Subgroups — C₂×#20276864 #27538718

Order 13568.

574 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
2	#20276864	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16810448	C₂	1	2	yes	maximal
4	C₂×C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic
5	C₂×#16810426	C₂	1	2	yes	maximal
6	C₂×C₅₃×#500	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₁₂	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₀₆	—	2	1	no	—
9	C₂²×#11719908	C₂²	1	2	yes	—
10	C₄×Dic₂₁₂	—	2	2	no	—
11	C₄×Dic₂₁₂	C₂²	1	4	yes	—
12	C₂×#13133151	C₂²	1	1	yes	characteristic
13	C₂×#13133142	—	2	1	no	—
14	#16810448	C₂²	1	8	yes	—
15	C₄⋊#11719908	C₂²	1	4	yes	—
16	#16810426	C₂²	1	8	yes	—
17	C₂×C₄⋊Q₈×C₅₃	—	2	1	no	—
18	C₅₃×#500	—	2	2	no	—
19	C₅₃×#500	C₂²	1	4	yes	—
20	C₂×#107×C₅₃	—	2	1	no	—
21	C₂×#107×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂²×Dic₁₀₆	—	2	1	no	—
24	C₂×Dic₂₁₂	—	2	1	no	—
25	C₂×Dic₂₁₂	—	2	2	no	—
26	C₂×Dic₂₁₂	Q₈	1	4	yes	complemented
27	C₂×Dic₂₁₂	C₂³	1	2	yes	—
28	C₄×Dic₁₀₆	—	2	4	no	—
29	C₄×Dic₁₀₆	—	4	2	no	—
30	C₂×C₄⋊Dic₅₃	—	2	1	no	—
31	C₂×#11719919	—	4	1	no	—
32	C₂×C₄×Dic₅₃	—	4	1	no	—
33	C₂×#11719908	—	2	4	no	—
34	C₂×#11719908	C₂³	1	4	yes	—
35	#13133151	—	2	1	no	—
36	#13133151	C₂³	1	2	yes	—
37	#13133142	—	2	4	no	—
38	C₂²×Q₈×C₅₃	—	4	1	no	—
39	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
41	C₄⋊Q₈×C₅₃	—	2	4	no	—
42	C₄⋊Q₈×C₅₃	—	4	2	no	—
43	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
44	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
45	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
46	#107×C₅₃	—	2	2	no	—
47	#107×C₅₃	—	2	4	no	—
48	#107×C₅₃	C₂³	1	4	yes	—
49	C₄×C₈×C₅₃	D₄	1	2	yes	—
50	C₄×C₈×C₅₃	C₂³	1	2	yes	—
51	C₂×Q₈×C₅₃	—	4	4	no	—
52	C₂×Q₈×C₅₃	—	8	4	no	—
53	C₂²×C₄×C₅₃	—	4	2	no	—
54	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₅₃	—	4	1	no	—
58	C₂²×Dic₅₃	—	4	1	no	—
59	C₂×Dic₁₀₆	—	2	4	no	—
60	C₂×Dic₁₀₆	—	2	2	no	—
61	C₂×Dic₁₀₆	—	4	1	no	—
62	C₂×Dic₁₀₆	—	4	2	no	—
63	C₂×C₈×C₅₃	—	2	2	no	—
64	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
65	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
67	C₂×C₈×C₅₃	#41	1	2	yes	—
68	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
69	C₄⋊C₄×C₅₃	—	2	4	no	—
70	C₄⋊C₄×C₅₃	—	4	2	no	—
71	C₄⋊C₄×C₅₃	—	4	4	no	—
72	C₄²×C₅₃	D₈	1	2	yes	—
73	C₄²×C₅₃	C₂×D₄	1	2	yes	—
74	C₄⋊Dic₅₃	—	2	2	no	—
75	C₄⋊Dic₅₃	—	4	1	no	—
76	#11719919	—	4	4	no	—
77	C₄×Dic₅₃	—	4	4	no	—
78	Dic₂₁₂	—	2	4	no	—
79	Dic₂₁₂	—	2	2	no	—
80	Dic₂₁₂	C₂×Q₈	1	4	yes	complemented
81	#11719908	—	2	4	no	—
82	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
83	Q₈×C₅₃	—	8	8	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	4	4	no	—
86	C₂×C₄×C₅₃	—	8	4	no	—
87	C₂×C₄×C₅₃	Q₃₂	1	4	yes	—
88	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
