Subgroups — #49096022

Order 35968.

260 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#14555886	C₂	1	2	yes	maximal
2	C₂×C₄⋊#14555886	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#14555886	C₂	1	1	yes	maximal, characteristic
4	#31823218	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₄×C₂₈₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×#14555886	C₂²	1	2	yes	—
7	C₂²×#14555886	C₂²	1	1	yes	characteristic
8	C₂²×#14555886	C₂²	1	1	yes	characteristic
9	C₄⋊#14555886	—	2	4	no	—
10	C₄⋊#14555886	—	2	2	no	—
11	C₄×#14555886	—	2	2	no	—
12	C₂×C₄⋊C₄×C₂₈₁	C₄	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₂₈₁	C₄	1	1	yes	characteristic
14	C₂×C₄²×C₂₈₁	C₂²	1	2	yes	—
15	C₂×C₄²×C₂₈₁	C₂²	1	1	yes	characteristic
16	#95×C₂₈₁	C₄	1	2	yes	—
17	C₂×#14555886	—	2	1	no	—
18	C₂×#14555886	—	2	1	no	—
19	C₂×#14555886	—	2	2	no	—
20	C₂×#14555886	—	2	2	no	—
21	C₂×#14555886	—	2	1	no	—
22	C₂×#14555886	—	2	1	no	—
23	C₂×#14555886	—	2	2	no	—
24	C₂×#14555886	D₄	1	2	yes	—
25	C₂×#14555886	Q₈	1	2	yes	—
26	C₂²×C₄×C₂₈₁	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₂₈₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₂₈₁	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₂₈₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₂₈₁	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₂₈₁	—	2	2	no	—
32	C₄⋊C₄×C₂₈₁	C₈	1	4	yes	complemented
33	C₄²×C₂₈₁	—	2	2	no	—
34	C₄²×C₂₈₁	—	2	2	no	—
35	C₄²×C₂₈₁	D₄	1	2	yes	—
36	C₄²×C₂₈₁	Q₈	1	2	yes	—
37	#14555886	—	2	2	no	—
38	#14555886	—	2	2	no	—
39	#14555886	—	4	2	no	—
40	C₂³×C₂₈₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₂₈₁	—	2	2	no	—
42	C₂×C₄×C₂₈₁	—	2	1	no	—
43	C₂×C₄×C₂₈₁	—	2	1	no	—
44	C₂×C₄×C₂₈₁	—	2	2	no	—
45	C₂×C₄×C₂₈₁	—	2	2	no	—
46	C₂×C₄×C₂₈₁	—	2	2	no	—
47	C₂×C₄×C₂₈₁	—	2	2	no	—
48	C₂×C₄×C₂₈₁	C₄⋊C₄	1	2	yes	—
49	C₂×C₄×C₂₈₁	C₄⋊C₄	1	2	yes	—
50	C₂×C₄×C₂₈₁	C₂×C₈	1	4	yes	—
51	C₂×C₄×C₂₈₁	C₂×C₈	1	2	yes	—
52	C₂×C₄×C₂₈₁	M₁₆	1	2	yes	—
53	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₂₈₁	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₄×C₂₈₁	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₄×C₂₈₁	#41	1	1	yes	characteristic
58	C₂×C₄×C₂₈₁	#41	1	1	yes	characteristic
59	C₂²×C₂₈₁	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₂₈₁	C₄×D₄	1	1	yes	characteristic
61	C₂²×C₂₈₁	C₄×Q₈	1	1	yes	characteristic
62	C₂²×C₂₈₁	C₂²×C₈	1	1	yes	characteristic
63	C₂²×C₂₈₁	C₂×M₁₆	1	1	yes	characteristic
64	C₂²×C₂₈₁	#131	1	1	yes	characteristic
65	C₂²×C₂₈₁	#131	1	1	yes	characteristic
66	C₄×C₂₈₁	—	2	2	no	—
67	C₄×C₂₈₁	—	2	4	no	—
68	C₄×C₂₈₁	—	4	2	no	—
69	C₄×C₂₈₁	C₄⋊C₈	1	4	yes	complemented
70	C₄×C₂₈₁	#121	1	1	yes	characteristic
71	C₄×C₂₈₁	C₂²⋊Q₈	1	1	yes	characteristic
72	C₄×C₂₈₁	#125	1	1	yes	characteristic
73	C₄×C₂₈₁	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₂₈₁	#389	1	1	yes	characteristic
75	C₂×C₂₈₁	C₂×C₄⋊C₈	1	1	yes	characteristic
76	C₂×C₂₈₁	#424	1	1	yes	characteristic
77	C₂×C₂₈₁	C₈×D₄	1	1	yes	characteristic
78	C₂×C₂₈₁	#436	1	1	yes	characteristic
79	C₂×C₂₈₁	C₈×Q₈	1	1	yes	characteristic
80	C₂×C₂₈₁	#446	1	1	yes	characteristic
