Group #27300024 — subgroups

Subgroups — #27300024

Order 13568.

454 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810580	C₂	1	1	yes	maximal, characteristic
2	#20276470	C₂	1	2	yes	maximal, complemented
3	C₂×#16810404	C₂	1	2	yes	maximal
4	#20272799	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1377	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#11719912	C₂²	1	1	yes	characteristic
7	C₂²×C₈×D₅₃	C₄	1	1	yes	characteristic
8	C₂×#13133174	C₄	1	1	yes	characteristic
9	C₂×#13133157	C₂²	1	2	yes	—
10	#16810580	—	2	4	no	—
11	C₂×#13133127	C₄	1	1	yes	characteristic
12	C₂×C₄×#88322	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
14	#16810441	—	2	4	no	—
15	#16810404	—	2	4	no	—
16	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
17	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×#11719912	—	2	2	no	—
20	C₂×#11719912	—	2	2	no	—
21	C₂×#11719912	D₄	1	2	yes	—
22	C₂×#11719912	Q₈	1	2	yes	—
23	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
24	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×D₅₃	—	2	2	no	—
26	C₂×C₈×D₅₃	—	2	2	no	—
27	C₂×C₈×D₅₃	—	2	2	no	—
28	#13133174	—	2	2	no	—
29	#13133174	—	2	2	no	—
30	C₂×#11719919	C₈	1	2	yes	—
31	C₂×#11719921	C₈	1	2	yes	—
32	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
33	#13133157	—	2	2	no	—
34	#13133157	—	2	2	no	—
35	C₂×#11719908	—	2	2	no	—
36	#13133127	—	2	2	no	—
37	C₄×#88322	—	2	2	no	—
38	C₈×Dic₅₃	D₄	1	4	yes	—
39	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
42	M₃₂×C₅₃	—	2	2	no	—
43	M₃₂×C₅₃	—	2	2	no	—
44	C₂×C₁₆×C₅₃	—	2	2	no	—
45	#97×C₅₃	—	2	2	no	—
46	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
50	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
51	C₂×C₄×D₅₃	—	2	1	no	—
52	C₂×C₄×D₅₃	—	2	1	no	—
53	C₂×C₄×D₅₃	—	2	1	no	—
54	C₂×C₄×D₅₃	—	2	1	no	—
55	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
56	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
57	C₂×C₈×C₅₃	—	2	1	no	—
58	C₂×C₈×C₅₃	—	2	1	no	—
59	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
60	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₈×C₅₃	#41	1	2	yes	—
64	C₄²×C₅₃	—	2	1	no	—
65	C₄²×C₅₃	—	2	1	no	—
66	#11719921	—	2	4	no	—
67	#11719919	—	2	4	no	—
68	C₄×Dic₅₃	C₂²⋊C₄	1	2	yes	—
69	C₄×Dic₅₃	M₁₆	1	2	yes	—
70	C₂×#88322	—	2	1	no	—
71	C₂×#88322	—	2	1	no	—
72	C₂×#88322	—	2	1	no	—
73	C₂×#88322	—	2	1	no	—
74	C₂×#88322	C₂²⋊C₄	1	2	yes	—
75	C₂×#88322	C₄⋊C₄	1	2	yes	—
76	#11719912	—	2	4	no	—
77	#11719912	—	2	4	no	—
78	C₈×D₅₃	—	2	8	no	—
79	C₁₆×C₅₃	—	4	2	no	—
