Subgroups — C₈×C₂²⋊#88324 #28301488

Order 13568.

726 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂²⋊#88324	C₂	1	2	yes	maximal
2	C₂×C₈×#88324	C₂	1	2	yes	maximal, complemented
3	#20289537	C₂	1	2	yes	maximal, complemented
4	C₂²×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₂²⋊#88324	C₄	1	4	yes	—
6	C₂×C₄×#88324	C₂²	1	4	yes	—
7	#16811489	C₄	1	4	yes	—
8	C₈×#88324	—	2	4	no	—
9	C₂³×#88322	C₄	1	1	yes	characteristic
10	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₅₃	—	2	4	no	—
12	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₅₃	C₄	1	1	yes	characteristic
17	C₂²×#88324	C₂×C₄	1	8	yes	—
18	C₂²⋊#88324	C₈	1	16	yes	direct product factor
19	C₄×#88324	—	2	4	no	—
20	C₄×#88324	—	2	4	no	—
21	C₂³×Dic₅₃	C₂×C₄	1	2	yes	—
22	C₂²×#88322	—	2	2	no	—
23	C₂²×#88322	—	2	2	no	—
24	C₂²×#88322	C₈	1	4	yes	complemented
25	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
26	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×Dic₅₃	—	2	2	no	—
28	C₂×C₄×Dic₅₃	—	2	2	no	—
29	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
32	C₈×Dic₅₃	—	2	2	no	—
33	C₈×Dic₅₃	—	2	2	no	—
34	C₈×Dic₅₃	—	2	2	no	—
35	C₈×Dic₅₃	D₄	1	4	yes	—
36	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	—	2	2	no	—
38	C₂²×C₈×C₅₃	—	2	2	no	—
39	C₂²×C₈×C₅₃	C₈	1	4	yes	complemented
40	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	—	2	1	no	—
44	C₂²×C₄×C₅₃	—	2	1	no	—
45	C₂²×C₄×C₅₃	—	2	1	no	—
46	C₂²×C₄×C₅₃	—	2	1	no	—
47	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×Dic₅₃	—	2	4	no	—
52	C₂²×Dic₅₃	—	2	4	no	—
53	C₂²×Dic₅₃	C₄²	1	2	yes	—
54	C₂²×Dic₅₃	C₂×C₈	1	8	yes	—
55	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
56	C₂×#88324	—	2	8	no	—
57	C₂×#88324	—	2	8	no	—
58	C₂×#88324	C₂×C₈	1	16	yes	complemented
59	C₂×C₈×C₅₃	—	2	1	no	—
60	C₂×C₈×C₅₃	—	2	1	no	—
61	C₂×C₈×C₅₃	—	2	8	no	—
62	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
64	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
65	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
66	C₄×Dic₅₃	—	2	1	no	—
67	C₄×Dic₅₃	—	2	1	no	—
68	C₄×Dic₅₃	—	2	4	no	—
69	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
70	C₄×Dic₅₃	#41	1	2	yes	—
71	C₂×#88322	—	2	1	no	—
72	C₂×#88322	—	2	1	no	—
73	C₂×#88322	—	2	8	no	—
74	C₂×#88322	C₂²⋊C₄	1	2	yes	—
75	C₂×#88322	C₂×C₈	1	2	yes	—
76	C₂×#88322	C₂×C₈	1	2	yes	—
77	C₂×#88322	M₁₆	1	2	yes	—
78	C₂³×C₅₃	—	2	1	no	—
79	C₂³×C₅₃	—	2	1	no	—
80	C₂³×C₅₃	—	2	2	no	—
81	C₂³×C₅₃	C₄×C₈	1	4	yes	—
82	C₂³×C₅₃	C₄×C₈	1	2	yes	—
83	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
84	C₂×C₄×C₅₃	—	2	1	no	—
85	C₂×C₄×C₅₃	—	2	1	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
91	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
97	C₂×Dic₅₃	—	2	2	no	—
98	C₂×Dic₅₃	—	2	2	no	—
99	C₂×Dic₅₃	—	2	16	no	—
100	C₂×Dic₅₃	C₄×C₈	1	4	yes	—
101	C₂×Dic₅₃	C₄×D₄	1	4	yes	—
102	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
