Subgroups — #28507256

Order 15104.

338 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21434610	C₂	1	1	yes	maximal, characteristic
2	C₄×#14273663	C₂	1	1	yes	maximal, characteristic
3	#21422192	C₂	1	4	yes	maximal
4	C₂×C₅₉×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	#16837959	—	2	1	no	—
6	#16837807	C₄	1	1	yes	characteristic
7	#16837807	C₄	1	1	yes	characteristic
8	#16837804	—	2	1	no	—
9	#16837804	C₄	1	2	yes	—
10	#16837800	—	2	1	no	—
11	C₂×#14273663	C₂²	1	2	yes	—
12	#16837440	—	2	1	no	—
13	#16837440	C₂²	1	2	yes	—
14	C₄×#11745664	—	2	1	no	—
15	#16837167	—	2	1	no	—
16	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₅₉×C₄⋊C₁₆	—	2	1	no	—
19	C₅₉×C₄⋊C₁₆	—	2	1	no	—
20	C₂×#11745664	—	2	2	no	—
21	#14273727	—	2	1	no	—
22	#14273727	—	2	1	no	—
23	#14273719	—	2	2	no	—
24	#14273706	—	4	1	no	—
25	#14273706	—	4	1	no	—
26	C₄×Dic₁₁₈	—	4	1	no	—
27	C₄×Dic₁₁₈	—	4	1	no	—
28	#14273686	—	2	2	no	—
29	#14273663	—	2	2	no	—
30	#14273663	Q₈	1	4	yes	complemented
31	#14273661	—	2	1	no	—
32	#14273661	—	2	1	no	—
33	#14273661	C₂×C₄	1	2	yes	—
34	#14273661	C₂×C₄	1	2	yes	—
35	#14273659	—	2	2	no	—
36	#14273659	—	2	2	no	—
37	#14273638	—	4	1	no	—
38	C₄×C₅₉⋊C₈	—	4	1	no	—
39	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
42	C₂×C₁₆×C₅₉	—	2	2	no	—
43	C₂×C₁₆×C₅₉	—	2	4	no	—
44	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
47	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
50	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
51	C₂×Dic₁₁₈	—	4	1	no	—
52	C₂×Dic₁₁₈	—	4	1	no	—
53	C₂×C₈×C₅₉	—	2	2	no	—
54	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
59	C₂×C₈×C₅₉	#41	1	2	yes	—
60	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₅₉	D₈	1	1	yes	characteristic
63	C₄²×C₅₉	D₈	1	1	yes	characteristic
64	C₂²⋊Dic₅₉	—	4	2	no	—
65	C₄⋊Dic₅₉	—	2	2	no	—
66	C₄⋊Dic₅₉	—	2	2	no	—
67	C₄⋊Dic₅₉	—	4	1	no	—
68	C₄⋊Dic₅₉	—	4	1	no	—
69	Dic₅₉⋊C₄	—	4	2	no	—
70	Dic₅₉⋊C₄	—	4	2	no	—
71	C₄×Dic₅₉	—	4	2	no	—
72	#11745664	—	2	4	no	—
73	#11745664	—	4	2	no	—
74	C₂×C₅₉⋊C₈	—	4	2	no	—
75	C₁₆×C₅₉	—	4	2	no	—
76	C₂³×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₅₉	—	2	2	no	—
78	C₂×C₄×C₅₉	—	4	1	no	—
79	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
80	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
81	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
82	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
83	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
85	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₅₉	#137	1	2	yes	—
87	C₂×Dic₅₉	—	4	2	no	—
88	C₂×Dic₅₉	—	4	2	no	—
89	Dic₁₁₈	—	8	2	no	—
90	Dic₁₁₈	—	8	2	no	—
91	C₈×C₅₉	—	2	1	no	—
92	C₈×C₅₉	—	2	1	no	—
93	C₈×C₅₉	C₄×Q₈	1	2	yes	—
94	C₈×C₅₉	#124	1	1	yes	characteristic
95	C₈×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
96	C₅₉⋊C₈	—	4	4	no	—
97	#13274	—	59	1	no	maximal, Sylow
98	C₂²×C₅₉	—	2	1	no	—
99	C₂²×C₅₉	—	2	1	no	—
100	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
