Group #20285322 — subgroups

Subgroups — #20285322

Order 6784.

364 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 125 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	#16811180	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811033	C₂	1	1	yes	maximal, characteristic, complemented
3	#16811018	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810993	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810982	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810879	C₂	1	2	yes	maximal, complemented
7	#16810867	C₂	1	2	yes	maximal
8	C₄×#11719925	C₂	1	2	yes	maximal, complemented
9	C₄×#11719923	C₂	1	2	yes	maximal, complemented
10	C₄×#11719917	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#529	C₂	1	1	yes	maximal, characteristic
12	C₂×#11719925	C₂²	1	2	yes	—
13	C₂×#11719923	C₂²	1	2	yes	—
14	C₂×#11719917	C₂²	1	2	yes	—
15	#13133217	C₂²	1	1	yes	characteristic
16	#13133217	C₂²	1	1	yes	characteristic
17	#13133209	—	2	1	no	—
18	#13133196	—	2	1	no	—
19	#13133196	C₂²	1	2	yes	—
20	C₄×Dic₁₀₆	—	2	1	no	—
21	C₄×Dic₁₀₆	C₂²	1	2	yes	—
22	#13133176	—	2	1	no	—
23	#13133150	C₂²	1	2	yes	—
24	#13133148	C₂²	1	2	yes	—
25	#13133141	C₂²	1	2	yes	—
26	#13133140	C₂²	1	2	yes	—
27	#13133140	C₂²	1	2	yes	—
28	#13133138	C₂²	1	2	yes	—
29	#13133128	C₂²	1	1	yes	characteristic
30	#13133128	C₂²	1	1	yes	characteristic
31	C₄×#88322	C₂²	1	2	yes	—
32	#126×C₅₃	—	2	1	no	—
33	#125×C₅₃	C₂²	1	1	yes	characteristic
34	#124×C₅₃	C₂²	1	1	yes	characteristic
35	#123×C₅₃	—	2	1	no	—
36	C₂²⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
37	#121×C₅₃	C₂²	1	1	yes	characteristic
38	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₄×D₄×C₅₃	—	2	1	no	—
40	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
41	#117×C₅₃	—	2	1	no	—
42	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
43	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
44	C₂×D₄×C₅₃	—	2	1	no	—
45	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	—	2	1	no	—
47	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
50	C₂×Dic₁₀₆	—	2	1	no	—
51	C₂×Dic₁₀₆	C₂³	1	2	yes	—
52	C₄⋊C₄×C₅₃	—	2	1	no	—
53	C₄⋊C₄×C₅₃	—	2	1	no	—
54	C₄⋊C₄×C₅₃	—	2	1	no	—
55	C₄⋊C₄×C₅₃	—	2	1	no	—
56	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
57	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
58	C₂²⋊C₄×C₅₃	—	2	1	no	—
59	C₂²⋊C₄×C₅₃	—	2	1	no	—
60	C₂²⋊C₄×C₅₃	—	2	1	no	—
61	C₂²⋊C₄×C₅₃	—	2	1	no	—
62	C₂²⋊C₄×C₅₃	—	2	1	no	—
63	C₄²×C₅₃	—	2	1	no	—
64	C₄²×C₅₃	D₄	1	1	yes	characteristic
65	C₄²×C₅₃	D₄	1	1	yes	characteristic
66	C₄²×C₅₃	C₂³	1	1	yes	characteristic
