Subgroups — C₂×Q₁₆×#88322 #27353463

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×Q₈×#88322	C₂	1	2	yes	maximal
2	Q₁₆×#88322	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16810376	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈⋊#88322	C₂	1	2	yes	maximal
5	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
7	Q₈×#88322	—	2	4	no	—
8	Q₈×#88322	C₂²	1	8	yes	—
9	C₂×C₄⋊#88322	—	2	2	no	—
10	C₂×C₄⋊#88322	C₂²	1	2	yes	—
11	C₂×C₄×#88322	—	2	2	no	—
12	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
13	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
14	#16810376	C₂²	1	4	yes	—
15	Q₈⋊#88322	C₂²	1	8	yes	—
16	C₈×#88322	C₂²	1	4	yes	—
17	C₂²×Q₁₆×C₅₃	C₄	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
19	C₄×Q₁₆×C₅₃	C₄	1	4	yes	—
20	C₄×Q₁₆×C₅₃	C₂²	1	4	yes	—
21	C₂×#107×C₅₃	C₄	1	1	yes	characteristic
22	C₂×#103×C₅₃	C₄	1	2	yes	—
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂²×#88322	—	2	2	no	—
25	C₂²×#88322	—	2	1	no	—
26	C₂²×#88322	D₄	1	1	yes	characteristic
27	C₂²×#88322	D₄	1	1	yes	characteristic
28	C₄⋊#88322	—	2	2	no	—
29	C₄⋊#88322	—	2	8	no	—
30	C₄⋊#88322	C₂³	1	4	yes	—
31	C₄×#88322	—	2	1	no	—
32	C₄×#88322	—	2	8	no	—
33	C₄×#88322	D₄	1	2	yes	—
34	C₄×#88322	C₂³	1	2	yes	—
35	C₄×#88322	C₂³	1	2	yes	—
36	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
37	C₂×Q₁₆×C₅₃	C₈	1	8	yes	complemented
38	C₂×Q₁₆×C₅₃	C₂×C₄	1	4	yes	—
39	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₄×Q₈×C₅₃	—	2	2	no	—
42	C₄×Q₈×C₅₃	—	2	2	no	—
43	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
44	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
45	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
46	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₅₃	—	2	2	no	—
48	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
49	#107×C₅₃	C₂×C₄	1	4	yes	—
50	#103×C₅₃	C₂×C₄	1	8	yes	—
51	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₅₃	C₂³	1	2	yes	—
53	C₂×Q₈×C₅₃	—	2	2	no	—
54	C₂×Q₈×C₅₃	—	2	4	no	—
55	C₂×Q₈×C₅₃	C₂×C₈	1	8	yes	—
56	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₅₃	—	2	2	no	—
58	C₂²×C₄×C₅₃	—	2	2	no	—
59	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
62	Q₁₆×C₅₃	C₂×C₈	1	16	yes	complemented
63	C₂×C₈×C₅₃	—	2	2	no	—
64	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
65	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
66	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
67	C₄⋊C₄×C₅₃	—	2	4	no	—
68	C₄⋊C₄×C₅₃	—	2	2	no	—
69	C₄⋊C₄×C₅₃	—	2	4	no	—
70	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
71	C₄²×C₅₃	—	2	4	no	—
72	C₄²×C₅₃	—	2	4	no	—
73	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₅₃	#41	1	1	yes	characteristic
76	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
77	C₂×#88322	—	2	4	no	—
78	C₂×#88322	—	2	2	no	—
79	C₂×#88322	—	2	2	no	—
80	C₂×#88322	—	2	4	no	—
81	C₂×#88322	—	4	4	no	—
82	C₂×#88322	Q₁₆	1	4	yes	direct product factor
