Group #27302952 — subgroups

Subgroups — #27302952

Order 13568.

558 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 129 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×#13133163	C₂	1	2	yes	maximal
2	C₂×#16810427	C₂	1	2	yes	maximal
3	#20275576	C₂	1	1	yes	maximal, characteristic
4	#20272793	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₃×#494	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₁₂	C₂²	1	1	yes	characteristic
7	C₂×#13133163	—	2	4	no	—
8	C₂×#13133163	—	2	4	no	—
9	C₂×#13133163	—	2	4	no	—
10	C₂²×#11719908	C₂²	1	2	yes	—
11	C₂×#13133151	C₂²	1	1	yes	characteristic
12	C₂×#13133142	—	2	1	no	—
13	#16810529	—	2	1	no	—
14	#16810427	—	2	4	no	—
15	C₂²×Q₁₆×C₅₃	—	2	1	no	—
16	C₂²×Q₁₆×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄⋊Q₈×C₅₃	—	2	1	no	—
18	C₅₃×#494	—	2	4	no	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×Dic₁₀₆	—	2	1	no	—
21	C₂×Dic₂₁₂	—	2	1	no	—
22	C₂×Dic₂₁₂	—	2	2	no	—
23	C₂×Dic₂₁₂	—	2	2	no	—
24	C₂×Dic₂₁₂	D₄	1	2	yes	—
25	C₂×C₄⋊Dic₅₃	—	2	1	no	—
26	#13133163	—	2	16	no	—
27	C₂×#11719908	—	2	2	no	—
28	C₂×#11719908	—	2	2	no	—
29	C₂×#11719908	D₄	1	4	yes	—
30	#13133151	—	2	2	no	—
31	#13133142	—	4	2	no	—
32	#13133124	—	4	2	no	—
33	C₂²×Q₈×C₅₃	—	2	2	no	—
34	C₂×Q₁₆×C₅₃	—	2	4	no	—
35	C₂×Q₁₆×C₅₃	—	2	2	no	—
36	C₂×Q₁₆×C₅₃	—	2	2	no	—
37	C₂×Q₁₆×C₅₃	—	2	2	no	—
38	C₂×Q₁₆×C₅₃	—	2	2	no	—
39	C₂×Q₁₆×C₅₃	—	2	2	no	—
40	C₂×Q₁₆×C₅₃	—	4	2	no	—
41	C₂×Q₁₆×C₅₃	—	4	2	no	—
42	C₂×Q₁₆×C₅₃	D₄	1	4	yes	—
43	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
45	C₄⋊Q₈×C₅₃	—	2	4	no	—
46	C₄⋊Q₈×C₅₃	—	4	2	no	—
47	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
48	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
49	C₄×C₈×C₅₃	—	2	2	no	—
50	C₂×Q₈×C₅₃	—	2	4	no	—
51	C₂×Q₈×C₅₃	—	4	2	no	—
52	C₂×Q₈×C₅₃	—	4	2	no	—
53	C₂×Q₈×C₅₃	—	4	2	no	—
54	C₂×Q₈×C₅₃	—	4	2	no	—
55	C₂×Q₈×C₅₃	—	4	2	no	—
56	C₂²×C₄×C₅₃	—	4	2	no	—
57	C₂²×C₄×C₅₃	D₈	1	2	yes	—
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×Dic₅₃	—	4	1	no	—
60	C₂²×Dic₅₃	—	4	1	no	—
61	C₂×Dic₁₀₆	—	2	2	no	—
62	C₂×Dic₁₀₆	—	4	1	no	—
63	C₂×Dic₁₀₆	—	4	1	no	—
64	C₂×Dic₁₀₆	—	4	1	no	—
65	C₂×Dic₁₀₆	—	4	1	no	—
66	C₂×Dic₁₀₆	—	4	1	no	—
67	Q₁₆×C₅₃	—	2	4	no	—
68	Q₁₆×C₅₃	—	2	4	no	—
69	Q₁₆×C₅₃	—	4	8	no	—
70	Q₁₆×C₅₃	—	4	2	no	—
71	Q₁₆×C₅₃	—	4	2	no	—
72	C₂×C₈×C₅₃	—	2	1	no	—
73	C₂×C₈×C₅₃	—	2	1	no	—
74	C₂×C₈×C₅₃	D₈	1	2	yes	—
75	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
79	C₄⋊C₄×C₅₃	—	4	8	no	—
80	C₄²×C₅₃	—	2	1	no	—
81	C₄²×C₅₃	—	2	1	no	—
82	C₄⋊Dic₅₃	—	4	1	no	—
83	C₄⋊Dic₅₃	—	4	1	no	—
84	Dic₂₁₂	—	2	4	no	—
85	Dic₂₁₂	—	4	2	no	—
86	#11719908	—	2	4	no	—