67	C₄²×C₅₃	C₂³	1	1	yes	characteristic
68	C₂²⋊Dic₅₃	—	2	2	no	—
69	#11719925	—	2	4	no	—
70	#11719923	—	2	4	no	—
71	C₄⋊Dic₅₃	—	2	1	no	—
72	C₄⋊Dic₅₃	—	2	1	no	—
73	C₄⋊Dic₅₃	—	2	1	no	—
74	C₄⋊Dic₅₃	C₂³	1	2	yes	—
75	#11719919	—	2	2	no	—
76	#11719919	—	2	2	no	—
77	C₄×Dic₅₃	—	2	2	no	—
78	#11719917	—	2	4	no	—
79	C₂×#88322	C₂³	1	2	yes	—
80	C₂×#88322	C₂³	1	2	yes	—
81	C₂³×C₅₃	—	2	1	no	—
82	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
83	Q₈×C₅₃	—	2	2	no	—
84	D₄×C₅₃	—	2	2	no	—
85	D₄×C₅₃	—	4	2	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	1	no	—
89	C₂×C₄×C₅₃	—	2	1	no	—
90	C₂×C₄×C₅₃	—	2	1	no	—
91	C₂×C₄×C₅₃	—	4	1	no	—
92	C₂×C₄×C₅₃	—	4	2	no	—
93	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
99	C₂×Dic₅₃	—	2	2	no	—
100	C₂×Dic₅₃	—	2	2	no	—
101	Dic₁₀₆	—	2	2	no	—
102	Dic₁₀₆	—	2	2	no	—
103	Dic₁₀₆	—	4	2	no	—
104	#88322	—	2	2	no	—
105	#88322	#41	1	4	yes	—
106	C₂²×C₅₃	—	2	1	no	—
107	C₂²×C₅₃	—	2	1	no	—
108	C₂²×C₅₃	—	2	1	no	—
109	C₂²×C₅₃	—	4	1	no	—
110	C₂²×C₅₃	—	4	1	no	—
111	C₂²×C₅₃	—	4	1	no	—
112	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
113	C₄×C₅₃	—	2	1	no	—
114	C₄×C₅₃	—	2	1	no	—
115	C₄×C₅₃	—	2	1	no	—
116	C₄×C₅₃	—	2	1	no	—
117	C₄×C₅₃	—	4	1	no	—
118	C₄×C₅₃	—	4	1	no	—
119	C₄×C₅₃	—	4	1	no	—
120	C₄×C₅₃	#137	1	2	yes	—
121	C₄×C₅₃	C₂×#41	1	2	yes	—
122	Dic₅₃	—	4	2	no	—
123	Dic₅₃	—	4	2	no	—
124	#3172	—	53	1	no	maximal, Sylow
125	C₂×C₅₃	—	4	1	no	—
126	C₂×C₅₃	—	8	1	no	—
127	C₂×C₅₃	#532	1	1	yes	characteristic
128	C₂×C₅₃	C₂×#137	1	1	yes	characteristic
129	C₂×C₅₃	#575	1	1	yes	characteristic
130	#533	—	53	1	no	—
131	#529	D₅₃	1	1	yes	characteristic
132	#501	—	53	1	no	—
133	#497	—	53	1	no	—
134	#491	—	53	1	no	—
135	#489	—	53	1	no	—
136	#470	—	53	2	no	—
137	#468	—	53	2	no	—
138	C₄×Q₁₆	—	53	2	no	—
139	C₄×SD₁₆	—	53	2	no	—
140	C₄×M₁₆	—	53	1	no	—
141	C₅₃	#3172	1	1	yes	characteristic, complemented, Sylow
142	C₂×Q₁₆	—	53	2	no	—
143	C₂×SD₁₆	—	53	2	no	—
144	C₂×M₁₆	—	53	2	no	—
145	C₄⋊Q₈	—	53	1	no	—
146	C₄⋊Q₈	—	53	1	no	—
147	#126	—	2	1	no	—
148	#125	—	106	1	no	—
149	#125	C₂×D₅₃	1	1	yes	characteristic
150	#124	C₂×D₅₃	1	1	yes	characteristic
151	#123	—	2	1	no	—
152	C₂²⋊Q₈	—	53	2	no	—
153	C₂²⋊Q₈	—	106	1	no	—
154	C₂²⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
155	#121	C₂×D₅₃	1	1	yes	characteristic
156	C₄×Q₈	—	53	2	no	—
157	C₄×Q₈	—	106	1	no	—
158	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
159	C₄×D₄	—	2	1	no	—
160	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
161	#117	—	2	1	no	—
162	#117	—	106	1	no	—
163	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
