Group #27538517 — subgroups

Subgroups — C₂×#20276807 #27538517

Order 13568.

702 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 195 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	C₂²×Dic₄₂₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810865	C₂	1	2	yes	maximal
4	#20276807	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16810426	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#366	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Dic₂₁₂	C₂²	1	2	yes	—
8	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
9	C₂×#13133197	—	2	2	no	—
10	C₂×Dic₄₂₄	—	2	4	no	—
11	C₂×Dic₄₂₄	C₂²	1	4	yes	complemented
12	C₂²×#11719908	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×D₅₃	—	2	4	no	—
14	C₂×C₁₆×D₅₃	C₂²	1	4	yes	—
15	#16810865	—	2	4	no	—
16	#16810865	C₂²	1	8	yes	—
17	C₂×#13133140	—	2	2	no	—
18	#16810426	C₂²	1	8	yes	complemented
19	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#107×C₅₃	C₂²	1	2	yes	—
21	C₅₃×#366	C₂²	1	4	yes	complemented
22	C₂²×Dic₁₀₆	—	2	2	no	—
23	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
24	C₂×Dic₂₁₂	—	2	2	no	—
25	C₂×Dic₂₁₂	—	2	4	no	—
26	C₂×Dic₂₁₂	—	2	4	no	—
27	C₂×Dic₂₁₂	C₂³	1	4	yes	—
28	C₂²×#88322	D₄	1	1	yes	characteristic
29	C₂×C₈×D₅₃	—	2	2	no	—
30	C₂×C₈×D₅₃	—	2	2	no	—
31	C₂×C₈×D₅₃	D₄	1	2	yes	—
32	C₂×C₈×D₅₃	C₂³	1	2	yes	—
33	#13133197	—	2	8	no	—
34	#13133197	—	4	4	no	—
35	C₂×C₄⋊Dic₅₃	—	4	2	no	—
36	C₂×#11719921	—	4	2	no	—
37	Dic₄₂₄	—	2	8	no	—
38	C₂×#11719908	—	2	2	no	—
39	C₂×#11719908	C₂³	1	2	yes	—
40	C₁₆×D₅₃	—	2	8	no	—
41	#13133140	—	2	8	no	—
42	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
43	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
44	C₂×C₁₆×C₅₃	D₄	1	4	yes	—
45	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
46	#107×C₅₃	—	2	2	no	—
47	#107×C₅₃	C₂³	1	4	yes	—
48	C₂³×D₅₃	D₈	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	—	4	2	no	—
50	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×Dic₅₃	—	4	2	no	—
52	C₂²×Dic₅₃	D₈	1	1	yes	characteristic
53	C₂×C₄×D₅₃	—	2	2	no	—
54	C₂×C₄×D₅₃	—	2	2	no	—
55	C₂×C₄×D₅₃	D₈	1	2	yes	—
56	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
57	C₂×Dic₁₀₆	—	2	4	no	—
58	C₂×Dic₁₀₆	—	4	4	no	—
59	C₂×Dic₁₀₆	—	4	4	no	—
60	C₂×Dic₁₀₆	—	4	2	no	—
61	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
63	C₂×C₈×C₅₃	#41	1	2	yes	—
64	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
65	C₄⋊C₄×C₅₃	—	2	4	no	—
66	C₄⋊C₄×C₅₃	—	4	2	no	—
67	#11719921	—	4	8	no	—
68	C₄⋊Dic₅₃	—	4	8	no	—
69	C₂×#88322	—	2	2	no	—
70	C₂×#88322	C₂×D₄	1	2	yes	—
71	Dic₂₁₂	—	2	4	no	—
72	Dic₂₁₂	—	2	4	no	—
73	Dic₂₁₂	—	4	4	no	—