89	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
90	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
92	C₂×C₄×C₅₃	#137	1	2	yes	—
93	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
94	C₂×Dic₅₃	—	4	2	no	—
95	C₂×Dic₅₃	—	4	4	no	—
96	C₂×Dic₅₃	—	4	2	no	—
97	C₂×Dic₅₃	—	8	2	no	—
98	Dic₁₀₆	—	2	4	no	—
99	Dic₁₀₆	—	4	2	no	—
100	Dic₁₀₆	—	4	4	no	—
101	C₈×C₅₃	—	2	2	no	—
102	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
103	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
104	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
105	C₂×#2227	—	53	1	no	maximal, Sylow
106	C₂²×C₅₃	#474	1	2	yes	—
107	C₂²×C₅₃	C₂×Q₃₂	1	2	yes	—
108	C₂²×C₅₃	#509	1	2	yes	—
109	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
110	C₄×C₅₃	—	2	2	no	—
111	C₄×C₅₃	—	8	4	no	—
112	C₄×C₅₃	#474	1	2	yes	—
113	C₄×C₅₃	C₂×Q₃₂	1	4	yes	—
114	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
115	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
116	Dic₅₃	—	4	4	no	—
117	Dic₅₃	—	8	2	no	—
118	C₂×#500	D₅₃	1	1	yes	characteristic
119	C₂×C₄×Q₁₆	—	53	1	no	—
120	#2227	—	53	8	no	—
121	C₂×#366	—	53	2	no	—
122	C₂×C₄⋊C₁₆	—	53	1	no	—
123	C₂×#358	—	53	2	no	—
124	C₂×C₅₃	#2227	1	4	yes	complemented
125	C₂×C₅₃	C₂×#474	1	1	yes	characteristic
126	C₂×C₅₃	C₂²×Q₃₂	1	1	yes	characteristic
127	C₂×C₅₃	C₂×#509	1	1	yes	characteristic
128	C₂²×Q₁₆	—	53	1	no	—
129	C₂×C₄⋊Q₈	—	2	1	no	—
130	C₂×C₄×Q₈	—	106	1	no	—
131	C₂²×C₁₆	—	53	2	no	—
132	#500	—	2	2	no	—
133	#500	C₂×D₅₃	1	4	yes	—
134	C₄×Q₁₆	—	53	4	no	—
135	C₄×Q₁₆	—	106	2	no	—
136	C₂×#107	—	2	1	no	—
137	C₂×#107	—	53	1	no	—
138	C₂×#107	C₂×D₅₃	1	2	yes	—
139	C₂×#103	—	106	1	no	—
140	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
141	#366	—	53	8	no	—
142	C₄⋊C₁₆	—	53	4	no	—
143	#358	—	53	8	no	—
144	C₅₃	C₂×#2227	1	1	yes	characteristic, complemented, Sylow
145	C₂²×Q₈	—	4	1	no	—
146	C₂²×Q₈	—	106	1	no	—
147	C₂×Q₁₆	—	53	4	no	—
148	C₂×Q₁₆	—	53	2	no	—
149	C₂×Q₁₆	—	106	2	no	—
150	C₂×Q₁₆	—	106	1	no	—
151	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
152	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
153	C₄⋊Q₈	—	2	4	no	—
154	C₄⋊Q₈	—	4	2	no	—
155	C₄×Q₈	—	106	4	no	—
156	C₄×Q₈	—	212	2	no	—
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	—	4	1	no	—
159	C₂×C₄⋊C₄	—	106	1	no	—
160	C₂×C₄⋊C₄	—	212	1	no	—
161	C₂×C₄²	—	212	1	no	—
162	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
163	C₂×C₁₆	—	53	4	no	—
164	C₂×C₁₆	—	106	4	no	—
165	#107	—	2	4	no	—
166	#107	—	2	2	no	—
167	#107	—	53	2	no	—
168	#107	—	106	1	no	—
169	#107	C₂²×D₅₃	1	4	yes	—
170	#103	—	106	4	no	—
171	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
172	C₄×C₈	C₂²×D₅₃	1	2	yes	—
173	C₂×Q₈	—	4	4	no	—
174	C₂×Q₈	—	8	4	no	—
175	C₂×Q₈	—	106	4	no	—
