Subgroups — C₂×#30656501 #45664043

Order 32896.

392 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#16861602	C₂	1	1	yes	maximal, characteristic
2	C₂³×#14523339	C₂	1	1	yes	maximal, characteristic
3	#30656501	C₂	1	4	yes	maximal, direct product factor
4	C₂²×M₁₆×C₂₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#16861602	—	2	1	no	—
6	C₂×#16861602	—	2	2	no	—
7	C₂×#16861602	C₄	1	4	yes	—
8	C₂×#16861602	C₂²	1	2	yes	—
9	C₂²×#14523339	—	2	2	no	—
10	C₂²×#14523339	—	2	2	no	—
11	C₂²×#14523339	—	2	1	no	—
12	C₂²×#14523339	C₄	1	2	yes	—
13	C₂²×#14523339	C₄	1	2	yes	—
14	C₂²×#14523339	C₂²	1	2	yes	—
15	C₂³×C₄×C₂₅₇	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₂₅₇	—	2	2	no	—
17	C₂×M₁₆×C₂₅₇	—	2	1	no	—
18	C₂×M₁₆×C₂₅₇	C₄	1	4	yes	—
19	C₂×M₁₆×C₂₅₇	C₂²	1	2	yes	—
20	C₂²×C₈×C₂₅₇	—	2	1	no	—
21	#16861602	—	2	4	no	—
22	#16861602	—	2	2	no	—
23	#16861602	C₂×C₄	1	4	yes	—
24	C₂×#14523339	—	2	4	no	—
25	C₂×#14523339	—	2	4	no	—
26	C₂×#14523339	—	2	4	no	—
27	C₂×#14523339	—	2	2	no	—
28	C₂×#14523339	—	2	2	no	—
29	C₂×#14523339	—	2	2	no	—
30	C₂×#14523339	C₂×C₄	1	4	yes	—
31	C₂⁴×C₂₅₇	Q₈	1	1	yes	characteristic
32	C₂²×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₂₅₇	C₂×C₄	1	2	yes	—
35	C₂²×C₄×C₂₅₇	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₂₅₇	D₄	1	2	yes	—
37	C₂²×C₄×C₂₅₇	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₂₅₇	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₂₅₇	D₄	1	2	yes	—
40	C₂²×C₄×C₂₅₇	Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
42	M₁₆×C₂₅₇	—	2	4	no	—
43	M₁₆×C₂₅₇	—	2	2	no	—
44	M₁₆×C₂₅₇	C₂×C₄	1	4	yes	—
45	C₂×C₈×C₂₅₇	—	2	4	no	—
46	C₂×C₈×C₂₅₇	—	2	2	no	—
47	#14523339	—	2	4	no	—
48	#14523339	—	2	4	no	—
49	C₂³×C₂₅₇	—	2	4	no	—
50	C₂³×C₂₅₇	C₄⋊C₄	1	2	yes	—
51	C₂³×C₂₅₇	C₄⋊C₄	1	1	yes	characteristic
52	C₂³×C₂₅₇	C₄⋊C₄	1	1	yes	characteristic
53	C₂³×C₂₅₇	C₄⋊C₄	1	2	yes	—
54	C₂³×C₂₅₇	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₄×C₂₅₇	C₄²	1	2	yes	—
56	C₂×C₄×C₂₅₇	C₄²	1	2	yes	—
57	C₂×C₄×C₂₅₇	C₂²⋊C₄	1	2	yes	—
58	C₂×C₄×C₂₅₇	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄×C₂₅₇	C₂²⋊C₄	1	2	yes	—
60	C₂×C₄×C₂₅₇	C₂²⋊C₄	1	2	yes	—
61	C₂×C₄×C₂₅₇	C₂²⋊C₄	1	2	yes	—
62	C₂×C₄×C₂₅₇	C₂²⋊C₄	1	2	yes	—
63	C₂×C₄×C₂₅₇	C₄⋊C₄	1	2	yes	—
64	C₂×C₄×C₂₅₇	C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₂₅₇	C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₂₅₇	C₄⋊C₄	1	2	yes	—