87	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
88	Q₈×C₅₃	—	4	4	no	—
89	Q₈×C₅₃	—	4	4	no	—
90	Q₈×C₅₃	—	8	2	no	—
91	Q₈×C₅₃	—	8	2	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	4	4	no	—
95	C₂×C₄×C₅₃	—	8	2	no	—
96	C₂×C₄×C₅₃	—	8	2	no	—
97	C₂×C₄×C₅₃	Q₃₂	1	4	yes	complemented
98	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
99	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
103	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
104	C₂×Dic₅₃	—	4	2	no	—
105	C₂×Dic₅₃	—	8	1	no	—
106	C₂×Dic₅₃	—	8	1	no	—
107	C₂×Dic₅₃	—	8	1	no	—
108	C₂×Dic₅₃	—	8	2	no	—
109	Dic₁₀₆	—	4	2	no	—
110	Dic₁₀₆	—	4	2	no	—
111	Dic₁₀₆	—	8	1	no	—
112	Dic₁₀₆	—	8	1	no	—
113	C₈×C₅₃	—	2	2	no	—
114	C₈×C₅₃	#120	1	2	yes	—
115	C₈×C₅₃	#121	1	2	yes	—
116	#13330	—	53	1	no	maximal, Sylow
117	C₂²×C₅₃	#447	1	1	yes	characteristic
118	C₂²×C₅₃	#459	1	1	yes	characteristic
119	C₂²×C₅₃	#493	1	1	yes	characteristic
120	C₂²×C₅₃	C₂×Q₃₂	1	2	yes	—
121	C₂²×C₅₃	#510	1	2	yes	—
122	C₄×C₅₃	—	2	4	no	—
123	C₄×C₅₃	—	8	4	no	—
124	C₄×C₅₃	#392	1	1	yes	characteristic
125	C₄×C₅₃	#447	1	1	yes	characteristic
126	C₄×C₅₃	#459	1	1	yes	characteristic
127	C₄×C₅₃	#496	1	1	yes	characteristic
128	Dic₅₃	—	8	2	no	—
129	Dic₅₃	—	16	1	no	—
130	C₂²×Q₃₂	—	53	2	no	—
131	C₂×#494	D₅₃	1	1	yes	characteristic
132	C₂×#358	—	53	2	no	—
133	#1969	—	53	1	no	—
134	#1374	—	53	1	no	—
135	C₂×C₅₃	#2011	1	1	yes	characteristic
136	C₂×C₅₃	#2188	1	2	yes	—
137	C₂×C₅₃	#2209	1	2	yes	—
138	C₂×C₅₃	#2247	1	1	yes	characteristic
139	C₂×C₅₃	#2251	1	1	yes	characteristic
140	C₂²×Q₁₆	—	2	1	no	—
141	C₂²×Q₁₆	—	53	1	no	—
142	C₂²×Q₁₆	C₂×D₅₃	1	2	yes	—
143	C₂×C₄⋊Q₈	—	2	1	no	—
144	C₂×Q₃₂	—	106	4	no	—
145	C₂×Q₃₂	—	106	4	no	—
146	C₂×Q₃₂	—	106	4	no	—
147	C₂²×C₁₆	—	53	2	no	—
148	#494	—	2	4	no	—
149	C₂×#107	—	53	1	no	—
150	C₂×#103	—	106	1	no	—
151	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
152	#391	—	106	1	no	—
153	#358	—	106	4	no	—
154	C₅₃	#13330	1	1	yes	characteristic, complemented, Sylow
155	C₂²×Q₈	—	2	2	no	—
156	C₂²×Q₈	—	106	1	no	—
157	C₂×Q₁₆	—	2	4	no	—
158	C₂×Q₁₆	—	2	2	no	—
159	C₂×Q₁₆	—	2	2	no	—
160	C₂×Q₁₆	—	2	2	no	—
161	C₂×Q₁₆	—	2	2	no	—
162	C₂×Q₁₆	—	2	2	no	—
163	C₂×Q₁₆	—	4	2	no	—
164	C₂×Q₁₆	—	4	2	no	—
165	C₂×Q₁₆	—	53	2	no	—
166	C₂×Q₁₆	—	106	2	no	—
167	C₂×Q₁₆	—	106	1	no	—
168	C₂×Q₁₆	—	106	2	no	—
169	C₂×Q₁₆	C₂²⋊D₅₃	1	4	yes	—
170	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
171	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
172	C₄⋊Q₈	—	2	4	no	—
173	C₄⋊Q₈	—	4	2	no	—
174	C₂×C₄⋊C₄	—	4	2	no	—
175	C₂×C₄⋊C₄	—	106	1	no	—
176	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
