Subgroups — C₂×#20290924 #28327896

Order 13568.

854 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133291	C₂	1	1	yes	maximal, characteristic
2	C₂³×#11719910	C₂	1	1	yes	maximal, characteristic
3	#20290924	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16811511	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×#11719910	C₂	1	2	yes	maximal, complemented
6	C₂×#16810579	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133291	C₄	1	8	yes	complemented
8	C₂×#13133291	C₂²	1	4	yes	—
9	C₂²×#11719910	C₄	1	8	yes	complemented
10	C₂²×#11719910	C₂²	1	1	yes	characteristic
11	C₂²×#11719910	C₂²	1	1	yes	characteristic
12	C₂²×#11719910	C₂²	1	2	yes	—
13	C₂²×#11719910	C₂²	1	4	yes	—
14	#16811511	C₂²	1	8	yes	complemented
15	C₄×#11719910	C₂²	1	8	yes	complemented
16	C₂²×#11719917	C₄	1	1	yes	characteristic
17	C₂³×#88322	C₂²	1	1	yes	characteristic
18	#16810579	C₄	1	4	yes	—
19	#16810579	C₂²	1	4	yes	—
20	C₂×#13133128	C₄	1	2	yes	—
21	C₂×C₄×#88322	C₂²	1	2	yes	—
22	C₂×#117×C₅₃	C₄	1	1	yes	characteristic
23	#13133291	C₂×C₄	1	16	yes	complemented
24	C₂×#11719910	C₂×C₄	1	4	yes	—
25	C₂×#11719910	C₂×C₄	1	4	yes	—
26	C₂×#11719910	C₂×C₄	1	8	yes	—
27	C₂×#11719910	C₂×C₄	1	16	yes	complemented
28	C₂×#11719910	C₂³	1	2	yes	—
29	C₂×#11719910	C₂³	1	2	yes	—
30	C₂×#11719910	C₂³	1	4	yes	—
31	C₂×#11719917	C₈	1	8	yes	—
32	C₂×#11719917	C₂×C₄	1	4	yes	—
33	C₂²×#88322	C₂×C₄	1	4	yes	—
34	C₂²×#88322	C₂×C₄	1	4	yes	—
35	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
36	C₂²×#88322	C₂×C₄	1	2	yes	—
37	C₂²×#88322	C₂×C₄	1	2	yes	—
38	C₂²×#88322	C₂³	1	1	yes	characteristic
39	C₂²×#88322	C₂³	1	2	yes	—
40	#13133128	C₂×C₄	1	8	yes	—
41	C₄×#88322	C₂×C₄	1	4	yes	—
42	C₄×#88322	C₂³	1	4	yes	—
43	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
44	#117×C₅₃	C₈	1	4	yes	—
45	#117×C₅₃	C₂×C₄	1	4	yes	—
46	C₂×C₄⋊C₄×C₅₃	C₈	1	2	yes	—
47	C₂×C₂²⋊C₄×C₅₃	C₈	1	2	yes	—
48	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
49	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
51	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
52	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₄⋊C₄×C₅₃	C₂×C₈	1	8	yes	—
58	C₂²⋊C₄×C₅₃	C₂×C₈	1	8	yes	—
59	C₄²×C₅₃	C₂×C₈	1	4	yes	—
60	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
61	#11719917	C₂×C₈	1	16	yes	—
62	C₂×#88322	C₄²	1	4	yes	—
63	C₂×#88322	C₄²	1	8	yes	—
64	C₂×#88322	C₂×C₈	1	8	yes	—
65	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
66	C₂×#88322	C₂²×C₄	1	2	yes	—
67	C₂×#88322	C₂²×C₄	1	2	yes	—
68	C₂×#88322	C₂²×C₄	1	2	yes	—
69	C₂×#88322	C₂²×C₄	1	4	yes	—
70	C₂×#88322	C₂²×C₄	1	4	yes	—
71	C₂×#88322	C₂²×C₄	1	4	yes	—
72	C₂×#88322	C₂⁴	1	1	yes	characteristic
73	#11719910	C₂²×C₄	1	4	yes	—
74	#11719910	C₂²×C₄	1	4	yes	—
75	#11719910	C₂²×C₄	1	8	yes	—
76	C₂³×C₅₃	—	2	4	no	—
