Subgroups — #26134534

Order 12032.

470 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689503	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688940	C₂	1	4	yes	maximal
3	#20243560	C₂	1	1	yes	maximal, characteristic
4	C₄×M₃₂×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₈₈	—	2	1	no	—
6	C₂²×Dic₁₈₈	C₂²	1	2	yes	—
7	C₂×#13094024	—	2	1	no	—
8	#15689503	—	2	1	no	—
9	#15689503	—	2	1	no	—
10	#15689503	C₄	1	2	yes	—
11	#15689503	C₄	1	2	yes	—
12	C₂×#13093960	—	2	1	no	—
13	C₂×#13093960	C₂²	1	2	yes	—
14	#15689147	—	2	1	no	—
15	#15688940	—	2	8	no	—
16	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#346	—	2	1	no	—
19	C₄×C₁₆×C₄₇	—	2	1	no	—
20	C₂²×Dic₉₄	—	2	2	no	—
21	C₂×Dic₁₈₈	—	2	4	no	—
22	C₂×Dic₁₈₈	—	2	4	no	—
23	C₂×Dic₁₈₈	—	2	2	no	—
24	C₂×Dic₁₈₈	—	2	2	no	—
25	C₂×Dic₁₈₈	—	2	2	no	—
26	C₂×Dic₁₈₈	—	4	2	no	—
27	C₂×Dic₁₈₈	—	4	2	no	—
28	C₂×Dic₁₈₈	C₂×C₄	1	4	yes	—
29	C₂×#10625329	—	2	2	no	—
30	#13094024	—	2	2	no	—
31	#13094024	—	2	2	no	—
32	#13094024	—	4	2	no	—
33	C₂×C₄⋊Dic₄₇	—	4	2	no	—
34	#13093960	—	2	4	no	—
35	#13093960	—	4	2	no	—
36	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
37	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
40	M₃₂×C₄₇	—	2	2	no	—
41	M₃₂×C₄₇	D₄	1	4	yes	—
42	C₂×C₁₆×C₄₇	—	2	2	no	—
43	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
46	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
48	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×Dic₄₇	—	4	2	no	—
50	C₂×Dic₉₄	—	2	4	no	—
51	C₂×Dic₉₄	—	4	2	no	—
52	C₂×Dic₉₄	—	4	2	no	—
53	C₂×Dic₉₄	—	4	2	no	—
54	C₂×Dic₉₄	—	4	2	no	—
55	C₂×Dic₉₄	—	4	2	no	—
56	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₄₇	D₈	1	2	yes	—
59	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
60	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₄₇	#41	1	2	yes	—
64	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₄²×C₄₇	D₈	1	2	yes	—
67	C₄⋊Dic₄₇	—	4	4	no	—
68	C₄⋊Dic₄₇	—	4	4	no	—
69	#10625329	—	4	2	no	—
70	#10625329	—	4	2	no	—
71	C₂×C₄₇⋊C₈	—	4	2	no	—
72	Dic₁₈₈	—	2	8	no	—
73	Dic₁₈₈	—	4	4	no	—
74	Dic₁₈₈	—	4	4	no	—
75	Dic₁₈₈	—	4	4	no	—
76	C₁₆×C₄₇	—	2	4	no	—
77	C₂³×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₄₇	—	2	4	no	—
79	C₂×C₄×C₄₇	#102	1	2	yes	—
80	C₂×C₄×C₄₇	#102	1	2	yes	—
81	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
83	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
84	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×Dic₄₇	—	4	4	no	—
88	C₂×Dic₄₇	—	8	2	no	—
89	C₂×Dic₄₇	—	8	2	no	—
90	Dic₉₄	—	4	4	no	—
91	Dic₉₄	—	4	4	no	—
92	Dic₉₄	—	8	2	no	—
93	Dic₉₄	—	8	2	no	—
94	C₈×C₄₇	#102	1	2	yes	—
95	C₈×C₄₇	#102	1	2	yes	—
96	C₈×C₄₇	C₄×D₄	1	2	yes	—
97	C₈×C₄₇	#121	1	2	yes	—
98	C₄₇⋊C₈	—	8	2	no	—
99	#13212	—	47	1	no	maximal, Sylow
100	C₂²×C₄₇	—	2	2	no	—
101	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
102	C₂²×C₄₇	#486	1	1	yes	characteristic
103	C₂²×C₄₇	#492	1	1	yes	characteristic
104	C₄×C₄₇	—	2	2	no	—
105	C₄×C₄₇	#362	1	4	yes	complemented
106	C₄×C₄₇	#390	1	1	yes	characteristic
107	C₄×C₄₇	C₂×#102	1	1	yes	characteristic
108	C₄×C₄₇	#486	1	1	yes	characteristic
109	C₄×C₄₇	#493	1	1	yes	characteristic
110	Dic₄₇	—	8	4	no	—
111	C₂×#494	—	47	1	no	—
112	C₂×#362	—	47	4	no	—
113	C₄×M₃₂	D₄₇	1	1	yes	characteristic
114	#1848	—	47	1	no	—
115	C₂×C₄₇	—	2	1	no	—
116	C₂×C₄₇	—	2	1	no	—
117	C₂×C₄₇	#1957	1	1	yes	characteristic
118	C₂×C₄₇	C₂×#362	1	2	yes	—
119	C₂²×Q₁₆	—	47	2	no	—
120	C₂²×Q₁₆	—	94	1	no	—
121	C₂×C₄⋊Q₈	—	94	1	no	—
122	C₂×M₃₂	C₂×D₄₇	1	2	yes	—
123	#494	—	47	2	no	—
124	#494	—	47	2	no	—
125	#494	—	94	1	no	—
126	#494	—	94	1	no	—
127	C₂×#108	—	47	2	no	—
128	C₂×#108	—	94	1	no	—
129	#423	—	94	1	no	—
130	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
131	#362	—	94	8	no	—
132	#346	—	2	1	no	—