177	Q₃₂	—	106	16	no	—
178	C₂×C₁₆	—	53	4	no	—
179	C₂×C₁₆	—	106	2	no	—
180	C₂×C₁₆	—	106	2	no	—
181	#107	—	106	2	no	—
182	#103	—	212	2	no	—
183	C₄×C₈	—	2	2	no	—
184	#95	—	212	2	no	—
185	C₂×Q₈	—	2	4	no	—
186	C₂×Q₈	—	4	2	no	—
187	C₂×Q₈	—	4	2	no	—
188	C₂×Q₈	—	4	2	no	—
189	C₂×Q₈	—	4	2	no	—
190	C₂×Q₈	—	4	2	no	—
191	C₂×Q₈	—	106	2	no	—
192	C₂×Q₈	—	212	1	no	—
193	C₂×Q₈	—	212	1	no	—
194	C₂×Q₈	—	212	1	no	—
195	C₂×Q₈	—	212	1	no	—
196	C₂×Q₈	—	212	1	no	—
197	C₂²×C₄	—	4	2	no	—
198	C₂²×C₄	—	212	1	no	—
199	C₂²×C₄	—	212	1	no	—
200	C₂²×C₄	#11719922	1	2	yes	—
201	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
202	Q₁₆	—	2	4	no	—
203	Q₁₆	—	2	4	no	—
204	Q₁₆	—	4	8	no	—
205	Q₁₆	—	4	2	no	—
206	Q₁₆	—	4	2	no	—
207	Q₁₆	—	106	4	no	—
208	Q₁₆	—	212	2	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	#11719922	1	2	yes	—
212	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
213	C₂×C₈	D₄×D₅₃	1	2	yes	—
214	C₂×C₈	#11719947	1	1	yes	characteristic
215	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
216	C₄⋊C₄	—	4	8	no	—
217	C₄⋊C₄	—	212	1	no	—
218	C₄⋊C₄	—	212	1	no	—
219	C₄²	—	2	1	no	—
220	C₄²	—	2	1	no	—
221	C₁₆	—	106	4	no	—
222	C₂³	C₂×#11719922	1	1	yes	characteristic
223	Q₈	—	4	4	no	—
224	Q₈	—	4	4	no	—
225	Q₈	—	8	2	no	—
226	Q₈	—	8	2	no	—
227	Q₈	—	424	1	no	—
228	Q₈	—	424	1	no	—
229	Q₈	—	212	2	no	—
230	Q₈	—	212	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	8	2	no	—
235	C₂×C₄	—	8	2	no	—
236	C₂×C₄	—	424	2	no	—
237	C₂×C₄	—	424	1	no	—
238	C₂×C₄	—	424	1	no	—
239	C₂×C₄	—	424	1	no	—
240	C₂×C₄	—	212	2	no	—
241	C₂×C₄	#13133163	1	4	yes	complemented
242	C₂×C₄	#13133186	1	1	yes	characteristic
243	C₂×C₄	#13133188	1	1	yes	characteristic
244	C₂×C₄	#13133213	1	1	yes	characteristic
245	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
246	C₂×C₄	#13133243	1	1	yes	characteristic
247	C₂×C₄	#13133243	1	1	yes	characteristic
248	C₈	—	2	2	no	—
249	C₈	#13133185	1	2	yes	—
250	C₈	#13133189	1	2	yes	—
251	C₂²	#16810757	1	1	yes	characteristic
252	C₂²	#16810815	1	1	yes	characteristic
253	C₂²	#16810999	1	1	yes	characteristic
254	C₂²	C₂×#13133163	1	2	yes	—
255	C₂²	#16811064	1	2	yes	—
256	C₄	—	2	4	no	—
257	C₄	—	8	4	no	—
258	C₄	—	848	1	no	—
259	C₄	—	424	2	no	—
260	C₄	#16810531	1	1	yes	characteristic
261	C₄	#16810757	1	1	yes	characteristic
262	C₄	#16810815	1	1	yes	characteristic
263	C₄	#16811010	1	1	yes	characteristic
264	C₂	#20275786	1	1	yes	characteristic
265	C₂	#20276649	1	2	yes	—
266	C₂	#20276762	1	2	yes	—
267	C₂	#20276967	1	1	yes	characteristic
268	C₂	#20276984	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#13133163

