Group #28308530 — subgroups

Subgroups — #28308530

Order 13568.

362 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 110 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	#20291048	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133292	C₂	1	1	yes	maximal, characteristic
3	#20289682	C₂	1	2	yes	maximal, complemented
4	Q₈⋊#11719910	C₂	1	2	yes	maximal, complemented
5	C₈×Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133292	—	2	1	no	—
7	C₂×#13133292	—	2	1	no	—
8	#16811544	—	2	1	no	—
9	C₄⋊#11719910	C₂²	1	2	yes	—
10	#16811512	—	2	1	no	—
11	C₄×#11719910	C₂²	1	2	yes	—
12	C₄×Q₈×D₅₃	C₄	1	1	yes	characteristic
13	Q₈×#88322	C₂²	1	1	yes	characteristic
14	C₈×Dic₁₀₆	—	2	1	no	—
15	C₈×Dic₁₀₆	C₄	1	1	yes	characteristic
16	#16810747	—	2	1	no	—
17	#16810747	C₂²	1	1	yes	characteristic
18	C₄⋊C₈×D₅₃	—	2	1	no	—
19	C₄⋊C₈×D₅₃	C₄	1	1	yes	characteristic
20	C₄×C₈×D₅₃	—	2	1	no	—
21	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
22	C₈×Q₈×C₅₃	C₄	1	1	yes	characteristic
23	#13133292	—	2	4	no	—
24	#13133292	—	4	2	no	—
25	C₂×#11719910	—	2	2	no	—
26	C₂×#11719910	—	2	2	no	—
27	C₂×Q₈×D₅₃	C₈	1	1	yes	characteristic
28	C₂×C₈×D₅₃	—	2	1	no	—
29	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
30	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
31	Q₈×Dic₅₃	C₈	1	1	yes	characteristic
32	C₄×Dic₁₀₆	—	2	1	no	—
33	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
34	#13133182	—	2	1	no	—
35	#13133182	C₈	1	1	yes	characteristic
36	C₄⋊C₄×D₅₃	—	2	1	no	—
37	C₄⋊C₄×D₅₃	C₈	1	1	yes	characteristic
38	C₄²×D₅₃	—	2	1	no	—
39	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
40	C₄⋊#88322	—	2	1	no	—
41	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
42	#13133146	—	2	1	no	—
43	#13133146	—	2	1	no	—
44	#13133146	—	2	1	no	—
45	C₄×#88322	—	2	1	no	—
46	C₄×#88322	C₂³	1	1	yes	characteristic
47	C₈×Dic₅₃	—	2	1	no	—
48	C₈×Dic₅₃	—	2	1	no	—
49	C₈×Dic₅₃	D₄	1	1	yes	characteristic
50	C₈×Dic₅₃	D₄	1	1	yes	characteristic
51	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₄⋊C₈×C₅₃	—	2	1	no	—
53	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₅₃	—	2	1	no	—
55	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
57	Q₈×D₅₃	—	2	2	no	—
58	Q₈×D₅₃	—	2	2	no	—
59	C₂×C₄×D₅₃	—	2	1	no	—
60	C₂×C₄×D₅₃	—	2	1	no	—
61	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
63	C₂×C₄×D₅₃	M₁₆	1	1	yes	characteristic
64	C₂×Dic₁₀₆	—	2	1	no	—
65	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
66	C₂×C₈×C₅₃	—	2	1	no	—
67	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₄⋊C₄×C₅₃	—	2	1	no	—
70	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
71	C₄²×C₅₃	—	2	1	no	—
72	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₄⋊Dic₅₃	—	2	1	no	—
74	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
75	#11719919	—	2	1	no	—
76	#11719919	—	2	1	no	—
77	#11719919	—	2	1	no	—
78	C₄×Dic₅₃	—	2	1	no	—
79	C₄×Dic₅₃	—	2	1	no	—