103	C₂×Dic₅₃	#131	1	4	yes	—
104	#88324	—	2	16	no	—
105	C₈×C₅₃	—	2	2	no	—
106	C₈×C₅₃	C₂²⋊C₈	1	4	yes	complemented
107	#88322	—	2	2	no	—
108	#88322	C₂²⋊C₈	1	4	yes	complemented
109	C₈×C₂²⋊C₈	—	53	1	no	maximal, Sylow
110	C₂²×C₅₃	—	2	4	no	—
111	C₂²×C₅₃	—	2	2	no	—
112	C₂²×C₅₃	—	2	2	no	—
113	C₂²×C₅₃	—	2	4	no	—
114	C₂²×C₅₃	C₈²	1	4	yes	complemented
115	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
116	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
117	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
118	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
119	C₂²×C₅₃	#405	1	2	yes	—
120	C₄×C₅₃	—	2	2	no	—
121	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
122	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
123	C₄×C₅₃	#431	1	1	yes	characteristic
124	C₄×C₅₃	#431	1	1	yes	characteristic
125	Dic₅₃	—	2	4	no	—
126	Dic₅₃	C₈×D₄	1	8	yes	—
127	C₂²×C₄×C₈	—	53	1	no	—
128	C₄×C₂²⋊C₈	—	53	2	no	—
129	C₂×C₈²	—	53	2	no	—
130	#1273	—	53	2	no	—
131	C₂×C₅₃	—	2	4	no	—
132	C₂×C₅₃	C₂×C₈²	1	1	yes	characteristic
133	C₂×C₅₃	C₈×M₁₆	1	2	yes	—
134	C₂×C₅₃	#1450	1	1	yes	characteristic
135	C₂×C₅₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
136	C₂×C₅₃	C₈×C₂²⋊C₄	1	2	yes	—
137	C₂³×C₈	—	53	1	no	—
138	C₂³×C₈	#6157	1	1	yes	characteristic
139	C₂²×C₄²	—	53	1	no	—
140	C₂×C₂²⋊C₈	—	53	4	no	—
141	C₂×C₄×C₈	—	53	1	no	—
142	C₂×C₄×C₈	—	53	4	no	—
143	C₂×C₄×C₈	—	53	1	no	—
144	C₂×C₄×C₈	—	53	1	no	—
145	C₂×C₄×C₈	—	53	1	no	—
146	C₂×C₄×C₈	—	106	4	no	—
147	#336	—	53	4	no	—
148	C₈²	—	106	4	no	—
149	C₅₃	C₈×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
150	C₂³×C₄	—	53	2	no	—
151	C₂³×C₄	C₂×#6157	1	1	yes	characteristic
152	C₂²×C₈	—	2	2	no	—
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	—	53	1	no	—
155	C₂²×C₈	—	53	8	no	—
156	C₂²×C₈	—	53	1	no	—
157	C₂²×C₈	—	53	4	no	—
158	C₂²×C₈	—	106	2	no	—
159	C₂²×C₈	—	106	2	no	—
160	C₂²×C₈	#88324	1	4	yes	complemented
161	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
162	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
163	C₂×C₄²	—	53	1	no	—
164	C₂×C₄²	—	53	2	no	—
165	C₂×C₄²	—	53	1	no	—
166	C₂×C₄²	—	106	2	no	—
167	C₂×C₄²	—	106	2	no	—
168	C₂²⋊C₈	—	53	16	no	—
169	C₄×C₈	—	53	4	no	—
170	C₄×C₈	—	106	2	no	—
171	C₄×C₈	—	106	4	no	—
172	C₄×C₈	—	106	4	no	—
173	C₄×C₈	—	106	2	no	—
174	C₄×C₈	—	106	2	no	—
175	C₂⁴	C₄×#6157	1	1	yes	characteristic
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	53	2	no	—
181	C₂²×C₄	—	53	8	no	—
182	C₂²×C₄	—	53	2	no	—
183	C₂²×C₄	—	106	4	no	—
184	C₂²×C₄	—	106	4	no	—
185	C₂²×C₄	D₅₃⋊C₈	1	2	yes	—
186	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
187	C₂²×C₄	C₂×#88324	1	2	yes	—
188	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	8	no	—
192	C₂×C₈	—	53	2	no	—
193	C₂×C₈	—	53	2	no	—
194	C₂×C₈	—	53	2	no	—
195	C₂×C₈	—	53	16	no	—
196	C₂×C₈	—	53	2	no	—
197	C₂×C₈	—	106	8	no	—
198	C₂×C₈	—	106	8	no	—