C₂

yes

maximal

C₂×#16810427

C₂

yes

maximal

#20275576

C₂

yes

maximal, characteristic

#20272793

C₂

yes

maximal, characteristic

C₂×C₅₃×#494

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₂

C₂²

yes

—

—

—

C₂²

yes

—

C₂×#13133151

C₂²

yes

—

—

—

C₂²×Q₁₆×C₅₃

—

C₂²×Q₁₆×C₅₃

C₂²

yes

—

C₂×C₄⋊Q₈×C₅₃

—

C₅₃×#494

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂²×Dic₁₀₆

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

D₄

yes

—

C₂×C₄⋊Dic₅₃

—

—

—

—

D₄

yes

—

#13133151

—

#13133142

—

#13133124

—

C₂²×Q₈×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

D₄

yes

—

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₈

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

#41

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

Dic₂₁₂

—

Dic₂₁₂

—

#11719908

—

C₂³×C₅₃

C₂×D₈

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

Q₃₂

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

100

C₂×C₄×C₅₃

#127

yes

characteristic

101

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

102

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

characteristic

103

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

characteristic

104

C₂×Dic₅₃

—

105

C₂×Dic₅₃

—

106

C₂×Dic₅₃

—

107

C₂×Dic₅₃

—

108

C₂×Dic₅₃

—

109

Dic₁₀₆

—

110

Dic₁₀₆

—

111

Dic₁₀₆

—

112

Dic₁₀₆

—

113

C₈×C₅₃

—

114

C₈×C₅₃

#120

yes

—

115

C₈×C₅₃

#121

yes

—

116

#13330

—

maximal, Sylow

117

C₂²×C₅₃

#447

yes

characteristic

118

C₂²×C₅₃

#459

yes

characteristic

119

C₂²×C₅₃

#493

yes

characteristic

120

C₂²×C₅₃

C₂×Q₃₂

yes

—

121

C₂²×C₅₃

#510

yes

—

122

C₄×C₅₃

—

123

C₄×C₅₃

—

124

C₄×C₅₃

#392

yes

characteristic

125

C₄×C₅₃

#447

yes

characteristic

126

C₄×C₅₃

#459

yes

characteristic

127

C₄×C₅₃

#496

yes

characteristic

128

Dic₅₃

—

129

Dic₅₃

—

130

C₂²×Q₃₂

—

131

C₂×#494

D₅₃

yes

characteristic

132

C₂×#358

—

133

#1969

—

134

#1374

—

135

C₂×C₅₃

#2011

yes

characteristic

136

C₂×C₅₃

#2188

yes

—

137

C₂×C₅₃

#2209

yes

—

138

C₂×C₅₃

#2247

yes

characteristic

139

C₂×C₅₃

#2251

yes

characteristic

140

C₂²×Q₁₆

—

141

C₂²×Q₁₆

—

142

C₂²×Q₁₆

C₂×D₅₃

yes

—

143

C₂×C₄⋊Q₈

—

144

C₂×Q₃₂

—

106

—

145

C₂×Q₃₂

—

106

—

146

C₂×Q₃₂

—

106

—

147

C₂²×C₁₆

—

148

#494

—

149

C₂×#107

—

150

C₂×#103

—

106

—

151

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

152

#391

—

106

—

153

#358

—

106

—

154

C₅₃

#13330

yes

characteristic, complemented, Sylow

155

C₂²×Q₈

—

156

C₂²×Q₈

—

106

—

157

C₂×Q₁₆

—

158

C₂×Q₁₆

—

159

C₂×Q₁₆

—

160

C₂×Q₁₆

—

161

C₂×Q₁₆

—

162

C₂×Q₁₆

—

163

C₂×Q₁₆

—

164

C₂×Q₁₆

—

165

C₂×Q₁₆

—

166

C₂×Q₁₆

—

106

—

167

C₂×Q₁₆

—

106

—

168

C₂×Q₁₆

—

106

—

169

C₂×Q₁₆

C₂²⋊D₅₃

yes

—

170

C₂²×C₈

C₂²⋊D₅₃

yes

characteristic

171

C₂²×C₈

C₂²×D₅₃

yes

characteristic

172

C₄⋊Q₈

—

173

C₄⋊Q₈

—

174

C₂×C₄⋊C₄

—

175

C₂×C₄⋊C₄

—

106

—

176

C₂×C₄²

C₂²⋊D₅₃

yes

characteristic

177

Q₃₂

—

106

—

178

C₂×C₁₆

—

179

C₂×C₁₆

—

106

—

180

C₂×C₁₆

—

106

—

181

#107

—

106

—

182

#103

—

212

—

183

C₄×C₈

—

184

#95