80	C₄×Dic₅₃	—	2	1	no	—
81	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
83	C₄×Dic₅₃	M₁₆	1	1	yes	characteristic
84	C₂×#88322	—	2	1	no	—
85	C₂×#88322	C₂×D₄	1	1	yes	characteristic
86	C₂×#88322	C₂×D₄	1	1	yes	characteristic
87	C₈×D₅₃	—	2	2	no	—
88	C₈×D₅₃	—	4	2	no	—
89	C₈×D₅₃	SD₁₆	1	2	yes	—
90	C₈×D₅₃	Q₁₆	1	2	yes	—
91	#11719910	—	2	4	no	—
92	#11719910	—	4	2	no	—
93	C₂²×D₅₃	C₂²⋊C₈	1	1	yes	characteristic
94	Q₈×C₅₃	—	2	1	no	—
95	Q₈×C₅₃	M₃₂	1	2	yes	complemented
96	C₂×C₄×C₅₃	—	2	1	no	—
97	C₂×C₄×C₅₃	—	2	1	no	—
98	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
101	C₂×Dic₅₃	—	2	1	no	—
102	C₂×Dic₅₃	—	2	1	no	—
103	C₂×Dic₅₃	C₂²⋊C₈	1	1	yes	characteristic
104	C₂×Dic₅₃	C₂²⋊C₈	1	1	yes	characteristic
105	C₂×Dic₅₃	C₂²⋊C₈	1	1	yes	characteristic
106	C₄×D₅₃	—	2	4	no	—
107	C₄×D₅₃	—	2	1	no	—
108	C₄×D₅₃	—	2	1	no	—
109	C₄×D₅₃	—	4	1	no	—
110	C₄×D₅₃	—	4	2	no	—
111	C₄×D₅₃	#103	1	2	yes	—
112	C₄×D₅₃	#104	1	2	yes	—
113	Dic₁₀₆	—	2	4	no	—
114	Dic₁₀₆	—	2	1	no	—
115	Dic₁₀₆	M₃₂	1	2	yes	complemented
116	C₈×C₅₃	—	2	1	no	—
117	C₈×C₅₃	—	4	1	no	—
118	C₈×C₅₃	#103	1	2	yes	—
119	#88322	—	2	1	no	—
120	#88322	—	4	1	no	—
121	#88322	C₂×SD₁₆	1	1	yes	characteristic
122	#88322	C₂×Q₁₆	1	1	yes	characteristic
123	#11607	—	53	1	no	maximal, Sylow
124	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
125	C₂×D₅₃	—	2	1	no	—
126	C₂×D₅₃	—	2	1	no	—
127	C₂×D₅₃	Q₈⋊C₈	1	2	yes	—
128	C₄×C₅₃	—	2	2	no	—
129	C₄×C₅₃	—	2	1	no	—
130	C₄×C₅₃	—	4	1	no	—
131	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
132	C₄×C₅₃	C₂×#104	1	1	yes	characteristic
133	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
134	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
135	Dic₅₃	—	2	1	no	—
136	Dic₅₃	—	2	2	no	—
137	Dic₅₃	—	2	1	no	—
138	Dic₅₃	—	4	1	no	—
139	Dic₅₃	Q₈⋊C₈	1	2	yes	—
140	C₂×C₈×Q₈	—	53	1	no	—
141	#2150	—	53	1	no	—
142	C₄×M₃₂	—	53	1	no	—
143	Q₈⋊C₁₆	—	53	2	no	—
144	Q₈⋊C₁₆	—	53	2	no	—
145	C₂×C₅₃	C₂×Q₈⋊C₈	1	1	yes	characteristic
146	C₂×C₅₃	#2112	1	1	yes	characteristic
147	C₂×C₅₃	#2117	1	1	yes	characteristic
148	D₅₃	—	2	2	no	—
149	C₂×C₄×Q₈	—	53	1	no	—
150	C₂×M₃₂	—	106	1	no	—
151	C₂×M₃₂	—	106	1	no	—
152	C₈×Q₈	—	53	1	no	—
153	C₈×Q₈	—	53	1	no	—
154	C₈×Q₈	—	53	1	no	—
155	C₈×Q₈	—	106	1	no	—
156	C₈×Q₈	—	106	1	no	—
157	C₈×Q₈	#6157	1	1	yes	characteristic
158	C₂×C₄⋊C₈	—	53	1	no	—
159	C₂×C₄⋊C₈	—	106	1	no	—
160	C₂×C₄×C₈	—	53	1	no	—
161	C₂×C₄×C₈	—	106	1	no	—
162	C₄⋊C₁₆	—	53	2	no	—
163	C₄⋊C₁₆	—	106	1	no	—
164	#346	—	106	1	no	—
165	C₄×C₁₆	—	53	2	no	—
166	C₅₃	#11607	1	1	yes	characteristic, complemented, Sylow
167	C₂²×Q₈	—	53	1	no	—
168	C₂²×C₈	—	53	1	no	—
169	C₂²×C₈	—	53	1	no	—
170	C₂²×C₈	—	106	1	no	—
171	C₄×Q₈	—	53	1	no	—
172	C₄×Q₈	—	53	1	no	—
173	C₄×Q₈	—	53	1	no	—
174	C₄×Q₈	—	106	1	no	—
175	C₄×Q₈	—	106	1	no	—
176	C₄×Q₈	C₂×#6157	1	1	yes	characteristic
177	C₂×C₄⋊C₄	—	53	1	no	—
178	C₂×C₄⋊C₄	—	106	1	no	—
179	C₂×C₄²	—	53	1	no	—