80	#11719908	—	4	2	no	—
81	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
82	C₂²×D₅₃	—	2	1	no	—
83	C₂²×D₅₃	—	2	1	no	—
84	C₂²×D₅₃	—	4	2	no	—
85	C₂²×D₅₃	C₄⋊C₈	1	2	yes	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	4	2	no	—
88	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
94	C₂×Dic₅₃	—	2	1	no	—
95	C₂×Dic₅₃	—	2	1	no	—
96	C₂×Dic₅₃	—	4	2	no	—
97	C₂×Dic₅₃	C₂²⋊C₈	1	4	yes	—
98	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
99	C₄×D₅₃	—	2	2	no	—
100	C₄×D₅₃	—	2	2	no	—
101	C₄×D₅₃	—	2	2	no	—
102	C₄×D₅₃	—	2	2	no	—
103	C₈×C₅₃	—	2	2	no	—
104	C₈×C₅₃	#121	1	2	yes	—
105	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
106	#88322	—	2	2	no	—
107	#88322	—	2	2	no	—
108	#12786	—	53	1	no	maximal, Sylow
109	C₂²×C₅₃	—	2	2	no	—
110	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
111	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
112	C₂²×C₅₃	#431	1	1	yes	characteristic
113	C₂×D₅₃	—	2	2	no	—
114	C₂×D₅₃	—	4	2	no	—
115	C₂×D₅₃	—	4	2	no	—
116	C₂×D₅₃	—	4	2	no	—
117	C₄×C₅₃	—	4	2	no	—
118	C₄×C₅₃	#380	1	1	yes	characteristic
119	C₄×C₅₃	#407	1	1	yes	characteristic
120	C₄×C₅₃	#423	1	1	yes	characteristic
121	C₄×C₅₃	#432	1	1	yes	characteristic
122	Dic₅₃	—	2	2	no	—
123	Dic₅₃	—	2	4	no	—
124	C₂×#405	—	53	1	no	—
125	#2150	—	53	2	no	—
126	C₂×#349	—	53	2	no	—
127	#1377	—	53	1	no	—
128	#1377	D₅₃	1	1	yes	characteristic, complemented
129	C₂×C₅₃	—	2	1	no	—
130	C₂×C₅₃	—	2	1	no	—
131	C₂×C₅₃	#1797	1	1	yes	characteristic
132	C₂×C₅₃	#2115	1	2	yes	—
133	D₅₃	—	4	2	no	—
134	C₂²×M₁₆	—	53	1	no	—
135	C₂³×C₈	—	53	1	no	—
136	C₂×#117	—	53	1	no	—
137	C₂×M₃₂	—	53	2	no	—
138	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
139	#405	—	106	4	no	—
140	C₂×#97	—	53	1	no	—
141	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
142	C₂×C₄×C₈	—	53	1	no	—
143	C₂×C₄×C₈	—	53	1	no	—
144	C₄⋊C₁₆	—	106	4	no	—
145	#349	—	106	4	no	—
146	C₅₃	#12786	1	1	yes	characteristic, complemented, Sylow
147	C₂³×C₄	—	53	1	no	—
148	C₂×M₁₆	—	53	2	no	—
149	C₂×M₁₆	—	53	2	no	—
150	C₂×M₁₆	—	106	2	no	—
151	C₂×M₁₆	—	106	2	no	—
152	C₂²×C₈	—	53	1	no	—
153	C₂²×C₈	—	53	1	no	—
154	C₂²×C₈	—	106	2	no	—
155	C₂²×C₈	—	106	2	no	—
156	C₂²×C₈	—	106	2	no	—
157	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
158	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
159	#117	—	106	2	no	—
160	#117	—	106	2	no	—
161	C₂×C₄⋊C₄	—	53	2	no	—
162	C₂×C₂²⋊C₄	—	53	2	no	—
163	C₂×C₄²	—	53	1	no	—
164	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
165	M₃₂	—	2	2	no	—
166	M₃₂	—	2	2	no	—
167	M₃₂	—	106	2	no	—
