Subgroups — #49095000

Order 35968.

416 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#17985122	C₂	1	1	yes	maximal, characteristic
2	C₂³×#14555886	C₂	1	1	yes	maximal, characteristic
3	#31823218	C₂	1	4	yes	maximal, complemented
4	C₂×#117×C₂₈₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#17985122	—	2	2	no	—
6	C₂×#17985122	—	2	2	no	—
7	C₂×#17985122	C₄	1	4	yes	complemented
8	C₂²×#14555886	—	2	2	no	—
9	C₂²×#14555886	—	2	2	no	—
10	C₂²×#14555886	C₄	1	4	yes	complemented
11	C₂²×#14555886	C₂²	1	2	yes	—
12	C₂²×#14555886	C₂²	1	2	yes	—
13	C₂³×C₄×C₂₈₁	C₂²	1	1	yes	characteristic
14	#117×C₂₈₁	—	2	2	no	—
15	#117×C₂₈₁	—	2	2	no	—
16	C₂×C₄⋊C₄×C₂₈₁	C₄	1	2	yes	—
17	C₂×C₂²⋊C₄×C₂₈₁	C₄	1	2	yes	—
18	C₂×C₄²×C₂₈₁	C₂²	1	2	yes	—
19	#17985122	—	2	8	no	—
20	C₂×#14555886	—	2	8	no	—
21	C₂×#14555886	—	2	2	no	—
22	C₂×#14555886	—	2	2	no	—
23	C₂×#14555886	—	2	2	no	—
24	C₂×#14555886	—	2	2	no	—
25	C₂×#14555886	C₂×C₄	1	4	yes	—
26	C₂×#14555886	C₂×C₄	1	4	yes	—
27	C₂⁴×C₂₈₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₂₈₁	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₂₈₁	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₂₈₁	C₂×C₄	1	4	yes	—
31	C₂²×C₄×C₂₈₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₂₈₁	D₄	1	2	yes	—
33	C₂²×C₄×C₂₈₁	D₄	1	2	yes	—
34	C₂²×C₄×C₂₈₁	D₄	1	2	yes	—
35	C₂²×C₄×C₂₈₁	Q₈	1	2	yes	—
36	C₂²×C₄×C₂₈₁	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×C₂₈₁	—	2	4	no	—
38	C₂²⋊C₄×C₂₈₁	—	2	4	no	—
39	C₄²×C₂₈₁	—	2	2	no	—
40	C₄²×C₂₈₁	—	2	2	no	—
41	#14555886	—	2	4	no	—
42	#14555886	—	2	4	no	—
43	C₂³×C₂₈₁	—	2	4	no	—
44	C₂³×C₂₈₁	C₄²	1	2	yes	—
45	C₂³×C₂₈₁	C₂²⋊C₄	1	2	yes	—
46	C₂³×C₂₈₁	C₄⋊C₄	1	2	yes	—
47	C₂³×C₂₈₁	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₂₈₁	—	2	4	no	—
49	C₂×C₄×C₂₈₁	—	2	8	no	—
50	C₂×C₄×C₂₈₁	C₄²	1	2	yes	—
51	C₂×C₄×C₂₈₁	C₂²⋊C₄	1	4	yes	—
52	C₂×C₄×C₂₈₁	C₂²⋊C₄	1	4	yes	—
53	C₂×C₄×C₂₈₁	C₂²⋊C₄	1	2	yes	—
54	C₂×C₄×C₂₈₁	C₄⋊C₄	1	4	yes	—
55	C₂×C₄×C₂₈₁	C₄⋊C₄	1	4	yes	—
56	C₂×C₄×C₂₈₁	C₄⋊C₄	1	2	yes	—
57	C₂×C₄×C₂₈₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₂₈₁	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₂₈₁	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₂₈₁	—	2	4	no	—
64	C₂²×C₂₈₁	—	2	4	no	—
65	C₂²×C₂₈₁	—	2	4	no	—
66	C₂²×C₂₈₁	#95	1	4	yes	—
67	C₂²×C₂₈₁	C₂×C₄²	1	1	yes	characteristic
68	C₂²×C₂₈₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₂₈₁	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₂₈₁	#131	1	4	yes	—
71	C₄×C₂₈₁	—	4	4	no	—
72	C₄×C₂₈₁	#95	1	4	yes	—
73	C₄×C₂₈₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₄×C₂₈₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₄×C₂₈₁	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₄×C₂₈₁	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₂₈₁	—	2	4	no	—
78	C₂×C₂₈₁	C₂×#95	1	1	yes	characteristic
79	C₂×C₂₈₁	#405	1	2	yes	—
80	C₂×C₂₈₁	#408	1	2	yes	—
81	C₂×C₂₈₁	#424	1	2	yes	—
82	C₂₈₁	#1729	1	1	yes	characteristic, complemented, Sylow
83	#1729	—	281	1	no	maximal, Sylow
