Group #27300033 — subgroups

Subgroups — #27300033

Order 13568.

538 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 139 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×C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810443	C₂	1	2	yes	maximal, complemented
3	#20276325	C₂	1	2	yes	maximal
4	#20272797	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1376	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₅₃	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
8	C₂×C₄²×D₅₃	C₄	1	1	yes	characteristic
9	C₂×#13133157	C₂²	1	2	yes	—
10	C₂×C₄×#88322	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
13	C₄×C₈×D₅₃	—	2	4	no	—
14	#16810443	—	2	4	no	—
15	#16810402	—	2	4	no	—
16	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²×C₄×D₅₃	—	2	1	no	—
19	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
21	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×D₅₃	—	2	2	no	—
23	C₂×C₈×D₅₃	—	2	2	no	—
24	C₂×C₈×D₅₃	—	2	2	no	—
25	C₂×C₈×D₅₃	—	2	2	no	—
26	C₂×C₈×D₅₃	C₈	1	4	yes	—
27	C₂×C₈×D₅₃	D₄	1	2	yes	—
28	C₂×C₈×D₅₃	Q₈	1	2	yes	—
29	C₂×C₄×Dic₅₃	—	2	1	no	—
30	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄²×D₅₃	—	2	2	no	—
32	C₄²×D₅₃	—	2	2	no	—
33	#13133157	—	2	2	no	—
34	#13133157	—	2	2	no	—
35	C₂×#11719908	—	2	2	no	—
36	C₄×#88322	—	2	2	no	—
37	C₈×Dic₅₃	—	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	C₄×C₈×C₅₃	—	2	2	no	—
46	C₂³×D₅₃	M₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	—	2	1	no	—
48	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×Dic₅₃	—	2	1	no	—
50	C₂²×Dic₅₃	M₁₆	1	1	yes	characteristic
51	C₂×C₄×D₅₃	—	2	4	no	—
52	C₂×C₄×D₅₃	—	2	8	no	—
53	C₂×C₄×D₅₃	—	2	1	no	—
54	C₂×C₄×D₅₃	—	2	1	no	—
55	C₂×C₄×D₅₃	—	2	1	no	—
56	C₂×C₄×D₅₃	—	2	1	no	—
57	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
58	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
59	C₂×C₈×C₅₃	—	2	1	no	—
60	C₂×C₈×C₅₃	—	2	1	no	—
61	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
62	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
64	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₈×C₅₃	#41	1	2	yes	—
66	C₄²×C₅₃	—	2	1	no	—
67	C₄²×C₅₃	M₁₆	1	2	yes	—
68	C₄×Dic₅₃	—	2	4	no	—
69	C₄×Dic₅₃	—	2	1	no	—
70	C₄×Dic₅₃	C₂²⋊C₄	1	2	yes	—
71	C₂×#88322	—	2	1	no	—
72	C₂×#88322	—	2	1	no	—
73	C₂×#88322	C₂²⋊C₄	1	2	yes	—
74	C₂×#88322	C₄⋊C₄	1	2	yes	—
75	C₂×#88322	C₂×C₈	1	2	yes	—
76	C₂×#88322	M₁₆	1	2	yes	—
77	C₈×D₅₃	—	2	4	no	—
78	C₈×D₅₃	—	2	4	no	—
79	C₈×D₅₃	—	2	8	no	—
80	C₁₆×C₅₃	—	4	2	no	—
81	#11719908	—	4	2	no	—