74	C₈×D₅₃	—	2	4	no	—
75	C₈×D₅₃	—	2	2	no	—
76	C₈×D₅₃	—	2	2	no	—
77	C₁₆×C₅₃	C₂×D₄	1	4	yes	—
78	#11719908	—	2	2	no	—
79	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
80	C₂²×D₅₃	—	2	2	no	—
81	C₂²×D₅₃	—	2	2	no	—
82	C₂²×D₅₃	Q₃₂	1	4	yes	complemented
83	C₂²×D₅₃	C₂×D₈	1	2	yes	—
84	C₂×C₄×C₅₃	—	4	4	no	—
85	C₂×C₄×C₅₃	—	8	4	no	—
86	C₂×C₄×C₅₃	#121	1	2	yes	—
87	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
88	C₂×C₄×C₅₃	#135	1	2	yes	—
89	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
90	C₂×Dic₅₃	—	2	2	no	—
91	C₂×Dic₅₃	—	4	4	no	—
92	C₂×Dic₅₃	—	8	4	no	—
93	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
94	C₄×D₅₃	—	2	2	no	—
95	C₄×D₅₃	—	2	2	no	—
96	C₄×D₅₃	—	2	4	no	—
97	Dic₁₀₆	—	4	4	no	—
98	Dic₁₀₆	—	4	4	no	—
99	Dic₁₀₆	—	8	4	no	—
100	C₈×C₅₃	#121	1	2	yes	—
101	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
102	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
103	#88322	—	2	2	no	—
104	C₂×#2216	—	53	1	no	maximal, Sylow
105	C₂²×C₅₃	#466	1	2	yes	—
106	C₂²×C₅₃	C₂×Q₃₂	1	2	yes	—
107	C₂²×C₅₃	#508	1	2	yes	—
108	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
109	C₂×D₅₃	—	2	2	no	—
110	C₂×D₅₃	—	2	8	no	—
111	C₂×D₅₃	—	2	2	no	—
112	C₂×D₅₃	C₂×Q₃₂	1	4	yes	complemented
113	C₄×C₅₃	—	8	4	no	—
114	C₄×C₅₃	#466	1	2	yes	—
115	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
116	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
117	Dic₅₃	—	2	2	no	—
118	Dic₅₃	—	8	4	no	—
119	C₂²×Q₃₂	—	53	1	no	—
120	C₂³×C₁₆	—	53	1	no	—
121	C₂×#467	—	53	2	no	—
122	#2216	—	53	8	no	—
123	C₂×#366	D₅₃	1	1	yes	characteristic, complemented
124	C₂×#358	—	53	2	no	—
125	C₂×C₅₃	#2216	1	4	yes	complemented
126	C₂×C₅₃	C₂×#466	1	1	yes	characteristic
127	C₂×C₅₃	C₂²×Q₃₂	1	1	yes	characteristic
128	C₂×C₅₃	C₂×#508	1	1	yes	characteristic
129	D₅₃	—	2	4	no	—
130	C₂²×Q₁₆	—	53	2	no	—
131	C₂³×C₈	—	53	1	no	—
132	C₂×C₂²⋊Q₈	—	106	2	no	—
133	C₂×Q₃₂	—	53	4	no	—
134	C₂×Q₃₂	—	106	4	no	—
135	C₂²×C₁₆	—	53	1	no	—
136	C₂²×C₁₆	—	53	4	no	—
137	C₂²×C₁₆	—	106	4	no	—
138	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
139	#467	—	53	8	no	—
140	#467	—	106	4	no	—
141	C₂×#107	C₂×D₅₃	1	2	yes	—
142	C₂×#103	—	106	2	no	—
143	#366	C₂×D₅₃	1	4	yes	complemented
144	#358	—	53	8	no	—
145	C₅₃	C₂×#2216	1	1	yes	characteristic, complemented, Sylow
146	C₂²×Q₈	—	106	2	no	—
147	C₂³×C₄	—	53	1	no	—
148	C₂×Q₁₆	—	53	4	no	—
149	C₂×Q₁₆	—	106	4	no	—
150	C₂×Q₁₆	—	106	2	no	—
151	C₂×Q₁₆	—	106	4	no	—
152	C₂²×C₈	—	53	1	no	—
153	C₂²×C₈	—	53	2	no	—
154	C₂²×C₈	—	53	2	no	—
155	C₂²×C₈	—	106	2	no	—
156	C₂²×C₈	—	106	2	no	—
157	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