180	C₂×C₄²	—	106	1	no	—
181	M₃₂	—	106	4	no	—
182	M₃₂	—	212	2	no	—
183	C₂×C₁₆	—	106	2	no	—
184	C₂×C₁₆	—	106	2	no	—
185	C₄⋊C₈	—	2	1	no	—
186	C₄⋊C₈	—	53	1	no	—
187	C₄⋊C₈	—	106	1	no	—
188	C₄⋊C₈	—	106	1	no	—
189	C₄⋊C₈	—	106	1	no	—
190	C₄⋊C₈	—	106	1	no	—
191	C₄⋊C₈	C₂×#6157	1	1	yes	characteristic
192	C₄×C₈	—	2	1	no	—
193	C₄×C₈	—	53	1	no	—
194	C₄×C₈	—	53	1	no	—
195	C₄×C₈	—	53	1	no	—
196	C₄×C₈	—	106	1	no	—
197	C₄×C₈	—	106	1	no	—
198	C₄×C₈	—	106	1	no	—
199	C₄×C₈	C₂×#6157	1	1	yes	characteristic
200	C₂×Q₈	—	53	1	no	—
201	C₂×Q₈	—	106	1	no	—
202	C₂×Q₈	—	106	2	no	—
203	C₂×Q₈	—	106	2	no	—
204	C₂×Q₈	D₅₃⋊C₈	1	1	yes	characteristic
205	C₂²×C₄	—	53	1	no	—
206	C₂²×C₄	—	53	1	no	—
207	C₂²×C₄	—	53	1	no	—
208	C₂²×C₄	—	106	1	no	—
209	C₂²×C₄	—	106	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	53	1	no	—
212	C₂×C₈	—	53	2	no	—
213	C₂×C₈	—	53	2	no	—
214	C₂×C₈	—	53	1	no	—
215	C₂×C₈	—	106	2	no	—
216	C₂×C₈	—	106	1	no	—
217	C₂×C₈	—	212	2	no	—
218	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
219	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
220	C₄⋊C₄	—	2	1	no	—
221	C₄⋊C₄	—	53	1	no	—
222	C₄⋊C₄	—	106	1	no	—
223	C₄⋊C₄	—	106	1	no	—
224	C₄⋊C₄	—	106	1	no	—
225	C₄⋊C₄	—	106	1	no	—
226	C₄⋊C₄	D₅₃⋊C₈	1	1	yes	characteristic
227	C₄²	—	2	1	no	—
228	C₄²	—	53	1	no	—
229	C₄²	—	53	1	no	—
230	C₄²	—	53	1	no	—
231	C₄²	—	106	1	no	—
232	C₄²	—	106	1	no	—
233	C₄²	—	106	1	no	—
234	C₄²	C₂²×#6157	1	1	yes	characteristic
235	C₁₆	—	106	4	no	—
236	C₁₆	—	212	2	no	—
237	C₂³	—	53	1	no	—
238	Q₈	—	2	1	no	—
239	Q₈	—	53	2	no	—
240	Q₈	—	106	4	no	—
241	Q₈	—	106	1	no	—
242	Q₈	#13133292	1	2	yes	complemented
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	53	1	no	—
246	C₂×C₄	—	53	2	no	—
247	C₂×C₄	—	53	2	no	—
248	C₂×C₄	—	53	1	no	—
249	C₂×C₄	—	53	1	no	—
250	C₂×C₄	—	106	4	no	—
251	C₂×C₄	—	106	1	no	—
252	C₂×C₄	—	106	1	no	—
253	C₂×C₄	—	106	1	no	—
254	C₂×C₄	—	106	1	no	—
255	C₂×C₄	—	212	1	no	—
256	C₂×C₄	—	212	2	no	—
257	C₂×C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
258	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
259	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
260	C₈	—	2	1	no	—
261	C₈	—	4	1	no	—
262	C₈	—	53	1	no	—
263	C₈	—	53	1	no	—
264	C₈	—	106	1	no	—
265	C₈	—	212	1	no	—
266	C₈	#13133280	1	2	yes	—
267	C₂²	—	53	2	no	—
268	C₂²	—	106	1	no	—
269	C₂²	—	106	1	no	—
270	C₂²	D₅₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	2	1	no	—
273	C₄	—	4	1	no	—
274	C₄	—	53	2	no	—
275	C₄	—	106	1	no	—
276	C₄	—	106	2	no	—
277	C₄	—	106	1	no	—
278	C₄	—	212	1	no	—
279	C₄	C₂×#13133280	1	1	yes	characteristic
280	C₄	D₅₃⋊#104	1	1	yes	characteristic
281	C₄	D₅₃⋊M₃₂	1	1	yes	characteristic
282	C₄	C₂×#13133292	1	1	yes	characteristic
283	C₂	—	106	2	no	—
284	C₂	D₅₃⋊(Q₈⋊C₈)	1	1	yes	characteristic
285	C₂	#20290943	1	1	yes	characteristic
286	C₂	#20290958	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20291048