—

212

—

185

C₂×Q₈

—

186

C₂×Q₈

—

187

C₂×Q₈

—

188

C₂×Q₈

—

189

C₂×Q₈

—

190

C₂×Q₈

—

191

C₂×Q₈

—

106

—

192

C₂×Q₈

—

212

—

193

C₂×Q₈

—

212

—

194

C₂×Q₈

—

212

—

195

C₂×Q₈

—

212

—

196

C₂×Q₈

—

212

—

197

C₂²×C₄

—

198

C₂²×C₄

—

212

—

199

C₂²×C₄

—

212

—

200

C₂²×C₄

#11719922

yes

—

201

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

202

Q₁₆

—

203

Q₁₆

—

204

Q₁₆

—

205

Q₁₆

—

206

Q₁₆

—

207

Q₁₆

—

106

—

208

Q₁₆

—

212

—

209

C₂×C₈

—

210

C₂×C₈

—

211

C₂×C₈

#11719922

yes

—

212

C₂×C₈

D₄×D₅₃

yes

characteristic

213

C₂×C₈

D₄×D₅₃

yes

—

214

C₂×C₈

#11719947

yes

characteristic

215

C₂×C₈

C₂×C₂²⋊D₅₃

yes

—

216

C₄⋊C₄

—

217

C₄⋊C₄

—

212

—

218

C₄⋊C₄

—

212

—

219

C₄²

—

220

C₄²

—

221

C₁₆

—

106

—

222

C₂³

C₂×#11719922

yes

characteristic

223

Q₈

—

224

Q₈

—

225

Q₈

—

226

Q₈

—

227

Q₈

—

424

—

228

Q₈

—

424

—

229

Q₈

—

212

—

230

Q₈

—

212

—

231

C₂×C₄

—

232

C₂×C₄

—

233

C₂×C₄

—

234

C₂×C₄

—

235

C₂×C₄

—

236

C₂×C₄

—

424

—

237

C₂×C₄

—

424

—

238

C₂×C₄

—

424

—

239

C₂×C₄

—

424

—

240

C₂×C₄

—

212

—

241

C₂×C₄

#13133163

yes

complemented

242

C₂×C₄

#13133186

yes

characteristic

243

C₂×C₄

#13133188

yes

characteristic

244

C₂×C₄

#13133213

yes

characteristic

245

C₂×C₄

C₂×#11719922

yes

characteristic

246

C₂×C₄

#13133243

yes

characteristic

247

C₂×C₄

#13133243

yes

characteristic

248

C₈

—

249

C₈

#13133185

yes

—

250

C₈

#13133189

yes

—

251

C₂²

#16810757

yes

characteristic

252

C₂²

#16810815

yes

characteristic

253

C₂²

#16810999

yes

characteristic

254

C₂²

C₂×#13133163

yes

—

255

C₂²

#16811064

yes

—

256

C₄

—

257

C₄

—

258

C₄

—

848

—

259

C₄

—

424

—

260

C₄

#16810531

yes

characteristic

261

C₄

#16810757

yes

characteristic

262

C₄

#16810815

yes

characteristic

263

C₄

#16811010

yes

characteristic

264

C₂

#20275786

yes

characteristic

265

C₂

#20276649

yes

—

266

C₂

#20276762

yes

—

267

C₂

#20276967

yes

characteristic

268

C₂

#20276984

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×#11719922
commutator, cosocle	C₂²×C₈×C₅₃	C₂³
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#494	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
hypercenter	C₂×#494	D₅₃
hypocenter	C₅₃	#13330
LCS	G ▷ C₂²×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

subgroup

G/H

center

C₂³

C₂×#11719922

commutator, cosocle

C₂²×C₈×C₅₃

C₂³

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₂×C₅₃×#494

C₂

socle, prime radical

C₂³×C₅₃

C₂×D₈

D₅₃

C₅₃

G ▷ C₂²×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

N	H ≅ G/N	type
97. C₂×C₄×C₅₃	Q₃₂	semidirect
154. C₅₃	#13330	semidirect
241. C₂×C₄	#13133163	semidirect

H ≅ G/N

type

97. C₂×C₄×C₅₃

Q₃₂

semidirect

154. C₅₃

#13330

semidirect

241. C₂×C₄

#13133163

semidirect

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

subgroup	index
C₂²×#13133163	2
C₂×#16810427	2
#20275576	2
#20272793	2
C₂×C₅₃×#494	2
#13330	53

subgroup

index

C₂×C₅₃×#494

Subgroups — #27302952

Conjugacy classes of subgroups

Complements

Maximal subgroups