158	C₂²⋊Q₈	—	106	8	no	—
159	C₂²⋊Q₈	—	212	4	no	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	—	212	2	no	—
162	C₂×C₂²⋊C₄	—	212	2	no	—
163	Q₃₂	—	106	8	no	—
164	C₂×C₁₆	—	53	2	no	—
165	C₂×C₁₆	—	106	2	no	—
166	C₂×C₁₆	—	106	8	no	—
167	C₂×C₁₆	D₂₁₂	1	4	yes	—
168	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
169	#107	—	2	2	no	—
170	#107	C₂²×D₅₃	1	4	yes	—
171	#103	—	106	8	no	—
172	C₂⁴	—	53	1	no	—
173	C₂×Q₈	—	106	4	no	—
174	C₂×Q₈	—	212	4	no	—
175	C₂×Q₈	—	212	4	no	—
176	C₂×Q₈	—	212	2	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	—	53	1	no	—
179	C₂²×C₄	—	53	2	no	—
180	C₂²×C₄	—	53	2	no	—
181	C₂²×C₄	—	106	2	no	—
182	C₂²×C₄	—	106	2	no	—
183	C₂²×C₄	—	212	2	no	—
184	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
185	Q₁₆	—	106	4	no	—
186	Q₁₆	—	106	4	no	—
187	Q₁₆	—	212	4	no	—
188	C₂×C₈	—	53	2	no	—
189	C₂×C₈	—	106	2	no	—
190	C₂×C₈	—	106	4	no	—
191	C₂×C₈	—	106	2	no	—
192	C₂×C₈	—	106	2	no	—
193	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
194	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
195	C₂×C₈	#11719949	1	2	yes	—
196	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	4	2	no	—
199	C₄⋊C₄	—	212	8	no	—
200	C₂²⋊C₄	—	212	8	no	—
201	C₁₆	—	106	2	no	—
202	C₁₆	C₂×D₂₁₂	1	4	yes	—
203	C₂³	—	53	4	no	—
204	C₂³	—	53	2	no	—
205	C₂³	—	106	2	no	—
206	C₂³	—	106	2	no	—
207	C₂³	D₈×D₅₃	1	1	yes	characteristic
208	Q₈	—	424	4	no	—
209	Q₈	—	212	4	no	—
210	Q₈	—	212	4	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	8	4	no	—
213	C₂×C₄	—	53	2	no	—
214	C₂×C₄	—	106	2	no	—
215	C₂×C₄	—	106	4	no	—
216	C₂×C₄	—	106	2	no	—
217	C₂×C₄	—	106	2	no	—
218	C₂×C₄	—	424	4	no	—
219	C₂×C₄	—	212	4	no	—
220	C₂×C₄	#13133192	1	2	yes	—
221	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
222	C₂×C₄	#13133240	1	2	yes	—
223	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
224	C₈	—	106	2	no	—
225	C₈	#13133192	1	2	yes	—
226	C₈	C₂²×D₂₁₂	1	1	yes	characteristic
227	C₈	C₂×#11719949	1	1	yes	characteristic
228	C₂²	—	53	4	no	—
229	C₂²	—	106	8	no	—
230	C₂²	—	106	2	no	—
231	C₂²	—	106	2	no	—
232	C₂²	#16810860	1	2	yes	—
233	C₂²	Q₃₂×D₅₃	1	2	yes	—
234	C₂²	#16811052	1	2	yes	—
235	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
236	C₄	—	8	4	no	—
237	C₄	—	106	2	no	—
238	C₄	—	424	4	no	—
239	C₄	#16810860	1	2	yes	—
240	C₄	C₂×#13133192	1	1	yes	characteristic
241	C₄	C₂×#13133240	1	1	yes	characteristic
242	C₂	—	106	4	no	—
243	C₂	#20276807	1	4	yes	direct product factor
244	C₂	C₂×#16810860	1	1	yes	characteristic
245	C₂	C₂×Q₃₂×D₅₃	1	1	yes	characteristic
246	C₂	C₂×#16811052	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₄₂₄

