Subgroups — C₄×Q₈×#88322 #27382700

Order 13568.

1070 conjugacy classes of subgroups.

The subgroup lattice has 184 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	C₄⋊C₄×#88322	C₂	1	6	yes	maximal
3	C₄×C₄⋊#88322	C₂	1	3	yes	maximal
4	C₄²×#88322	C₂	1	3	yes	maximal
5	C₄²×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	Q₈×#88322	C₄	1	16	yes	direct product factor
7	C₂×C₄⋊#88322	C₂²	1	6	yes	—
8	C₂×C₄×#88322	C₂²	1	3	yes	—
9	C₂×C₄×#88322	C₂²	1	6	yes	—
10	C₂×C₄×#88322	C₂²	1	6	yes	—
11	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
12	#16810378	C₂²	1	6	yes	—
13	C₂×C₄×Q₈×C₅₃	C₄	1	2	yes	—
14	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₅₃	C₄	1	6	yes	—
16	C₄×C₄⋊C₄×C₅₃	C₂²	1	3	yes	—
17	C₄³×C₅₃	C₂²	1	3	yes	—
18	C₂²×#88322	C₂³	1	6	yes	—
19	C₂²×#88322	C₂³	1	2	yes	—
20	C₄⋊#88322	C₂×C₄	1	24	yes	—
21	C₄×#88322	—	2	6	no	—
22	C₄×#88322	—	2	12	no	—
23	C₄×#88322	C₂×C₄	1	24	yes	—
24	C₄×#88322	Q₈	1	4	yes	direct product factor
25	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₄×Q₈×C₅₃	C₈	1	16	yes	complemented
27	C₄×Q₈×C₅₃	C₂×C₄	1	8	yes	—
28	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	3	yes	—
29	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	6	yes	—
30	C₂×C₄²×C₅₃	C₂×C₄	1	6	yes	—
31	C₂×C₄²×C₅₃	C₂×C₄	1	6	yes	—
32	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
33	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
34	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
35	#95×C₅₃	C₂×C₄	1	6	yes	—
36	C₂×Q₈×C₅₃	C₄²	1	4	yes	—
37	C₂×Q₈×C₅₃	C₂×C₈	1	8	yes	—
38	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
39	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
40	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
41	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
42	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
43	C₄⋊C₄×C₅₃	C₄²	1	12	yes	—
44	C₄⋊C₄×C₅₃	C₂×C₈	1	24	yes	—
45	C₄²×C₅₃	—	2	12	no	—
46	C₄²×C₅₃	—	2	6	no	—
47	C₄²×C₅₃	C₂×C₈	1	24	yes	—
48	C₄²×C₅₃	C₂²×C₄	1	12	yes	—
49	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
50	C₄²×C₅₃	#41	1	2	yes	—
51	C₂×#88322	—	2	6	no	—
52	C₂×#88322	—	2	6	no	—
53	C₂×#88322	C₂²×C₄	1	12	yes	—
54	C₂×#88322	C₂²×C₄	1	4	yes	—
55	C₂×#88322	C₂×Q₈	1	4	yes	—
56	C₂×#88322	#41	1	4	yes	—
57	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
58	Q₈×C₅₃	C₄×C₈	1	16	yes	complemented
59	C₂×C₄×C₅₃	—	2	6	no	—
60	C₂×C₄×C₅₃	—	2	6	no	—
61	C₂×C₄×C₅₃	—	2	6	no	—
62	C₂×C₄×C₅₃	C₂×C₄²	1	6	yes	—
63	C₂×C₄×C₅₃	C₂×C₄²	1	6	yes	—
64	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
65	C₂×C₄×C₅₃	C₂²×C₈	1	12	yes	—
66	C₂×C₄×C₅₃	C₂²×C₈	1	12	yes	—
67	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
68	C₂×C₄×C₅₃	#131	1	4	yes	—
69	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
70	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
72	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
73	#88322	—	2	12	no	—
74	#88322	C₄×Q₈	1	8	yes	direct product factor
75	C₄×C₈×Q₈	—	53	1	no	maximal, Sylow
76	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
77	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
78	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
79	C₂²×C₅₃	C₂³×C₈	1	2	yes	—
80	C₂²×C₅₃	C₂×#131	1	2	yes	—
81	C₄×C₅₃	—	2	6	no	—
82	C₄×C₅₃	—	2	12	no	—
83	C₄×C₅₃	C₂×C₄×C₈	1	12	yes	—
84	C₄×C₅₃	C₈×Q₈	1	8	yes	complemented
85	C₄×C₅₃	C₂×C₄×Q₈	1	2	yes	—
86	C₄×C₅₃	C₄×#41	1	2	yes	—
87	C₂×C₈×Q₈	—	53	2	no	—
88	C₄²×Q₈	D₅₃	1	1	yes	characteristic
89	C₈×C₄⋊C₄	—	53	6	no	—
90	C₄×C₄⋊C₈	—	53	3	no	—
91	C₄²×C₈	—	53	3	no	—
92	C₂×C₅₃	C₄²×Q₈	1	1	yes	characteristic
93	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