C₂

yes

maximal, characteristic

C₄×#13133292

C₂

yes

maximal, characteristic

#20289682

C₂

yes

maximal, complemented

Q₈⋊#11719910

C₂

yes

maximal, complemented

C₈×Q₈×D₅₃

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

#16811512

—

C₄×#11719910

C₂²

yes

—

C₄×Q₈×D₅₃

C₄

yes

characteristic

Q₈×#88322

C₂²

yes

characteristic

C₈×Dic₁₀₆

—

C₈×Dic₁₀₆

C₄

yes

characteristic

#16810747

—

#16810747

C₂²

yes

characteristic

C₄⋊C₈×D₅₃

—

C₄⋊C₈×D₅₃

C₄

yes

characteristic

C₄×C₈×D₅₃

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

C₈×Q₈×C₅₃

C₄

yes

—

—

—

—

C₂×Q₈×D₅₃

C₈

yes

characteristic

C₂×C₈×D₅₃

—

C₂×C₈×D₅₃

D₄

yes

characteristic

C₂×C₈×D₅₃

D₄

yes

characteristic

Q₈×Dic₅₃

C₈

yes

characteristic

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

C₂×C₄

yes

characteristic

#13133182

—

#13133182

C₈

yes

characteristic

C₄⋊C₄×D₅₃

—

C₄⋊C₄×D₅₃

C₈

yes

characteristic

C₄²×D₅₃

—

C₄²×D₅₃

C₂×C₄

yes

characteristic

C₄⋊#88322

—

C₄⋊#88322

C₂×C₄

yes

—

—

—

—

C₂³

yes

characteristic

C₈×Dic₅₃

—

C₈×Dic₅₃

—

C₈×Dic₅₃

D₄

yes

characteristic

C₈×Dic₅₃

D₄

yes

characteristic

C₄×Q₈×C₅₃

C₂×C₄

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×Q₈×C₅₃

C₂×C₈

yes

characteristic

Q₈×D₅₃

—

Q₈×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×D₅₃

C₂×C₈

yes

characteristic

C₂×C₄×D₅₃

M₁₆

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂×C₈

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₈

yes

characteristic

C₄²×C₅₃

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₈

yes

—

—

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₅₃

C₂×C₈

yes

characteristic

C₄×Dic₅₃

M₁₆

yes

characteristic

C₂×#88322

—

C₂×#88322

C₂×D₄

yes

characteristic

C₂×#88322

C₂×D₄

yes

characteristic

C₈×D₅₃

—

C₈×D₅₃

—

C₈×D₅₃

SD₁₆

yes

—

C₈×D₅₃

Q₁₆

yes

—

#11719910

—

#11719910

—

C₂²×D₅₃

C₂²⋊C₈

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

M₃₂

yes

complemented

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

100

C₂×C₄×C₅₃

C₂×M₁₆

yes

characteristic

101

C₂×Dic₅₃

—

102

C₂×Dic₅₃

—

103

C₂×Dic₅₃

C₂²⋊C₈

yes

characteristic

104

C₂×Dic₅₃

C₂²⋊C₈

yes

characteristic

105

C₂×Dic₅₃

C₂²⋊C₈

yes

characteristic

106

C₄×D₅₃

—

107

C₄×D₅₃

—

108

C₄×D₅₃

—

109

C₄×D₅₃

—

110

C₄×D₅₃

—

111

C₄×D₅₃

#103

yes

—

112

C₄×D₅₃

#104

yes

—

113

Dic₁₀₆

—

114

Dic₁₀₆

—

115

Dic₁₀₆

M₃₂

yes

complemented

116

C₈×C₅₃

—

117

C₈×C₅₃

—

118

C₈×C₅₃