C₂

yes

maximal, characteristic, complemented

C₂²×C₁₆×D₅₃

C₂

yes

maximal, characteristic

C₂×#16810865

C₂

yes

maximal

#20276807

C₂

yes

maximal, direct product factor

C₂×#16810426

C₂

yes

maximal, complemented

C₂×C₅₃×#366

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₂₁₂

C₂²

yes

—

C₂²×C₈×D₅₃

C₂²

yes

characteristic

C₂×#13133197

—

C₂×Dic₄₂₄

—

C₂×Dic₄₂₄

C₂²

yes

complemented

C₂²×#11719908

C₂²

yes

characteristic

C₂×C₁₆×D₅₃

—

C₂×C₁₆×D₅₃

C₂²

yes

—

#16810865

—

#16810865

C₂²

yes

—

C₂×#13133140

—

#16810426

C₂²

yes

complemented

C₂²×C₁₆×C₅₃

C₂²

yes

characteristic

C₂×#107×C₅₃

C₂²

yes

—

C₅₃×#366

C₂²

yes

complemented

C₂²×Dic₁₀₆

—

C₂²×C₄×D₅₃

D₄

yes

characteristic

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

C₂³

yes

—

C₂²×#88322

D₄

yes

characteristic

C₂×C₈×D₅₃

—

C₂×C₈×D₅₃

—

C₂×C₈×D₅₃

D₄

yes

—

C₂×C₈×D₅₃

C₂³

yes

—

#13133197

—

#13133197

—

C₂×C₄⋊Dic₅₃

—

C₂×#11719921

—

Dic₄₂₄

—

C₂×#11719908

—

C₂×#11719908

C₂³

yes

—

C₁₆×D₅₃

—

#13133140

—

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₅₃

—

C₂×C₁₆×C₅₃

D₄

yes

—

C₂×C₁₆×C₅₃

C₂³

yes

—

#107×C₅₃

—

#107×C₅₃

C₂³

yes

—

C₂³×D₅₃

D₈

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

D₈

yes

characteristic

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

D₈

yes

—

C₂×C₄×D₅₃

C₂×D₄

yes

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

#41

yes

—

C₂×C₈×C₅₃

C₂⁴

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

#11719921

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

C₂×D₄

yes

—

Dic₂₁₂

—

Dic₂₁₂

—

Dic₂₁₂

—

C₈×D₅₃

—

C₈×D₅₃

—

C₈×D₅₃

—

C₁₆×C₅₃

C₂×D₄

yes

—

#11719908

—

C₂³×C₅₃

C₂×D₈

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

Q₃₂

yes

complemented

C₂²×D₅₃

C₂×D₈

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#121

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

#135

yes

—

C₂×C₄×C₅₃

C₂²×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₈

yes

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

100

C₈×C₅₃

#121

yes

—

101

C₈×C₅₃

C₂²×D₄

yes

characteristic

102

C₈×C₅₃

C₂×#41

yes

characteristic

103

#88322

—

104

C₂×#2216

—

maximal, Sylow

105

C₂²×C₅₃

#466

yes

—

106

C₂²×C₅₃

C₂×Q₃₂

yes

—

107

C₂²×C₅₃

#508

yes

—

108

C₂²×C₅₃

C₂²×D₈

yes

characteristic

109

C₂×D₅₃

—

110

C₂×D₅₃

—

111

C₂×D₅₃

—

112

C₂×D₅₃

C₂×Q₃₂

yes

complemented

113

C₄×C₅₃

—

114

C₄×C₅₃

#466

yes

—

115

C₄×C₅₃

C₂×#121

yes

characteristic