83	C₂×#88322	C₂×D₄	1	2	yes	—
84	C₂×#88322	C₂×D₄	1	2	yes	—
85	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
86	Q₈×C₅₃	—	2	4	no	—
87	Q₈×C₅₃	—	2	8	no	—
88	Q₈×C₅₃	C₂²×C₈	1	8	yes	—
89	C₂×C₄×C₅₃	—	2	4	no	—
90	C₂×C₄×C₅₃	—	2	8	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	2	4	no	—
93	C₂×C₄×C₅₃	—	4	4	no	—
94	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
98	C₂×C₄×C₅₃	#131	1	2	yes	—
99	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
100	C₂×C₄×C₅₃	#135	1	2	yes	—
101	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
104	C₈×C₅₃	—	2	2	no	—
105	C₈×C₅₃	C₂²×C₈	1	4	yes	—
106	#88322	—	2	4	no	—
107	#88322	—	2	2	no	—
108	#88322	—	4	4	no	—
109	#88322	C₂×Q₁₆	1	4	yes	direct product factor
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	—	2	8	no	—
116	C₄×C₅₃	—	2	1	no	—
117	C₄×C₅₃	—	2	1	no	—
118	C₄×C₅₃	—	4	4	no	—
119	C₄×C₅₃	C₈×D₄	1	2	yes	—
120	C₄×C₅₃	C₄×Q₁₆	1	2	yes	—
121	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
122	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
123	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
124	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
125	C₂×C₈×Q₈	—	53	2	no	—
126	C₂×C₄×Q₁₆	D₅₃	1	1	yes	characteristic
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	Dic₅₃	1	1	yes	characteristic
136	C₂×C₄×Q₈	C₂×D₅₃	1	2	yes	—
137	C₈×Q₈	—	53	8	no	—
138	C₈×Q₈	—	106	4	no	—
139	C₄×Q₁₆	Dic₅₃	1	4	yes	—
140	C₄×Q₁₆	C₂×D₅₃	1	4	yes	—
141	C₂×#107	Dic₅₃	1	1	yes	characteristic
142	C₂×C₄⋊C₈	—	53	2	no	—
143	C₂×C₄⋊C₈	—	106	2	no	—
144	C₂×#103	Dic₅₃	1	2	yes	—
145	C₂×C₄×C₈	—	53	1	no	—
146	C₂×C₄×C₈	—	53	1	no	—
147	C₂×C₄×C₈	—	106	2	no	—
148	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
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	C₂×Dic₅₃	1	2	yes	—
154	C₂×Q₁₆	#88322	1	8	yes	direct product factor
155	C₂×Q₁₆	C₂×Dic₅₃	1	4	yes	—
156	C₂²×C₈	—	53	1	no	—
157	C₂²×C₈	—	53	1	no	—
158	C₂²×C₈	—	106	1	no	—
159	C₂²×C₈	—	106	2	no	—
160	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
161	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
162	C₄×Q₈	—	2	2	no	—
163	C₄×Q₈	—	2	2	no	—
164	C₄×Q₈	C₂×Dic₅₃	1	4	yes	—
165	C₄×Q₈	C₂²×D₅₃	1	4	yes	—
166	C₂×C₄⋊C₄	—	2	2	no	—
167	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
168	C₂×C₄²	—	2	2	no	—
169	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
170	#107	C₂×Dic₅₃	1	4	yes	—
171	C₄⋊C₈	—	53	4	no	—
172	C₄⋊C₈	—	106	8	no	—
173	C₄⋊C₈	—	106	2	no	—
174	#103	C₂×Dic₅₃	1	8	yes	—
175	C₄×C₈	—	53	2	no	—
176	C₄×C₈	—	53	2	no	—
177	C₄×C₈	—	53	2	no	—
178	C₄×C₈	—	106	8	no	—
179	C₄×C₈	—	106	1	no	—
180	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
181	C₄×C₈	C₂²×D₅₃	1	2	yes	—
182	C₂×Q₈	—	2	2	no	—
183	C₂×Q₈	—	2	4	no	—
184	C₂×Q₈	C₂×#88322	1	8	yes	—
185	C₂×Q₈	C₂²×Dic₅₃	1	4	yes	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
189	C₂²×C₄	#11719947	1	1	yes	characteristic
