Subgroups — #28503911

Order 15104.

462 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₃₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837205	C₂	1	2	yes	maximal
3	C₂×#16837178	C₂	1	1	yes	maximal, characteristic
4	#21422187	C₂	1	2	yes	maximal
5	C₄×#107×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₃₆	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₁₁₈	—	2	1	no	—
8	C₂²×C₅₉⋊C₁₆	C₂²	1	2	yes	—
9	C₄×Dic₂₃₆	—	2	2	no	—
10	C₄×Dic₂₃₆	C₄	1	4	yes	complemented
11	C₂×#14273661	C₂²	1	1	yes	characteristic
12	C₂×#14273652	—	2	1	no	—
13	#16837205	C₄	1	8	yes	complemented
14	#16837178	—	2	2	no	—
15	C₂×#107×C₅₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₅₉	—	2	1	no	—
18	C₅₉×#340	—	2	1	no	—
19	C₂²×Dic₁₁₈	—	2	1	no	—
20	C₂×Dic₂₃₆	—	2	1	no	—
21	C₂×Dic₂₃₆	—	2	2	no	—
22	C₂×Dic₂₃₆	C₂×C₄	1	4	yes	—
23	C₂×Dic₂₃₆	C₂×C₄	1	2	yes	—
24	C₄×Dic₁₁₈	—	2	4	no	—
25	C₄×Dic₁₁₈	—	4	2	no	—
26	C₂×C₄⋊Dic₅₉	—	2	1	no	—
27	C₂×Dic₅₉⋊C₄	—	4	1	no	—
28	C₂×C₄×Dic₅₉	—	4	1	no	—
29	C₂×C₅₉⋊C₁₆	—	2	2	no	—
30	C₂×C₅₉⋊C₁₆	—	2	2	no	—
31	C₂×C₅₉⋊C₁₆	C₂×C₄	1	4	yes	—
32	#14273661	—	2	1	no	—
33	#14273661	C₂×C₄	1	2	yes	—
34	#14273652	—	2	4	no	—
35	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
38	C₂×C₄²×C₅₉	—	4	1	no	—
39	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
40	#107×C₅₉	—	2	2	no	—
41	#107×C₅₉	C₂×C₄	1	4	yes	—
42	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₅₉	D₄	1	2	yes	—
44	#95×C₅₉	—	4	1	no	—
45	C₂²×C₄×C₅₉	—	4	2	no	—
46	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
47	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
48	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×Dic₅₉	—	4	1	no	—
50	C₂²×Dic₅₉	—	4	1	no	—
51	C₂×Dic₁₁₈	—	2	2	no	—
52	C₂×Dic₁₁₈	—	2	4	no	—
53	C₂×Dic₁₁₈	—	4	1	no	—
54	C₂×Dic₁₁₈	—	4	2	no	—
55	C₂×C₈×C₅₉	—	2	2	no	—
56	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
59	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₅₉	#41	1	2	yes	—
61	C₄⋊C₄×C₅₉	—	2	4	no	—
62	C₄⋊C₄×C₅₉	—	4	2	no	—
63	C₄²×C₅₉	—	8	2	no	—
64	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₅₉	SD₁₆	1	2	yes	—
66	C₄⋊Dic₅₉	—	2	2	no	—
67	C₄⋊Dic₅₉	—	4	1	no	—
68	Dic₅₉⋊C₄	—	4	4	no	—
69	C₄×Dic₅₉	—	4	4	no	—
70	Dic₂₃₆	—	2	2	no	—
71	Dic₂₃₆	—	2	4	no	—
72	Dic₂₃₆	C₄²	1	4	yes	complemented
73	C₅₉⋊C₁₆	—	2	4	no	—
74	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₅₉	—	2	1	no	—
76	C₂×C₄×C₅₉	—	2	2	no	—
77	C₂×C₄×C₅₉	—	2	1	no	—
78	C₂×C₄×C₅₉	—	4	4	no	—
79	C₂×C₄×C₅₉	—	8	2	no	—
80	C₂×C₄×C₅₉	—	8	2	no	—
81	C₂×C₄×C₅₉	#102	1	2	yes	—
