Subgroups — C₂×#20288342 #28291197

Order 13568.

8418 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20288342	C₂	1	32	yes	maximal, direct product factor
2	C₂×#16811339	C₂	1	9	yes	maximal, complemented
3	C₂×#16811295	C₂	1	6	yes	maximal, complemented
4	C₂×#16811174	C₂	1	9	yes	maximal, complemented
5	C₂×C₄×#11719949	C₂	1	6	yes	maximal, complemented
6	C₂×Q₈²×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#11719949	C₂²	1	6	yes	—
8	C₂²×#11719949	C₂²	1	6	yes	—
9	#16811339	C₂²	1	144	yes	complemented
10	#16811295	C₂²	1	96	yes	complemented
11	#16811174	C₂²	1	144	yes	complemented
12	C₂×#13133214	C₂²	1	9	yes	—
13	C₂×#13133206	C₂²	1	18	yes	—
14	C₂×#13133192	C₂²	1	36	yes	—
15	C₄×#11719949	C₂²	1	96	yes	complemented
16	C₂×Q₈×Dic₅₃	C₂²	1	2	yes	—
17	C₂×C₄×D₂₁₂	C₂²	1	18	yes	—
18	C₂×#13133179	C₂²	1	18	yes	—
19	C₂×#13133175	C₂²	1	18	yes	—
20	C₂×C₄²×D₅₃	C₂²	1	9	yes	—
21	Q₈²×C₅₃	C₂²	1	16	yes	complemented
22	C₂×C₄⋊Q₈×C₅₃	C₂²	1	9	yes	—
23	C₂×C₄×Q₈×C₅₃	C₂²	1	6	yes	—
24	C₂×#11719949	—	2	48	no	—
25	C₂×#11719949	—	4	24	no	—
26	C₂×#11719949	C₂³	1	48	yes	—
27	C₂×#11719949	C₂³	1	48	yes	—
28	C₂²×D₂₁₂	C₂³	1	18	yes	—
29	C₂²×D₂₁₂	C₂³	1	18	yes	—
30	C₂²×C₄×D₅₃	C₂³	1	18	yes	—
31	C₂²×C₄×D₅₃	C₂³	1	9	yes	—
32	#13133214	C₂³	1	72	yes	—
33	#13133206	C₂³	1	144	yes	—
34	#13133192	C₂³	1	288	yes	—
35	Q₈×Dic₅₃	C₂³	1	16	yes	—
36	C₄×D₂₁₂	C₂³	1	144	yes	—
37	#13133179	C₂³	1	144	yes	—
38	#13133175	C₂³	1	144	yes	—
39	C₂×C₄⋊Dic₅₃	C₂³	1	6	yes	—
40	C₂×#11719921	C₂³	1	36	yes	—
41	C₂×C₄×Dic₅₃	C₂³	1	6	yes	—
42	C₂×C₄×Dic₅₃	C₂³	1	9	yes	—
43	C₄²×D₅₃	C₂³	1	72	yes	—
44	C₂²×Q₈×C₅₃	C₂³	1	6	yes	—
45	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
46	C₄⋊Q₈×C₅₃	C₂³	1	72	yes	—
47	C₄×Q₈×C₅₃	C₂³	1	48	yes	—
48	C₂×C₄⋊C₄×C₅₃	C₂³	1	18	yes	—
49	C₂×C₄²×C₅₃	C₂³	1	9	yes	—
50	C₂³×D₅₃	C₂⁴	1	9	yes	—
51	C₂×Q₈×C₅₃	—	4	12	no	—
52	C₂×Q₈×C₅₃	#41	1	16	yes	complemented
53	C₂×Q₈×C₅₃	C₂⁴	1	24	yes	—
54	C₂×Q₈×C₅₃	C₂⁴	1	8	yes	—
55	C₂²×C₄×C₅₃	C₂⁴	1	9	yes	—
56	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
57	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
58	C₂²×Dic₅₃	C₂⁴	1	1	yes	characteristic
59	#11719949	—	2	192	no	—
60	#11719949	—	4	96	no	—
61	C₂×D₂₁₂	—	2	72	no	—
62	C₂×D₂₁₂	—	4	36	no	—
63	C₂×D₂₁₂	C₂⁴	1	72	yes	—
64	C₂×D₂₁₂	C₂⁴	1	72	yes	—
65	C₂×C₄×D₅₃	—	2	72	no	—
66	C₂×C₄×D₅₃	—	2	36	no	—
67	C₂×C₄×D₅₃	C₂⁴	1	72	yes	—
68	C₂×C₄×D₅₃	C₂⁴	1	36	yes	—
69	C₄⋊C₄×C₅₃	C₂⁴	1	72	yes	—
70	C₄²×C₅₃	C₂⁴	1	36	yes	—
71	#11719921	C₂⁴	1	144	yes	—
72	C₄⋊Dic₅₃	C₂⁴	1	24	yes	—
73	C₄×Dic₅₃	C₂⁴	1	24	yes	—
74	C₄×Dic₅₃	C₂⁴	1	36	yes	—
75	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
76	C₂²×D₅₃	—	2	36	no	—
77	C₂²×D₅₃	—	2	18	no	—
78	C₂²×D₅₃	C₂⁵	1	18	yes	—
79	Q₈×C₅₃	—	4	24	no	—
80	Q₈×C₅₃	C₂×#41	1	32	yes	complemented
81	C₂×C₄×C₅₃	—	2	18	no	—
82	C₂×C₄×C₅₃	C₂×#41	1	24	yes	—
83	C₂×C₄×C₅₃	C₂⁵	1	18	yes	—
84	C₂×C₄×C₅₃	C₂⁵	1	12	yes	—
85	C₂×Dic₅₃	—	2	12	no	—
86	C₂×Dic₅₃	H(5,𝔽₂)	1	4	yes	—
87	C₂×Dic₅₃	C₂⁵	1	12	yes	—
88	C₂×Dic₅₃	C₂⁵	1	2	yes	—
89	D₂₁₂	—	2	144	no	—
90	D₂₁₂	—	4	72	no	—
91	C₄×D₅₃	—	2	144	no	—
92	C₄×D₅₃	—	2	72	no	—
93	C₂×#3483	—	53	1	no	maximal, Sylow
94	C₂²×C₅₃	C₂²×#41	1	4	yes	—
95	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
96	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
97	C₂×D₅₃	—	2	36	no	—
98	C₂×D₅₃	—	2	18	no	—
99	C₂×D₅₃	—	4	36	no	—
100	C₄×C₅₃	—	2	18	no	—
101	C₄×C₅₃	C₂²×#41	1	24	yes	—
102	Dic₅₃	—	2	12	no	—