168	M₃₂	—	106	2	no	—
169	C₂×C₁₆	—	2	2	no	—
170	C₂×C₁₆	—	106	2	no	—
171	#97	—	2	2	no	—
172	#97	—	106	2	no	—
173	C₄×C₈	—	53	4	no	—
174	C₄×C₈	—	106	2	no	—
175	C₂⁴	—	53	1	no	—
176	C₂²×C₄	—	53	1	no	—
177	C₂²×C₄	—	53	2	no	—
178	C₂²×C₄	—	53	2	no	—
179	C₂²×C₄	—	53	2	no	—
180	C₂²×C₄	—	106	1	no	—
181	C₂²×C₄	—	106	1	no	—
182	C₂²×C₄	—	106	1	no	—
183	C₂²×C₄	—	106	1	no	—
184	C₂²×C₄	C₈×D₅₃	1	2	yes	—
185	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
186	M₁₆	—	106	4	no	—
187	M₁₆	—	106	4	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	53	2	no	—
191	C₂×C₈	—	53	2	no	—
192	C₂×C₈	—	106	8	no	—
193	C₂×C₈	—	106	1	no	—
194	C₂×C₈	—	106	1	no	—
195	C₂×C₈	—	106	1	no	—
196	C₂×C₈	—	106	1	no	—
197	C₂×C₈	#11719912	1	2	yes	—
198	C₂×C₈	#11719945	1	2	yes	—
199	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
200	C₂×C₈	D₄×D₅₃	1	2	yes	—
201	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
202	C₄⋊C₄	—	106	4	no	—
203	C₂²⋊C₄	—	106	4	no	—
204	C₄²	—	2	1	no	—
205	C₄²	—	2	1	no	—
206	C₄²	—	53	2	no	—
207	C₄²	—	53	2	no	—
208	C₁₆	—	4	2	no	—
209	C₁₆	—	212	2	no	—
210	C₂³	—	53	2	no	—
211	C₂³	—	106	1	no	—
212	C₂³	—	106	1	no	—
213	C₂³	—	212	2	no	—
214	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	53	4	no	—
218	C₂×C₄	—	53	2	no	—
219	C₂×C₄	—	106	1	no	—
220	C₂×C₄	—	106	1	no	—
221	C₂×C₄	—	106	2	no	—
222	C₂×C₄	—	106	2	no	—
223	C₂×C₄	—	106	2	no	—
224	C₂×C₄	—	106	2	no	—
225	C₂×C₄	—	212	2	no	—
226	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
227	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
228	C₂×C₄	#13133174	1	1	yes	characteristic
229	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
230	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
231	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
232	C₈	—	2	2	no	—
233	C₈	—	106	2	no	—
234	C₈	—	106	2	no	—
235	C₈	#13133190	1	2	yes	—
236	C₈	#13133195	1	2	yes	—
237	C₂²	—	2	2	no	—
238	C₂²	—	106	2	no	—
239	C₂²	—	212	2	no	—
240	C₂²	—	212	2	no	—
241	C₂²	—	212	2	no	—
242	C₂²	C₂²⋊C₈×D₅₃	1	1	yes	characteristic
243	C₂²	C₄⋊C₈×D₅₃	1	1	yes	characteristic
244	C₂²	#16810674	1	1	yes	characteristic
245	C₄	—	4	2	no	—
246	C₄	—	106	2	no	—
247	C₄	—	106	4	no	—
248	C₄	#16810482	1	1	yes	characteristic
249	C₄	#16810585	1	1	yes	characteristic
250	C₄	#16810645	1	1	yes	characteristic
251	C₄	#16810678	1	1	yes	characteristic
252	C₂	—	2	1	no	—
253	C₂	—	2	1	no	—
254	C₂	—	212	2	no	—
255	C₂	#20274694	1	1	yes	characteristic
256	C₂	#20276334	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16810580

