Subgroups — #26216665

Order 12032.

1118 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689147	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689146	C₂	1	4	yes	maximal
3	C₂×#15688969	C₂	1	1	yes	maximal, characteristic
4	#20243541	C₂	1	8	yes	maximal, complemented
5	C₂×C₄₇×#423	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
8	#15689147	—	2	4	no	—
9	#15689146	—	2	16	no	—
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
11	C₂×Dic₄₇⋊C₈	C₂²	1	16	yes	—
12	#15688969	C₄	1	4	yes	—
13	#15688969	C₂²	1	4	yes	—
14	C₂×C₄₇⋊#95	C₄	1	4	yes	—
15	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
16	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#423	—	2	4	no	—
18	C₂×C₄⋊C₈×C₄₇	C₂²	1	4	yes	—
19	C₂³×Dic₄₇	C₂×C₄	1	4	yes	—
20	C₂×#10625329	—	2	8	no	—
21	C₂×#10625329	—	2	4	no	—
22	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
23	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
24	C₂×C₄×Dic₄₇	D₄	1	8	yes	—
25	C₂×C₄×Dic₄₇	Q₈	1	8	yes	—
26	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
27	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
28	Dic₄₇⋊C₈	—	2	32	no	—
29	C₄₇⋊#95	C₂×C₄	1	16	yes	—
30	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
31	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
32	C₂×M₁₆×C₄₇	—	2	8	no	—
33	C₂×M₁₆×C₄₇	—	2	4	no	—
34	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
35	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₄₇	D₄	1	4	yes	—
37	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
38	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
39	C₄⋊C₈×C₄₇	—	2	8	no	—
40	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	—	4	4	no	—
42	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
43	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
44	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
46	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
48	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
49	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
50	C₂²×Dic₄₇	—	4	8	no	—
51	C₂²×Dic₄₇	C₄⋊C₄	1	16	yes	—
52	C₂²×Dic₄₇	C₂²×C₄	1	8	yes	—
53	M₁₆×C₄₇	—	4	8	no	—
54	C₂×C₈×C₄₇	—	2	4	no	—
55	C₂×C₈×C₄₇	—	2	8	no	—
56	C₄²×C₄₇	C₄⋊C₄	1	8	yes	—
57	C₄²×C₄₇	C₂×D₄	1	4	yes	—
58	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
59	C₄×Dic₄₇	C₄⋊C₄	1	16	yes	—
60	C₄×Dic₄₇	C₂×D₄	1	8	yes	—
61	C₄×Dic₄₇	C₂×Q₈	1	8	yes	—
62	#10625329	—	4	8	no	—
63	C₂×C₄₇⋊C₈	—	2	8	no	—
64	C₂×C₄₇⋊C₈	—	2	4	no	—
65	C₂³×C₄₇	—	2	4	no	—
66	C₂³×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
67	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
68	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
69	C₂×C₄×C₄₇	—	2	8	no	—
70	C₂×C₄×C₄₇	—	4	8	no	—
71	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
72	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
73	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
74	C₂×C₄×C₄₇	#127	1	4	yes	—
75	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
76	C₂×C₄×C₄₇	C₄⋊Q₈	1	8	yes	—
77	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
78	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
79	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
80	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
81	C₂×Dic₄₇	—	2	16	no	—
82	C₂×Dic₄₇	—	4	16	no	—
83	C₂×Dic₄₇	C₂×C₄⋊C₄	1	16	yes	—
84	C₈×C₄₇	—	4	4	no	—
85	C₄₇⋊C₈	—	4	4	no	—
86	#20792	—	47	1	no	maximal, Sylow
87	C₂²×C₄₇	—	2	8	no	—
88	C₂²×C₄₇	—	2	4	no	—
89	C₂²×C₄₇	#384	1	4	yes	—
90	C₂²×C₄₇	C₂²×C₄⋊C₄	1	2	yes	—
91	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₄₇	#568	1	4	yes	—
93	C₄×C₄₇	—	2	8	no	—
94	C₄×C₄₇	#384	1	4	yes	—
95	C₄×C₄₇	C₂×#127	1	1	yes	characteristic
96	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
97	C₄×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
98	Dic₄₇	—	2	16	no	—
99	C₂×#423	—	47	1	no	—
100	C₂×#423	—	47	4	no	—