82	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
83	C₂²×D₅₃	—	2	1	no	—
84	C₂²×D₅₃	—	2	1	no	—
85	C₂²×D₅₃	—	2	4	no	—
86	C₂²×D₅₃	C₄⋊C₈	1	2	yes	—
87	C₂×C₄×C₅₃	—	2	4	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
95	C₂×Dic₅₃	—	2	4	no	—
96	C₂×Dic₅₃	—	2	2	no	—
97	C₂×Dic₅₃	—	2	1	no	—
98	C₂×Dic₅₃	—	2	1	no	—
99	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
100	C₄×D₅₃	—	2	16	no	—
101	C₄×D₅₃	—	2	2	no	—
102	C₄×D₅₃	—	2	2	no	—
103	C₄×D₅₃	—	2	2	no	—
104	C₄×D₅₃	—	2	2	no	—
105	C₈×C₅₃	—	2	2	no	—
106	C₈×C₅₃	#121	1	2	yes	—
107	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
108	#88322	—	2	2	no	—
109	#88322	C₂²⋊C₈	1	4	yes	—
110	#12788	—	53	1	no	maximal, Sylow
111	C₂²×C₅₃	—	2	2	no	—
112	C₂²×C₅₃	#407	1	1	yes	characteristic
113	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
114	C₂²×C₅₃	#432	1	1	yes	characteristic
115	C₂×D₅₃	—	2	4	no	—
116	C₂×D₅₃	—	2	4	no	—
117	C₂×D₅₃	—	2	2	no	—
118	C₂×D₅₃	—	2	2	no	—
119	C₂×D₅₃	#364	1	4	yes	complemented
120	C₄×C₅₃	—	2	4	no	—
121	C₄×C₅₃	#380	1	1	yes	characteristic
122	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
123	C₄×C₅₃	#423	1	1	yes	characteristic
124	C₄×C₅₃	#431	1	1	yes	characteristic
125	Dic₅₃	—	2	4	no	—
126	Dic₅₃	—	2	2	no	—
127	C₂²×C₄×C₈	—	53	1	no	—
128	C₂×#364	—	53	2	no	—
129	#2112	—	53	2	no	—
130	#1376	—	53	1	no	—
131	#1376	D₅₃	1	1	yes	characteristic, complemented
132	C₂×C₅₃	—	2	1	no	—
133	C₂×C₅₃	—	2	1	no	—
134	C₂×C₅₃	#1797	1	1	yes	characteristic
135	C₂×C₅₃	C₂×#364	1	2	yes	—
136	D₅₃	—	2	4	no	—
137	C₂³×C₈	—	53	1	no	—
138	C₂³×C₈	—	53	1	no	—
139	C₂²×C₄²	—	53	1	no	—
140	C₂×M₃₂	—	53	2	no	—
141	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
142	C₂×C₄×C₈	—	53	1	no	—
143	C₂×C₄×C₈	—	53	1	no	—
144	C₂×C₄×C₈	—	53	1	no	—
145	C₂×C₄×C₈	—	106	4	no	—
146	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
147	#364	—	106	4	no	—
148	C₂²⋊C₁₆	—	106	4	no	—
149	C₅₃	#12788	1	1	yes	characteristic, complemented, Sylow
150	C₂³×C₄	—	53	1	no	—
151	C₂³×C₄	—	106	1	no	—
152	C₂²×C₈	—	53	1	no	—
153	C₂²×C₈	—	53	1	no	—
154	C₂²×C₈	—	53	2	no	—
155	C₂²×C₈	—	53	2	no	—
156	C₂²×C₈	—	53	4	no	—
157	C₂²×C₈	—	106	2	no	—
158	C₂²×C₈	—	106	2	no	—
159	C₂²×C₈	—	106	2	no	—
160	C₂²×C₈	—	106	2	no	—
161	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
162	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
163	C₂×C₄²	—	53	1	no	—
164	C₂×C₄²	—	106	2	no	—
165	C₂×C₄²	—	106	1	no	—
166	C₂×C₄²	—	106	2	no	—
167	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
168	M₃₂	—	2	2	no	—
169	M₃₂	—	2	2	no	—
170	M₃₂	—	106	2	no	—
171	M₃₂	—	106	2	no	—
