4	no	—
108	C₂×C₄₇	—	2	2	no	—
109	C₂×C₄₇	#1988	1	1	yes	characteristic
110	C₂×C₄₇	C₂×#361	1	1	yes	characteristic
111	C₂×C₄₇	C₂×#362	1	1	yes	characteristic
112	C₂×C₄⋊Q₈	—	2	1	no	—
113	C₂×M₃₂	—	47	2	no	—
114	#500	—	2	1	no	—
115	#500	—	2	1	no	—
116	#500	Dic₄₇	1	2	yes	—
117	#500	Dic₄₇	1	2	yes	—
118	C₂×#108	—	47	2	no	—
119	C₂×#107	—	2	1	no	—
120	C₂×#107	C₂×D₄₇	1	2	yes	—
121	C₄×M₁₆	—	94	1	no	—
122	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
123	C₄⋊C₁₆	—	94	1	no	—
124	C₄⋊C₁₆	—	94	1	no	—
125	#341	—	94	1	no	—
126	C₄₇	#13279	1	1	yes	characteristic, complemented, Sylow
127	C₂²×Q₈	—	4	1	no	—
128	C₂×M₁₆	—	94	2	no	—
129	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
130	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
131	C₄⋊Q₈	—	2	2	no	—
132	C₄⋊Q₈	—	2	2	no	—
133	C₄⋊Q₈	—	4	1	no	—
134	C₄⋊Q₈	—	4	1	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	4	1	no	—
137	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
138	M₃₂	—	94	4	no	—
139	C₂×C₁₆	—	94	2	no	—
140	#108	—	47	4	no	—
141	#108	—	94	2	no	—
142	#107	—	2	2	no	—
143	#107	—	2	2	no	—
144	#107	—	2	2	no	—
145	#107	C₂×Dic₄₇	1	4	yes	—
146	#97	—	188	1	no	—
147	C₄×C₈	—	188	1	no	—
148	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
149	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
150	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
151	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
152	C₂×Q₈	—	4	2	no	—
153	C₂×Q₈	—	4	2	no	—
154	C₂×Q₈	—	8	4	no	—
155	C₂²×C₄	—	4	2	no	—
156	C₂²×C₄	#10625335	1	1	yes	characteristic
157	C₂²×C₄	#10625335	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
159	M₁₆	—	94	4	no	—
160	M₁₆	—	188	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	188	2	no	—
163	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
164	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
165	C₂×C₈	Q₈×D₄₇	1	2	yes	—
166	C₂×C₈	#10625354	1	2	yes	—
167	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
168	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
169	C₄⋊C₄	—	2	4	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄²	#10625334	1	1	yes	characteristic
174	C₄²	#10625334	1	1	yes	characteristic
175	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
176	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
177	C₁₆	—	188	2	no	—
178	C₂³	C₂×#10625335	1	1	yes	characteristic
179	Q₈	—	8	4	no	—
180	Q₈	—	8	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	8	4	no	—
185	C₂×C₄	#13093945	1	1	yes	characteristic
186	C₂×C₄	#13093945	1	1	yes	characteristic
187	C₂×C₄	#13093945	1	1	yes	characteristic
188	C₂×C₄	#13093945	1	1	yes	characteristic
189	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
190	C₂×C₄	#13094000	1	2	yes	—
191	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
192	C₂×C₄	#13094055	1	2	yes	—
193	C₈	—	2	1	no	—
194	C₈	—	2	1	no	—
195	C₈	—	188	4	no	—
196	C₈	Q₈×Dic₄₇	1	2	yes	—
197	C₈	#13094013	1	1	yes	characteristic
198	C₈	#13094022	1	1	yes	characteristic
199	C₂²	—	2	1	no	—
200	C₂²	—	2	1	no	—
201	C₂²	C₂×#13093945	1	1	yes	characteristic
202	C₂²	#15689405	1	2	yes	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	8	4	no	—
206	C₄	#15688936	1	2	yes	—
207	C₄	#15688939	1	2	yes	—
208	C₄	#15689030	1	1	yes	characteristic
209	C₄	#15689132	1	1	yes	characteristic
210	C₄	#15689400	1	2	yes	—
211	C₂	—	2	2	no	—
212	C₂	#20244341	1	1	yes	characteristic
213	C₂	C₂×#15688936	1	1	yes	characteristic
214	C₂	C₂×#15688939	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13093945
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#500	C₂
socle	C₂²×C₄₇	C₂×#102
prime radical	C₂³×C₄₇	C₂×SD₁₆
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×#500	D₄₇
hypocenter	C₄₇	#13279
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
23. #13093960	Q₈	semidirect
126. C₄₇	#13279	semidirect

Maximal subgroups

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

subgroup	index
#20245144	2
C₄×#13093960	2
#20241496	2
C₂×C₄₇×#500	2
#13279	47