Subgroups — #107×#88324 #28302115

Order 13568.

446 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊C₄×#88324	C₂	1	2	yes	maximal
2	C₂×#16811486	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×#88324	C₂	1	1	yes	maximal, characteristic
4	#20289551	C₂	1	2	yes	maximal
5	#107×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88324	C₂²	1	2	yes	—
7	C₂×C₄×#88324	—	2	2	no	—
8	C₂×C₄×#88324	C₂²	1	1	yes	characteristic
9	C₂×C₄×#88324	C₂²	1	1	yes	characteristic
10	#16811489	—	2	2	no	—
11	#16811486	C₄	1	4	yes	complemented
12	C₈×#88324	C₄	1	4	yes	complemented
13	C₂×#13133151	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×Dic₅₃	C₂²	1	2	yes	—
16	#16810385	C₄	1	2	yes	—
17	C₂×#107×C₅₃	C₄	1	1	yes	characteristic
18	C₂²×#88324	—	2	1	no	—
19	C₂²×#88324	—	2	2	no	—
20	C₂²×#88324	D₄	1	1	yes	characteristic
21	C₂²×#88324	D₄	1	1	yes	characteristic
22	C₄⋊#88324	—	2	2	no	—
23	C₄⋊#88324	C₂×C₄	1	4	yes	—
24	C₄×#88324	—	2	1	no	—
25	C₄×#88324	—	4	4	no	—
26	C₄×#88324	C₂×C₄	1	2	yes	—
27	C₄×#88324	C₂×C₄	1	2	yes	—
28	C₄×#88324	Q₈	1	2	yes	—
29	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₄×Dic₅₃	—	2	2	no	—
32	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
33	#13133151	C₈	1	4	yes	complemented
34	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
35	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
36	#13133124	—	2	2	no	—
37	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
39	#107×C₅₃	C₈	1	4	yes	complemented
40	C₂²×C₄×C₅₃	—	2	2	no	—
41	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×Dic₅₃	—	2	2	no	—
43	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×Dic₅₃	#41	1	1	yes	characteristic
45	C₂×#88324	—	2	2	no	—
46	C₂×#88324	—	2	2	no	—
47	C₂×#88324	—	2	2	no	—
48	C₂×#88324	—	2	1	no	—
49	C₂×#88324	—	2	1	no	—
50	C₂×#88324	—	2	4	no	—
51	C₂×#88324	—	4	2	no	—
52	C₂×#88324	—	4	2	no	—
53	C₂×#88324	C₄⋊C₄	1	2	yes	—
54	C₂×#88324	C₄⋊C₄	1	2	yes	—
55	C₂×#88324	D₈	1	2	yes	—
56	C₂×#88324	Q₁₆	1	2	yes	—
57	C₂×C₈×C₅₃	C₄²	1	2	yes	—
58	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
59	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
60	C₄⋊C₄×C₅₃	—	2	2	no	—
61	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
62	C₄⋊Dic₅₃	—	2	2	no	—
63	C₄⋊Dic₅₃	C₂×C₈	1	4	yes	—
64	C₄×Dic₅₃	—	4	2	no	—
65	C₄×Dic₅₃	—	4	2	no	—
66	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
69	C₄×Dic₅₃	#41	1	1	yes	characteristic
70	C₂×#88322	—	2	1	no	—
71	C₂×#88322	—	2	1	no	—
72	C₂×#88322	C₄²	1	2	yes	—
73	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	—	2	4	no	—
75	C₂×C₄×C₅₃	—	4	2	no	—
76	C₂×C₄×C₅₃	—	4	2	no	—
77	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
83	C₂×Dic₅₃	—	2	1	no	—
84	C₂×Dic₅₃	—	2	1	no	—
85	C₂×Dic₅₃	—	2	4	no	—
86	C₂×Dic₅₃	—	4	2	no	—
87	C₂×Dic₅₃	—	4	2	no	—
88	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×Dic₅₃	#117	1	1	yes	characteristic
90	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