172	C₂×C₁₆	—	2	2	no	—
173	C₂×C₁₆	—	106	2	no	—
174	C₄×C₈	—	2	2	no	—
175	C₄×C₈	—	53	4	no	—
176	C₄×C₈	—	106	2	no	—
177	C₄×C₈	—	106	2	no	—
178	C₂⁴	—	53	1	no	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	53	1	no	—
181	C₂²×C₄	—	53	2	no	—
182	C₂²×C₄	—	53	2	no	—
183	C₂²×C₄	—	106	4	no	—
184	C₂²×C₄	—	106	8	no	—
185	C₂²×C₄	—	106	1	no	—
186	C₂²×C₄	—	106	1	no	—
187	C₂²×C₄	—	106	1	no	—
188	C₂²×C₄	—	106	1	no	—
189	C₂²×C₄	—	106	1	no	—
190	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	53	2	no	—
194	C₂×C₈	—	53	2	no	—
195	C₂×C₈	—	53	2	no	—
196	C₂×C₈	—	53	2	no	—
197	C₂×C₈	—	106	8	no	—
198	C₂×C₈	—	106	1	no	—
199	C₂×C₈	—	106	1	no	—
200	C₂×C₈	—	106	4	no	—
201	C₂×C₈	—	106	4	no	—
202	C₂×C₈	C₈×D₅₃	1	2	yes	—
203	C₂×C₈	#11719945	1	2	yes	—
204	C₂×C₈	D₄×D₅₃	1	2	yes	—
205	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
206	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
207	C₄²	—	2	1	no	—
208	C₄²	—	53	2	no	—
209	C₄²	—	106	4	no	—
210	C₄²	—	106	1	no	—
211	C₄²	#11719912	1	2	yes	—
212	C₁₆	—	4	2	no	—
213	C₁₆	—	212	2	no	—
214	C₂³	—	53	2	no	—
215	C₂³	—	106	4	no	—
216	C₂³	—	106	1	no	—
217	C₂³	—	106	1	no	—
218	C₂³	M₁₆×D₅₃	1	1	yes	characteristic
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	53	2	no	—
222	C₂×C₄	—	106	16	no	—
223	C₂×C₄	—	106	4	no	—
224	C₂×C₄	—	106	2	no	—
225	C₂×C₄	—	106	1	no	—
226	C₂×C₄	—	106	1	no	—
227	C₂×C₄	—	106	2	no	—
228	C₂×C₄	—	106	2	no	—
229	C₂×C₄	—	106	2	no	—
230	C₂×C₄	—	106	2	no	—
231	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
232	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
233	C₂×C₄	#13133174	1	1	yes	characteristic
234	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
235	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
236	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	—	53	4	no	—
239	C₈	—	106	2	no	—
240	C₈	#13133190	1	2	yes	—
241	C₈	#13133195	1	2	yes	—
242	C₂²	—	2	2	no	—
243	C₂²	—	53	4	no	—
244	C₂²	—	106	4	no	—
245	C₂²	—	106	4	no	—
246	C₂²	—	106	2	no	—
247	C₂²	—	106	2	no	—
248	C₂²	#16810585	1	1	yes	characteristic
249	C₂²	C₄⋊C₈×D₅₃	1	1	yes	characteristic
250	C₂²	#16810678	1	1	yes	characteristic
251	C₄	—	2	4	no	—
252	C₄	—	106	4	no	—
253	C₄	—	106	2	no	—
254	C₄	#16810482	1	1	yes	characteristic
255	C₄	C₂²⋊C₈×D₅₃	1	1	yes	characteristic
256	C₄	#16810645	1	1	yes	characteristic
257	C₄	#16810674	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	—	106	4	no	—
261	C₂	#20274694	1	1	yes	characteristic
262	C₂	#364×D₅₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₈×D₅₃