101	C₂×#423	D₄₇	1	1	yes	characteristic
102	C₂×#376	—	47	1	no	—
103	#1843	—	47	8	no	—
104	C₂×C₄₇	—	2	4	no	—
105	C₂×C₄₇	C₂×#384	1	1	yes	characteristic
106	C₂×C₄₇	#2905	1	4	yes	—
107	C₂×C₄₇	#2905	1	2	yes	—
108	C₂²×M₁₆	—	47	2	no	—
109	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
110	C₂²×C₄²	—	47	2	no	—
111	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
112	#423	—	2	4	no	—
113	#423	—	94	16	no	—
114	#423	—	94	4	no	—
115	C₂×C₄⋊C₈	—	47	16	no	—
116	C₂×C₄⋊C₈	—	47	4	no	—
117	C₂×C₄⋊C₈	C₂×D₄₇	1	4	yes	—
118	#376	—	47	4	no	—
119	#376	—	47	4	no	—
120	C₂×#95	—	47	4	no	—
121	C₄₇	#20792	1	1	yes	characteristic, complemented, Sylow
122	C₂³×C₄	—	47	4	no	—
123	C₂³×C₄	C₄×D₄₇	1	2	yes	—
124	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
125	C₂×M₁₆	—	2	8	no	—
126	C₂×M₁₆	—	2	4	no	—
127	C₂×M₁₆	—	94	8	no	—
128	C₂×M₁₆	—	94	4	no	—
129	C₂²×C₈	—	47	4	no	—
130	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
131	C₂×C₄²	—	47	4	no	—
132	C₂×C₄²	—	47	8	no	—
133	C₂×C₄²	—	47	8	no	—
134	C₂×C₄²	—	47	4	no	—
135	C₂×C₄²	Dic₉₄	1	4	yes	—
136	C₂×C₄²	C₄×D₄₇	1	2	yes	—
137	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
138	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
139	C₄⋊C₈	—	2	8	no	—
140	C₄⋊C₈	—	94	32	no	—
141	C₄⋊C₈	—	94	8	no	—
142	#95	—	47	16	no	—
143	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
144	C₂²×C₄	—	4	4	no	—
145	C₂²×C₄	—	47	8	no	—
146	C₂²×C₄	—	47	16	no	—
147	C₂²×C₄	—	188	8	no	—
148	C₂²×C₄	Dic₄₇⋊C₄	1	8	yes	—
149	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
150	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
151	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
152	C₂²×C₄	D₄×D₄₇	1	4	yes	—
153	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
154	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
155	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
156	M₁₆	—	4	8	no	—
157	M₁₆	—	188	8	no	—
158	C₂×C₈	—	2	8	no	—
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	—	94	8	no	—
161	C₂×C₈	—	94	4	no	—
162	C₄²	—	47	16	no	—
163	C₄²	—	47	8	no	—
164	C₄²	—	47	8	no	—
165	C₄²	Dic₄₇⋊C₄	1	8	yes	—
166	C₄²	C₂×Dic₉₄	1	4	yes	—
167	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
168	C₂³	—	2	4	no	—
169	C₂³	C₄⋊C₄×D₄₇	1	4	yes	—
170	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
171	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
172	C₂×C₄	—	2	8	no	—
173	C₂×C₄	—	4	8	no	—
174	C₂×C₄	—	47	16	no	—
175	C₂×C₄	—	94	16	no	—
176	C₂×C₄	—	188	16	no	—
177	C₂×C₄	C₄⋊C₄×D₄₇	1	4	yes	—
178	C₂×C₄	C₂×Dic₄₇⋊C₄	1	8	yes	—
179	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
180	C₂×C₄	#13094020	1	4	yes	—
181	C₂×C₄	#13094022	1	4	yes	—
182	C₂×C₄	#13094023	1	8	yes	—
183	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
184	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
185	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
186	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
187	C₈	—	4	4	no	—
188	C₈	—	188	4	no	—
189	C₂²	—	2	8	no	—
190	C₂²	—	2	4	no	—
191	C₂²	#15688992	1	4	yes	—
192	C₂²	C₂×C₄⋊C₄×D₄₇	1	2	yes	—
193	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
194	C₂²	#15689889	1	4	yes	—
195	C₄	—	2	8	no	—
196	C₄	—	94	16	no	—
197	C₄	#15688992	1	4	yes	—
198	C₄	C₂×#13094020	1	1	yes	characteristic
199	C₄	C₂×#13094022	1	1	yes	characteristic
200	C₄	C₂×#13094023	1	2	yes	—
201	C₂	—	2	4	no	—
202	C₂	C₂×#15688992	1	1	yes	characteristic
203	C₂	#20251950	1	4	yes	—
204	C₂	#20251951	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₂×C₄₇×#423	C₂
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#423	D₄₇
hypocenter	C₄₇	#20792
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
4. #20243541	C₂	semidirect
121. C₄₇	#20792	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689147	2
C₂×#15689146	2
C₂×#15688969	2
#20243541	2
C₂×C₄₇×#423	2
#20792	47