C₂

yes

maximal, characteristic

#20276470

C₂

yes

maximal, complemented

C₂×#16810404

C₂

yes

maximal

#20272799

C₂

yes

maximal, characteristic, complemented

C₅₃×#1377

C₂

yes

maximal, characteristic, complemented

C₂²×#11719912

C₂²

yes

characteristic

C₂²×C₈×D₅₃

C₄

yes

characteristic

C₂×#13133174

C₄

yes

characteristic

C₂×#13133157

C₂²

yes

—

#16810580

—

C₂×#13133127

C₄

yes

characteristic

C₂×C₄×#88322

C₄

yes

characteristic

C₂×C₈×Dic₅₃

C₂²

yes

characteristic

#16810441

—

#16810404

—

C₂×M₃₂×C₅₃

C₂²

yes

—

C₂×#97×C₅₃

C₂²

yes

characteristic

C₂²×C₄×D₅₃

C₂×C₄

yes

—

—

D₄

yes

—

C₂×#11719912

Q₈

yes

—

C₂²×#88322

C₂×C₄

yes

characteristic

C₂²×#88322

C₂×C₄

yes

characteristic

C₂×C₈×D₅₃

—

C₂×C₈×D₅₃

—

C₂×C₈×D₅₃

—

#13133174

—

#13133174

—

C₂×#11719919

C₈

yes

—

C₂×#11719921

C₈

yes

—

C₂×C₄×Dic₅₃

C₂×C₄

yes

—

—

—

—

—

C₈×Dic₅₃

D₄

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

M₃₂×C₅₃

—

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

—

#97×C₅₃

—

C₂³×D₅₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₈

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×Dic₅₃

C₂×C₈

yes

characteristic

C₂²×Dic₅₃

C₂×C₈

yes

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

C₄⋊C₄

yes

—

C₂×C₄×D₅₃

M₁₆

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

M₁₆

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

#41

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

—

#11719921

—

#11719919

—

C₄×Dic₅₃

C₂²⋊C₄

yes

—

C₄×Dic₅₃

M₁₆

yes

—

—

—

—

—

C₂²⋊C₄

yes

—

C₂×#88322

C₄⋊C₄

yes

—

#11719912

—

#11719912

—

C₈×D₅₃

—

C₁₆×C₅₃

—

#11719908

—

C₂³×C₅₃

C₂²×C₈

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

C₄⋊C₈

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#117

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂²⋊C₈

yes

—

C₂×Dic₅₃

C₄⋊C₈

yes

—

C₄×D₅₃

—

100

C₄×D₅₃

—

101

C₄×D₅₃

—

102

C₄×D₅₃

—

103

C₈×C₅₃

—

104

C₈×C₅₃

#121

yes

—

105

C₈×C₅₃

C₂²⋊Q₈

yes

—

106

#88322

—

107

#88322

—

108

#12786

—

maximal, Sylow

109

C₂²×C₅₃

—

110

C₂²×C₅₃

C₂×C₂²⋊C₈

yes

characteristic

111

C₂²×C₅₃

C₂×C₄⋊C₈

yes

characteristic

112

C₂²×C₅₃

#431

yes

characteristic

113

C₂×D₅₃

—

114

C₂×D₅₃

—

115

C₂×D₅₃

—

116

C₂×D₅₃

—

117

C₄×C₅₃

—

118

C₄×C₅₃

#380

yes

characteristic

119

C₄×C₅₃

#407

yes

characteristic

120

C₄×C₅₃

#423

yes

characteristic

121

C₄×C₅₃

#432

yes

characteristic

122

Dic₅₃

—

123

Dic₅₃

—

124

C₂×#405

—

125

#2150

—

126

C₂×#349

—

127

#1377

—

128

#1377

D₅₃

yes

characteristic, complemented

129

C₂×C₅₃

—

130

C₂×C₅₃

—

131

C₂×C₅₃

#1797

yes

characteristic

132

C₂×C₅₃

#2115

yes

—

133

D₅₃

—

134

C₂²×M₁₆

—

135

C₂³×C₈

—

136

C₂×#117

—

137

C₂×M₃₂

—

138

C₂×M₃₂

C₂×D₅₃

yes

—

139

#405

—

106

—

140

C₂×#97

—

141

C₂×#97

C₂×D₅₃

yes

characteristic

142

C₂×C₄×C₈

—

143

C₂×C₄×C₈

—

144

C₄⋊C₁₆

—

106

—

145

#349

—

106

—

146

C₅₃

#12786

yes

characteristic, complemented, Sylow

147

C₂³×C₄

—

148

C₂×M₁₆

—

149

C₂×M₁₆

—

150

C₂×M₁₆

—

106

—

151

C₂×M₁₆

—

106

—

152

C₂²×C₈

—

153

C₂²×C₈

—

154

C₂²×C₈

—

106

—

155

C₂²×C₈

—

106

—

156

C₂²×C₈

—

106

—

157

C₂²×C₈

C₄×D₅₃

yes

characteristic

158

C₂²×C₈

C₂²×D₅₃

yes

characteristic

159

#117

—

106

—

160

#117

—

106

—

161

C₂×C₄⋊C₄