81	C₂₈₁	#1939	1	1	yes	characteristic, complemented, Sylow
82	#1939	—	281	1	no	maximal, Sylow
83	C₂×C₄⋊C₈	—	281	2	no	—
84	C₂×C₄⋊C₈	—	281	1	no	—
85	C₂×C₄×C₈	—	281	1	no	—
86	C₄×C₄⋊C₄	D₂₈₁	1	1	yes	characteristic
87	#336	—	281	2	no	—
88	C₂²×C₈	—	281	1	no	—
89	C₂²×C₈	—	281	1	no	—
90	C₂²×C₈	—	281	2	no	—
91	C₂×C₄⋊C₄	Dic₂₈₁	1	1	yes	characteristic
92	C₂×C₄⋊C₄	Dic₂₈₁	1	1	yes	characteristic
93	C₂×C₄²	C₂×D₂₈₁	1	2	yes	—
94	C₂×C₄²	C₂×D₂₈₁	1	1	yes	characteristic
95	C₄⋊C₈	—	562	4	no	—
96	C₄⋊C₈	—	562	2	no	—
97	C₄×C₈	—	562	2	no	—
98	#95	Dic₂₈₁	1	2	yes	—
99	C₂²×C₄	C₂×Dic₂₈₁	1	2	yes	—
100	C₂²×C₄	C₂×Dic₂₈₁	1	1	yes	characteristic
101	C₂²×C₄	C₂×Dic₂₈₁	1	2	yes	—
102	C₂²×C₄	C₂×Dic₂₈₁	1	1	yes	characteristic
103	C₂²×C₄	C₂²×D₂₈₁	1	1	yes	characteristic
104	C₂×C₈	—	281	2	no	—
105	C₂×C₈	—	281	2	no	—
106	C₂×C₈	—	562	1	no	—
107	C₂×C₈	—	562	1	no	—
108	C₂×C₈	—	562	1	no	—
109	C₂×C₈	—	562	1	no	—
110	C₂×C₈	—	562	2	no	—
111	C₂×C₈	—	562	2	no	—
112	C₂×C₈	—	562	2	no	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	#14555886	1	4	yes	complemented
115	C₄²	—	2	2	no	—
116	C₄²	—	2	2	no	—
117	C₄²	D₁₁₂₄	1	2	yes	—
118	C₄²	Dic₅₆₂	1	2	yes	—
119	C₂³	C₂²×Dic₂₈₁	1	1	yes	characteristic
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	C₄⋊Dic₂₈₁	1	2	yes	—
128	C₂×C₄	C₄⋊Dic₂₈₁	1	2	yes	—
129	C₂×C₄	C₂×#14555886	1	4	yes	—
130	C₂×C₄	C₂×#14555886	1	2	yes	—
131	C₂×C₄	#17985122	1	2	yes	—
132	C₂×C₄	D₄×D₂₈₁	1	1	yes	characteristic
133	C₂×C₄	C₂×D₁₁₂₄	1	1	yes	characteristic
134	C₂×C₄	Q₈×D₂₈₁	1	1	yes	characteristic
135	C₂×C₄	C₂×Dic₅₆₂	1	1	yes	characteristic
136	C₂×C₄	#17985137	1	1	yes	characteristic
137	C₂×C₄	#17985139	1	1	yes	characteristic
138	C₈	—	562	2	no	—
139	C₈	—	562	2	no	—
140	C₈	—	1124	2	no	—
141	C₂²	C₂×C₄⋊Dic₂₈₁	1	1	yes	characteristic
142	C₂²	D₄×Dic₂₈₁	1	1	yes	characteristic
143	C₂²	Q₈×Dic₂₈₁	1	1	yes	characteristic
144	C₂²	C₂²×#14555886	1	1	yes	characteristic
145	C₂²	C₂×#17985122	1	1	yes	characteristic
146	C₂²	#22582886	1	1	yes	characteristic
147	C₂²	#22582886	1	1	yes	characteristic
148	C₄	—	2	2	no	—
149	C₄	—	2	4	no	—
150	C₄	—	4	2	no	—
151	C₄	C₄⋊#14555886	1	4	yes	complemented
152	C₄	#22582854	1	1	yes	characteristic
153	C₄	#22582859	1	1	yes	characteristic
154	C₄	#22582870	1	1	yes	characteristic
155	C₄	#22582878	1	1	yes	characteristic
156	C₂	#31823362	1	1	yes	characteristic
157	C₂	C₂×C₄⋊#14555886	1	1	yes	characteristic
158	C₂	#31823490	1	1	yes	characteristic
159	C₂	D₄×#14555886	1	1	yes	characteristic
160	C₂	#31823541	1	1	yes	characteristic
161	C₂	Q₈×#14555886	1	1	yes	characteristic
162	C₂	#31823594	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₈₁
commutator	C₂²×C₂₈₁	C₂²×C₈
Fitting	C₄×C₄⋊C₄×C₂₈₁	C₂
socle, prime radical	C₂³×C₂₈₁	C₂²×C₄
cosocle	C₂²×C₄×C₂₈₁	C₂³
hypercenter	C₄×C₄⋊C₄	D₂₈₁
hypocenter	C₂₈₁	#1939
LCS	G ▷ C₂²×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
32. C₄⋊C₄×C₂₈₁	C₈	semidirect
69. C₄×C₂₈₁	C₄⋊C₈	semidirect
81. C₂₈₁	#1939	semidirect
114. C₄⋊C₄	#14555886	semidirect
151. C₄	C₄⋊#14555886	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#14555886	2
C₂×C₄×#14555886	2
#31823218	2
C₄×C₄⋊C₄×C₂₈₁	2
#1939	281