Subgroups — #26216895

Order 12032.

758 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252021	C₂	1	2	yes	maximal
2	#20243563	C₂	1	2	yes	maximal, complemented
3	#20243529	C₂	1	2	yes	maximal, complemented
4	#20243430	C₂	1	4	yes	maximal
5	C₄×#13093960	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
7	#15689180	—	2	4	no	—
8	#15689176	—	2	4	no	—
9	C₂×#13093960	C₂²	1	8	yes	—
10	#15689147	C₂²	1	4	yes	—
11	#15689087	C₂²	1	4	yes	—
12	C₄×#10625329	C₂²	1	4	yes	—
13	#15688888	C₂²	1	8	yes	—
14	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
15	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₄×M₁₆×C₄₇	C₄	1	8	yes	—
17	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
18	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
20	C₂×#10625329	C₂³	1	8	yes	—
21	#13093960	—	2	16	no	—
22	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
23	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
24	#13093935	—	2	4	no	—
25	C₄×C₄₇⋊C₈	—	2	4	no	—
26	C₂³×C₄×C₄₇	Q₈	1	2	yes	—
27	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
28	C₂×M₁₆×C₄₇	—	2	8	no	—
29	C₂×M₁₆×C₄₇	C₂×C₄	1	8	yes	—
30	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
31	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
32	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₄₇	D₄	1	2	yes	—
34	C₂×C₄²×C₄₇	D₄	1	2	yes	—
35	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
36	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
38	#97×C₄₇	C₂×C₄	1	4	yes	—
39	#97×C₄₇	C₂×C₄	1	4	yes	—
40	C₄×C₈×C₄₇	C₂×C₄	1	8	yes	—
41	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	—	2	4	no	—
43	C₂²×C₄×C₄₇	—	4	4	no	—
44	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
45	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
46	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
50	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
51	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
52	M₁₆×C₄₇	—	2	16	no	—
53	C₂×C₈×C₄₇	—	2	4	no	—
54	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
55	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
56	C₂×C₈×C₄₇	#41	1	8	yes	—
57	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
58	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
59	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
60	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
61	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₄₇	C₂×D₄	1	2	yes	—
63	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
64	#10625329	—	2	16	no	—
65	C₂×C₄₇⋊C₈	—	2	8	no	—
66	C₂³×C₄₇	—	2	2	no	—
67	C₂³×C₄₇	—	4	2	no	—
68	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
69	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₄₇	—	2	2	no	—
71	C₂×C₄×C₄₇	—	2	4	no	—
72	C₂×C₄×C₄₇	—	2	4	no	—
73	C₂×C₄×C₄₇	—	2	8	no	—
74	C₂×C₄×C₄₇	—	2	4	no	—
75	C₂×C₄×C₄₇	—	4	4	no	—
76	C₂×C₄×C₄₇	—	4	2	no	—
77	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
78	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
79	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
80	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
81	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
82	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
83	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
84	C₈×C₄₇	—	2	4	no	—
85	C₈×C₄₇	#117	1	8	yes	—
86	C₄₇⋊C₈	—	4	8	no	—
87	#20827	—	47	1	no	maximal, Sylow
88	C₂²×C₄₇	—	2	2	no	—
89	C₂²×C₄₇	—	2	4	no	—
90	C₂²×C₄₇	—	2	2	no	—
91	C₂²×C₄₇	—	4	4	no	—
92	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
93	C₂²×C₄₇	#533	1	2	yes	—
94	C₄×C₄₇	—	2	4	no	—
95	C₄×C₄₇	—	2	2	no	—
96	C₄×C₄₇	—	2	2	no	—
97	C₄×C₄₇	#430	1	4	yes	—
98	C₄×C₄₇	C₂×#117	1	2	yes	—
99	C₄×C₄₇	#532	1	2	yes	—
100	#2920	—	47	2	no	—
101	C₂×C₄×M₁₆	D₄₇	1	1	yes	characteristic
102	#1849	—	47	2	no	—
103	#1838	—	47	2	no	—
104	#1811	—	47	4	no	—
105	C₄×#108	—	47	4	no	—
106	C₂×C₄₇	—	2	2	no	—
107	C₂×C₄₇	—	4	2	no	—
108	C₂×C₄₇	#2285	1	1	yes	characteristic