82	C₂×C₄×C₅₉	#102	1	2	yes	—
83	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
86	C₂×C₄×C₅₉	#135	1	2	yes	—
87	C₂×Dic₅₉	—	4	2	no	—
88	C₂×Dic₅₉	—	4	4	no	—
89	C₂×Dic₅₉	—	4	2	no	—
90	C₂×Dic₅₉	—	8	1	no	—
91	C₂×Dic₅₉	—	8	1	no	—
92	Dic₁₁₈	—	2	4	no	—
93	Dic₁₁₈	—	4	2	no	—
94	Dic₁₁₈	—	4	4	no	—
95	C₈×C₅₉	—	2	2	no	—
96	C₈×C₅₉	C₂×C₄²	1	1	yes	characteristic
97	C₈×C₅₉	#117	1	1	yes	characteristic
98	C₈×C₅₉	C₄×D₄	1	2	yes	—
99	#12638	—	59	1	no	maximal, Sylow
100	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
101	C₂²×C₅₉	C₄×D₈	1	2	yes	—
102	C₂²×C₅₉	#509	1	1	yes	characteristic
103	C₂²×C₅₉	#510	1	2	yes	—
104	C₂²×C₅₉	#510	1	1	yes	characteristic
105	C₄×C₅₉	—	2	2	no	—
106	C₄×C₅₉	—	2	2	no	—
107	C₄×C₅₉	—	8	4	no	—
108	C₄×C₅₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
109	C₄×C₅₉	#416	1	1	yes	characteristic
110	C₄×C₅₉	C₄×SD₁₆	1	2	yes	—
111	Dic₅₉	—	4	4	no	—
112	Dic₅₉	—	8	2	no	—
113	C₂×C₄×Q₁₆	—	59	1	no	—
114	C₂×#358	—	59	2	no	—
115	C₂×#346	—	59	1	no	—
116	C₄×#107	D₅₉	1	1	yes	characteristic
117	#1380	—	59	2	no	—
118	C₂×C₅₉	C₄×#102	1	1	yes	characteristic
119	C₂×C₅₉	#2139	1	1	yes	characteristic
120	C₂×C₅₉	#2141	1	1	yes	characteristic
121	C₂×C₅₉	#2175	1	2	yes	—
122	C₂×C₅₉	#2176	1	2	yes	—
123	C₂²×Q₁₆	—	59	1	no	—
124	C₂×C₄×Q₈	—	118	1	no	—
125	C₂²×C₁₆	—	59	2	no	—
126	C₄×Q₁₆	—	59	4	no	—
127	C₄×Q₁₆	—	118	2	no	—
128	C₂×#107	—	59	1	no	—
129	C₂×#107	C₂×D₅₉	1	2	yes	—
130	C₂×#103	—	118	1	no	—
131	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
132	C₄×C₄⋊C₄	—	2	1	no	—
133	#358	—	59	8	no	—
134	#346	—	118	2	no	—
135	#340	—	2	1	no	—
136	C₅₉	#12638	1	1	yes	characteristic, complemented, Sylow
137	C₂²×Q₈	—	118	1	no	—
138	C₂×Q₁₆	—	59	4	no	—
139	C₂×Q₁₆	—	59	2	no	—
140	C₂×Q₁₆	—	118	1	no	—
141	C₂×Q₁₆	—	118	2	no	—
142	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
143	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
144	C₄×Q₈	—	118	4	no	—
145	C₄×Q₈	—	236	2	no	—
146	C₂×C₄⋊C₄	—	2	2	no	—
147	C₂×C₄⋊C₄	—	118	1	no	—
148	C₂×C₄⋊C₄	—	236	1	no	—
149	C₂×C₄²	—	4	1	no	—
150	C₂×C₄²	—	236	1	no	—
151	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
152	C₂×C₁₆	—	59	4	no	—
153	C₂×C₁₆	—	118	2	no	—
154	C₂×C₁₆	—	118	2	no	—
155	#107	—	2	2	no	—
156	#107	—	59	2	no	—
157	#107	—	118	1	no	—
158	#107	C₄×D₅₉	1	4	yes	—
159	#103	—	118	4	no	—
160	C₄×C₈	C₄×D₅₉	1	2	yes	—
161	C₄×C₈	D₂₃₆	1	2	yes	—
162	#95	—	4	1	no	—
163	C₂×Q₈	—	118	2	no	—
164	C₂×Q₈	—	118	4	no	—
165	C₂×Q₈	—	236	1	no	—
166	C₂×Q₈	—	236	2	no	—
167	C₂²×C₄	—	4	2	no	—
168	C₂²×C₄	—	236	1	no	—
169	C₂²×C₄	—	236	1	no	—
