Subgroups — C₂²×#88324⋊C₈ #28313738

Order 13568.

1182 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×#88324	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×#88324	C₂	1	1	yes	maximal, characteristic
3	C₂×#88324⋊C₈	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×#88324	C₄	1	2	yes	—
6	C₂³×#88324	C₄	1	2	yes	—
7	C₂×C₄×#88324	C₄	1	6	yes	—
8	C₂×C₄×#88324	C₄	1	6	yes	—
9	C₂×C₄×#88324	C₂²	1	6	yes	—
10	C₂×C₄×#88324	C₂²	1	6	yes	—
11	#88324⋊C₈	C₂²	1	16	yes	direct product factor
12	C₂³×#88322	C₄	1	1	yes	characteristic
13	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₅₃	C₄	1	6	yes	—
15	C₂×C₈×Dic₅₃	C₂²	1	6	yes	—
16	C₂³×C₈×C₅₃	C₄	1	1	yes	characteristic
17	C₂²×#88324	—	2	8	no	—
18	C₂²×#88324	C₈	1	16	yes	complemented
19	C₂²×#88324	C₂×C₄	1	12	yes	—
20	C₂²×#88324	C₂×C₄	1	12	yes	—
21	C₄×#88324	C₂×C₄	1	12	yes	—
22	C₄×#88324	C₂×C₄	1	12	yes	—
23	C₄×#88324	C₂³	1	4	yes	—
24	C₄×#88324	C₂³	1	4	yes	—
25	C₂³×Dic₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂³×Dic₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×#88322	—	2	4	no	—
28	C₂²×#88322	C₂×C₄	1	6	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	C₂³	1	3	yes	—
33	C₈×Dic₅₃	C₂×C₄	1	12	yes	—
34	C₈×Dic₅₃	C₂³	1	4	yes	—
35	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	—	2	4	no	—
37	C₂²×C₈×C₅₃	C₂×C₄	1	6	yes	—
38	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₄²	1	3	yes	—
40	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
41	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
42	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
43	C₂²×Dic₅₃	C₄²	1	3	yes	—
44	C₂²×Dic₅₃	C₄²	1	3	yes	—
45	C₂²×Dic₅₃	C₂×C₈	1	8	yes	—
46	C₂²×Dic₅₃	M₁₆	1	8	yes	—
47	C₂²×Dic₅₃	C₂²×C₄	1	3	yes	—
48	C₂²×Dic₅₃	C₂²×C₄	1	3	yes	—
49	C₂×#88324	—	2	24	no	—
50	C₂×#88324	C₂×C₈	1	48	yes	complemented
51	C₂×#88324	C₂²×C₄	1	8	yes	—
52	C₂×#88324	C₂²×C₄	1	8	yes	—
53	C₂×C₈×C₅₃	—	2	12	no	—
54	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
55	C₄×Dic₅₃	C₄²	1	6	yes	—
56	C₄×Dic₅₃	C₂²×C₄	1	3	yes	—
57	C₄×Dic₅₃	C₂²×C₄	1	3	yes	—
58	C₄×Dic₅₃	C₂²×C₄	1	3	yes	—
59	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
60	C₂×#88322	—	2	12	no	—
61	C₂×#88322	C₂²×C₄	1	4	yes	—
62	C₂³×C₅₃	C₄×C₈	1	4	yes	—
63	C₂³×C₅₃	#97	1	4	yes	—
64	C₂³×C₅₃	#97	1	4	yes	—
65	C₂³×C₅₃	C₂×C₄²	1	3	yes	—
66	C₂×C₄×C₅₃	#97	1	12	yes	—
67	C₂×C₄×C₅₃	C₂×C₄²	1	3	yes	—