—

162

C₂×C₂²⋊C₄

—

163

C₂×C₄²

—

164

C₂×C₄²

C₄×D₅₃

yes

characteristic

165

M₃₂

—

166

M₃₂

—

167

M₃₂

—

106

—

168

M₃₂

—

106

—

169

C₂×C₁₆

—

170

C₂×C₁₆

—

106

—

171

#97

—

172

#97

—

106

—

173

C₄×C₈

—

174

C₄×C₈

—

106

—

175

C₂⁴

—

176

C₂²×C₄

—

177

C₂²×C₄

—

178

C₂²×C₄

—

179

C₂²×C₄

—

180

C₂²×C₄

—

106

—

181

C₂²×C₄

—

106

—

182

C₂²×C₄

—

106

—

183

C₂²×C₄

—

106

—

184

C₂²×C₄

C₈×D₅₃

yes

—

185

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

186

M₁₆

—

106

—

187

M₁₆

—

106

—

188

C₂×C₈

—

189

C₂×C₈

—

190

C₂×C₈

—

191

C₂×C₈

—

192

C₂×C₈

—

106

—

193

C₂×C₈

—

106

—

194

C₂×C₈

—

106

—

195

C₂×C₈

—

106

—

196

C₂×C₈

—

106

—

197

C₂×C₈

#11719912

yes

—

198

C₂×C₈

#11719945

yes

—

199

C₂×C₈

D₄×D₅₃

yes

characteristic

200

C₂×C₈

D₄×D₅₃

yes

—

201

C₂×C₈

Q₈×D₅₃

yes

characteristic

202

C₄⋊C₄

—

106

—

203

C₂²⋊C₄

—

106

—

204

C₄²

—

205

C₄²

—

206

C₄²

—

207

C₄²

—

208

C₁₆

—

209

C₁₆

—

212

—

210

C₂³

—

211

C₂³

—

106

—

212

C₂³

—

106

—

213

C₂³

—

212

—

214

C₂³

C₂×C₈×D₅₃

yes

characteristic

215

C₂×C₄

—

216

C₂×C₄

—

217

C₂×C₄

—

218

C₂×C₄

—

219

C₂×C₄

—

106

—

220

C₂×C₄

—

106

—

221

C₂×C₄

—

106

—

222

C₂×C₄

—

106

—

223

C₂×C₄

—

106

—

224

C₂×C₄

—

106

—

225

C₂×C₄

—

212

—

226

C₂×C₄

C₂²⋊C₄×D₅₃

yes

characteristic

227

C₂×C₄

C₄⋊C₄×D₅₃

yes

characteristic

228

C₂×C₄

#13133174

yes

characteristic

229

C₂×C₄

M₁₆×D₅₃

yes

characteristic

230

C₂×C₄

M₁₆×D₅₃

yes

characteristic

231

C₂×C₄

C₂×#11719912

yes

characteristic

232

C₈

—

233

C₈

—

106

—

234

C₈

—

106

—

235

C₈

#13133190

yes

—

236

C₈

#13133195

yes

—

237

C₂²

—

238

C₂²

—

106

—

239

C₂²

—

212

—

240

C₂²

—

212

—

241

C₂²

—

212

—

242

C₂²

C₂²⋊C₈×D₅₃

yes

characteristic

243

C₂²

C₄⋊C₈×D₅₃

yes

characteristic

244

C₂²

#16810674

yes

characteristic

245

C₄

—

246

C₄

—

106

—

247

C₄

—

106

—

248

C₄

#16810482

yes

characteristic

249

C₄

#16810585

yes

characteristic

250

C₄

#16810645

yes

characteristic

251

C₄

#16810678

yes

characteristic

252

C₂

—

253

C₂

—

254

C₂

—

212

—

255

C₂

#20274694

yes

characteristic

256

C₂

#20276334

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₅₃
commutator	C₂³×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1377	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1377	D₅₃
hypocenter	C₅₃	#12786
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1377

subgroup

G/H

center

C₂×C₄

C₂²⋊C₄×D₅₃

commutator

C₂³×C₅₃

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₅₃×#1377

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₈

prime radical

C₂³×D₅₃

C₂×C₈

cosocle

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1377

N	H ≅ G/N	type
2. #20276470	C₂	semidirect
4. #20272799	C₂	semidirect
5. C₅₃×#1377	C₂	semidirect
128. #1377	D₅₃	semidirect
146. C₅₃	#12786	semidirect

H ≅ G/N

type

2. #20276470

C₂

semidirect

4. #20272799

C₂

semidirect

5. C₅₃×#1377

C₂

semidirect

128. #1377

D₅₃

semidirect

146. C₅₃

#12786

semidirect

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

subgroup	index
C₂×#16810580	2
#20276470	2
C₂×#16810404	2
#20272799	2
C₅₃×#1377	2
#12786	53

subgroup

index

C₅₃×#1377

Subgroups — #27300024

Conjugacy classes of subgroups

Complements

Maximal subgroups