#103

yes

—

119

#88322

—

120

#88322

—

121

#88322

C₂×SD₁₆

yes

characteristic

122

#88322

C₂×Q₁₆

yes

characteristic

123

#11607

—

maximal, Sylow

124

C₂²×C₅₃

C₂×C₂²⋊C₈

yes

characteristic

125

C₂×D₅₃

—

126

C₂×D₅₃

—

127

C₂×D₅₃

Q₈⋊C₈

yes

—

128

C₄×C₅₃

—

129

C₄×C₅₃

—

130

C₄×C₅₃

—

131

C₄×C₅₃

C₂×#103

yes

characteristic

132

C₄×C₅₃

C₂×#104

yes

characteristic

133

C₄×C₅₃

C₂×M₃₂

yes

characteristic

134

C₄×C₅₃

C₂×M₃₂

yes

characteristic

135

Dic₅₃

—

136

Dic₅₃

—

137

Dic₅₃

—

138

Dic₅₃

—

139

Dic₅₃

Q₈⋊C₈

yes

—

140

C₂×C₈×Q₈

—

141

#2150

—

142

C₄×M₃₂

—

143

Q₈⋊C₁₆

—

144

Q₈⋊C₁₆

—

145

C₂×C₅₃

C₂×Q₈⋊C₈

yes

characteristic

146

C₂×C₅₃

#2112

yes

characteristic

147

C₂×C₅₃

#2117

yes

characteristic

148

D₅₃

—

149

C₂×C₄×Q₈

—

150

C₂×M₃₂

—

106

—

151

C₂×M₃₂

—

106

—

152

C₈×Q₈

—

153

C₈×Q₈

—

154

C₈×Q₈

—

155

C₈×Q₈

—

106

—

156

C₈×Q₈

—

106

—

157

C₈×Q₈

#6157

yes

characteristic

158

C₂×C₄⋊C₈

—

159

C₂×C₄⋊C₈

—

106

—

160

C₂×C₄×C₈

—

161

C₂×C₄×C₈

—

106

—

162

C₄⋊C₁₆

—

163

C₄⋊C₁₆

—

106

—

164

#346

—

106

—

165

C₄×C₁₆

—

166

C₅₃

#11607

yes

characteristic, complemented, Sylow

167

C₂²×Q₈

—

168

C₂²×C₈

—

169

C₂²×C₈

—

170

C₂²×C₈

—

106

—

171

C₄×Q₈

—

172

C₄×Q₈

—

173

C₄×Q₈

—

174

C₄×Q₈

—

106

—

175

C₄×Q₈

—

106

—

176

C₄×Q₈

C₂×#6157

yes

characteristic

177

C₂×C₄⋊C₄

—

178

C₂×C₄⋊C₄

—

106

—

179

C₂×C₄²

—

180

C₂×C₄²

—

106

—

181

M₃₂

—

106

—

182

M₃₂

—

212

—

183

C₂×C₁₆

—

106

—

184

C₂×C₁₆

—

106

—

185

C₄⋊C₈

—

186

C₄⋊C₈

—

187

C₄⋊C₈

—

106

—

188

C₄⋊C₈

—

106

—

189

C₄⋊C₈

—

106

—

190

C₄⋊C₈

—

106

—

191

C₄⋊C₈

C₂×#6157

yes

characteristic

192

C₄×C₈

—

193

C₄×C₈

—

194

C₄×C₈

—

195

C₄×C₈

—

196

C₄×C₈

—

106

—

197

C₄×C₈

—

106

—

198

C₄×C₈

—

106

—

199

C₄×C₈

C₂×#6157

yes

characteristic

200

C₂×Q₈

—

201

C₂×Q₈

—

106

—

202

C₂×Q₈

—

106

—

203

C₂×Q₈

—

106

—

204

C₂×Q₈

D₅₃⋊C₈

yes

characteristic

205

C₂²×C₄

—

206

C₂²×C₄

—

207

C₂²×C₄

—

208

C₂²×C₄

—

106

—

209

C₂²×C₄

—

106

—

210

C₂×C₈

—

211

C₂×C₈

—

212

C₂×C₈

—

213

C₂×C₈

—

214

C₂×C₈

—

215

C₂×C₈

—

106

—

216

C₂×C₈

—

106

—

217

C₂×C₈

—

212

—

218

C₂×C₈

C₂²⋊#6157

yes

characteristic

219

C₂×C₈

C₂²⋊#6157

yes

characteristic

220

C₄⋊C₄

—

221

C₄⋊C₄

—

222

C₄⋊C₄

—

106

—

223

C₄⋊C₄

—

106

—

224

C₄⋊C₄

—

106

—

225

C₄⋊C₄

—

106

—

226