68	C₂×C₄×C₅₃	C₂×M₁₆	1	6	yes	—
69	C₂×C₄×C₅₃	C₂×M₁₆	1	6	yes	—
70	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
71	C₂×Dic₅₃	C₄×C₈	1	12	yes	—
72	C₂×Dic₅₃	#97	1	12	yes	—
73	C₂×Dic₅₃	C₂×C₄²	1	3	yes	—
74	C₂×Dic₅₃	C₂×C₄²	1	3	yes	—
75	C₂×Dic₅₃	C₂²×C₈	1	12	yes	—
76	C₂×Dic₅₃	C₂×M₁₆	1	12	yes	—
77	C₂×Dic₅₃	C₂³×C₄	1	1	yes	characteristic
78	C₂×Dic₅₃	C₂³×C₄	1	1	yes	characteristic
79	#88324	—	2	8	no	—
80	#88324	C₂²×C₈	1	16	yes	complemented
81	C₈×C₅₃	—	2	4	no	—
82	#88322	—	2	4	no	—
83	C₂²×#322	—	53	1	no	maximal, Sylow
84	C₂²×C₅₃	#322	1	16	yes	complemented
85	C₂²×C₅₃	C₂×C₄×C₈	1	6	yes	—
86	C₂²×C₅₃	C₂×#97	1	6	yes	—
87	C₂²×C₅₃	C₂×#97	1	6	yes	—
88	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
89	C₄×C₅₃	C₂×#97	1	6	yes	—
90	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
91	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
92	Dic₅₃	C₂×C₄×C₈	1	6	yes	—
93	Dic₅₃	C₂×#97	1	6	yes	—
94	Dic₅₃	C₂³×C₈	1	2	yes	—
95	Dic₅₃	C₂²×M₁₆	1	2	yes	—
96	C₂²×C₄×C₈	—	53	1	no	—
97	C₂²×C₄×C₈	—	53	1	no	—
98	C₂²×C₄×C₈	—	53	1	no	—
99	C₂×#322	—	53	12	no	—
100	C₂×C₅₃	C₂×#322	1	12	yes	complemented
101	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
102	C₂×C₅₃	C₂²×#97	1	1	yes	characteristic
103	C₂×C₅₃	C₂²×#97	1	1	yes	characteristic
104	C₂³×C₈	—	53	2	no	—
105	C₂³×C₈	—	53	2	no	—
106	C₂³×C₈	—	53	1	no	—
107	C₂³×C₈	#6157	1	1	yes	characteristic
108	C₂²×C₄²	—	53	1	no	—
109	C₂×C₄×C₈	—	53	6	no	—
110	C₂×C₄×C₈	—	53	6	no	—
111	C₂×C₄×C₈	—	53	6	no	—
112	C₂×C₄×C₈	—	53	6	no	—
113	C₂×C₄×C₈	—	53	6	no	—
114	C₂×C₄×C₈	—	53	6	no	—
115	#322	—	53	16	no	—
116	C₅₃	C₂²×#322	1	1	yes	characteristic, complemented, Sylow
117	C₂³×C₄	—	53	1	no	—
118	C₂³×C₄	—	53	1	no	—
119	C₂³×C₄	C₂×#6157	1	1	yes	characteristic
120	C₂²×C₈	—	2	4	no	—
121	C₂²×C₈	—	53	12	no	—
122	C₂²×C₈	—	53	16	no	—
123	C₂²×C₈	—	53	12	no	—
124	C₂²×C₈	—	53	6	no	—
125	C₂²×C₈	—	106	8	no	—
126	C₂²×C₈	—	106	4	no	—
127	C₂²×C₈	C₂×#6157	1	6	yes	—
128	C₂×C₄²	—	53	3	no	—
129	C₂×C₄²	—	53	3	no	—
130	C₂×C₄²	—	53	3	no	—
131	C₂×C₄²	—	53	3	no	—
132	C₄×C₈	—	53	4	no	—
133	C₄×C₈	—	53	4	no	—
134	C₄×C₈	—	53	12	no	—
135	C₄×C₈	—	53	12	no	—
136	C₄×C₈	—	53	4	no	—
137	C₄×C₈	—	53	12	no	—
138	C₂⁴	C₄×#6157	1	1	yes	characteristic
139	C₂²×C₄	—	53	8	no	—
140	C₂²×C₄	—	53	8	no	—
141	C₂²×C₄	—	53	3	no	—
142	C₂²×C₄	—	53	3	no	—
143	C₂²×C₄	—	53	3	no	—
144	C₂²×C₄	—	53	3	no	—
145	C₂²×C₄	#11719936	1	4	yes	—
