Subgroups — Q₈×C₄⋊#88322 #27402886

Order 13568.

896 conjugacy classes of subgroups.

The subgroup lattice has 250 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	#20275434	C₂	1	6	yes	maximal
3	#20274867	C₂	1	3	yes	maximal
4	C₄×C₄⋊#88322	C₂	1	3	yes	maximal
5	C₄²×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	Q₈×#88322	—	2	8	no	—
7	C₂×C₄⋊#88322	C₂²	1	3	yes	—
8	C₂×C₄⋊#88322	C₂²	1	6	yes	—
9	C₂×C₄⋊#88322	C₂²	1	6	yes	—
10	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
11	C₂×C₄×#88322	C₂²	1	6	yes	—
12	#16810378	C₂²	1	6	yes	—
13	C₂×C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
15	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₅₃	C₄	1	3	yes	—
17	C₄×C₄⋊C₄×C₅₃	C₄	1	3	yes	—
18	C₄×C₄⋊C₄×C₅₃	C₂²	1	3	yes	—
19	C₄³×C₅₃	C₂²	1	3	yes	—
20	C₂²×#88322	C₂³	1	6	yes	—
21	C₂²×#88322	C₂³	1	2	yes	—
22	C₄⋊#88322	—	2	12	no	—
23	C₄⋊#88322	—	2	6	no	—
24	C₄⋊#88322	—	4	6	no	—
25	C₄⋊#88322	Q₈	1	4	yes	direct product factor
26	C₄×#88322	—	2	12	no	—
27	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄×Q₈×C₅₃	—	2	4	no	—
29	C₄×Q₈×C₅₃	C₈	1	8	yes	complemented
30	C₄×Q₈×C₅₃	D₄	1	4	yes	—
31	C₄×Q₈×C₅₃	Q₈	1	4	yes	—
32	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	3	yes	—
33	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	3	yes	—
34	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	3	yes	—
35	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
36	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
37	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
38	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
39	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
40	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
41	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
42	#95×C₅₃	C₂×C₄	1	6	yes	—
43	C₂×Q₈×C₅₃	C₄⋊C₄	1	4	yes	—
44	C₂×Q₈×C₅₃	C₂×C₈	1	4	yes	—
45	C₂×Q₈×C₅₃	M₁₆	1	4	yes	—
46	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
47	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
48	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
50	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
51	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
53	C₄⋊C₄×C₅₃	—	2	6	no	—
54	C₄⋊C₄×C₅₃	C₄⋊C₄	1	12	yes	—
55	C₄⋊C₄×C₅₃	C₂×C₈	1	12	yes	—
56	C₄²×C₅₃	—	2	6	no	—
57	C₄²×C₅₃	—	2	6	no	—
58	C₄²×C₅₃	—	2	3	no	—
59	C₄²×C₅₃	—	2	3	no	—
60	C₄²×C₅₃	—	4	3	no	—
61	C₄²×C₅₃	C₂×C₈	1	12	yes	—
62	C₄²×C₅₃	C₂×D₄	1	6	yes	—
63	C₄²×C₅₃	C₂×Q₈	1	6	yes	—
64	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
65	C₄²×C₅₃	#41	1	2	yes	—
66	C₂×#88322	—	2	2	no	—
67	C₂×#88322	—	2	2	no	—
68	C₂×#88322	—	2	6	no	—
69	C₂×#88322	—	2	6	no	—
70	C₂×#88322	—	2	6	no	—
71	C₂×#88322	C₂×Q₈	1	4	yes	—
72	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
73	Q₈×C₅₃	C₄⋊C₈	1	16	yes	complemented
74	C₂×C₄×C₅₃	—	2	3	no	—
75	C₂×C₄×C₅₃	—	2	1	no	—
76	C₂×C₄×C₅₃	—	2	3	no	—
77	C₂×C₄×C₅₃	—	2	3	no	—
78	C₂×C₄×C₅₃	—	2	3	no	—
79	C₂×C₄×C₅₃	—	2	1	no	—
80	C₂×C₄×C₅₃	—	2	3	no	—
81	C₂×C₄×C₅₃	—	2	3	no	—
82	C₂×C₄×C₅₃	—	2	3	no	—
83	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
84	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
85	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