103	Dic₅₃	C₂×H(5,𝔽₂)	1	4	yes	—
104	#3483	—	53	32	no	—
105	C₂×Q₈²	D₅₃	1	1	yes	characteristic, complemented
106	C₂×#555	—	53	9	no	—
107	C₂×#550	—	53	6	no	—
108	C₂×#532	—	53	9	no	—
109	C₂×C₄×#41	—	53	6	no	—
110	C₂×C₅₃	#3483	1	4	yes	complemented
111	C₂×C₅₃	C₂³×#41	1	2	yes	—
112	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
113	D₅₃	—	4	18	no	—
114	C₂²×#41	—	53	6	no	—
115	C₂²×#41	—	53	6	no	—
116	Q₈²	C₂×D₅₃	1	16	yes	complemented
117	#555	—	53	144	no	—
118	#550	—	53	96	no	—
119	#532	—	53	144	no	—
120	C₂×C₄⋊Q₈	C₂×D₅₃	1	9	yes	—
121	C₂×#127	—	53	9	no	—
122	C₂×#124	—	53	18	no	—
123	C₂×#121	—	53	36	no	—
124	C₄×#41	—	53	96	no	—
125	C₂×C₄×Q₈	—	53	2	no	—
126	C₂×C₄×Q₈	C₂×D₅₃	1	6	yes	—
127	C₂×C₄×D₄	—	53	18	no	—
128	C₂×C₄×D₄	—	53	18	no	—
129	C₂×#117	—	53	18	no	—
130	C₂²×C₄²	—	53	9	no	—
131	C₅₃	C₂×#3483	1	1	yes	characteristic, complemented, Sylow
132	C₂×#41	—	53	48	no	—
133	C₂×#41	—	53	48	no	—
134	C₂×#41	—	106	48	no	—
135	C₂×#41	—	212	24	no	—
136	C₂²×Q₈	C₂²×D₅₃	1	6	yes	—
137	C₂²×Q₈	C₂²×D₅₃	1	2	yes	—
138	C₂²×D₄	—	53	18	no	—
139	C₂²×D₄	—	53	18	no	—
140	C₂³×C₄	—	53	18	no	—
141	C₂³×C₄	—	53	9	no	—
142	C₄⋊Q₈	C₂²×D₅₃	1	72	yes	—
143	#127	—	53	72	no	—
144	#124	—	53	144	no	—
145	#121	—	53	288	no	—
146	C₄×Q₈	—	53	16	no	—
147	C₄×Q₈	C₂²×D₅₃	1	48	yes	—
148	C₄×D₄	—	53	144	no	—
149	C₄×D₄	—	53	144	no	—
150	#117	—	53	144	no	—
151	C₂×C₄⋊C₄	—	53	6	no	—
152	C₂×C₄⋊C₄	C₂²×D₅₃	1	18	yes	—
153	C₂×C₂²⋊C₄	—	53	36	no	—
154	C₂×C₄²	—	53	9	no	—
155	C₂×C₄²	—	53	6	no	—
156	C₂×C₄²	—	53	72	no	—
157	C₂×C₄²	C₂²×D₅₃	1	9	yes	—
158	C₂⁴	—	53	9	no	—
159	#41	—	106	192	no	—
160	#41	—	212	96	no	—
161	C₂×Q₈	—	4	12	no	—
162	C₂×Q₈	#11719949	1	16	yes	complemented
163	C₂×Q₈	C₂³×D₅₃	1	24	yes	—
164	C₂×Q₈	C₂³×D₅₃	1	8	yes	—
165	C₂×D₄	—	53	72	no	—
166	C₂×D₄	—	53	72	no	—
167	C₂×D₄	—	106	72	no	—
168	C₂×D₄	—	212	36	no	—
169	C₂²×C₄	—	53	1	no	—
170	C₂²×C₄	—	53	6	no	—
171	C₂²×C₄	—	53	72	no	—
172	C₂²×C₄	—	53	36	no	—
173	C₂²×C₄	—	106	36	no	—
174	C₂²×C₄	—	106	72	no	—
175	C₂²×C₄	C₂³×D₅₃	1	9	yes	—
176	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
177	C₄⋊C₄	—	53	24	no	—
178	C₄⋊C₄	C₂³×D₅₃	1	72	yes	—
179	C₂²⋊C₄	—	53	144	no	—
180	C₄²	—	53	24	no	—
181	C₄²	—	53	36	no	—
182	C₄²	C₂³×D₅₃	1	36	yes	—
183	C₂³	—	53	18	no	—
184	C₂³	—	106	36	no	—
185	C₂³	—	106	18	no	—
186	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
187	Q₈	—	4	24	no	—
188	Q₈	C₂×#11719949	1	32	yes	complemented
189	D₄	—	106	144	no	—
190	D₄	—	212	72	no	—
191	C₂×C₄	—	2	18	no	—
192	C₂×C₄	—	53	4	no	—
193	C₂×C₄	—	53	2	no	—
194	C₂×C₄	—	53	12	no	—
195	C₂×C₄	—	106	72	no	—
196	C₂×C₄	—	106	12	no	—
197	C₂×C₄	—	106	144	no	—
198	C₂×C₄	C₂×#11719949	1	24	yes	—
199	C₂×C₄	C₂⁴×D₅₃	1	18	yes	—
200	C₂×C₄	C₂⁴×D₅₃	1	12	yes	—
201	C₂²	—	106	36	no	—
202	C₂²	—	106	18	no	—
203	C₂²	—	212	36	no	—
204	C₂²	C₂²×#11719949	1	4	yes	—
205	C₂²	H(5,𝔽₂)×D₅₃	1	2	yes	—
206	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
207	C₄	—	2	18	no	—
208	C₄	—	53	4	no	—
209	C₄	—	106	12	no	—
210	C₄	C₂²×#11719949	1	24	yes	—
211	C₂	—	212	18	no	—
212	C₂	#20288342	1	4	yes	direct product factor
213	C₂	C₂³×#11719949	1	2	yes	—
214	C₂	C₂×H(5,𝔽₂)×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂⁴×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂×Q₈²×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁵
hypercenter	C₂×Q₈²	D₅₃
hypocenter	C₅₃	C₂×#3483
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×Q₈²

