Subgroups — C₄×#2253 #496149

Order 512.

532 conjugacy classes of subgroups.

The subgroup lattice has 205 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2253	C₂	1	1	yes	maximal, characteristic
2	C₄×#499	C₂	1	2	yes	maximal
3	#13281	C₂	1	2	yes	maximal
4	#13166	C₂	1	2	yes	maximal
5	#13165	C₂	1	2	yes	maximal
6	#12907	C₂	1	1	yes	maximal, characteristic
7	C₄×#367	C₂	1	2	yes	maximal
8	C₄×#366	C₂	1	2	yes	maximal
9	C₄²×C₁₆	C₂	1	1	yes	maximal, characteristic
10	C₂×#499	C₂²	1	2	yes	—
11	C₄×#125	—	2	2	no	—
12	#2253	C₄	1	8	yes	direct product factor
13	C₂×#367	C₂²	1	2	yes	—
14	C₂×#367	C₂²	1	2	yes	—
15	C₂×#366	C₂²	1	2	yes	—
16	C₂×#366	C₂²	1	2	yes	—
17	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
23	#1991	—	2	2	no	—
24	#1942	C₂²	1	4	yes	—
25	#1941	—	2	2	no	—
26	#1846	C₂²	1	2	yes	—
27	C₄×#107	C₂²	1	4	yes	—
28	C₄×#106	—	2	2	no	—
29	C₄²×C₈	C₂²	1	1	yes	characteristic
30	#1380	C₂²	1	4	yes	—
31	#1380	C₂²	1	4	yes	—
32	C₂×#125	—	2	2	no	—
33	C₂²×C₁₆	C₂³	1	1	yes	characteristic
34	C₂²×C₁₆	C₂³	1	1	yes	characteristic
35	C₂²×C₁₆	C₂³	1	1	yes	characteristic
36	C₂²×C₁₆	C₂³	1	1	yes	characteristic
37	#499	—	2	4	no	—
38	#499	C₂×C₄	1	8	yes	—
39	C₂×#107	C₂³	1	4	yes	—
40	C₂×#107	C₂³	1	4	yes	—
41	C₂×#106	—	2	2	no	—
42	C₂×#106	—	2	2	no	—
43	C₂×C₄×C₈	D₄	1	1	yes	characteristic
44	C₂×C₄×C₈	D₄	1	1	yes	characteristic
45	C₂×C₄×C₈	D₄	1	1	yes	characteristic
46	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
47	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
48	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
49	#389	—	2	4	no	—
50	#387	—	4	4	no	—
51	#382	—	2	2	no	—
52	C₄×C₄⋊C₄	—	2	4	no	—
53	C₄×C₄⋊C₄	—	4	4	no	—
54	C₄³	D₄	1	1	yes	characteristic
55	#367	—	2	4	no	—
56	#367	C₂×C₄	1	8	yes	—
57	#366	—	2	4	no	—
58	#366	C₂×C₄	1	8	yes	—
59	C₄×C₁₆	—	2	2	no	—
60	C₄×C₁₆	—	2	2	no	—
61	C₄×C₁₆	—	2	2	no	—
62	C₄×C₁₆	C₂×C₄	1	4	yes	—
63	C₄×C₁₆	Q₈	1	4	yes	—
64	C₄×C₁₆	Q₈	1	4	yes	—
65	#340	—	2	4	no	—
66	#340	—	2	4	no	—
67	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₈	C₂⁴	1	1	yes	characteristic
71	#125	—	2	8	no	—
72	#125	—	4	4	no	—
73	C₂×C₄⋊C₄	—	2	4	no	—
74	C₂×C₄⋊C₄	—	2	4	no	—
75	C₂×C₄⋊C₄	—	4	4	no	—
76	C₂×C₄⋊C₄	—	4	4	no	—
77	C₂×C₄²	—	4	4	no	—
78	C₂×C₄²	D₈	1	2	yes	—
79	C₂×C₄²	D₈	1	2	yes	—
80	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
83	C₂×C₁₆	—	2	1	no	—
84	C₂×C₁₆	—	2	1	no	—
85	C₂×C₁₆	—	2	1	no	—
86	C₂×C₁₆	—	2	1	no	—
87	C₂×C₁₆	C₂²×C₄	1	2	yes	—
88	C₂×C₁₆	C₂²×C₄	1	2	yes	—
89	C₂×C₁₆	C₂×Q₈	1	2	yes	—
90	C₂×C₁₆	C₂×Q₈	1	2	yes	—
91	C₂×C₁₆	C₂×Q₈	1	2	yes	—
92	C₂×C₁₆	C₂×Q₈	1	2	yes	—
93	C₂×C₁₆	#41	1	2	yes	—
94	C₂×C₁₆	#41	1	2	yes	—
95	#107	—	2	4	no	—
96	#107	—	2	4	no	—
97	#107	—	2	4	no	—
98	#107	C₂²×C₄	1	8	yes	—
99	#106	—	2	8	no	—
100	#106	—	4	4	no	—
101	C₄×C₈	—	2	2	no	—
102	C₄×C₈	—	2	1	no	—
103	C₄×C₈	—	2	1	no	—
104	C₄×C₈	—	2	2	no	—
105	C₄×C₈	Q₁₆	1	4	yes	—
106	C₄×C₈	C₂²×C₄	1	2	yes	—
107	C₄×C₈	C₂×Q₈	1	2	yes	—
108	C₄×C₈	C₂×Q₈	1	2	yes	—
109	C₄×C₈	#41	1	2	yes	—
110	#95	—	4	4	no	—
111	#95	—	4	4	no	—