116

C₄×C₅₃

C₂×#135

yes

characteristic

117

Dic₅₃

—

118

Dic₅₃

—

119

C₂²×Q₃₂

—

120

C₂³×C₁₆

—

121

C₂×#467

—

122

#2216

—

123

C₂×#366

D₅₃

yes

characteristic, complemented

124

C₂×#358

—

125

C₂×C₅₃

#2216

yes

complemented

126

C₂×C₅₃

C₂×#466

yes

characteristic

127

C₂×C₅₃

C₂²×Q₃₂

yes

characteristic

128

C₂×C₅₃

C₂×#508

yes

characteristic

129

D₅₃

—

130

C₂²×Q₁₆

—

131

C₂³×C₈

—

132

C₂×C₂²⋊Q₈

—

106

—

133

C₂×Q₃₂

—

134

C₂×Q₃₂

—

106

—

135

C₂²×C₁₆

—

136

C₂²×C₁₆

—

137

C₂²×C₁₆

—

106

—

138

C₂²×C₁₆

C₂×D₅₃

yes

characteristic

139

#467

—

140

#467

—

106

—

141

C₂×#107

C₂×D₅₃

yes

—

142

C₂×#103

—

106

—

143

#366

C₂×D₅₃

yes

complemented

144

#358

—

145

C₅₃

C₂×#2216

yes

characteristic, complemented, Sylow

146

C₂²×Q₈

—

106

—

147

C₂³×C₄

—

148

C₂×Q₁₆

—

149

C₂×Q₁₆

—

106

—

150

C₂×Q₁₆

—

106

—

151

C₂×Q₁₆

—

106

—

152

C₂²×C₈

—

153

C₂²×C₈

—

154

C₂²×C₈

—

155

C₂²×C₈

—

106

—

156

C₂²×C₈

—

106

—

157

C₂²×C₈

C₂²×D₅₃

yes

characteristic

158

C₂²⋊Q₈

—

106

—

159

C₂²⋊Q₈

—

212

—

160

C₂×C₄⋊C₄

—

161

C₂×C₄⋊C₄

—

212

—

162

C₂×C₂²⋊C₄

—

212

—

163

Q₃₂

—

106

—

164

C₂×C₁₆

—

165

C₂×C₁₆

—

106

—

166

C₂×C₁₆

—

106

—

167

C₂×C₁₆

D₂₁₂

yes

—

168

C₂×C₁₆

C₂²×D₅₃

yes

—

169

#107

—

170

#107

C₂²×D₅₃

yes

—

171

#103

—

106

—

172

C₂⁴

—

173

C₂×Q₈

—

106

—

174

C₂×Q₈

—

212

—

175

C₂×Q₈

—

212

—

176

C₂×Q₈

—

212

—

177

C₂²×C₄

—

178

C₂²×C₄

—

179

C₂²×C₄

—

180

C₂²×C₄

—

181

C₂²×C₄

—

106

—

182

C₂²×C₄

—

106

—

183

C₂²×C₄

—

212

—

184

C₂²×C₄

D₄×D₅₃

yes

characteristic

185

Q₁₆

—

106

—

186

Q₁₆

—

106

—

187

Q₁₆

—

212

—

188

C₂×C₈

—

189

C₂×C₈

—

106

—

190

C₂×C₈

—

106

—

191

C₂×C₈

—

106

—

192

C₂×C₈

—

106

—

193

C₂×C₈

C₂×D₂₁₂

yes

—

194

C₂×C₈

D₄×D₅₃

yes

characteristic

195

C₂×C₈

#11719949

yes

—

196

C₂×C₈

C₂³×D₅₃

yes

characteristic

197

C₄⋊C₄

—

198

C₄⋊C₄

—

199

C₄⋊C₄

—

212

—

200

C₂²⋊C₄

—

212

—

201

C₁₆

—

106

—

202

C₁₆

C₂×D₂₁₂

yes

—

203

C₂³

—

204

C₂³

—

205

C₂³

—

106

—

206

C₂³

—

106

—

207

C₂³

D₈×D₅₃

yes

characteristic

208

Q₈

—

424

—

209

Q₈

—

212

—

210

Q₈

—

212

—

211

C₂×C₄

—

212

C₂×C₄

—

213

C₂×C₄

—

214

C₂×C₄

—

106

—

215

C₂×C₄

—

106

—

216

C₂×C₄

—

106

—

217

C₂×C₄

—

106

—

218

C₂×C₄

—

424

—

219

C₂×C₄

—

212

—

220

C₂×C₄

#13133192

yes

—

221

C₂×C₄

D₈×D₅₃

yes

characteristic