199	C₂×C₈	—	106	8	no	—
200	C₂×C₈	—	106	1	no	—
201	C₂×C₈	—	106	1	no	—
202	C₂×C₈	C₂×#88324	1	2	yes	—
203	C₂×C₈	C₂×#88324	1	2	yes	—
204	C₂×C₈	#11719940	1	2	yes	—
205	C₂×C₈	C₂²⋊#6157	1	2	yes	—
206	C₄²	—	53	2	no	—
207	C₄²	—	53	2	no	—
208	C₄²	—	106	4	no	—
209	C₄²	—	106	1	no	—
210	C₄²	—	106	1	no	—
211	C₂³	—	2	1	no	—
212	C₂³	—	2	1	no	—
213	C₂³	—	2	2	no	—
214	C₂³	C₈×#6157	1	4	yes	—
215	C₂³	C₄×#88324	1	2	yes	—
216	C₂³	C₂×C₄×#6157	1	1	yes	characteristic
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	53	4	no	—
224	C₂×C₄	—	53	4	no	—
225	C₂×C₄	—	53	4	no	—
226	C₂×C₄	—	53	4	no	—
227	C₂×C₄	—	106	16	no	—
228	C₂×C₄	—	106	2	no	—
229	C₂×C₄	—	106	2	no	—
230	C₂×C₄	C₄×#88324	1	2	yes	—
231	C₂×C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
232	C₂×C₄	#13133296	1	1	yes	characteristic
233	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
234	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
235	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
236	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	—	53	4	no	—
239	C₈	—	106	16	no	—
240	C₈	—	106	2	no	—
241	C₈	C₂²⋊#88324	1	4	yes	direct product factor
242	C₂²	—	2	4	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	4	no	—
246	C₂²	C₈×#88324	1	4	yes	complemented
247	C₂²	C₄×C₂²⋊#6157	1	1	yes	characteristic
248	C₂²	C₂×C₈×#6157	1	2	yes	—
249	C₂²	C₂×C₄×#88324	1	1	yes	characteristic
250	C₂²	C₄×#11719940	1	1	yes	characteristic
251	C₂²	#16811586	1	2	yes	—
252	C₄	—	2	2	no	—
253	C₄	—	53	8	no	—
254	C₄	—	106	4	no	—
255	C₄	C₂×C₂²⋊#88324	1	1	yes	characteristic
256	C₄	D₅₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
257	C₄	#16811639	1	1	yes	characteristic
258	C₄	#16811640	1	1	yes	characteristic
259	C₂	—	2	4	no	—
260	C₂	C₂×C₈×#88324	1	1	yes	characteristic
261	C₂	C₈×#11719940	1	2	yes	—
262	C₂	#20289918	1	1	yes	characteristic
263	C₂	C₄×C₂²⋊#88324	1	1	yes	characteristic
264	C₂	C₈×C₂²⋊#6157	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂×#6157
commutator	C₂×C₅₃	C₂×C₈²
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂³×C₈×C₅₃	C₄
socle	C₂³×C₅₃	C₂×C₄²
prime radical	C₂⁴×C₅₃	C₄²
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	C₂³×C₈	#6157
hypocenter	C₅₃	C₈×C₂²⋊C₈
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×#88324	C₂	semidirect
3. #20289537	C₂	semidirect
18. C₂²⋊#88324	C₈	direct (N × H)
24. C₂²×#88322	C₈	semidirect
39. C₂²×C₈×C₅₃	C₈	semidirect
58. C₂×#88324	C₂×C₈	semidirect
106. C₈×C₅₃	C₂²⋊C₈	semidirect
108. #88322	C₂²⋊C₈	semidirect
114. C₂²×C₅₃	C₈²	semidirect
149. C₅₃	C₈×C₂²⋊C₈	semidirect
160. C₂²×C₈	#88324	semidirect
241. C₈	C₂²⋊#88324	direct (N × H)
246. C₂²	C₈×#88324	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₂²⋊#88324	2
C₂×C₈×#88324	2
#20289537	2
C₂²×C₈×Dic₅₃	2
C₈×C₂²⋊C₈	53

Subgroups — C8×C22⋊#88324 #28301488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×C₂²⋊#88324 #28301488