164	#107	—	53	2	no	—
165	#106	—	53	2	no	—
166	#103	—	53	2	no	—
167	#103	—	53	2	no	—
168	#103	—	53	2	no	—
169	#102	—	53	2	no	—
170	#97	—	53	1	no	—
171	#97	—	53	1	no	—
172	C₄×C₈	—	53	2	no	—
173	C₂×Q₈	—	53	2	no	—
174	C₂×Q₈	—	106	1	no	—
175	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
176	C₂×D₄	—	2	1	no	—
177	C₂×D₄	C₂²×D₅₃	1	1	yes	characteristic
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
180	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
181	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
182	Q₁₆	—	106	4	no	—
183	SD₁₆	—	106	4	no	—
184	M₁₆	—	106	4	no	—
185	C₂×C₈	—	53	2	no	—
186	C₂×C₈	—	53	2	no	—
187	C₄⋊C₄	—	2	1	no	—
188	C₄⋊C₄	—	2	1	no	—
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	—	53	2	no	—
192	C₄⋊C₄	—	106	1	no	—
193	C₄⋊C₄	—	106	1	no	—
194	C₄⋊C₄	—	106	2	no	—
195	C₄⋊C₄	—	106	2	no	—
196	C₄⋊C₄	—	106	1	no	—
197	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
198	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
199	C₂²⋊C₄	—	2	1	no	—
200	C₂²⋊C₄	—	2	1	no	—
201	C₂²⋊C₄	—	2	1	no	—
202	C₂²⋊C₄	—	2	1	no	—
203	C₂²⋊C₄	—	2	1	no	—
204	C₂²⋊C₄	—	106	2	no	—
205	C₄²	—	2	1	no	—
206	C₄²	—	106	2	no	—
207	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
208	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
209	C₄²	C₂²×D₅₃	1	1	yes	characteristic
210	C₄²	C₂²×D₅₃	1	1	yes	characteristic
211	C₂³	—	2	1	no	—
212	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
213	Q₈	—	2	2	no	—
214	Q₈	—	106	2	no	—
215	Q₈	—	106	2	no	—
216	Q₈	—	212	2	no	—
217	D₄	—	2	2	no	—
218	D₄	—	4	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	106	2	no	—
227	C₂×C₄	—	106	2	no	—
228	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
229	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
230	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
231	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
232	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
233	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
234	C₈	—	53	4	no	—
235	C₈	—	106	2	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	4	1	no	—
240	C₂²	—	4	1	no	—
241	C₂²	—	4	1	no	—
242	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	2	1	no	—
246	C₄	—	2	1	no	—
247	C₄	—	4	1	no	—
248	C₄	—	4	1	no	—
249	C₄	—	4	1	no	—
250	C₄	—	212	2	no	—
251	C₄	—	212	2	no	—
252	C₄	#13133247	1	2	yes	—
253	C₄	#41×D₅₃	1	2	yes	—
254	C₂	—	4	1	no	—
255	C₂	—	8	1	no	—
256	C₂	#16811169	1	1	yes	characteristic
257	C₂	C₂×#13133247	1	1	yes	characteristic
258	C₂	#16811421	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811180

