Subgroups — C₂×#20290303 #28316356

Order 13568.

734 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811672	C₂	1	2	yes	maximal
2	#20290303	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16811486	C₂	1	1	yes	maximal, characteristic
4	C₂×#16811473	C₂	1	2	yes	maximal
5	C₂×C₈×#88324	C₂	1	1	yes	maximal, characteristic
6	C₂×#16810717	C₂	1	1	yes	maximal, characteristic
7	#16811672	—	2	4	no	—
8	#16811672	C₂²	1	8	yes	—
9	C₂×#13133284	—	2	2	no	—
10	C₂×C₄⋊#88324	C₂²	1	2	yes	—
11	C₂×C₄×#88324	—	2	2	no	—
12	C₂×C₄×#88324	C₂²	1	1	yes	characteristic
13	C₂×C₄×#88324	C₂²	1	1	yes	characteristic
14	#16811486	C₂²	1	4	yes	—
15	#16811473	C₂²	1	8	yes	—
16	C₈×#88324	C₂²	1	4	yes	—
17	C₂²×Dic₂₁₂	C₄	1	1	yes	characteristic
18	C₂×#13133182	C₂²	1	2	yes	—
19	#16810717	C₄	1	4	yes	—
20	#16810717	C₂²	1	4	yes	—
21	C₂×#13133140	C₄	1	2	yes	—
22	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
23	C₂×#107×C₅₃	C₄	1	1	yes	characteristic
24	C₂²×#88324	—	2	1	no	—
25	C₂²×#88324	—	2	2	no	—
26	C₂²×#88324	D₄	1	1	yes	characteristic
27	C₂²×#88324	D₄	1	1	yes	characteristic
28	#13133284	—	2	8	no	—
29	C₄⋊#88324	—	2	2	no	—
30	C₄⋊#88324	C₂³	1	4	yes	—
31	C₄×#88324	—	2	1	no	—
32	C₄×#88324	—	2	8	no	—
33	C₄×#88324	D₄	1	2	yes	—
34	C₄×#88324	C₂³	1	2	yes	—
35	C₄×#88324	C₂³	1	2	yes	—
36	C₂²×Dic₁₀₆	C₂×C₄	1	2	yes	—
37	C₂×Dic₂₁₂	C₈	1	8	yes	complemented
38	C₂×Dic₂₁₂	C₂×C₄	1	4	yes	—
39	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
40	#13133182	—	2	2	no	—
41	#13133182	—	2	2	no	—
42	#13133182	C₂×C₄	1	4	yes	—
43	#13133182	C₂³	1	4	yes	—
44	C₂×#11719919	—	2	2	no	—
45	C₂×C₄×Dic₅₃	—	2	2	no	—
46	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
47	#13133140	C₂×C₄	1	8	yes	—
48	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
49	C₈×Dic₅₃	C₂³	1	2	yes	—
50	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
52	#107×C₅₃	C₂×C₄	1	4	yes	—
53	C₂²×C₄×C₅₃	—	2	2	no	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×Dic₅₃	—	2	2	no	—
56	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₅₃	#41	1	1	yes	characteristic
58	C₂×Dic₁₀₆	—	2	2	no	—
59	C₂×Dic₁₀₆	—	2	4	no	—
60	C₂×Dic₁₀₆	C₂×C₈	1	8	yes	—
61	C₂×Dic₁₀₆	C₂²×C₄	1	4	yes	—
62	C₂×#88324	—	2	4	no	—
63	C₂×#88324	—	2	2	no	—
64	C₂×#88324	—	2	2	no	—
65	C₂×#88324	—	2	4	no	—
66	C₂×#88324	—	4	4	no	—
67	C₂×#88324	Q₁₆	1	4	yes	complemented
68	C₂×#88324	C₂×D₄	1	2	yes	—
69	C₂×#88324	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
71	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
72	C₄⋊C₄×C₅₃	—	2	2	no	—
73	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
74	#11719919	—	2	4	no	—
75	#11719919	—	2	4	no	—
76	C₄×Dic₅₃	—	2	4	no	—
77	C₄×Dic₅₃	—	2	4	no	—
78	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
80	C₄×Dic₅₃	#41	1	1	yes	characteristic
81	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
82	C₂×#88322	—	2	2	no	—
83	C₂×#88322	C₂²×C₄	1	2	yes	—
84	Dic₂₁₂	C₂×C₈	1	16	yes	complemented