C₂

yes

maximal, characteristic

C₂×#16810443

C₂

yes

maximal, complemented

#20276325

C₂

yes

maximal

#20272797

C₂

yes

maximal, characteristic, complemented

C₅₃×#1376

C₂

yes

maximal, characteristic, complemented

C₂²×C₈×D₅₃

C₄

yes

characteristic

C₂²×C₈×D₅₃

C₂²

yes

characteristic

C₂×C₄²×D₅₃

C₄

yes

characteristic

C₂×#13133157

C₂²

yes

—

C₂×C₄×#88322

C₄

yes

characteristic

C₂×C₈×Dic₅₃

C₄

yes

characteristic

C₂×C₈×Dic₅₃

C₂²

yes

characteristic

C₄×C₈×D₅₃

—

#16810443

—

#16810402

—

C₂×M₃₂×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂²×C₄×D₅₃

—

C₂²×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂²×#88322

C₂×C₄

yes

characteristic

C₂²×#88322

C₂×C₄

yes

characteristic

C₂×C₈×D₅₃

—

C₂×C₈×D₅₃

—

C₂×C₈×D₅₃

—

C₂×C₈×D₅₃

—

C₂×C₈×D₅₃

C₈

yes

—

C₂×C₈×D₅₃

D₄

yes

—

C₂×C₈×D₅₃

Q₈

yes

—

C₂×C₄×Dic₅₃

—

C₂×C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄²×D₅₃

—

C₄²×D₅₃

—

—

—

—

—

C₈×Dic₅₃

—

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

—

C₄×C₈×C₅₃

—

C₂³×D₅₃

M₁₆

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

M₁₆

yes

characteristic

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

C₄⋊C₄

yes

—

C₂×C₄×D₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₈

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

M₁₆

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂²⋊C₄

yes

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

C₂²⋊C₄

yes

—

C₂×#88322

C₄⋊C₄

yes

—

C₂×#88322

C₂×C₈

yes

—

C₂×#88322

M₁₆

yes

—

C₈×D₅₃

—

C₈×D₅₃

—

C₈×D₅₃

—

C₁₆×C₅₃

—

#11719908

—

C₂³×C₅₃

C₂×M₁₆

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

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₄⋊C₈

yes

—

100

C₄×D₅₃

—

101

C₄×D₅₃

—

102

C₄×D₅₃

—

103

C₄×D₅₃

—

104

C₄×D₅₃

—

105

C₈×C₅₃

—

106

C₈×C₅₃

#121

yes

—

107

C₈×C₅₃

C₂²⋊Q₈

yes

—

108

#88322

—

109

#88322

C₂²⋊C₈

yes

—

110

#12788

—

maximal, Sylow

111

C₂²×C₅₃

—

112

C₂²×C₅₃

#407

yes

characteristic

113

C₂²×C₅₃

C₂×C₄⋊C₈

yes

characteristic

114

C₂²×C₅₃

#432

yes

characteristic

115

C₂×D₅₃

—

116

C₂×D₅₃

—

117

C₂×D₅₃

—

118

C₂×D₅₃

—

119

C₂×D₅₃

#364

yes

complemented

120

C₄×C₅₃

—

121

C₄×C₅₃

#380

yes

characteristic

122

C₄×C₅₃

C₂×C₂²⋊C₈

yes

characteristic

123

C₄×C₅₃

#423

yes

characteristic

124

C₄×C₅₃

#431

yes

characteristic

125

Dic₅₃

—

126

Dic₅₃

—

127

C₂²×C₄×C₈

—

128

C₂×#364

—

129

#2112

—

130

#1376

—

131

#1376

D₅₃

yes

characteristic, complemented

132

C₂×C₅₃

—

133

C₂×C₅₃

—

134

C₂×C₅₃

#1797

yes

characteristic

135

C₂×C₅₃

C₂×#364

yes

—

136

D₅₃

—

137

C₂³×C₈

—

138

C₂³×C₈

—

139

C₂²×C₄²

—

140

C₂×M₃₂

—

141

C₂×M₃₂

C₂×D₅₃

yes

—

142

C₂×C₄×C₈

—

143

C₂×C₄×C₈

—

144

C₂×C₄×C₈

—

145

C₂×C₄×C₈

—

106

—

146

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

147

#364

—

106

—

148

C₂²⋊C₁₆

—

106

—

149

C₅₃

#12788

yes

characteristic, complemented, Sylow

150

C₂³×C₄

—

151

C₂³×C₄

—

106

—

152

C₂²×C₈

—

153

C₂²×C₈

—

154

C₂²×C₈

—

155

C₂²×C₈

—

156

C₂²×C₈

—

157

C₂²×C₈

—

106

—

158

C₂²×C₈

—

106

—

159

C₂²×C₈

—

106

—

160

C₂²×C₈

—

106

—

161

C₂²×C₈

C₄×D₅₃

yes

characteristic

162

C₂²×C₈

C₂²×D₅₃

yes

characteristic

163

C₂×C₄²

—

164

C₂×C₄²

—

106

—

165

C₂×C₄²

—

106

—

166

C₂×C₄²

—

106

—

167

C₂×C₄²

C₄×D₅₃

yes

characteristic

168

M₃₂

—

