Subgroups — C₄×Q₈×#88324 #28318621

Order 13568.

1070 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×#88324	C₂	1	2	yes	maximal
2	C₄⋊C₄×#88324	C₂	1	6	yes	maximal
3	C₄×C₄⋊#88324	C₂	1	3	yes	maximal
4	C₄²×#88324	C₂	1	3	yes	maximal
5	C₄×Q₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	Q₈×#88324	C₄	1	16	yes	direct product factor
7	C₂×C₄⋊#88324	C₂²	1	6	yes	—
8	C₂×C₄×#88324	C₂²	1	3	yes	—
9	C₂×C₄×#88324	C₂²	1	6	yes	—
10	C₂×C₄×#88324	C₂²	1	6	yes	—
11	C₂×C₄×#88324	C₂²	1	1	yes	characteristic
12	#16811489	C₂²	1	6	yes	—
13	C₂×Q₈×Dic₅₃	C₄	1	1	yes	characteristic
14	C₂×Q₈×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×Dic₅₃	C₄	1	3	yes	—
16	C₄⋊C₄×Dic₅₃	C₂²	1	3	yes	—
17	C₄×C₄⋊Dic₅₃	C₄	1	3	yes	—
18	C₄²×Dic₅₃	C₂²	1	3	yes	—
19	C₂×C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
20	C₂²×#88324	C₂³	1	6	yes	—
21	C₂²×#88324	C₂³	1	2	yes	—
22	C₄⋊#88324	C₂×C₄	1	24	yes	—
23	C₄×#88324	—	2	6	no	—
24	C₄×#88324	—	2	12	no	—
25	C₄×#88324	C₂×C₄	1	24	yes	—
26	C₄×#88324	Q₈	1	4	yes	direct product factor
27	Q₈×Dic₅₃	C₈	1	8	yes	complemented
28	Q₈×Dic₅₃	C₂×C₄	1	8	yes	—
29	C₂×C₄⋊Dic₅₃	C₂×C₄	1	3	yes	—
30	C₂×C₄⋊Dic₅₃	C₂×C₄	1	3	yes	—
31	C₂×C₄×Dic₅₃	C₂×C₄	1	3	yes	—
32	C₂×C₄×Dic₅₃	C₂×C₄	1	3	yes	—
33	C₂×C₄×Dic₅₃	C₂×C₄	1	3	yes	—
34	C₂×C₄×Dic₅₃	C₂³	1	3	yes	—
35	C₂×C₄×Dic₅₃	C₂³	1	3	yes	—
36	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
37	#13133124	C₂×C₄	1	6	yes	—
38	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₄×Q₈×C₅₃	C₈	1	8	yes	complemented
40	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	3	yes	—
41	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
42	C₂×Q₈×C₅₃	C₄²	1	4	yes	—
43	C₂×Q₈×C₅₃	C₂×C₈	1	4	yes	—
44	C₂×Q₈×C₅₃	C₂×C₈	1	4	yes	—
45	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
46	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
47	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂²×Dic₅₃	C₂²×C₄	1	3	yes	—
49	C₂²×Dic₅₃	C₂²×C₄	1	3	yes	—
50	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×Dic₅₃	C₂⁴	1	1	yes	characteristic
52	C₂×#88324	—	2	6	no	—
53	C₂×#88324	—	2	6	no	—
54	C₂×#88324	C₂²×C₄	1	12	yes	—
55	C₂×#88324	C₂²×C₄	1	4	yes	—
56	C₂×#88324	C₂×Q₈	1	4	yes	—
57	C₂×#88324	#41	1	4	yes	—
58	C₄⋊C₄×C₅₃	C₂×C₈	1	12	yes	—
59	C₄²×C₅₃	C₂×C₈	1	12	yes	—
60	C₄⋊Dic₅₃	C₄²	1	12	yes	—
61	C₄⋊Dic₅₃	C₂×C₈	1	12	yes	—
62	C₄×Dic₅₃	—	2	6	no	—
63	C₄×Dic₅₃	—	2	6	no	—
64	C₄×Dic₅₃	—	2	6	no	—
65	C₄×Dic₅₃	C₂×C₈	1	12	yes	—
66	C₄×Dic₅₃	C₂²×C₄	1	12	yes	—
67	C₄×Dic₅₃	C₂×Q₈	1	2	yes	—
68	C₄×Dic₅₃	#41	1	2	yes	—
69	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
70	Q₈×C₅₃	C₄×C₈	1	16	yes	complemented
71	C₂×C₄×C₅₃	—	2	3	no	—
72	C₂×C₄×C₅₃	—	2	3	no	—
73	C₂×C₄×C₅₃	C₂×C₄²	1	6	yes	—
74	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
75	C₂×C₄×C₅₃	C₂²×C₈	1	6	yes	—
76	C₂×C₄×C₅₃	C₂²×C₈	1	6	yes	—
77	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
78	C₂×C₄×C₅₃	C₂²×C₈	1	6	yes	—