Complements

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

N	H ≅ G/N	type
1. #20288342	C₂	direct (N × H)
2. C₂×#16811339	C₂	semidirect
3. C₂×#16811295	C₂	semidirect
4. C₂×#16811174	C₂	semidirect
5. C₂×C₄×#11719949	C₂	semidirect
6. C₂×Q₈²×C₅₃	C₂	semidirect
9. #16811339	C₂²	semidirect
10. #16811295	C₂²	semidirect
11. #16811174	C₂²	semidirect
15. C₄×#11719949	C₂²	semidirect
21. Q₈²×C₅₃	C₂²	semidirect
52. C₂×Q₈×C₅₃	#41	semidirect
80. Q₈×C₅₃	C₂×#41	semidirect
105. C₂×Q₈²	D₅₃	semidirect
110. C₂×C₅₃	#3483	semidirect
116. Q₈²	C₂×D₅₃	semidirect
131. C₅₃	C₂×#3483	semidirect
162. C₂×Q₈	#11719949	semidirect
188. Q₈	C₂×#11719949	semidirect
212. C₂	#20288342	direct (N × H)

Maximal subgroups

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

subgroup	index
#20288342	2
C₂×#16811339	2
C₂×#16811295	2
C₂×#16811174	2
C₂×C₄×#11719949	2
C₂×Q₈²×C₅₃	2
C₂×#3483	53

Subgroups — C2×#20288342 #28291197

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20288342 #28291197