169

M₃₂

—

170

M₃₂

—

106

—

171

M₃₂

—

106

—

172

C₂×C₁₆

—

173

C₂×C₁₆

—

106

—

174

C₄×C₈

—

175

C₄×C₈

—

176

C₄×C₈

—

106

—

177

C₄×C₈

—

106

—

178

C₂⁴

—

179

C₂²×C₄

—

180

C₂²×C₄

—

181

C₂²×C₄

—

182

C₂²×C₄

—

183

C₂²×C₄

—

106

—

184

C₂²×C₄

—

106

—

185

C₂²×C₄

—

106

—

186

C₂²×C₄

—

106

—

187

C₂²×C₄

—

106

—

188

C₂²×C₄

—

106

—

189

C₂²×C₄

—

106

—

190

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

191

C₂×C₈

—

192

C₂×C₈

—

193

C₂×C₈

—

194

C₂×C₈

—

195

C₂×C₈

—

196

C₂×C₈

—

197

C₂×C₈

—

106

—

198

C₂×C₈

—

106

—

199

C₂×C₈

—

106

—

200

C₂×C₈

—

106

—

201

C₂×C₈

—

106

—

202

C₂×C₈

C₈×D₅₃

yes

—

203

C₂×C₈

#11719945

yes

—

204

C₂×C₈

D₄×D₅₃

yes

—

205

C₂×C₈

D₄×D₅₃

yes

characteristic

206

C₂×C₈

Q₈×D₅₃

yes

characteristic

207

C₄²

—

208

C₄²

—

209

C₄²

—

106

—

210

C₄²

—

106

—

211

C₄²

#11719912

yes

—

212

C₁₆

—

213

C₁₆

—

212

—

214

C₂³

—

215

C₂³

—

106

—

216

C₂³

—

106

—

217

C₂³

—

106

—

218

C₂³

M₁₆×D₅₃

yes

characteristic

219

C₂×C₄

—

220

C₂×C₄

—

221

C₂×C₄

—

222

C₂×C₄

—

106

—

223

C₂×C₄

—

106

—

224

C₂×C₄

—

106

—

225

C₂×C₄

—

106

—

226

C₂×C₄

—

106

—

227

C₂×C₄

—

106

—

228

C₂×C₄

—

106

—

229

C₂×C₄

—

106

—

230

C₂×C₄

—

106

—

231

C₂×C₄

C₂²⋊C₄×D₅₃

yes

characteristic

232

C₂×C₄

C₄⋊C₄×D₅₃

yes

characteristic

233

C₂×C₄

#13133174

yes

characteristic

234

C₂×C₄

C₂×C₈×D₅₃

yes

characteristic

235

C₂×C₄

M₁₆×D₅₃

yes

characteristic

236

C₂×C₄

C₂×#11719912

yes

characteristic

237

C₈

—

238

C₈

—

239

C₈

—

106

—

240

C₈

#13133190

yes

—

241

C₈

#13133195

yes

—

242

C₂²

—

243

C₂²

—

244

C₂²

—

106

—

245

C₂²

—

106

—

246

C₂²

—

106

—

247

C₂²

—

106

—

248

C₂²

#16810585

yes

characteristic

249

C₂²

C₄⋊C₈×D₅₃

yes

characteristic

250

C₂²

#16810678

yes

characteristic

251

C₄

—

252

C₄

—

106

—

253

C₄

—

106

—

254

C₄

#16810482

yes

characteristic

255

C₄

C₂²⋊C₈×D₅₃

yes

characteristic

256

C₄

#16810645

yes

characteristic

257

C₄

#16810674

yes

characteristic

258

C₂

—

259

C₂

—

260

C₂

—

106

—

261

C₂

#20274694

yes

characteristic

262

C₂

#364×D₅₃

yes

—

	subgroup	G/H
center	C₂×C₈	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1376	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₈
prime radical	C₂³×D₅₃	M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1376	D₅₃
hypocenter	C₅₃	#12788
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1376

subgroup

G/H

center

C₂×C₈

D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₅₃×#1376

C₂

socle

C₂²×C₅₃

C₂×C₄⋊C₈

prime radical

C₂³×D₅₃

M₁₆

cosocle

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1376

N	H ≅ G/N	type
2. C₂×#16810443	C₂	semidirect
4. #20272797	C₂	semidirect
5. C₅₃×#1376	C₂	semidirect
119. C₂×D₅₃	#364	semidirect
131. #1376	D₅₃	semidirect
149. C₅₃	#12788	semidirect

H ≅ G/N

type

2. C₂×#16810443

C₂

semidirect

4. #20272797

C₂

semidirect

5. C₅₃×#1376

C₂

semidirect

119. C₂×D₅₃

#364

semidirect

131. #1376

D₅₃

semidirect

149. C₅₃

#12788

semidirect

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

subgroup	index
C₂×C₄×C₈×D₅₃	2
C₂×#16810443	2
#20276325	2
#20272797	2
C₅₃×#1376	2
#12788	53

subgroup

index

C₂×C₄×C₈×D₅₃

C₅₃×#1376

Subgroups — #27300033

Conjugacy classes of subgroups

Complements

Maximal subgroups