94	C₂×C₅₃	C₄×#131	1	1	yes	characteristic
95	C₂×C₅₃	C₂×C₈×Q₈	1	2	yes	—
96	C₂×C₅₃	C₈×#41	1	2	yes	—
97	C₂×C₄×Q₈	Dic₅₃	1	2	yes	—
98	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
99	C₈×Q₈	—	53	16	no	—
100	C₂×C₄⋊C₈	—	53	6	no	—
101	C₂×C₄×C₈	—	53	6	no	—
102	C₂×C₄×C₈	—	53	1	no	—
103	C₂×C₄×C₈	—	53	6	no	—
104	C₂×C₄×C₈	—	53	3	no	—
105	C₄×C₄⋊C₄	Dic₅₃	1	6	yes	—
106	C₄×C₄⋊C₄	C₂×D₅₃	1	3	yes	—
107	C₄³	C₂×D₅₃	1	3	yes	—
108	#336	—	53	6	no	—
109	C₅₃	C₄×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
110	C₂²×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
111	C₂²×C₈	—	53	2	no	—
112	C₂²×C₈	—	53	6	no	—
113	C₄×Q₈	#88322	1	16	yes	direct product factor
114	C₄×Q₈	C₄×D₅₃	1	8	yes	—
115	C₂×C₄⋊C₄	C₂×Dic₅₃	1	6	yes	—
116	C₂×C₄⋊C₄	C₂×Dic₅₃	1	3	yes	—
117	C₂×C₄²	C₂×Dic₅₃	1	6	yes	—
118	C₂×C₄²	C₂×Dic₅₃	1	6	yes	—
119	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
120	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
121	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
122	C₄⋊C₈	—	53	24	no	—
123	C₄×C₈	—	53	24	no	—
124	C₄×C₈	—	53	4	no	—
125	C₄×C₈	—	106	12	no	—
126	C₄×C₈	—	106	6	no	—
127	#95	C₂×Dic₅₃	1	6	yes	—
128	C₂×Q₈	C₂×#88322	1	8	yes	—
129	C₂×Q₈	C₄×Dic₅₃	1	4	yes	—
130	C₂²×C₄	C₂²×Dic₅₃	1	3	yes	—
131	C₂²×C₄	C₂²×Dic₅₃	1	6	yes	—
132	C₂²×C₄	C₂²×Dic₅₃	1	3	yes	—
133	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
134	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
135	C₂×C₈	—	53	4	no	—
136	C₂×C₈	—	53	4	no	—
137	C₂×C₈	—	53	4	no	—
138	C₂×C₈	—	53	12	no	—
139	C₂×C₈	—	106	6	no	—
140	C₂×C₈	—	106	6	no	—
141	C₄⋊C₄	C₂×#88322	1	24	yes	—
142	C₄⋊C₄	C₄×Dic₅₃	1	12	yes	—
143	C₄²	—	2	12	no	—
144	C₄²	—	2	6	no	—
145	C₄²	C₂×#88322	1	24	yes	—
146	C₄²	C₂×C₄×D₅₃	1	12	yes	—
147	C₄²	Q₈×D₅₃	1	2	yes	—
148	C₄²	#11719949	1	2	yes	—
149	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
150	Q₈	C₄×#88322	1	16	yes	direct product factor
151	C₂×C₄	—	2	6	no	—
152	C₂×C₄	—	2	6	no	—
153	C₂×C₄	—	2	6	no	—
154	C₂×C₄	C₂×C₄×Dic₅₃	1	6	yes	—
155	C₂×C₄	C₂×C₄×Dic₅₃	1	6	yes	—
156	C₂×C₄	Q₈×Dic₅₃	1	4	yes	—
157	C₂×C₄	C₂²×#88322	1	12	yes	—
158	C₂×C₄	C₂²×#88322	1	12	yes	—
159	C₂×C₄	C₂²×#88322	1	4	yes	—
160	C₂×C₄	#13133224	1	4	yes	—
161	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
162	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
163	C₂×C₄	#41×D₅₃	1	2	yes	—
164	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
165	C₈	—	53	8	no	—
166	C₈	—	106	12	no	—
167	C₂²	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
168	C₂²	C₂×Q₈×Dic₅₃	1	1	yes	characteristic
169	C₂²	#41×Dic₅₃	1	1	yes	characteristic
170	C₂²	C₂³×#88322	1	2	yes	—
171	C₂²	C₂×#13133224	1	2	yes	—
172	C₄	—	2	6	no	—
173	C₄	—	2	12	no	—
174	C₄	C₂×C₄×#88322	1	12	yes	—
175	C₄	Q₈×#88322	1	8	yes	direct product factor
176	C₄	C₄×Q₈×D₅₃	1	2	yes	—
177	C₄	C₄×#11719949	1	2	yes	—
178	C₂	C₄×Q₈×Dic₅₃	1	1	yes	characteristic
179	C₂	C₂²×C₄×#88322	1	1	yes	characteristic
180	C₂	C₄×#13133224	1	1	yes	characteristic
181	C₂	C₂×Q₈×#88322	1	2	yes	—
182	C₂	#41×#88322	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
6. Q₈×#88322	C₄	direct (N × H)
24. C₄×#88322	Q₈	direct (N × H)
26. C₄×Q₈×C₅₃	C₈	semidirect
58. Q₈×C₅₃	C₄×C₈	semidirect
74. #88322	C₄×Q₈	direct (N × H)
84. C₄×C₅₃	C₈×Q₈	semidirect
109. C₅₃	C₄×C₈×Q₈	semidirect
113. C₄×Q₈	#88322	direct (N × H)
150. Q₈	C₄×#88322	direct (N × H)
175. C₄	Q₈×#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
C₄⋊C₄×#88322	2
C₄×C₄⋊#88322	2
C₄²×#88322	2
C₄²×Q₈×C₅₃	2
C₄×C₈×Q₈	53

Subgroups — C4×Q8×#88322 #27382700

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×Q₈×#88322 #27382700