133	C₄×C₁₆	—	2	1	no	—
134	C₄₇	#13212	1	1	yes	characteristic, complemented, Sylow
135	C₂²×Q₈	—	94	2	no	—
136	C₂×Q₁₆	—	47	4	no	—
137	C₂×Q₁₆	—	94	2	no	—
138	C₂×Q₁₆	—	94	2	no	—
139	C₂×Q₁₆	—	94	4	no	—
140	C₂×Q₁₆	—	94	4	no	—
141	C₂×Q₁₆	—	94	2	no	—
142	C₂×Q₁₆	—	188	2	no	—
143	C₂×Q₁₆	—	188	2	no	—
144	C₂×M₁₆	—	94	2	no	—
145	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
146	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
147	C₄⋊Q₈	—	94	2	no	—
148	C₄⋊Q₈	—	94	2	no	—
149	C₄⋊Q₈	—	188	2	no	—
150	C₂×C₄⋊C₄	—	188	2	no	—
151	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
152	M₃₂	—	2	2	no	—
153	M₃₂	D₁₈₈	1	4	yes	—
154	C₂×C₁₆	—	2	2	no	—
155	#108	—	94	4	no	—
156	#108	—	188	2	no	—
157	C₄⋊C₈	—	188	2	no	—
158	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
159	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
160	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
161	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
162	C₂×Q₈	—	94	4	no	—
163	C₂×Q₈	—	188	2	no	—
164	C₂×Q₈	—	188	2	no	—
165	C₂×Q₈	—	188	2	no	—
166	C₂×Q₈	—	188	2	no	—
167	C₂×Q₈	—	188	2	no	—
168	C₂²×C₄	—	188	2	no	—
169	C₂²×C₄	#10625325	1	2	yes	—
170	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
171	Q₁₆	—	94	8	no	—
172	Q₁₆	—	188	4	no	—
173	Q₁₆	—	188	4	no	—
174	Q₁₆	—	188	4	no	—
175	M₁₆	—	188	2	no	—
176	M₁₆	—	188	2	no	—
177	C₂×C₈	—	188	2	no	—
178	C₂×C₈	#10625325	1	2	yes	—
179	C₂×C₈	D₃₇₆	1	2	yes	—
180	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
181	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
182	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
183	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
184	C₂×C₈	#10625350	1	2	yes	—
185	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
186	C₄⋊C₄	—	188	4	no	—
187	C₄⋊C₄	—	188	4	no	—
188	C₄²	D₃₇₆	1	2	yes	—
189	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
190	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
191	C₁₆	—	2	4	no	—
192	C₂³	C₂×#10625325	1	1	yes	characteristic
193	Q₈	—	376	2	no	—
194	Q₈	—	376	2	no	—
195	Q₈	—	188	4	no	—
196	Q₈	—	188	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	376	2	no	—
199	C₂×C₄	—	376	2	no	—
200	C₂×C₄	—	188	4	no	—
201	C₂×C₄	#13093946	1	2	yes	—
202	C₂×C₄	#13093946	1	2	yes	—
203	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
204	C₂×C₄	#13094012	1	1	yes	characteristic
205	C₂×C₄	#13094021	1	1	yes	characteristic
206	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
207	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
208	C₂×C₄	C₂×#10625325	1	1	yes	characteristic
209	C₈	—	376	2	no	—
210	C₈	#13093946	1	2	yes	—
211	C₈	#13093946	1	2	yes	—
212	C₈	C₄×D₁₈₈	1	2	yes	—
213	C₈	#13093998	1	2	yes	—
214	C₂²	—	2	2	no	—
215	C₂²	C₂×#13093946	1	1	yes	characteristic
216	C₂²	#15689470	1	1	yes	characteristic
217	C₂²	#15689496	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	376	4	no	—
220	C₄	#15688940	1	4	yes	complemented
221	C₄	#15689025	1	1	yes	characteristic
222	C₄	C₂×#13093946	1	1	yes	characteristic
223	C₄	#15689470	1	1	yes	characteristic
224	C₄	#15689500	1	1	yes	characteristic
225	C₂	—	2	1	no	—
226	C₂	—	2	1	no	—
227	C₂	#20244194	1	1	yes	characteristic
228	C₂	C₂×#15688940	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#13093946
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄×M₃₂×C₄₇	C₂
socle	C₂²×C₄₇	C₂×#102
prime radical	C₂³×C₄₇	C₂×SD₁₆
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₄×M₃₂	D₄₇
hypocenter	C₄₇	#13212
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
105. C₄×C₄₇	#362	semidirect
134. C₄₇	#13212	semidirect
220. C₄	#15688940	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689503	2
C₂×#15688940	2
#20243560	2
C₄×M₃₂×C₄₇	2
#13212	47