91	C₂×Dic₅₃	C₄×Q₈	1	1	yes	characteristic
92	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
93	C₂×Dic₅₃	C₂×Q₁₆	1	1	yes	characteristic
94	C₂×Dic₅₃	#135	1	1	yes	characteristic
95	C₂×Dic₅₃	#135	1	1	yes	characteristic
96	#88324	—	2	4	no	—
97	#88324	—	2	2	no	—
98	#88324	—	4	4	no	—
99	#88324	#107	1	4	yes	direct product factor
100	C₈×C₅₃	C₄×C₈	1	4	yes	complemented
101	#88322	—	2	2	no	—
102	C₈×#107	—	53	1	no	maximal, Sylow
103	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
104	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
105	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
106	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
107	C₂²×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
108	C₂²×C₅₃	#443	1	1	yes	characteristic
109	C₂²×C₅₃	#443	1	1	yes	characteristic
110	C₄×C₅₃	—	4	4	no	—
111	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
112	C₄×C₅₃	#405	1	1	yes	characteristic
113	C₄×C₅₃	C₈×Q₈	1	1	yes	characteristic
114	C₄×C₅₃	C₈×Q₈	1	1	yes	characteristic
115	Dic₅₃	—	2	1	no	—
116	Dic₅₃	—	2	1	no	—
117	Dic₅₃	—	4	4	no	—
118	Dic₅₃	C₂×#107	1	1	yes	characteristic
119	Dic₅₃	#427	1	1	yes	characteristic
120	Dic₅₃	C₄×D₈	1	1	yes	characteristic
121	Dic₅₃	C₄×Q₁₆	1	1	yes	characteristic
122	C₄×#107	—	53	1	no	—
123	C₈×C₄⋊C₄	—	53	2	no	—
124	C₂×#335	—	53	1	no	—
125	C₂×C₈²	—	53	1	no	—
126	#1279	—	53	2	no	—
127	C₂×C₅₃	C₈×D₈	1	1	yes	characteristic
128	C₂×C₅₃	C₈×D₈	1	1	yes	characteristic
129	C₂×C₅₃	C₈×Q₁₆	1	1	yes	characteristic
130	C₂×C₅₃	C₈×Q₁₆	1	1	yes	characteristic
131	C₂×C₅₃	C₈×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₅₃	#1772	1	1	yes	characteristic
133	C₂×C₅₃	C₄×#107	1	1	yes	characteristic
134	C₂×#107	—	53	1	no	—
135	C₂×#107	#6157	1	1	yes	characteristic
136	C₂×C₄⋊C₈	—	53	2	no	—
137	C₂×C₄×C₈	—	53	1	no	—
138	C₂×C₄×C₈	—	53	1	no	—
139	C₂×C₄×C₈	—	53	1	no	—
140	C₂×C₄×C₈	—	106	2	no	—
141	C₄×C₄⋊C₄	—	53	2	no	—
142	#340	—	53	2	no	—
143	#336	—	106	2	no	—
144	#335	—	53	4	no	—
145	C₈²	—	53	4	no	—
146	C₅₃	C₈×#107	1	1	yes	characteristic, complemented, Sylow
147	C₂²×C₈	—	53	1	no	—
148	C₂²×C₈	—	53	1	no	—
149	C₂²×C₈	—	53	1	no	—
150	C₂²×C₈	—	106	1	no	—
151	C₂²×C₈	—	106	2	no	—
152	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
153	C₂×C₄⋊C₄	—	53	2	no	—
154	C₂×C₄⋊C₄	C₂×#6157	1	2	yes	—
155	C₂×C₄²	—	53	1	no	—
156	C₂×C₄²	—	106	2	no	—
157	#107	—	53	4	no	—
158	#107	#88324	1	4	yes	direct product factor
159	C₄⋊C₈	—	53	4	no	—
160	C₄⋊C₈	—	106	2	no	—
161	C₄×C₈	—	53	2	no	—
162	C₄×C₈	—	53	2	no	—
163	C₄×C₈	—	53	2	no	—
164	C₄×C₈	—	53	2	no	—
165	C₄×C₈	—	53	2	no	—
166	C₄×C₈	—	106	1	no	—
167	C₄×C₈	—	212	4	no	—
168	#95	—	106	2	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	53	1	no	—
171	C₂²×C₄	—	53	1	no	—
172	C₂²×C₄	—	106	2	no	—
173	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
174	C₂×C₈	—	53	2	no	—
175	C₂×C₈	—	53	2	no	—
176	C₂×C₈	—	53	2	no	—
177	C₂×C₈	—	53	2	no	—
178	C₂×C₈	—	53	2	no	—
179	C₂×C₈	—	106	2	no	—