C₂

yes

maximal, characteristic, complemented

#16811033

C₂

yes

maximal, characteristic, complemented

#16811018

C₂

yes

maximal, characteristic, complemented

#16810993

C₂

yes

maximal, characteristic, complemented

#16810982

C₂

yes

maximal, characteristic, complemented

#16810879

C₂

yes

maximal, complemented

#16810867

C₂

yes

maximal

C₄×#11719925

C₂

yes

maximal, complemented

C₄×#11719923

C₂

yes

maximal, complemented

C₄×#11719917

C₂

yes

maximal, characteristic, complemented

C₅₃×#529

C₂

yes

maximal, characteristic

C₂×#11719925

C₂²

yes

—

C₂×#11719923

C₂²

yes

—

C₂×#11719917

C₂²

yes

—

#13133217

C₂²

yes

characteristic

#13133217

C₂²

yes

—

—

C₂²

yes

—

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

C₂²

yes

—

#13133176

—

#13133150

C₂²

yes

—

#13133148

C₂²

yes

—

#13133141

C₂²

yes

—

#13133140

C₂²

yes

—

#13133140

C₂²

yes

—

#13133138

C₂²

yes

—

#13133128

C₂²

yes

characteristic

#13133128

C₂²

yes

characteristic

C₄×#88322

C₂²

yes

—

#126×C₅₃

—

#125×C₅₃

C₂²

yes

characteristic

#124×C₅₃

C₂²

yes

characteristic

#123×C₅₃

—

C₂²⋊Q₈×C₅₃

C₂²

yes

characteristic

#121×C₅₃

C₂²

yes

characteristic

C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₄×D₄×C₅₃

—

C₄×D₄×C₅₃

C₂²

yes

characteristic

#117×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

C₂³

yes

characteristic

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂³

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

C₂³

yes

characteristic

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₂²⋊Dic₅₃

—

#11719925

—

#11719923

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

#11719919

—

C₄×Dic₅₃

—

#11719917

—

C₂×#88322

C₂³

yes

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

100

C₂×Dic₅₃

—

101

Dic₁₀₆

—

102

Dic₁₀₆

—

103

Dic₁₀₆

—

104

#88322

—

105

#88322

#41

yes

—

106

C₂²×C₅₃

—

107

C₂²×C₅₃

—

108

C₂²×C₅₃

—

109

C₂²×C₅₃

—

110

C₂²×C₅₃

—

111

C₂²×C₅₃

—

112

C₂²×C₅₃

C₂²×D₄

yes

characteristic

113

C₄×C₅₃

—

114

C₄×C₅₃

—

115

C₄×C₅₃

—

116

C₄×C₅₃

—

117

C₄×C₅₃

—

118

C₄×C₅₃

—

119

C₄×C₅₃

—

120

C₄×C₅₃

#137

yes

—

121

C₄×C₅₃

C₂×#41

yes

—

122

Dic₅₃

—

123

Dic₅₃

—

124

#3172

—

maximal, Sylow

125

C₂×C₅₃

—

126

C₂×C₅₃

—

127

C₂×C₅₃

#532

yes

characteristic

128

C₂×C₅₃

C₂×#137

yes

characteristic

129

C₂×C₅₃

#575

yes

characteristic

130

#533

—

131

#529

D₅₃

yes

characteristic

132

#501

—

133

#497

—

134

#491

—

135

#489

—

136

#470

—

137

#468

—

138

C₄×Q₁₆

—

139

C₄×SD₁₆

—

140

C₄×M₁₆

—

141

C₅₃

#3172

yes

characteristic, complemented, Sylow

142

C₂×Q₁₆

—

143

C₂×SD₁₆

—

144

C₂×M₁₆

—

145

C₄⋊Q₈

—

146

C₄⋊Q₈

—

147

#126

—

148

#125

—

106

—

149

#125

C₂×D₅₃

yes

characteristic

150

#124

C₂×D₅₃

yes

characteristic

151

#123

—

152

C₂²⋊Q₈

—

153

C₂²⋊Q₈

—

106

—

154

C₂²⋊Q₈

C₂×D₅₃

yes

characteristic

155

#121

C₂×D₅₃

yes

characteristic

156

C₄×Q₈

—

157

C₄×Q₈

—

106

—

158

C₄×Q₈

C₂×D₅₃

yes

characteristic

159

C₄×D₄

—

160

C₄×D₄

C₂×D₅₃

yes

characteristic

161

#117

—

162

#117

—

106

—

163

C₂×C₄²

C₂×D₅₃

yes

characteristic

164