85	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₅₃	—	2	4	no	—
87	C₂×C₄×C₅₃	—	4	4	no	—
88	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
90	C₂×C₄×C₅₃	#131	1	2	yes	—
91	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
92	C₂×Dic₅₃	—	2	4	no	—
93	C₂×Dic₅₃	—	2	8	no	—
94	C₂×Dic₅₃	—	2	2	no	—
95	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
96	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
97	C₂×Dic₅₃	C₂×Q₁₆	1	2	yes	—
98	C₂×Dic₅₃	#135	1	2	yes	—
99	C₂×Dic₅₃	C₂²×D₄	1	1	yes	characteristic
100	C₂×Dic₅₃	C₂×#41	1	1	yes	characteristic
101	Dic₁₀₆	—	2	4	no	—
102	Dic₁₀₆	—	2	8	no	—
103	Dic₁₀₆	C₂²×C₈	1	8	yes	—
104	#88324	—	2	4	no	—
105	#88324	—	2	2	no	—
106	#88324	—	4	4	no	—
107	#88324	C₂×Q₁₆	1	4	yes	complemented
108	C₈×C₅₃	C₂²×C₈	1	4	yes	—
109	#88322	—	2	2	no	—
110	C₂×C₈×Q₁₆	—	53	1	no	maximal, Sylow
111	C₂²×C₅₃	C₈×D₄	1	2	yes	—
112	C₂²×C₅₃	C₄×Q₁₆	1	2	yes	—
113	C₂²×C₅₃	#443	1	2	yes	—
114	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
115	C₄×C₅₃	—	4	4	no	—
116	C₄×C₅₃	C₈×D₄	1	2	yes	—
117	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
118	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
119	Dic₅₃	—	2	8	no	—
120	Dic₅₃	—	2	1	no	—
121	Dic₅₃	—	2	1	no	—
122	Dic₅₃	C₄×Q₁₆	1	2	yes	—
123	Dic₅₃	C₂²×Q₁₆	1	1	yes	characteristic
124	Dic₅₃	C₂×#135	1	1	yes	characteristic
125	C₂×C₈×Q₈	—	53	2	no	—
126	C₂×C₄×Q₁₆	—	53	1	no	—
127	C₈×Q₁₆	—	53	8	no	—
128	C₂×#335	—	53	1	no	—
129	C₂×Q₈⋊C₈	—	53	2	no	—
130	C₂×C₈²	—	53	1	no	—
131	C₂×C₅₃	C₈×Q₁₆	1	4	yes	complemented
132	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
133	C₂×C₅₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
134	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
135	C₂²×Q₁₆	—	53	1	no	—
136	C₂×C₄×Q₈	—	53	2	no	—
137	C₈×Q₈	—	53	8	no	—
138	C₈×Q₈	—	106	4	no	—
139	C₄×Q₁₆	—	53	4	no	—
140	C₄×Q₁₆	—	53	4	no	—
141	C₂×#107	#6157	1	1	yes	characteristic
142	C₂×C₄⋊C₈	—	53	2	no	—
143	C₂×C₄⋊C₈	—	106	2	no	—
144	C₂×#103	—	53	2	no	—
145	C₂×C₄×C₈	—	53	1	no	—
146	C₂×C₄×C₈	—	53	1	no	—
147	C₂×C₄×C₈	—	53	1	no	—
148	C₂×C₄×C₈	—	106	2	no	—
149	#335	—	53	4	no	—
150	Q₈⋊C₈	—	53	8	no	—
151	C₈²	—	53	4	no	—
152	C₅₃	C₂×C₈×Q₁₆	1	1	yes	characteristic, complemented, Sylow
153	C₂²×Q₈	—	53	2	no	—
154	C₂×Q₁₆	—	53	4	no	—
155	C₂×Q₁₆	—	53	8	no	—
156	C₂²×C₈	—	53	1	no	—
157	C₂²×C₈	—	53	1	no	—
158	C₂²×C₈	—	53	1	no	—
159	C₂²×C₈	—	106	1	no	—
160	C₂²×C₈	—	106	2	no	—
161	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
162	C₄×Q₈	—	53	4	no	—
163	C₄×Q₈	—	53	4	no	—
164	C₄×Q₈	—	106	2	no	—
165	C₄×Q₈	—	106	2	no	—
166	C₂×C₄⋊C₄	—	106	2	no	—
167	C₂×C₄⋊C₄	C₂×#6157	1	2	yes	—
168	C₂×C₄²	—	53	1	no	—
169	C₂×C₄²	—	106	2	no	—
170	#107	C₂×#6157	1	4	yes	—
171	C₄⋊C₈	—	53	4	no	—
172	C₄⋊C₈	—	106	2	no	—
173	C₄⋊C₈	—	106	8	no	—
174	#103	—	53	8	no	—
175	C₄×C₈	—	53	2	no	—
176	C₄×C₈	—	53	2	no	—
177	C₄×C₈	—	53	2	no	—
178	C₄×C₈	—	53	2	no	—
179	C₄×C₈	—	53	2	no	—
180	C₄×C₈	—	106	1	no	—
181	C₄×C₈	—	106	8	no	—