180	C₂×C₈	—	106	1	no	—
181	C₂×C₈	—	106	1	no	—
182	C₂×C₈	—	106	2	no	—
183	C₂×C₈	—	106	2	no	—
184	C₂×C₈	—	106	1	no	—
185	C₂×C₈	—	106	1	no	—
186	C₂×C₈	—	106	4	no	—
187	C₂×C₈	—	212	2	no	—
188	C₂×C₈	—	212	2	no	—
189	C₂×C₈	D₅₃⋊C₈	1	2	yes	—
190	C₂×C₈	C₄×#6157	1	2	yes	—
191	C₂×C₈	C₂×#88324	1	2	yes	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	53	4	no	—
194	C₄⋊C₄	—	106	2	no	—
195	C₄⋊C₄	C₂×#88324	1	4	yes	—
196	C₄²	—	53	1	no	—
197	C₄²	—	53	1	no	—
198	C₄²	—	53	1	no	—
199	C₄²	—	53	1	no	—
200	C₄²	—	212	2	no	—
201	C₄²	—	212	2	no	—
202	C₂³	D₄×#6157	1	1	yes	characteristic
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	—	53	1	no	—
207	C₂×C₄	—	53	1	no	—
208	C₂×C₄	—	53	1	no	—
209	C₂×C₄	—	53	1	no	—
210	C₂×C₄	—	53	1	no	—
211	C₂×C₄	—	53	1	no	—
212	C₂×C₄	—	53	1	no	—
213	C₂×C₄	—	53	1	no	—
214	C₂×C₄	—	106	1	no	—
215	C₂×C₄	—	106	1	no	—
216	C₂×C₄	—	106	4	no	—
217	C₂×C₄	—	212	2	no	—
218	C₂×C₄	—	212	2	no	—
219	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
220	C₂×C₄	Q₈×#6157	1	1	yes	characteristic
221	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
222	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
223	C₂×C₄	#13133304	1	1	yes	characteristic
224	C₂×C₄	#13133305	1	1	yes	characteristic
225	C₈	—	53	4	no	—
226	C₈	—	106	2	no	—
227	C₈	—	106	4	no	—
228	C₈	—	106	2	no	—
229	C₈	—	212	4	no	—
230	C₈	C₄×#88324	1	4	yes	complemented
231	C₂²	C₄⋊C₄×#6157	1	1	yes	characteristic
232	C₂²	D₄×#88324	1	1	yes	characteristic
233	C₂²	#16811651	1	1	yes	characteristic
234	C₂²	D₈×#6157	1	1	yes	characteristic
235	C₂²	Q₁₆×#6157	1	1	yes	characteristic
236	C₂²	#16811665	1	1	yes	characteristic
237	C₂²	#16811667	1	1	yes	characteristic
238	C₄	—	4	4	no	—
239	C₄	—	53	1	no	—
240	C₄	—	53	1	no	—
241	C₄	—	53	1	no	—
242	C₄	—	53	1	no	—
243	C₄	—	106	1	no	—
244	C₄	—	106	1	no	—
245	C₄	—	212	4	no	—
246	C₄	C₂×C₄×#88324	1	1	yes	characteristic
247	C₄	#16811588	1	1	yes	characteristic
248	C₄	Q₈×#88324	1	1	yes	characteristic
249	C₄	#16811672	1	1	yes	characteristic
250	C₂	D₈×#88324	1	1	yes	characteristic
251	C₂	#20290296	1	1	yes	characteristic
252	C₂	Q₁₆×#88324	1	1	yes	characteristic
253	C₂	#20290303	1	1	yes	characteristic
254	C₂	C₄⋊C₄×#88324	1	1	yes	characteristic
255	C₂	#20290519	1	1	yes	characteristic
256	C₂	#107×#6157	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×#6157
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂×#107×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	C₂×#107	#6157
hypocenter	C₅₃	C₈×#107
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
11. #16811486	C₄	semidirect
12. C₈×#88324	C₄	semidirect
33. #13133151	C₈	semidirect
39. #107×C₅₃	C₈	semidirect
99. #88324	#107	direct (N × H)
100. C₈×C₅₃	C₄×C₈	semidirect
146. C₅₃	C₈×#107	semidirect
158. #107	#88324	direct (N × H)
230. C₈	C₄×#88324	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊C₄×#88324	2
C₂×#16811486	2
C₂×C₈×#88324	2
#20289551	2
#107×Dic₅₃	2
C₈×#107	53