67	C₂×C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
73	C₈×C₂₅₇	—	2	4	no	—
74	C₂²×C₂₅₇	—	2	4	no	—
75	C₂²×C₂₅₇	—	2	4	no	—
76	C₂²×C₂₅₇	—	2	2	no	—
77	C₂²×C₂₅₇	—	2	2	no	—
78	C₂²×C₂₅₇	#95	1	2	yes	—
79	C₂²×C₂₅₇	#95	1	2	yes	—
80	C₂²×C₂₅₇	#108	1	4	yes	—
81	C₂²×C₂₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₂₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₂₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₄×C₂₅₇	#95	1	2	yes	—
85	C₄×C₂₅₇	#95	1	2	yes	—
86	C₄×C₂₅₇	C₂×C₄²	1	1	yes	characteristic
87	C₄×C₂₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₄×C₂₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₄×C₂₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₂₅₇	—	2	2	no	—
91	C₂×C₂₅₇	—	2	1	no	—
92	C₂×C₂₅₇	—	2	1	no	—
93	C₂×C₂₅₇	#341	1	4	yes	complemented
94	C₂×C₂₅₇	C₂×#95	1	1	yes	characteristic
95	C₂×C₂₅₇	C₂×#108	1	2	yes	—
96	C₂₅₇	C₂×#341	1	1	yes	characteristic, complemented, Sylow
97	C₂×#341	—	257	1	no	maximal, Sylow
98	C₂²×M₁₆	—	257	1	no	—
99	C₂²×M₁₆	D₂₅₇	1	1	yes	characteristic
100	C₂³×C₈	—	257	1	no	—
101	#341	—	257	4	no	—
102	C₂³×C₄	C₂×D₂₅₇	1	1	yes	characteristic
103	C₂×M₁₆	—	257	4	no	—
104	C₂×M₁₆	—	257	2	no	—
105	C₂×M₁₆	—	2	2	no	—
106	C₂×M₁₆	—	2	1	no	—
107	C₂×M₁₆	—	514	1	no	—
108	C₂×M₁₆	—	514	2	no	—
109	C₂×M₁₆	Dic₂₅₇	1	4	yes	—
110	C₂×M₁₆	C₂×D₂₅₇	1	2	yes	—
111	C₂²×C₈	—	257	2	no	—
112	C₂²×C₈	—	257	2	no	—
113	C₂²×C₈	—	257	2	no	—
114	C₂²×C₈	—	2	1	no	—
115	C₂²×C₈	—	514	2	no	—
116	C₂²×C₈	—	514	1	no	—
117	C₂²×C₈	—	514	2	no	—
118	C₂⁴	Dic₅₁₄	1	1	yes	characteristic
119	C₂²×C₄	C₄×D₂₅₇	1	1	yes	characteristic
120	C₂²×C₄	C₄×D₂₅₇	1	1	yes	characteristic
121	C₂²×C₄	C₄×D₂₅₇	1	2	yes	—
122	C₂²×C₄	C₂×Dic₂₅₇	1	2	yes	—
123	C₂²×C₄	C₂²⋊D₂₅₇	1	2	yes	—
124	C₂²×C₄	C₂²⋊D₂₅₇	1	1	yes	characteristic
125	C₂²×C₄	C₂²⋊D₂₅₇	1	1	yes	characteristic
126	C₂²×C₄	D₁₀₂₈	1	2	yes	—
127	C₂²×C₄	Dic₅₁₄	1	1	yes	characteristic
128	C₂²×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
129	M₁₆	—	257	4	no	—
130	M₁₆	—	2	4	no	—
131	M₁₆	—	2	2	no	—
132	M₁₆	—	514	4	no	—
133	M₁₆	—	514	2	no	—
134	M₁₆	C₂×Dic₂₅₇	1	4	yes	—
135	C₂×C₈	—	257	4	no	—
136	C₂×C₈	—	2	4	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	514	4	no	—
139	C₂×C₈	—	514	4	no	—
140	C₂×C₈	—	514	4	no	—
141	C₂×C₈	—	514	2	no	—
142	C₂×C₈	—	514	2	no	—
143	C₂×C₈	—	514	2	no	—
144	C₂³	—	2	4	no	—