84	C₂²×M₁₆	—	281	1	no	—
85	C₂³×C₈	—	281	1	no	—
86	C₂×#117	D₂₈₁	1	1	yes	characteristic
87	#336	—	281	4	no	—
88	C₂³×C₄	C₂×D₂₈₁	1	1	yes	characteristic
89	C₂×M₁₆	—	281	4	no	—
90	C₂×M₁₆	—	562	2	no	—
91	C₂×M₁₆	—	562	2	no	—
92	C₂²×C₈	—	281	4	no	—
93	C₂²×C₈	—	281	2	no	—
94	C₂²×C₈	—	281	2	no	—
95	C₂²×C₈	—	562	2	no	—
96	C₂²×C₈	—	562	2	no	—
97	#117	—	2	2	no	—
98	#117	—	2	2	no	—
99	C₂×C₄⋊C₄	Dic₂₈₁	1	2	yes	—
100	C₂×C₂²⋊C₄	Dic₂₈₁	1	2	yes	—
101	C₂×C₄²	C₂×D₂₈₁	1	2	yes	—
102	C₂⁴	C₂×Dic₂₈₁	1	1	yes	characteristic
103	C₂²×C₄	C₄×D₂₈₁	1	2	yes	—
104	C₂²×C₄	C₄×D₂₈₁	1	2	yes	—
105	C₂²×C₄	C₂×Dic₂₈₁	1	4	yes	—
106	C₂²×C₄	C₂×Dic₂₈₁	1	1	yes	characteristic
107	C₂²×C₄	C₂²⋊D₂₈₁	1	2	yes	—
108	C₂²×C₄	C₂²⋊D₂₈₁	1	2	yes	—
109	C₂²×C₄	D₁₁₂₄	1	2	yes	—
110	C₂²×C₄	Dic₅₆₂	1	2	yes	—
111	C₂²×C₄	C₂²×D₂₈₁	1	1	yes	characteristic
112	M₁₆	—	562	8	no	—
113	C₂×C₈	—	281	4	no	—
114	C₂×C₈	—	281	4	no	—
115	C₂×C₈	—	562	8	no	—
116	C₂×C₈	—	562	2	no	—
117	C₂×C₈	—	562	2	no	—
118	C₂×C₈	—	562	2	no	—
119	C₂×C₈	—	562	2	no	—
120	C₄⋊C₄	—	2	4	no	—
121	C₂²⋊C₄	—	2	4	no	—
122	C₄²	—	2	2	no	—
123	C₄²	—	2	2	no	—
124	C₂³	—	2	4	no	—
125	C₂³	C₄×Dic₂₈₁	1	2	yes	—
126	C₂³	C₂²⋊Dic₂₈₁	1	2	yes	—
127	C₂³	C₄⋊Dic₂₈₁	1	2	yes	—
128	C₂³	C₂²×Dic₂₈₁	1	1	yes	characteristic
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	2	8	no	—
131	C₂×C₄	C₄×Dic₂₈₁	1	2	yes	—
132	C₂×C₄	#17985115	1	4	yes	—
133	C₂×C₄	#17985115	1	4	yes	—
134	C₂×C₄	C₂²⋊Dic₂₈₁	1	2	yes	—
135	C₂×C₄	#17985117	1	4	yes	—
136	C₂×C₄	#17985117	1	4	yes	—
137	C₂×C₄	C₄⋊Dic₂₈₁	1	2	yes	—
138	C₂×C₄	C₂×C₄×D₂₈₁	1	1	yes	characteristic
139	C₂×C₄	C₂×C₄×D₂₈₁	1	1	yes	characteristic
140	C₂×C₄	C₂×C₂²⋊D₂₈₁	1	1	yes	characteristic
141	C₂×C₄	C₂×C₂²⋊D₂₈₁	1	1	yes	characteristic
142	C₂×C₄	C₂×D₁₁₂₄	1	1	yes	characteristic
143	C₂×C₄	C₂×Dic₅₆₂	1	1	yes	characteristic
144	C₈	—	562	4	no	—
145	C₈	—	562	4	no	—
146	C₂²	—	2	4	no	—
147	C₂²	—	2	4	no	—
148	C₂²	—	2	4	no	—
149	C₂²	#22582786	1	4	yes	—
150	C₂²	C₂×C₄×Dic₂₈₁	1	1	yes	characteristic
151	C₂²	C₂×C₂²⋊Dic₂₈₁	1	1	yes	characteristic
152	C₂²	C₂×C₄⋊Dic₂₈₁	1	1	yes	characteristic
153	C₂²	#22582886	1	4	yes	—
154	C₄	—	4	4	no	—
155	C₄	#22582786	1	4	yes	—
156	C₄	C₂×#17985115	1	1	yes	characteristic
157	C₄	C₂×#17985115	1	1	yes	characteristic
158	C₄	C₂×#17985117	1	1	yes	characteristic
159	C₄	C₂×#17985117	1	1	yes	characteristic
160	C₂	—	2	4	no	—
161	C₂	C₂×#22582786	1	1	yes	characteristic
162	C₂	#31823419	1	2	yes	—
163	C₂	#31823430	1	2	yes	—
164	C₂	#31823490	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₈₁
commutator	C₂²×C₂₈₁	C₂×C₄²
Fitting	C₂×#117×C₂₈₁	C₂
socle	C₂³×C₂₈₁	C₂²×C₄
prime radical	C₂⁴×C₂₈₁	C₂×C₄
cosocle	C₂²×C₄×C₂₈₁	C₂³
hypercenter	C₂×#117	D₂₈₁
hypocenter	C₂₈₁	#1729
LCS	G ▷ C₂²×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #31823218	C₂	semidirect
7. C₂×#17985122	C₄	semidirect
10. C₂²×#14555886	C₄	semidirect
82. C₂₈₁	#1729	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#17985122	2
C₂³×#14555886	2
#31823218	2
C₂×#117×C₂₈₁	2
#1729	281