#107

—

165

#106

—

166

#103

—

167

#103

—

168

#103

—

169

#102

—

170

#97

—

171

#97

—

172

C₄×C₈

—

173

C₂×Q₈

—

174

C₂×Q₈

—

106

—

175

C₂×Q₈

C₂²×D₅₃

yes

characteristic

176

C₂×D₄

—

177

C₂×D₄

C₂²×D₅₃

yes

characteristic

178

C₂²×C₄

—

179

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

180

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

181

C₂²×C₄

C₂²×D₅₃

yes

characteristic

182

Q₁₆

—

106

—

183

SD₁₆

—

106

—

184

M₁₆

—

106

—

185

C₂×C₈

—

186

C₂×C₈

—

187

C₄⋊C₄

—

188

C₄⋊C₄

—

189

C₄⋊C₄

—

190

C₄⋊C₄

—

191

C₄⋊C₄

—

192

C₄⋊C₄

—

106

—

193

C₄⋊C₄

—

106

—

194

C₄⋊C₄

—

106

—

195

C₄⋊C₄

—

106

—

196

C₄⋊C₄

—

106

—

197

C₄⋊C₄

C₂²×D₅₃

yes

characteristic

198

C₄⋊C₄

C₂²×D₅₃

yes

characteristic

199

C₂²⋊C₄

—

200

C₂²⋊C₄

—

201

C₂²⋊C₄

—

202

C₂²⋊C₄

—

203

C₂²⋊C₄

—

204

C₂²⋊C₄

—

106

—

205

C₄²

—

206

C₄²

—

106

—

207

C₄²

C₂²⋊D₅₃

yes

characteristic

208

C₄²

C₂²⋊D₅₃

yes

characteristic

209

C₄²

C₂²×D₅₃

yes

characteristic

210

C₄²

C₂²×D₅₃

yes

characteristic

211

C₂³

—

212

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

213

Q₈

—

214

Q₈

—

106

—

215

Q₈

—

106

—

216

Q₈

—

212

—

217

D₄

—

218

D₄

—

219

C₂×C₄

—

220

C₂×C₄

—

221

C₂×C₄

—

222

C₂×C₄

—

223

C₂×C₄

—

224

C₂×C₄

—

225

C₂×C₄

—

226

C₂×C₄

—

106

—

227

C₂×C₄

—

106

—

228

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

229

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

230

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

231

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

232

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

233

C₂×C₄

C₂³×D₅₃

yes

characteristic

234

C₈

—

235

C₈

—

106

—

236

C₂²

—

237

C₂²

—

238

C₂²

—

239

C₂²

—

240

C₂²

—

241

C₂²

—

242

C₂²

C₂²×C₂²⋊D₅₃

yes

characteristic

243

C₄

—

244

C₄

—

245

C₄

—

246

C₄

—

247

C₄

—

248

C₄

—

249

C₄

—

250

C₄

—

212

—

251

C₄

—

212

—

252

C₄

#13133247

yes

—

253

C₄

#41×D₅₃

yes

—

254

C₂

—

255

C₂

—

256

C₂

#16811169

yes

characteristic

257

C₂

C₂×#13133247

yes

characteristic

258

C₂

#16811421

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#529	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#529	D₅₃
hypocenter	C₅₃	#3172
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #529

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#529

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #529

N	H ≅ G/N	type
1. #16811180	C₂	semidirect
2. #16811033	C₂	semidirect
3. #16811018	C₂	semidirect
4. #16810993	C₂	semidirect
5. #16810982	C₂	semidirect
6. #16810879	C₂	semidirect
8. C₄×#11719925	C₂	semidirect
9. C₄×#11719923	C₂	semidirect
10. C₄×#11719917	C₂	semidirect
141. C₅₃	#3172	semidirect

H ≅ G/N

type

1. #16811180

C₂

semidirect

2. #16811033

C₂

semidirect

3. #16811018

C₂

semidirect

4. #16810993

C₂

semidirect

5. #16810982

C₂

semidirect

6. #16810879

C₂

semidirect

8. C₄×#11719925

C₂

semidirect

9. C₄×#11719923

C₂

semidirect

10. C₄×#11719917

C₂

semidirect

141. C₅₃

#3172

semidirect

Subgroups — #20285322

Conjugacy classes of subgroups

Complements