190	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
191	Q₁₆	C₂×#88322	1	16	yes	direct product factor
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	53	4	no	—
194	C₂×C₈	—	53	2	no	—
195	C₂×C₈	—	53	2	no	—
196	C₂×C₈	—	106	4	no	—
197	C₂×C₈	—	106	2	no	—
198	C₂×C₈	—	106	2	no	—
199	C₂×C₈	—	106	4	no	—
200	C₂×C₈	—	212	4	no	—
201	C₂×C₈	C₂×#88322	1	4	yes	—
202	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
203	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	C₂²×Dic₅₃	1	4	yes	—
208	C₄²	—	2	4	no	—
209	C₄²	—	2	4	no	—
210	C₄²	D₄×D₅₃	1	1	yes	characteristic
211	C₄²	#11719947	1	1	yes	characteristic
212	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
213	C₄²	C₂³×D₅₃	1	1	yes	characteristic
214	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
215	Q₈	—	2	4	no	—
216	Q₈	—	2	8	no	—
217	Q₈	C₂²×#88322	1	8	yes	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	8	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
224	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
225	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
226	C₂×C₄	C₂²×#88322	1	2	yes	—
227	C₂×C₄	#13133224	1	2	yes	—
228	C₂×C₄	Q₁₆×D₅₃	1	2	yes	—
229	C₂×C₄	#13133240	1	2	yes	—
230	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
231	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
232	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
233	C₈	—	2	2	no	—
234	C₈	—	53	4	no	—
235	C₈	—	106	4	no	—
236	C₈	—	106	2	no	—
237	C₈	—	212	4	no	—
238	C₈	C₂²×#88322	1	4	yes	—
239	C₂²	D₄×#88322	1	2	yes	—
240	C₂²	Q₁₆×Dic₅₃	1	2	yes	—
241	C₂²	#16810740	1	2	yes	—
242	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	2	8	no	—
246	C₄	—	4	4	no	—
247	C₄	D₄×#88322	1	2	yes	—
248	C₄	Q₁₆×Dic₅₃	1	2	yes	—
249	C₄	C₂³×#88322	1	1	yes	characteristic
250	C₄	C₂×#13133224	1	1	yes	characteristic
251	C₄	C₂×Q₁₆×D₅₃	1	1	yes	characteristic
252	C₄	C₂×#13133240	1	1	yes	characteristic
253	C₂	Q₁₆×#88322	1	4	yes	direct product factor
254	C₂	C₂×D₄×#88322	1	1	yes	characteristic
255	C₂	C₂×Q₁₆×Dic₅₃	1	1	yes	characteristic
256	C₂	C₂×#16810740	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₃
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₄×Q₁₆×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₅₃
hypocenter	C₅₃	C₂×C₈×Q₁₆
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. Q₁₆×#88322	C₂	direct (N × H)
37. C₂×Q₁₆×C₅₃	C₈	semidirect
62. Q₁₆×C₅₃	C₂×C₈	semidirect
82. C₂×#88322	Q₁₆	direct (N × H)
109. #88322	C₂×Q₁₆	direct (N × H)
131. C₂×C₅₃	C₈×Q₁₆	semidirect
152. C₅₃	C₂×C₈×Q₁₆	semidirect
154. C₂×Q₁₆	#88322	direct (N × H)
191. Q₁₆	C₂×#88322	direct (N × H)
253. C₂	Q₁₆×#88322	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×Q₈×#88322	2
Q₁₆×#88322	2
C₂×#16810376	2
C₂×Q₈⋊#88322	2
C₂×C₈×#88322	2
C₂×C₄×Q₁₆×C₅₃	2
C₂×C₈×Q₁₆	53

Subgroups — C2×Q16×#88322 #27353463

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₁₆×#88322 #27353463