77	C₂³×C₅₃	C₄×C₈	1	4	yes	—
78	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
79	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₅₃	—	2	8	no	—
81	C₂×C₄×C₅₃	C₄×C₈	1	8	yes	—
82	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
83	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
84	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
85	C₂×C₄×C₅₃	C₂²×C₈	1	8	yes	—
86	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
88	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
89	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
90	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
91	#88322	C₂×C₄²	1	2	yes	—
92	#88322	C₂×C₄²	1	4	yes	—
93	#88322	C₂²×C₈	1	8	yes	—
94	#88322	C₂³×C₄	1	1	yes	characteristic
95	#88322	C₂³×C₄	1	1	yes	characteristic
96	C₂×#2108	—	53	1	no	maximal, Sylow
97	C₂²×C₅₃	—	2	8	no	—
98	C₂²×C₅₃	—	2	4	no	—
99	C₂²×C₅₃	C₂×C₄×C₈	1	4	yes	—
100	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
101	C₂²×C₅₃	#504	1	4	yes	—
102	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
103	C₄×C₅₃	—	2	8	no	—
104	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
105	C₄×C₅₃	C₂×C₄×C₈	1	4	yes	—
106	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
107	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
108	C₂²×M₃₂	—	53	1	no	—
109	C₂³×C₁₆	—	53	1	no	—
110	C₂×#405	—	53	1	no	—
111	#2108	—	53	8	no	—
112	C₂×#346	—	53	2	no	—
113	C₂×C₄×C₁₆	—	53	2	no	—
114	C₂×C₅₃	—	2	4	no	—
115	C₂×C₅₃	#2108	1	4	yes	complemented
116	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
117	C₂×C₅₃	C₂×#504	1	2	yes	—
118	C₂²×M₁₆	—	53	1	no	—
119	C₂³×C₈	—	53	1	no	—
120	C₂×#117	#6157	1	1	yes	characteristic
121	C₂×M₃₂	—	53	4	no	—
122	C₂×M₃₂	—	53	8	no	—
123	C₂²×C₁₆	—	53	1	no	—
124	C₂²×C₁₆	—	53	2	no	—
125	C₂²×C₁₆	—	53	1	no	—
126	C₂²×C₁₆	—	53	4	no	—
127	C₂²×C₁₆	—	53	8	no	—
128	#405	—	53	4	no	—
129	#405	—	53	4	no	—
130	C₂×#97	—	53	2	no	—
131	C₂×C₄×C₈	—	53	2	no	—
132	#346	—	53	8	no	—
133	C₄×C₁₆	—	53	8	no	—
134	C₅₃	C₂×#2108	1	1	yes	characteristic, complemented, Sylow
135	C₂³×C₄	C₂×#6157	1	1	yes	characteristic
136	C₂×M₁₆	—	53	4	no	—
137	C₂×M₁₆	—	53	8	no	—
138	C₂²×C₈	—	53	4	no	—
139	C₂²×C₈	—	53	4	no	—
140	C₂²×C₈	—	53	2	no	—
141	C₂²×C₈	—	53	2	no	—
142	C₂²×C₈	—	53	1	no	—
143	C₂²×C₈	—	53	1	no	—
144	C₂²×C₈	—	53	2	no	—
145	#117	#88324	1	4	yes	—
146	#117	C₂×#6157	1	4	yes	—
147	C₂×C₄⋊C₄	#88324	1	2	yes	—
148	C₂×C₂²⋊C₄	#88324	1	2	yes	—
149	C₂×C₄²	C₂×#6157	1	2	yes	—
150	M₃₂	—	53	16	no	—
151	C₂×C₁₆	—	53	2	no	—
152	C₂×C₁₆	—	53	4	no	—
153	C₂×C₁₆	—	53	2	no	—
154	C₂×C₁₆	—	53	4	no	—
155	C₂×C₁₆	—	53	4	no	—
156	C₂×C₁₆	—	53	8	no	—
157	C₂×C₁₆	—	53	16	no	—
158	#97	—	53	8	no	—
159	C₄×C₈	—	53	4	no	—
160	C₄×C₈	—	53	4	no	—
161	C₂⁴	C₂×#88324	1	1	yes	characteristic
162	C₂²×C₄	D₅₃⋊C₈	1	4	yes	—
