Subgroups — #26134535

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×#15688939	C₂	1	4	yes	maximal
2	C₄×#13093964	C₂	1	1	yes	maximal, characteristic
3	#20243560	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄₇×#494	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093964	C₂²	1	2	yes	—
6	C₂×#13093960	—	2	1	no	—
7	C₂×#13093960	C₂²	1	2	yes	—
8	#15689147	—	2	1	no	—
9	#15688939	—	2	8	no	—
10	#15688899	—	2	1	no	—
11	C₄×C₄₇⋊C₁₆	—	2	1	no	—
12	C₂²×Q₁₆×C₄₇	—	2	1	no	—
13	C₂²×Q₁₆×C₄₇	C₂²	1	2	yes	—
14	C₂×C₄⋊Q₈×C₄₇	—	2	1	no	—
15	C₄₇×#494	—	2	1	no	—
16	C₄₇×#494	—	2	1	no	—
17	C₄₇×#494	C₄	1	2	yes	—
18	C₄₇×#494	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×#10625329	—	2	2	no	—
21	#13093964	—	2	2	no	—
22	#13093964	D₄	1	4	yes	—
23	C₂×C₄₇⋊C₁₆	—	2	2	no	—
24	#13093960	—	2	4	no	—
25	#13093960	—	4	2	no	—
26	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
27	C₂²×Q₈×C₄₇	—	2	2	no	—
28	C₂×Q₁₆×C₄₇	—	2	2	no	—
29	C₂×Q₁₆×C₄₇	—	2	2	no	—
30	C₂×Q₁₆×C₄₇	—	2	4	no	—
31	C₂×Q₁₆×C₄₇	—	2	2	no	—
32	C₂×Q₁₆×C₄₇	—	2	4	no	—
33	C₂×Q₁₆×C₄₇	—	4	2	no	—
34	C₂×Q₁₆×C₄₇	—	4	2	no	—
35	C₂×Q₁₆×C₄₇	C₂×C₄	1	4	yes	—
36	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
38	C₄⋊Q₈×C₄₇	—	2	2	no	—
39	C₄⋊Q₈×C₄₇	—	2	2	no	—
40	C₄⋊Q₈×C₄₇	—	4	2	no	—
41	C₂×C₄⋊C₄×C₄₇	—	4	2	no	—
42	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
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	C₂×Q₈×C₄₇	—	2	4	no	—
48	C₂×Q₈×C₄₇	—	4	2	no	—
49	C₂×Q₈×C₄₇	—	4	2	no	—
50	C₂×Q₈×C₄₇	—	4	2	no	—
51	C₂×Q₈×C₄₇	—	4	2	no	—
52	C₂×Q₈×C₄₇	—	4	2	no	—
53	C₂²×C₄×C₄₇	—	4	2	no	—
54	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
56	Q₁₆×C₄₇	—	2	8	no	—
57	Q₁₆×C₄₇	—	4	4	no	—
58	Q₁₆×C₄₇	—	4	4	no	—
59	Q₁₆×C₄₇	—	4	4	no	—
60	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₄₇	D₈	1	2	yes	—
63	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
64	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₄₇	#41	1	2	yes	—
68	C₄⋊C₄×C₄₇	—	4	4	no	—
69	C₄⋊C₄×C₄₇	—	4	4	no	—
70	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₄₇	D₈	1	2	yes	—
73	#10625329	—	4	2	no	—
74	#10625329	—	4	2	no	—
75	C₂×C₄₇⋊C₈	—	4	2	no	—
76	C₄₇⋊C₁₆	—	2	4	no	—
77	C₂³×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
78	Q₈×C₄₇	—	4	4	no	—
79	Q₈×C₄₇	—	4	4	no	—
80	Q₈×C₄₇	—	8	2	no	—
81	Q₈×C₄₇	—	8	2	no	—
82	C₂×C₄×C₄₇	—	2	4	no	—
83	C₂×C₄×C₄₇	—	4	4	no	—
84	C₂×C₄×C₄₇	—	8	2	no	—
85	C₂×C₄×C₄₇	—	8	2	no	—
86	C₂×C₄×C₄₇	#102	1	2	yes	—
87	C₂×C₄×C₄₇	#102	1	2	yes	—
88	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
90	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
91	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
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	—	8	4	no	—
106	C₄×C₄₇	#362	1	4	yes	complemented
107	C₄×C₄₇	#390	1	1	yes	characteristic
108	C₄×C₄₇	C₂×#102	1	1	yes	characteristic
109	C₄×C₄₇	#486	1	1	yes	characteristic
110	C₄×C₄₇	#493	1	1	yes	characteristic
111	C₂×#494	D₄₇	1	1	yes	characteristic
112	C₂×#362	—	47	4	no	—
113	C₄×M₃₂	—	47	1	no	—
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	—	2	1	no	—
120	C₂²×Q₁₆	C₂×D₄₇	1	2	yes	—
121	C₂×C₄⋊Q₈	—	2	1	no	—
122	C₂×M₃₂	—	47	2	no	—
123	#494	—	2	1	no	—
124	#494	—	2	1	no	—
125	#494	Dic₄₇	1	2	yes	—
126	#494	Dic₄₇	1	2	yes	—
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	—	94	1	no	—
133	C₄×C₁₆	—	94	1	no	—
134	C₄₇	#13212	1	1	yes	characteristic, complemented, Sylow
135	C₂²×Q₈	—	2	2	no	—