C₄⋊C₄

D₅₃⋊C₈

yes

characteristic

227

C₄²

—

228

C₄²

—

229

C₄²

—

230

C₄²

—

231

C₄²

—

106

—

232

C₄²

—

106

—

233

C₄²

—

106

—

234

C₄²

C₂²×#6157

yes

characteristic

235

C₁₆

—

106

—

236

C₁₆

—

212

—

237

C₂³

—

238

Q₈

—

239

Q₈

—

240

Q₈

—

106

—

241

Q₈

—

106

—

242

Q₈

#13133292

yes

complemented

243

C₂×C₄

—

244

C₂×C₄

—

245

C₂×C₄

—

246

C₂×C₄

—

247

C₂×C₄

—

248

C₂×C₄

—

249

C₂×C₄

—

250

C₂×C₄

—

106

—

251

C₂×C₄

—

106

—

252

C₂×C₄

—

106

—

253

C₂×C₄

—

106

—

254

C₂×C₄

—

106

—

255

C₂×C₄

—

212

—

256

C₂×C₄

—

212

—

257

C₂×C₄

C₂×C₂²⋊#6157

yes

characteristic

258

C₂×C₄

C₂×D₅₃⋊C₈

yes

characteristic

259

C₂×C₄

D₅₃⋊M₁₆

yes

characteristic

260

C₈

—

261

C₈

—

262

C₈

—

263

C₈

—

264

C₈

—

106

—

265

C₈

—

212

—

266

C₈

#13133280

yes

—

267

C₂²

—

268

C₂²

—

106

—

269

C₂²

—

106

—

270

C₂²

D₅₃⋊(C₂²⋊C₈)

yes

characteristic

271

C₄

—

272

C₄

—

273

C₄

—

274

C₄

—

275

C₄

—

106

—

276

C₄

—

106

—

277

C₄

—

106

—

278

C₄

—

212

—

279

C₄

C₂×#13133280

yes

characteristic

280

C₄

D₅₃⋊#104

yes

characteristic

281

C₄

D₅₃⋊M₃₂

yes

characteristic

282

C₄

C₂×#13133292

yes

characteristic

283

C₂

—

106

—

284

C₂

D₅₃⋊(Q₈⋊C₈)

yes

characteristic

285

C₂

#20290943

yes

characteristic

286

C₂

#20290958

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₄²	C₂²×#6157
Fitting	C₈×Q₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂²×D₅₃	C₂²⋊C₈
cosocle	C₄×#88322	C₂³
hypercenter	C₈×Q₈	#6157
hypocenter	C₅₃	#11607
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊#6157

commutator

C₂×C₄×C₅₃

C₂²×C₈

Frattini

C₄²

C₂²×#6157

Fitting

C₈×Q₈×C₅₃

C₄

socle

C₂²×C₅₃

C₂×C₂²⋊C₈

prime radical

C₂²×D₅₃

C₂²⋊C₈

C₂³

C₈×Q₈

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈×Q₈

N	H ≅ G/N	type
3. #20289682	C₂	semidirect
4. Q₈⋊#11719910	C₂	semidirect
95. Q₈×C₅₃	M₃₂	semidirect
115. Dic₁₀₆	M₃₂	semidirect
166. C₅₃	#11607	semidirect
242. Q₈	#13133292	semidirect

H ≅ G/N

type

3. #20289682

C₂

semidirect

4. Q₈⋊#11719910

C₂

semidirect

95. Q₈×C₅₃

M₃₂

semidirect

115. Dic₁₀₆

M₃₂

semidirect

166. C₅₃

#11607

semidirect

242. Q₈

#13133292

semidirect

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

subgroup	index
#20291048	2
C₄×#13133292	2
#20289682	2
Q₈⋊#11719910	2
C₈×Q₈×D₅₃	2
#11607	53

subgroup

index

C₈×Q₈×D₅₃

Subgroups — #28308530

Conjugacy classes of subgroups

Complements

Maximal subgroups