79	C₂×C₄×C₅₃	#131	1	2	yes	—
80	C₂×Dic₅₃	—	2	3	no	—
81	C₂×Dic₅₃	—	2	3	no	—
82	C₂×Dic₅₃	—	2	3	no	—
83	C₂×Dic₅₃	—	2	3	no	—
84	C₂×Dic₅₃	C₂×C₄²	1	6	yes	—
85	C₂×Dic₅₃	C₄×Q₈	1	2	yes	—
86	C₂×Dic₅₃	C₂²×C₈	1	6	yes	—
87	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
88	C₂×Dic₅₃	#131	1	2	yes	—
89	C₂×Dic₅₃	C₂³×C₄	1	2	yes	—
90	C₂×Dic₅₃	C₂²×Q₈	1	1	yes	characteristic
91	C₂×Dic₅₃	C₂×#41	1	2	yes	—
92	C₂×Dic₅₃	C₂×#41	1	1	yes	characteristic
93	#88324	—	2	12	no	—
94	#88324	C₄×Q₈	1	8	yes	direct product factor
95	C₄×C₈×Q₈	—	53	1	no	maximal, Sylow
96	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
97	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
98	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
99	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
100	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
101	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
102	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
103	C₄×C₅₃	—	2	6	no	—
104	C₄×C₅₃	C₂×C₄×C₈	1	12	yes	—
105	C₄×C₅₃	C₈×Q₈	1	4	yes	complemented
106	Dic₅₃	—	2	6	no	—
107	Dic₅₃	—	2	6	no	—
108	Dic₅₃	C₈×Q₈	1	4	yes	complemented
109	Dic₅₃	C₂×C₄×Q₈	1	2	yes	—
110	Dic₅₃	C₄×#41	1	2	yes	—
111	C₂×C₈×Q₈	—	53	2	no	—
112	C₄²×Q₈	—	53	1	no	—
113	C₈×C₄⋊C₄	—	53	6	no	—
114	C₄×C₄⋊C₈	—	53	3	no	—
115	C₄²×C₈	—	53	3	no	—
116	C₂×C₅₃	C₄²×Q₈	1	1	yes	characteristic
117	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
118	C₂×C₅₃	C₄×#131	1	1	yes	characteristic
119	C₂×C₅₃	C₂×C₈×Q₈	1	1	yes	characteristic
120	C₂×C₅₃	C₂×C₈×Q₈	1	1	yes	characteristic
121	C₂×C₅₃	C₈×#41	1	1	yes	characteristic
122	C₂×C₅₃	C₈×#41	1	1	yes	characteristic
123	C₂×C₄×Q₈	—	53	1	no	—
124	C₂×C₄×Q₈	—	53	1	no	—
125	C₂×C₄×Q₈	#6157	1	1	yes	characteristic
126	C₈×Q₈	—	53	16	no	—
127	C₂×C₄⋊C₈	—	53	6	no	—
128	C₂×C₄×C₈	—	53	6	no	—
129	C₂×C₄×C₈	—	53	1	no	—
130	C₂×C₄×C₈	—	53	3	no	—
131	C₂×C₄×C₈	—	53	6	no	—
132	C₄×C₄⋊C₄	—	53	3	no	—
133	C₄×C₄⋊C₄	—	53	3	no	—
134	C₄×C₄⋊C₄	—	53	3	no	—
135	C₄³	—	53	3	no	—
136	#336	—	53	6	no	—
137	C₅₃	C₄×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
138	C₂²×Q₈	C₂×#6157	1	1	yes	characteristic
139	C₂²×C₈	—	53	2	no	—
140	C₂²×C₈	—	53	6	no	—
141	C₄×Q₈	—	53	8	no	—
142	C₄×Q₈	—	53	8	no	—
143	C₄×Q₈	#88324	1	8	yes	direct product factor
144	C₂×C₄⋊C₄	—	53	3	no	—
145	C₂×C₄⋊C₄	—	53	3	no	—
146	C₂×C₄⋊C₄	C₂×#6157	1	3	yes	—
147	C₂×C₄²	—	53	3	no	—
148	C₂×C₄²	—	53	3	no	—
149	C₂×C₄²	—	53	1	no	—
150	C₂×C₄²	—	53	3	no	—
151	C₂×C₄²	—	53	3	no	—
152	C₂×C₄²	—	53	3	no	—
153	C₂×C₄²	C₂×#6157	1	3	yes	—
154	C₄⋊C₈	—	53	24	no	—
155	C₄×C₈	—	53	24	no	—
156	C₄×C₈	—	53	4	no	—
157	C₄×C₈	—	106	6	no	—
158	C₄×C₈	—	106	12	no	—
159	#95	—	53	6	no	—
160	C₂×Q₈	D₅₃⋊C₈	1	4	yes	—
161	C₂×Q₈	C₄×#6157	1	4	yes	—
162	C₂×Q₈	C₂×#88324	1	4	yes	—
163	C₂²×C₄	—	53	1	no	—
164	C₂²×C₄	—	53	1	no	—
165	C₂²×C₄	—	53	3	no	—
166	C₂²×C₄	—	53	3	no	—
167	C₂²×C₄	C₂²×#6157	1	3	yes	—
168	C₂²×C₄	C₂²×#6157	1	3	yes	—