136	C₂×Q₁₆	—	2	2	no	—
137	C₂×Q₁₆	—	2	2	no	—
138	C₂×Q₁₆	—	2	4	no	—
139	C₂×Q₁₆	—	2	4	no	—
140	C₂×Q₁₆	—	2	2	no	—
141	C₂×Q₁₆	—	4	2	no	—
142	C₂×Q₁₆	—	4	2	no	—
143	C₂×Q₁₆	C₂×Dic₄₇	1	4	yes	—
144	C₂×M₁₆	—	94	2	no	—
145	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
146	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
147	C₄⋊Q₈	—	2	2	no	—
148	C₄⋊Q₈	—	2	2	no	—
149	C₄⋊Q₈	—	4	2	no	—
150	C₂×C₄⋊C₄	—	4	2	no	—
151	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
152	M₃₂	—	47	4	no	—
153	M₃₂	—	94	2	no	—
154	C₂×C₁₆	—	94	2	no	—
155	#108	—	94	4	no	—
156	#108	—	188	2	no	—
157	C₄⋊C₈	—	188	2	no	—
158	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
159	C₄×C₈	C₂×Dic₄₇	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	—	2	4	no	—
163	C₂×Q₈	—	4	2	no	—
164	C₂×Q₈	—	4	2	no	—
165	C₂×Q₈	—	4	2	no	—
166	C₂×Q₈	—	4	2	no	—
167	C₂×Q₈	—	4	2	no	—
168	C₂²×C₄	—	4	2	no	—
169	C₂²×C₄	#10625335	1	2	yes	—
170	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
171	Q₁₆	—	2	8	no	—
172	Q₁₆	—	4	4	no	—
173	Q₁₆	—	4	4	no	—
174	Q₁₆	—	4	4	no	—
175	M₁₆	—	188	2	no	—
176	M₁₆	—	188	2	no	—
177	C₂×C₈	—	188	2	no	—
178	C₂×C₈	#10625334	1	2	yes	—
179	C₂×C₈	#10625335	1	2	yes	—
180	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
181	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
182	C₂×C₈	D₄×D₄₇	1	2	yes	—
183	C₂×C₈	#10625352	1	2	yes	—
184	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
185	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
186	C₄⋊C₄	—	4	4	no	—
187	C₄⋊C₄	—	4	4	no	—
188	C₄²	#10625334	1	2	yes	—
189	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
190	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
191	C₁₆	—	94	4	no	—
192	C₂³	C₂×#10625335	1	1	yes	characteristic
193	Q₈	—	4	4	no	—
194	Q₈	—	4	4	no	—
195	Q₈	—	8	2	no	—
196	Q₈	—	8	2	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	—	8	2	no	—
200	C₂×C₄	—	8	2	no	—
201	C₂×C₄	#13093945	1	2	yes	—
202	C₂×C₄	#13093945	1	2	yes	—
203	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
204	C₂×C₄	#13094011	1	1	yes	characteristic
205	C₂×C₄	#13094020	1	1	yes	characteristic
206	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
207	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
208	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
209	C₈	—	376	2	no	—
210	C₈	#13093945	1	2	yes	—
211	C₈	#13093945	1	2	yes	—
212	C₈	D₄×Dic₄₇	1	2	yes	—
213	C₈	#13093995	1	2	yes	—
214	C₂²	—	2	2	no	—
215	C₂²	C₂×#13093945	1	1	yes	characteristic
216	C₂²	#15689469	1	1	yes	characteristic
217	C₂²	#15689495	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	8	4	no	—
220	C₄	#15688939	1	4	yes	complemented
221	C₄	#15689026	1	1	yes	characteristic
222	C₄	C₂×#13093945	1	1	yes	characteristic
223	C₄	#15689469	1	1	yes	characteristic
224	C₄	#15689499	1	1	yes	characteristic
225	C₂	—	2	1	no	—
226	C₂	—	2	1	no	—
227	C₂	#20244195	1	1	yes	characteristic
228	C₂	C₂×#15688939	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#13093945
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#494	C₂
socle	C₂²×C₄₇	C₂×#102
prime radical	C₂³×C₄₇	C₂×SD₁₆
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×#494	D₄₇
hypocenter	C₄₇	#13212
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
106. C₄×C₄₇	#362	semidirect
134. C₄₇	#13212	semidirect
220. C₄	#15688939	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688939	2
C₄×#13093964	2
#20243560	2
C₂×C₄₇×#494	2
#13212	47