163	C₂²×C₄	C₄×#6157	1	4	yes	—
164	C₂²×C₄	C₂×#88324	1	4	yes	—
165	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
166	C₂²×C₄	C₂×#88324	1	2	yes	—
167	C₂²×C₄	C₂²×#6157	1	2	yes	—
168	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
169	M₁₆	—	53	16	no	—
170	C₂×C₈	—	53	4	no	—
171	C₂×C₈	—	53	2	no	—
172	C₂×C₈	—	53	2	no	—
173	C₂×C₈	—	53	4	no	—
174	C₂×C₈	—	53	8	no	—
175	C₂×C₈	—	53	4	no	—
176	C₂×C₈	—	53	8	no	—
177	C₂×C₈	—	53	1	no	—
178	C₂×C₈	—	53	1	no	—
179	C₂×C₈	—	53	4	no	—
180	C₂×C₈	—	53	2	no	—
181	C₄⋊C₄	C₂×#88324	1	8	yes	—
182	C₂²⋊C₄	C₂×#88324	1	8	yes	—
183	C₄²	C₂×#88324	1	4	yes	—
184	C₄²	C₂²×#6157	1	4	yes	—
185	C₁₆	—	53	4	no	—
186	C₁₆	—	53	4	no	—
187	C₁₆	—	53	8	no	—
188	C₂³	—	2	4	no	—
189	C₂³	C₄×#88324	1	4	yes	—
190	C₂³	C₂²×#88324	1	2	yes	—
191	C₂³	C₂²×#88324	1	1	yes	characteristic
192	C₂×C₄	—	2	8	no	—
193	C₂×C₄	C₄×#88324	1	8	yes	—
194	C₂×C₄	C₄×#88324	1	4	yes	—
195	C₂×C₄	C₂×C₄×#6157	1	4	yes	—
196	C₂×C₄	C₂×C₄×#6157	1	2	yes	—
197	C₂×C₄	C₂²×#88324	1	8	yes	—
198	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
199	C₂×C₄	C₂²×#88324	1	2	yes	—
200	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
201	C₂×C₄	C₂×D₅₃⋊C₈	1	4	yes	—
202	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
203	C₈	—	53	2	no	—
204	C₈	—	53	4	no	—
205	C₈	—	53	1	no	—
206	C₈	—	53	1	no	—
207	C₈	—	53	8	no	—
208	C₂²	—	2	4	no	—
209	C₂²	—	2	8	no	—
210	C₂²	C₂×C₄×#88324	1	4	yes	—
211	C₂²	C₂×C₄×#88324	1	2	yes	—
212	C₂²	#16811681	1	4	yes	—
213	C₂²	C₂³×#88324	1	1	yes	characteristic
214	C₄	—	2	8	no	—
215	C₄	C₂×C₄×#88324	1	4	yes	—
216	C₄	C₂×C₄×#88324	1	2	yes	—
217	C₄	C₂²×C₄×#6157	1	1	yes	characteristic
218	C₄	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
219	C₂	—	2	4	no	—
220	C₂	#20290924	1	4	yes	direct product factor
221	C₂	C₂²×C₄×#88324	1	1	yes	characteristic
222	C₂	C₂×#16811681	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×#6157
commutator	C₂×C₅₃	C₂²×C₄×C₈
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₂×#117×C₅₃	C₄
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	C₂×#117	#6157
hypocenter	C₅₃	C₂×#2108
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #20290924	C₂	direct (N × H)
4. C₂×#16811511	C₂	semidirect
5. C₂×C₄×#11719910	C₂	semidirect
7. C₂×#13133291	C₄	semidirect
9. C₂²×#11719910	C₄	semidirect
14. #16811511	C₂²	semidirect
15. C₄×#11719910	C₂²	semidirect
23. #13133291	C₂×C₄	semidirect
27. C₂×#11719910	C₂×C₄	semidirect
115. C₂×C₅₃	#2108	semidirect
134. C₅₃	C₂×#2108	semidirect
220. C₂	#20290924	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×#13133291	2
C₂³×#11719910	2
#20290924	2
C₂×#16811511	2
C₂×C₄×#11719910	2
C₂×#16810579	2
C₂×#2108	53

Subgroups — C2×#20290924 #28327896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20290924 #28327896