146	C₂²×C₄	C₄×#6157	1	3	yes	—
147	C₂²×C₄	#11719940	1	4	yes	—
148	C₂²×C₄	C₂²×#6157	1	3	yes	—
149	C₂×C₈	—	2	12	no	—
150	C₂×C₈	—	53	48	no	—
151	C₂×C₈	—	53	8	no	—
152	C₂×C₈	—	53	8	no	—
153	C₂×C₈	—	53	4	no	—
154	C₂×C₈	—	106	24	no	—
155	C₂×C₈	—	106	12	no	—
156	C₂×C₈	C₂²×#6157	1	4	yes	—
157	C₄²	—	53	3	no	—
158	C₄²	—	53	3	no	—
159	C₄²	—	53	6	no	—
160	C₄²	—	53	1	no	—
161	C₄²	—	53	3	no	—
162	C₂³	C₈×#6157	1	4	yes	—
163	C₂³	#13133271	1	4	yes	—
164	C₂³	#13133272	1	4	yes	—
165	C₂³	C₂×C₄×#6157	1	3	yes	—
166	C₂×C₄	—	53	12	no	—
167	C₂×C₄	—	53	12	no	—
168	C₂×C₄	—	53	12	no	—
169	C₂×C₄	—	53	12	no	—
170	C₂×C₄	—	53	3	no	—
171	C₂×C₄	—	53	3	no	—
172	C₂×C₄	—	53	1	no	—
173	C₂×C₄	—	53	1	no	—
174	C₂×C₄	#13133272	1	12	yes	—
175	C₂×C₄	C₂×C₄×#6157	1	3	yes	—
176	C₂×C₄	C₂×#11719940	1	6	yes	—
177	C₂×C₄	C₂×#11719936	1	6	yes	—
178	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
179	C₈	—	2	4	no	—
180	C₈	—	53	16	no	—
181	C₈	—	106	8	no	—
182	C₈	—	106	4	no	—
183	C₂²	#88324⋊C₈	1	16	yes	direct product factor
184	C₂²	C₂×C₈×#6157	1	6	yes	—
185	C₂²	C₂×#13133271	1	6	yes	—
186	C₂²	C₂×#13133272	1	6	yes	—
187	C₂²	C₂²×C₄×#6157	1	1	yes	characteristic
188	C₄	—	53	2	no	—
189	C₄	—	53	6	no	—
190	C₄	—	53	2	no	—
191	C₄	—	53	6	no	—
192	C₄	C₂×#13133272	1	6	yes	—
193	C₄	C₂²×#11719940	1	1	yes	characteristic
194	C₄	C₂²×#11719936	1	1	yes	characteristic
195	C₂	C₂×#88324⋊C₈	1	12	yes	direct product factor
196	C₂	C₂²×C₈×#6157	1	1	yes	characteristic
197	C₂	C₂²×#13133271	1	1	yes	characteristic
198	C₂	C₂²×#13133272	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂×#6157
commutator	C₂×C₅₃	C₂²×C₄×C₈
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₂³×C₈×C₅₃	C₄
socle, prime radical	C₂⁴×C₅₃	C₄²
cosocle	C₄×Dic₅₃	C₂⁴
hypercenter	C₂³×C₈	#6157
hypocenter	C₅₃	C₂²×#322
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
3. C₂×#88324⋊C₈	C₂	direct (N × H)
11. #88324⋊C₈	C₂²	direct (N × H)
18. C₂²×#88324	C₈	semidirect
50. C₂×#88324	C₂×C₈	semidirect
80. #88324	C₂²×C₈	semidirect
84. C₂²×C₅₃	#322	semidirect
100. C₂×C₅₃	C₂×#322	semidirect
116. C₅₃	C₂²×#322	semidirect
183. C₂²	#88324⋊C₈	direct (N × H)
195. C₂	C₂×#88324⋊C₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×#88324	2
C₂×#88324⋊C₈	2
C₂²×C₈×Dic₅₃	2
C₂²×#322	53

Subgroups — C22×#88324⋊C8 #28313738

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#88324⋊C₈ #28313738