176	C₂×Q₈	—	106	2	no	—
177	C₂×Q₈	—	212	1	no	—
178	C₂×Q₈	—	212	2	no	—
179	C₂²×C₄	—	4	2	no	—
180	C₂²×C₄	—	212	1	no	—
181	C₂²×C₄	—	212	1	no	—
182	C₂²×C₄	#11719922	1	1	yes	characteristic
183	C₂²×C₄	#11719922	1	1	yes	characteristic
184	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
185	Q₁₆	—	53	4	no	—
186	Q₁₆	—	106	4	no	—
187	Q₁₆	—	106	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	Q₈×D₅₃	1	2	yes	—
190	C₂×C₈	#11719949	1	2	yes	—
191	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
192	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
193	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	4	no	—
197	C₄⋊C₄	—	106	2	no	—
198	C₄⋊C₄	—	212	1	no	—
199	C₄⋊C₄	—	212	4	no	—
200	C₄²	—	212	4	no	—
201	C₄²	#11719922	1	2	yes	—
202	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
203	C₁₆	—	106	4	no	—
204	C₂³	C₂×#11719922	1	1	yes	characteristic
205	Q₈	—	8	8	no	—
206	Q₈	—	106	4	no	—
207	Q₈	—	212	4	no	—
208	Q₈	—	212	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	8	4	no	—
212	C₂×C₄	—	424	2	no	—
213	C₂×C₄	—	212	2	no	—
214	C₂×C₄	—	212	2	no	—
215	C₂×C₄	—	212	4	no	—
216	C₂×C₄	#13133163	1	4	yes	—
217	C₂×C₄	#13133193	1	2	yes	—
218	C₂×C₄	C₂×#11719922	1	2	yes	—
219	C₂×C₄	C₂×#11719922	1	2	yes	—
220	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
221	C₂×C₄	#13133248	1	2	yes	—
222	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	#13133193	1	2	yes	—
225	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
226	C₈	C₂×#11719949	1	1	yes	characteristic
227	C₂²	#16810900	1	2	yes	—
228	C₂²	C₂×#13133163	1	2	yes	—
229	C₂²	#16811059	1	2	yes	—
230	C₂²	C₂²×#11719922	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	8	4	no	—
233	C₄	—	424	2	no	—
234	C₄	—	212	4	no	—
235	C₄	#16810900	1	2	yes	—
236	C₄	C₂×#13133163	1	4	yes	—
237	C₄	C₂×#13133193	1	1	yes	characteristic
238	C₄	C₂×#13133248	1	1	yes	characteristic
239	C₂	#20276864	1	4	yes	direct product factor
240	C₂	C₂×#16810900	1	1	yes	characteristic
241	C₂	C₂²×#13133163	1	1	yes	characteristic
242	C₂	C₂×#16811059	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11719922
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#500	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
hypercenter	C₂×#500	D₅₃
hypocenter	C₅₃	C₂×#2227
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
2. #20276864	C₂	direct (N × H)
26. C₂×Dic₂₁₂	Q₈	semidirect
80. Dic₂₁₂	C₂×Q₈	semidirect
124. C₂×C₅₃	#2227	semidirect
144. C₅₃	C₂×#2227	semidirect
239. C₂	#20276864	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×C₄×Dic₂₁₂	2
#20276864	2
C₂×#16810448	2
C₂×C₄⋊#11719908	2
C₂×#16810426	2
C₂×C₅₃×#500	2
C₂×#2227	53

Subgroups — C2×#20276864 #27538718

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276864 #27538718