182	C₂×Q₈	—	53	8	no	—
183	C₂×Q₈	—	53	4	no	—
184	C₂×Q₈	—	106	2	no	—
185	C₂×Q₈	—	106	4	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	53	1	no	—
188	C₂²×C₄	—	53	1	no	—
189	C₂²×C₄	—	106	2	no	—
190	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
191	Q₁₆	—	53	16	no	—
192	C₂×C₈	—	53	4	no	—
193	C₂×C₈	—	53	2	no	—
194	C₂×C₈	—	53	2	no	—
195	C₂×C₈	—	53	2	no	—
196	C₂×C₈	—	106	2	no	—
197	C₂×C₈	—	106	2	no	—
198	C₂×C₈	—	106	4	no	—
199	C₂×C₈	—	106	2	no	—
200	C₂×C₈	—	106	4	no	—
201	C₂×C₈	—	212	4	no	—
202	C₂×C₈	D₅₃⋊C₈	1	4	yes	—
203	C₂×C₈	C₂²×#6157	1	2	yes	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	106	4	no	—
206	C₄⋊C₄	—	106	4	no	—
207	C₄⋊C₄	C₂²×#6157	1	4	yes	—
208	C₄²	—	53	1	no	—
209	C₄²	—	53	1	no	—
210	C₄²	—	53	1	no	—
211	C₄²	—	53	1	no	—
212	C₄²	—	106	4	no	—
213	C₄²	—	106	4	no	—
214	C₂³	D₄×#6157	1	1	yes	characteristic
215	Q₈	—	53	8	no	—
216	Q₈	—	106	8	no	—
217	Q₈	—	106	4	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	—	53	2	no	—
221	C₂×C₄	—	53	2	no	—
222	C₂×C₄	—	53	1	no	—
223	C₂×C₄	—	53	1	no	—
224	C₂×C₄	—	53	1	no	—
225	C₂×C₄	—	53	1	no	—
226	C₂×C₄	—	106	8	no	—
227	C₂×C₄	—	106	2	no	—
228	C₂×C₄	—	106	4	no	—
229	C₂×C₄	D₄×#6157	1	1	yes	characteristic
230	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
231	C₂×C₄	#13133305	1	2	yes	—
232	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
233	C₈	—	53	4	no	—
234	C₈	—	106	2	no	—
235	C₈	—	106	4	no	—
236	C₈	—	106	2	no	—
237	C₈	—	212	4	no	—
238	C₈	C₂×D₅₃⋊C₈	1	4	yes	—
239	C₂²	#16811651	1	2	yes	—
240	C₂²	Q₁₆×#6157	1	2	yes	—
241	C₂²	#16811665	1	2	yes	—
242	C₂²	C₂×D₄×#6157	1	1	yes	characteristic
243	C₄	—	4	4	no	—
244	C₄	—	53	1	no	—
245	C₄	—	53	1	no	—
246	C₄	—	53	2	no	—
247	C₄	—	106	8	no	—
248	C₄	—	106	1	no	—
249	C₄	—	106	1	no	—
250	C₄	#16811651	1	2	yes	—
251	C₄	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
252	C₄	C₂×#13133305	1	1	yes	characteristic
253	C₂	#20290303	1	4	yes	direct product factor
254	C₂	C₂×#16811651	1	1	yes	characteristic
255	C₂	C₂×Q₁₆×#6157	1	1	yes	characteristic
256	C₂	C₂×#16811665	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×#6157
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₂×#107×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₄×Dic₅₃	C₂⁴
hypercenter	C₂×#107	#6157
hypocenter	C₅₃	C₂×C₈×Q₁₆
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

Complements

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

N	H ≅ G/N	type
2. #20290303	C₂	direct (N × H)
37. C₂×Dic₂₁₂	C₈	semidirect
67. C₂×#88324	Q₁₆	semidirect
84. Dic₂₁₂	C₂×C₈	semidirect
107. #88324	C₂×Q₁₆	semidirect
131. C₂×C₅₃	C₈×Q₁₆	semidirect
152. C₅₃	C₂×C₈×Q₁₆	semidirect
253. C₂	#20290303	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×#16811672	2
#20290303	2
C₂×#16811486	2
C₂×#16811473	2
C₂×C₈×#88324	2
C₂×#16810717	2
C₂×C₈×Q₁₆	53

Subgroups — C2×#20290303 #28316356

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20290303 #28316356