145	C₂³	#16861597	1	1	yes	characteristic
146	C₂³	#16861597	1	2	yes	—
147	C₂³	#16861597	1	1	yes	characteristic
148	C₂³	C₄⋊Dic₂₅₇	1	2	yes	—
149	C₂³	C₂×Dic₅₁₄	1	1	yes	characteristic
150	C₂×C₄	C₄×Dic₂₅₇	1	2	yes	—
151	C₂×C₄	C₄×Dic₂₅₇	1	2	yes	—
152	C₂×C₄	#16861595	1	2	yes	—
153	C₂×C₄	#16861595	1	2	yes	—
154	C₂×C₄	#16861595	1	2	yes	—
155	C₂×C₄	#16861595	1	2	yes	—
156	C₂×C₄	C₂²⋊Dic₂₅₇	1	2	yes	—
157	C₂×C₄	C₂²⋊Dic₂₅₇	1	2	yes	—
158	C₂×C₄	#16861597	1	2	yes	—
159	C₂×C₄	#16861597	1	1	yes	characteristic
160	C₂×C₄	#16861597	1	1	yes	characteristic
161	C₂×C₄	C₄⋊Dic₂₅₇	1	2	yes	—
162	C₂×C₄	C₂×C₄×D₂₅₇	1	1	yes	characteristic
163	C₂×C₄	C₂×C₄×D₂₅₇	1	1	yes	characteristic
164	C₂×C₄	C₂²×Dic₂₅₇	1	1	yes	characteristic
165	C₂×C₄	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
166	C₂×C₄	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
167	C₂×C₄	C₂×D₁₀₂₈	1	1	yes	characteristic
168	C₈	—	2	4	no	—
169	C₈	—	514	4	no	—
170	C₈	—	514	4	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	2	2	no	—
173	C₂²	—	2	4	no	—
174	C₂²	—	2	4	no	—
175	C₂²	#22551499	1	2	yes	—
176	C₂²	#22551499	1	2	yes	—
177	C₂²	#22551527	1	4	yes	—
178	C₂²	C₂×#16861597	1	1	yes	characteristic
179	C₂²	C₂×#16861597	1	1	yes	characteristic
180	C₂²	C₂×C₄⋊Dic₂₅₇	1	1	yes	characteristic
181	C₄	#22551499	1	2	yes	—
182	C₄	#22551499	1	2	yes	—
183	C₄	C₂×C₄×Dic₂₅₇	1	1	yes	characteristic
184	C₄	C₂×#16861595	1	1	yes	characteristic
185	C₄	C₂×#16861595	1	1	yes	characteristic
186	C₄	C₂×C₂²⋊Dic₂₅₇	1	1	yes	characteristic
187	C₂	—	2	2	no	—
188	C₂	—	2	1	no	—
189	C₂	—	2	1	no	—
190	C₂	#30656501	1	4	yes	direct product factor
191	C₂	C₂×#22551499	1	1	yes	characteristic
192	C₂	C₂×#22551527	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²⋊D₂₅₇
commutator	C₄×C₂₅₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₅₇
Fitting	C₂²×M₁₆×C₂₅₇	C₂
socle	C₂³×C₂₅₇	C₄⋊C₄
prime radical	C₂⁴×C₂₅₇	Q₈
cosocle	C₂²×C₄×C₂₅₇	C₂³
hypercenter	C₂²×M₁₆	D₂₅₇
hypocenter	C₂₅₇	C₂×#341
LCS	G ▷ C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #30656501	C₂	direct (N × H)
93. C₂×C₂₅₇	#341	semidirect
96. C₂₅₇	C₂×#341	semidirect
190. C₂	#30656501	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#16861602	2
C₂³×#14523339	2
#30656501	2
C₂²×M₁₆×C₂₅₇	2
C₂×#341	257

Subgroups — C2×#30656501 #45664043

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#30656501 #45664043