86	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
87	C₂×C₄×C₅₃	C₂²×C₈	1	6	yes	—
88	C₂×C₄×C₅₃	C₂²×C₈	1	6	yes	—
89	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
90	C₂×C₄×C₅₃	C₂×M₁₆	1	6	yes	—
91	C₂×C₄×C₅₃	#131	1	2	yes	—
92	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
96	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
97	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
98	#88322	—	2	4	no	—
99	#88322	—	4	6	no	—
100	Q₈×C₄⋊C₈	—	53	1	no	maximal, Sylow
101	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
102	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
103	C₂²×C₅₃	#520	1	1	yes	characteristic
104	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
105	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
106	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
107	C₂²×C₅₃	#568	1	1	yes	characteristic
108	C₄×C₅₃	—	2	6	no	—
109	C₄×C₅₃	—	2	2	no	—
110	C₄×C₅₃	—	2	6	no	—
111	C₄×C₅₃	—	4	3	no	—
112	C₄×C₅₃	C₂×C₄⋊C₈	1	12	yes	—
113	C₄×C₅₃	C₈×Q₈	1	4	yes	complemented
114	C₄×C₅₃	D₄×Q₈	1	1	yes	characteristic
115	C₄×C₅₃	#550	1	1	yes	characteristic
116	C₄×C₅₃	#557	1	1	yes	characteristic
117	C₄×C₅₃	Q₈²	1	1	yes	characteristic
118	C₂×C₈×Q₈	—	53	2	no	—
119	C₄²×Q₈	D₅₃	1	1	yes	characteristic
120	#1939	—	53	6	no	—
121	#1842	—	53	3	no	—
122	C₄×C₄⋊C₈	—	53	3	no	—
123	C₂×C₅₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
125	C₂×C₅₃	#2906	1	1	yes	characteristic
126	C₂×C₅₃	C₂×C₈×Q₈	1	1	yes	characteristic
127	C₂×C₅₃	Q₈×M₁₆	1	1	yes	characteristic
128	C₂×C₅₃	#2988	1	1	yes	characteristic
129	C₂×C₅₃	#2991	1	1	yes	characteristic
130	C₂×C₄×Q₈	Dic₅₃	1	1	yes	characteristic
131	C₂×C₄×Q₈	Dic₅₃	1	1	yes	characteristic
132	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
133	C₈×Q₈	—	106	8	no	—
134	C₂×C₄⋊C₈	—	53	1	no	—
135	C₂×C₄⋊C₈	—	53	3	no	—
136	C₂×C₄⋊C₈	—	53	6	no	—
137	C₂×C₄⋊C₈	—	53	6	no	—
138	C₂×C₄×C₈	—	53	6	no	—
139	C₄×C₄⋊C₄	Dic₅₃	1	3	yes	—
140	C₄×C₄⋊C₄	Dic₅₃	1	3	yes	—
141	C₄×C₄⋊C₄	C₂×D₅₃	1	3	yes	—
142	C₄³	C₂×D₅₃	1	3	yes	—
143	#336	—	53	6	no	—
144	C₅₃	Q₈×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
145	C₂²×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
146	C₂²×C₈	—	53	2	no	—
147	C₂²×C₈	—	53	6	no	—
148	C₄×Q₈	—	2	4	no	—
149	C₄×Q₈	#88322	1	8	yes	complemented
150	C₄×Q₈	Dic₁₀₆	1	4	yes	—
151	C₄×Q₈	D₂₁₂	1	4	yes	—
152	C₂×C₄⋊C₄	C₂×Dic₅₃	1	3	yes	—
153	C₂×C₄⋊C₄	C₂×Dic₅₃	1	3	yes	—
154	C₂×C₄⋊C₄	C₂×Dic₅₃	1	3	yes	—
155	C₂×C₄²	C₂×Dic₅₃	1	3	yes	—
156	C₂×C₄²	C₂×Dic₅₃	1	3	yes	—
157	C₂×C₄²	C₂×Dic₅₃	1	3	yes	—
158	C₂×C₄²	C₂×Dic₅₃	1	3	yes	—
159	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
160	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
161	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
162	C₄⋊C₈	—	53	4	no	—
163	C₄⋊C₈	—	106	12	no	—
164	C₄⋊C₈	—	106	6	no	—
165	C₄⋊C₈	—	212	6	no	—
166	C₄×C₈	—	106	12	no	—
167	#95	C₂×Dic₅₃	1	6	yes	—
168	C₂×Q₈	C₂×#88322	1	4	yes	—
169	C₂×Q₈	#11719917	1	4	yes	—
170	C₂×Q₈	C₄⋊Dic₅₃	1	4	yes	—
171	C₂²×C₄	C₂²×Dic₅₃	1	3	yes	—
172	C₂²×C₄	C₂²×Dic₅₃	1	3	yes	—
173	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
174	C₂²×C₄	C₂²×Dic₅₃	1	3	yes	—
175	C₂²×C₄	C₂²×Dic₅₃	1	3	yes	—
176	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