112	#95	—	4	4	no	—
113	C₂²×C₄	—	4	4	no	—
114	C₂²×C₄	—	4	4	no	—
115	C₂²×C₄	C₂×D₈	1	2	yes	—
116	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
117	C₂²×C₄	C₂×D₈	1	2	yes	—
118	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
119	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	C₄×D₄	1	2	yes	—
125	C₂×C₈	C₄⋊Q₈	1	2	yes	—
126	C₂×C₈	C₂×Q₁₆	1	2	yes	—
127	C₂×C₈	C₂×Q₁₆	1	2	yes	—
128	C₂×C₈	#135	1	2	yes	—
129	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
131	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
132	C₂×C₈	C₂×#41	1	1	yes	characteristic
133	C₂×C₈	C₂×#41	1	1	yes	characteristic
134	C₂×C₈	C₂×#41	1	1	yes	characteristic
135	C₄⋊C₄	—	2	8	no	—
136	C₄⋊C₄	—	4	4	no	—
137	C₄⋊C₄	—	4	16	no	—
138	C₄⋊C₄	—	4	4	no	—
139	C₄⋊C₄	—	4	4	no	—
140	C₄⋊C₄	—	8	8	no	—
141	C₄²	—	2	1	no	—
142	C₄²	—	2	1	no	—
143	C₄²	—	2	4	no	—
144	C₄²	—	2	4	no	—
145	C₄²	—	8	8	no	—
146	C₄²	C₄×D₄	1	2	yes	—
147	C₄²	C₄⋊Q₈	1	2	yes	—
148	C₄²	C₂×Q₁₆	1	2	yes	—
149	C₄²	#135	1	2	yes	—
150	C₁₆	—	2	2	no	—
151	C₁₆	—	2	2	no	—
152	C₁₆	C₄×Q₈	1	4	yes	—
153	C₁₆	C₄×Q₈	1	4	yes	—
154	C₂³	C₂²×D₈	1	1	yes	characteristic
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	4	8	no	—
161	C₂×C₄	—	8	4	no	—
162	C₂×C₄	—	8	4	no	—
163	C₂×C₄	—	8	4	no	—
164	C₂×C₄	—	8	4	no	—
165	C₂×C₄	—	8	4	no	—
166	C₂×C₄	C₄×D₈	1	4	yes	—
167	C₂×C₄	#500	1	2	yes	—
168	C₂×C₄	#500	1	2	yes	—
169	C₂×C₄	#508	1	4	yes	—
170	C₂×C₄	#508	1	4	yes	—
171	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
172	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
173	C₂×C₄	#532	1	1	yes	characteristic
174	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
175	C₂×C₄	C₂×#135	1	1	yes	characteristic
176	C₂×C₄	C₂×#135	1	1	yes	characteristic
177	C₈	—	2	2	no	—
178	C₈	—	2	2	no	—
179	C₈	C₄×Q₁₆	1	4	yes	—
180	C₈	C₂×C₄×Q₈	1	1	yes	characteristic
181	C₈	C₂×C₄×Q₈	1	1	yes	characteristic
182	C₈	#533	1	1	yes	characteristic
183	C₈	#533	1	1	yes	characteristic
184	C₂²	C₂×C₄×D₈	1	1	yes	characteristic
185	C₂²	C₂×#500	1	1	yes	characteristic
186	C₂²	#3169	1	1	yes	characteristic
187	C₂²	C₂×#508	1	2	yes	—
188	C₂²	C₂×#508	1	2	yes	—
189	C₄	—	2	4	no	—
190	C₄	—	2	2	no	—
191	C₄	—	2	4	no	—
192	C₄	—	8	8	no	—
193	C₄	—	16	4	no	—
194	C₄	#2253	1	4	yes	direct product factor
195	C₄	C₄×C₄⋊Q₈	1	1	yes	characteristic
196	C₄	C₂×C₄×Q₁₆	1	1	yes	characteristic
197	C₄	#3160	1	1	yes	characteristic
198	C₄	#3170	1	1	yes	characteristic
199	C₂	C₄×#500	1	1	yes	characteristic
200	C₂	C₄×#508	1	2	yes	—
201	C₂	C₂×#2253	1	1	yes	characteristic
202	C₂	#33726	1	1	yes	characteristic
203	C₂	#33735	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle, prime radical	C₂³	C₂²×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
12. #2253	C₄	direct (N × H)
194. C₄	#2253	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#2253	2
C₄×#499	2
#13281	2
#13166	2
#13165	2
#12907	2
C₄×#367	2
C₄×#366	2
C₄²×C₁₆	2

Subgroups — C4×#2253 #496149

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#2253 #496149