170	C₂²×C₄	#11745669	1	1	yes	characteristic
171	C₂²×C₄	#11745669	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
173	Q₁₆	—	59	4	no	—
174	Q₁₆	—	118	2	no	—
175	Q₁₆	—	118	4	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
178	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
179	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
180	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
181	C₂×C₈	#11745685	1	2	yes	—
182	C₄⋊C₄	—	2	4	no	—
183	C₄⋊C₄	—	4	2	no	—
184	C₄⋊C₄	—	118	2	no	—
185	C₄⋊C₄	—	236	4	no	—
186	C₄⋊C₄	—	236	1	no	—
187	C₄²	—	8	2	no	—
188	C₄²	—	236	4	no	—
189	C₄²	(C₂×D₅₉)⋊C₄	1	2	yes	—
190	C₄²	#11745671	1	2	yes	—
191	C₁₆	—	118	4	no	—
192	C₂³	C₂×#11745669	1	1	yes	characteristic
193	Q₈	—	118	4	no	—
194	Q₈	—	236	2	no	—
195	Q₈	—	236	4	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	8	2	no	—
201	C₂×C₄	—	8	2	no	—
202	C₂×C₄	—	472	1	no	—
203	C₂×C₄	—	472	1	no	—
204	C₂×C₄	—	236	2	no	—
205	C₂×C₄	—	236	2	no	—
206	C₂×C₄	—	236	4	no	—
207	C₂×C₄	#14273647	1	2	yes	—
208	C₂×C₄	#14273647	1	2	yes	—
209	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
210	C₂×C₄	C₄×C₂²⋊D₅₉	1	2	yes	—
211	C₂×C₄	C₂×#11745671	1	1	yes	characteristic
212	C₂×C₄	#14273747	1	2	yes	—
213	C₈	—	2	2	no	—
214	C₈	C₄²×D₅₉	1	1	yes	characteristic
215	C₈	#14273684	1	1	yes	characteristic
216	C₈	C₄×D₂₃₆	1	2	yes	—
217	C₂²	C₂×#14273647	1	1	yes	characteristic
218	C₂²	C₄×#11745669	1	2	yes	—
219	C₂²	#16837838	1	1	yes	characteristic
220	C₂²	#16837842	1	2	yes	—
221	C₂²	#16837843	1	1	yes	characteristic
222	C₄	—	2	2	no	—
223	C₄	—	2	2	no	—
224	C₄	—	8	4	no	—
225	C₄	—	472	2	no	—
226	C₄	—	236	4	no	—
227	C₄	C₄×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
228	C₄	#16837396	1	1	yes	characteristic
229	C₄	C₄×#11745671	1	2	yes	—
230	C₂	C₄×#14273647	1	1	yes	characteristic
231	C₂	#21425806	1	1	yes	characteristic
232	C₂	#21425815	1	1	yes	characteristic
233	C₂	#21425960	1	2	yes	—
234	C₂	#21425965	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11745669
commutator	C₈×C₅₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₄×#107×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	C₂×D₈
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₄×#107	D₅₉
hypocenter	C₅₉	#12638
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
10. C₄×Dic₂₃₆	C₄	semidirect
13. #16837205	C₄	semidirect
72. Dic₂₃₆	C₄²	semidirect
136. C₅₉	#12638	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₃₆	2
C₂×#16837205	2
C₂×#16837178	2
#21422187	2
C₄×#107×C₅₉	2
#12638	59