177	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
178	C₂×C₈	—	53	4	no	—
179	C₂×C₈	—	106	2	no	—
180	C₂×C₈	—	106	2	no	—
181	C₂×C₈	—	106	6	no	—
182	C₂×C₈	—	106	6	no	—
183	C₂×C₈	—	106	6	no	—
184	C₄⋊C₄	—	2	6	no	—
185	C₄⋊C₄	C₂×#88322	1	12	yes	—
186	C₄⋊C₄	C₄⋊Dic₅₃	1	12	yes	—
187	C₄²	—	2	6	no	—
188	C₄²	—	2	6	no	—
189	C₄²	—	2	3	no	—
190	C₄²	—	2	3	no	—
191	C₄²	—	4	3	no	—
192	C₄²	C₂×#88322	1	12	yes	—
193	C₄²	C₂×Dic₁₀₆	1	6	yes	—
194	C₄²	C₂×D₂₁₂	1	6	yes	—
195	C₄²	Q₈×D₅₃	1	2	yes	—
196	C₄²	#11719949	1	2	yes	—
197	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
198	Q₈	C₄⋊#88322	1	16	yes	direct product factor
199	C₂×C₄	—	2	3	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	3	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	3	no	—
204	C₂×C₄	—	2	3	no	—
205	C₂×C₄	—	2	3	no	—
206	C₂×C₄	—	2	3	no	—
207	C₂×C₄	—	2	3	no	—
208	C₂×C₄	C₂×C₄⋊Dic₅₃	1	6	yes	—
209	C₂×C₄	C₂×C₄⋊Dic₅₃	1	6	yes	—
210	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
211	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
212	C₂×C₄	C₂²×#88322	1	6	yes	—
213	C₂×C₄	C₂²×#88322	1	6	yes	—
214	C₂×C₄	C₂²×#88322	1	2	yes	—
215	C₂×C₄	C₂×#11719917	1	6	yes	—
216	C₂×C₄	#13133224	1	2	yes	—
217	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
218	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
219	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
220	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
221	C₂×C₄	#13133263	1	1	yes	characteristic
222	C₂×C₄	#13133264	1	1	yes	characteristic
223	C₈	—	106	4	no	—
224	C₈	—	212	6	no	—
225	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
226	C₂²	C₂×Q₈×Dic₅₃	1	1	yes	characteristic
227	C₂²	#16811106	1	1	yes	characteristic
228	C₂²	C₂³×#88322	1	1	yes	characteristic
229	C₂²	C₂²×#11719917	1	1	yes	characteristic
230	C₂²	C₂×#13133224	1	1	yes	characteristic
231	C₂²	#16811390	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	2	6	no	—
234	C₄	—	2	6	no	—
235	C₄	—	4	3	no	—
236	C₄	C₂×C₄⋊#88322	1	12	yes	—
237	C₄	Q₈×#88322	1	4	yes	complemented
238	C₄	Q₈×D₂₁₂	1	1	yes	characteristic
239	C₄	#16811295	1	1	yes	characteristic
240	C₄	#16811352	1	1	yes	characteristic
241	C₄	Q₈×Dic₁₀₆	1	1	yes	characteristic
242	C₂	Q₈×C₄⋊Dic₅₃	1	1	yes	characteristic
243	C₂	C₂²×C₄⋊#88322	1	1	yes	characteristic
244	C₂	#20283283	1	1	yes	characteristic
245	C₂	C₂×Q₈×#88322	1	1	yes	characteristic
246	C₂	Q₈×#11719917	1	1	yes	characteristic
247	C₂	#20283826	1	1	yes	characteristic
248	C₂	#20283847	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂³×C₈
Fitting	C₄²×Q₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂³×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄²×Q₈	D₅₃
hypocenter	C₅₃	Q₈×C₄⋊C₈
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
25. C₄⋊#88322	Q₈	direct (N × H)
29. C₄×Q₈×C₅₃	C₈	semidirect
73. Q₈×C₅₃	C₄⋊C₈	semidirect
113. C₄×C₅₃	C₈×Q₈	semidirect
144. C₅₃	Q₈×C₄⋊C₈	semidirect
149. C₄×Q₈	#88322	semidirect
198. Q₈	C₄⋊#88322	direct (N × H)
237. C₄	Q₈×#88322	semidirect

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
#20275434	2
#20274867	2
C₄×C₄⋊#88322	2
C₄²×Q₈×C₅₃	2
Q₈×C₄⋊C₈	53

Subgroups — Q8×C4⋊#88322 #27402886

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₄⋊#88322 #27402886