169	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
170	C₂×C₈	—	53	4	no	—
171	C₂×C₈	—	53	4	no	—
172	C₂×C₈	—	53	4	no	—
173	C₂×C₈	—	53	12	no	—
174	C₂×C₈	—	106	6	no	—
175	C₂×C₈	—	106	6	no	—
176	C₄⋊C₄	—	53	12	no	—
177	C₄⋊C₄	—	53	12	no	—
178	C₄⋊C₄	C₂×#88324	1	12	yes	—
179	C₄²	—	53	12	no	—
180	C₄²	—	53	2	no	—
181	C₄²	—	53	2	no	—
182	C₄²	—	53	12	no	—
183	C₄²	—	106	6	no	—
184	C₄²	—	106	6	no	—
185	C₄²	—	106	6	no	—
186	C₄²	C₂×#88324	1	12	yes	—
187	C₂³	C₂³×#6157	1	1	yes	characteristic
188	Q₈	C₄×#88324	1	16	yes	direct product factor
189	C₂×C₄	—	2	3	no	—
190	C₂×C₄	—	2	3	no	—
191	C₂×C₄	—	53	2	no	—
192	C₂×C₄	—	53	2	no	—
193	C₂×C₄	—	53	1	no	—
194	C₂×C₄	—	53	1	no	—
195	C₂×C₄	—	53	2	no	—
196	C₂×C₄	—	53	2	no	—
197	C₂×C₄	—	53	2	no	—
198	C₂×C₄	—	53	6	no	—
199	C₂×C₄	—	53	6	no	—
200	C₂×C₄	—	106	3	no	—
201	C₂×C₄	—	106	3	no	—
202	C₂×C₄	—	106	3	no	—
203	C₂×C₄	—	106	3	no	—
204	C₂×C₄	C₂×C₄×#6157	1	6	yes	—
205	C₂×C₄	Q₈×#6157	1	2	yes	—
206	C₂×C₄	C₂²×#88324	1	6	yes	—
207	C₂×C₄	C₂²×#88324	1	6	yes	—
208	C₂×C₄	C₂²×#88324	1	2	yes	—
209	C₂×C₄	C₂×D₅₃⋊C₈	1	6	yes	—
210	C₂×C₄	#13133305	1	2	yes	—
211	C₈	—	53	8	no	—
212	C₈	—	106	12	no	—
213	C₂²	C₂²×C₄×#6157	1	1	yes	characteristic
214	C₂²	C₂×Q₈×#6157	1	1	yes	characteristic
215	C₂²	#41×#6157	1	1	yes	characteristic
216	C₂²	C₂³×#88324	1	1	yes	characteristic
217	C₂²	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
218	C₂²	D₅₃⋊#131	1	1	yes	characteristic
219	C₂²	C₂×#13133305	1	1	yes	characteristic
220	C₄	—	2	6	no	—
221	C₄	—	53	2	no	—
222	C₄	—	53	2	no	—
223	C₄	—	53	4	no	—
224	C₄	—	106	6	no	—
225	C₄	—	106	6	no	—
226	C₄	C₂×C₄×#88324	1	12	yes	—
227	C₄	Q₈×#88324	1	4	yes	direct product factor
228	C₂	C₄×Q₈×#6157	1	1	yes	characteristic
229	C₂	C₂²×C₄×#88324	1	1	yes	characteristic
230	C₂	C₄×#13133305	1	1	yes	characteristic
231	C₂	C₂×Q₈×#88324	1	1	yes	characteristic
232	C₂	Q₈×D₅₃⋊C₈	1	1	yes	characteristic
233	C₂	#41×#88324	1	1	yes	characteristic
234	C₂	#20291521	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×#6157
commutator	C₂×C₅₃	C₂²×C₄×C₈
Frattini	C₂³	C₂³×#6157
Fitting	C₂×C₄×Q₈×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₂³×C₄
cosocle	C₂²×Dic₅₃	C₂⁴
hypercenter	C₂×C₄×Q₈	#6157
hypocenter	C₅₃	C₄×C₈×Q₈
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
6. Q₈×#88324	C₄	direct (N × H)
26. C₄×#88324	Q₈	direct (N × H)
27. Q₈×Dic₅₃	C₈	semidirect
39. C₄×Q₈×C₅₃	C₈	semidirect
70. Q₈×C₅₃	C₄×C₈	semidirect
94. #88324	C₄×Q₈	direct (N × H)
105. C₄×C₅₃	C₈×Q₈	semidirect
108. Dic₅₃	C₈×Q₈	semidirect
137. C₅₃	C₄×C₈×Q₈	semidirect
143. C₄×Q₈	#88324	direct (N × H)
188. Q₈	C₄×#88324	direct (N × H)
227. C₄	Q₈×#88324	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×#88324	2
C₄⋊C₄×#88324	2
C₄×C₄⋊#88324	2
C₄²×#88324	2
C₄×Q₈×Dic₅₃	2
C₄×C₈×Q₈	53

Subgroups — C4×Q8×#88324 #28318621

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×Q₈×#88324 #28318621