109	C₂×C₄₇	C₂×#430	1	2	yes	—
110	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
111	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
112	#432	—	94	4	no	—
113	#431	—	94	4	no	—
114	C₂×#108	—	47	8	no	—
115	#423	—	47	4	no	—
116	#407	—	47	4	no	—
117	C₄×M₁₆	—	47	4	no	—
118	C₄×M₁₆	Dic₄₇	1	8	yes	—
119	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
120	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
121	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
122	#341	—	47	8	no	—
123	C₄₇	#20827	1	1	yes	characteristic, complemented, Sylow
124	C₂³×C₄	Dic₉₄	1	2	yes	—
125	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
126	C₂×M₁₆	—	2	8	no	—
127	C₂×M₁₆	—	47	8	no	—
128	C₂×M₁₆	C₂×Dic₄₇	1	8	yes	—
129	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
130	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
131	C₂×C₄²	Dic₉₄	1	2	yes	—
132	C₂×C₄²	D₁₈₈	1	2	yes	—
133	C₂×C₄²	D₁₈₈	1	2	yes	—
134	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
135	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
136	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
137	#108	—	94	16	no	—
138	C₄⋊C₈	—	94	8	no	—
139	C₂²⋊C₈	—	94	8	no	—
140	#97	—	94	4	no	—
141	#97	C₂×Dic₄₇	1	4	yes	—
142	#97	C₂×Dic₄₇	1	4	yes	—
143	C₄×C₈	—	94	4	no	—
144	C₄×C₈	C₂×Dic₄₇	1	8	yes	—
145	C₂⁴	C₂×Dic₉₄	1	1	yes	characteristic
146	C₂²×C₄	—	2	4	no	—
147	C₂²×C₄	—	4	4	no	—
148	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
149	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
150	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
151	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
152	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
153	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
154	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
155	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
156	M₁₆	—	2	16	no	—
157	M₁₆	—	94	16	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	94	8	no	—
160	C₂×C₈	#10625350	1	8	yes	—
161	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
162	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
163	C₄²	C₄⋊Dic₄₇	1	4	yes	—
164	C₄²	C₄⋊Dic₄₇	1	4	yes	—
165	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
166	C₄²	C₂×D₁₈₈	1	2	yes	—
167	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
168	C₄²	C₂²×Dic₄₇	1	2	yes	—
169	C₄²	C₂²×Dic₄₇	1	2	yes	—
170	C₂³	—	2	2	no	—
171	C₂³	—	4	2	no	—
172	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
173	C₂³	C₂²×Dic₉₄	1	1	yes	characteristic
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
182	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
183	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
184	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
185	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
186	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
187	C₂×C₄	C₂×#10625350	1	4	yes	—
188	C₈	—	2	4	no	—
189	C₈	—	188	8	no	—
190	C₈	#13093983	1	8	yes	—
191	C₂²	—	2	2	no	—
192	C₂²	—	2	2	no	—
193	C₂²	—	2	4	no	—
194	C₂²	—	4	4	no	—
195	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
196	C₂²	#15689680	1	2	yes	—
197	C₄	—	2	2	no	—
198	C₄	—	2	4	no	—
199	C₄	—	2	2	no	—
200	C₄	#15689172	1	4	yes	—
201	C₄	C₂×#13093983	1	2	yes	—
202	C₄	#15689673	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	—	4	2	no	—
205	C₂	#20245795	1	1	yes	characteristic
206	C₂	C₂×#15689172	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄×M₁₆×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₄₇	C₂×Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₄₇
hypocenter	C₄₇	#20827
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #20243563	C₂	semidirect
3. #20243529	C₂	semidirect
5. C₄×#13093960	C₂	semidirect
123. C₄₇	#20827	semidirect

Maximal subgroups

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

subgroup	index
#20252021	2
#20243563	2
#20243529	2
#20243430	2
C₄×#13093960	2
C₂×C₄×M₁₆×C₄₇	2
#20827	47