222

C₂×C₄

#13133240

yes

—

223

C₂×C₄

C₂×D₄×D₅₃

yes

characteristic

224

C₈

—

106

—

225

C₈

#13133192

yes

—

226

C₈

C₂²×D₂₁₂

yes

characteristic

227

C₈

C₂×#11719949

yes

characteristic

228

C₂²

—

229

C₂²

—

106

—

230

C₂²

—

106

—

231

C₂²

—

106

—

232

C₂²

#16810860

yes

—

233

C₂²

Q₃₂×D₅₃

yes

—

234

C₂²

#16811052

yes

—

235

C₂²

C₂×D₈×D₅₃

yes

characteristic

236

C₄

—

237

C₄

—

106

—

238

C₄

—

424

—

239

C₄

#16810860

yes

—

240

C₄

C₂×#13133192

yes

characteristic

241

C₄

C₂×#13133240

yes

characteristic

242

C₂

—

106

—

243

C₂

#20276807

yes

direct product factor

244

C₂

C₂×#16810860

yes

characteristic

245

C₂

C₂×Q₃₂×D₅₃

yes

characteristic

246

C₂

C₂×#16811052

yes

characteristic

	subgroup	G/H
center	C₂³	D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#366	C₂
socle	C₂³×C₅₃	C₂×D₈
prime radical	C₂³×D₅₃	D₈
hypercenter	C₂×#366	D₅₃
hypocenter	C₅₃	C₂×#2216
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

subgroup

G/H

center

C₂³

D₈×D₅₃

commutator, cosocle

C₂×C₈×C₅₃

C₂⁴

Frattini

C₂×C₈

C₂³×D₅₃

Fitting

C₂×C₅₃×#366

C₂

socle

C₂³×C₅₃

C₂×D₈

prime radical

C₂³×D₅₃

D₈

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

N	H ≅ G/N	type
1. C₂²×Dic₄₂₄	C₂	semidirect
4. #20276807	C₂	direct (N × H)
5. C₂×#16810426	C₂	semidirect
6. C₂×C₅₃×#366	C₂	semidirect
11. C₂×Dic₄₂₄	C₂²	semidirect
18. #16810426	C₂²	semidirect
21. C₅₃×#366	C₂²	semidirect
82. C₂²×D₅₃	Q₃₂	semidirect
112. C₂×D₅₃	C₂×Q₃₂	semidirect
123. C₂×#366	D₅₃	semidirect
125. C₂×C₅₃	#2216	semidirect
143. #366	C₂×D₅₃	semidirect
145. C₅₃	C₂×#2216	semidirect
243. C₂	#20276807	direct (N × H)

H ≅ G/N

type

1. C₂²×Dic₄₂₄

C₂

semidirect

4. #20276807

C₂

direct (N × H)

5. C₂×#16810426

C₂

semidirect

6. C₂×C₅₃×#366

C₂

semidirect

11. C₂×Dic₄₂₄

C₂²

semidirect

18. #16810426

C₂²

semidirect

21. C₅₃×#366

C₂²

semidirect

82. C₂²×D₅₃

Q₃₂

semidirect

112. C₂×D₅₃

C₂×Q₃₂

semidirect

123. C₂×#366

D₅₃

semidirect

125. C₂×C₅₃

#2216

semidirect

143. #366

C₂×D₅₃

semidirect

145. C₅₃

C₂×#2216

semidirect

243. C₂

#20276807

direct (N × H)

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

subgroup	index
C₂²×Dic₄₂₄	2
C₂²×C₁₆×D₅₃	2
C₂×#16810865	2
#20276807	2
C₂×#16810426	2
C₂×C₅₃×#366	2
C₂×#2216	53

subgroup

index

C₂²×Dic₄₂₄

C₂²×C₁₆×D₅₃

C₂×C₅₃×#366

Subgroups — C₂×#20276807 #27538517

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20276807 #27538517

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276807 #27538517