101	C₂²×C₅₉	#475	1	2	yes	—
102	C₄×C₅₉	—	2	1	no	—
103	C₄×C₅₉	—	2	1	no	—
104	C₄×C₅₉	—	2	1	no	—
105	C₄×C₅₉	#359	1	2	yes	—
106	C₄×C₅₉	#391	1	1	yes	characteristic
107	C₄×C₅₉	#419	1	1	yes	characteristic
108	C₄×C₅₉	#474	1	2	yes	—
109	Dic₅₉	—	8	2	no	—
110	Dic₅₉	—	8	2	no	—
111	#3160	—	59	1	no	—
112	C₂×C₄⋊C₁₆	D₅₉	1	1	yes	characteristic
113	C₄×#108	—	59	1	no	—
114	#1382	—	59	4	no	—
115	C₂×C₅₉	—	2	2	no	—
116	C₂×C₅₉	#1988	1	1	yes	characteristic
117	C₂×C₅₉	C₂×#359	1	1	yes	characteristic
118	C₂×C₅₉	#2143	1	1	yes	characteristic
119	#533	—	118	1	no	—
120	C₂²×C₁₆	C₂×D₅₉	1	2	yes	—
121	#500	—	59	1	no	—
122	#500	—	59	1	no	—
123	#499	—	59	2	no	—
124	#499	—	118	1	no	—
125	#498	—	118	1	no	—
126	C₂×#108	—	59	2	no	—
127	#427	—	59	2	no	—
128	#427	—	118	1	no	—
129	C₄×M₁₆	—	118	1	no	—
130	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
131	C₄⋊C₁₆	—	2	1	no	—
132	C₄⋊C₁₆	—	2	1	no	—
133	#341	—	118	1	no	—
134	C₅₉	#13274	1	1	yes	characteristic, complemented, Sylow
135	C₂×M₁₆	—	118	2	no	—
136	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
137	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
138	C₄⋊Q₈	—	118	1	no	—
139	C₄⋊Q₈	—	118	1	no	—
140	#125	—	118	2	no	—
141	C₂²⋊Q₈	—	236	1	no	—
142	C₂²⋊Q₈	—	236	1	no	—
143	C₄×Q₈	—	236	1	no	—
144	C₄×Q₈	—	236	1	no	—
145	#117	—	118	2	no	—
146	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
147	C₂×C₁₆	—	2	4	no	—
148	C₂×C₁₆	—	2	2	no	—
149	#108	—	59	4	no	—
150	#108	—	118	2	no	—
151	#107	—	59	2	no	—
152	#107	—	59	2	no	—
153	#107	—	118	1	no	—
154	#107	—	118	1	no	—
155	#106	—	118	2	no	—
156	#106	—	118	2	no	—
157	#97	—	236	1	no	—
158	C₄×C₈	—	236	1	no	—
159	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
160	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
161	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
162	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
163	C₂×Q₈	—	236	1	no	—
164	C₂×Q₈	—	236	1	no	—
165	C₂²×C₄	#11745660	1	1	yes	characteristic
166	C₂²×C₄	#11745660	1	1	yes	characteristic
167	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
168	M₁₆	—	118	4	no	—
169	M₁₆	—	236	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	236	2	no	—
172	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
173	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
174	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
175	C₂×C₈	#11745687	1	2	yes	—
176	C₂×C₈	Q₈×D₅₉	1	2	yes	—
177	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
178	C₄⋊C₄	—	118	2	no	—
179	C₄⋊C₄	—	118	2	no	—
180	C₄⋊C₄	—	236	2	no	—
181	C₄⋊C₄	—	236	2	no	—
182	C₄⋊C₄	—	236	1	no	—
183	C₄⋊C₄	—	236	1	no	—
184	C₂²⋊C₄	—	236	2	no	—
185	C₄²	—	236	2	no	—
186	C₄²	D₄₇₂	1	1	yes	characteristic
187	C₄²	D₄₇₂	1	1	yes	characteristic
188	C₄²	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
189	C₄²	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
190	C₁₆	—	4	2	no	—
191	C₂³	C₂×#11745660	1	1	yes	characteristic
192	Q₈	—	472	2	no	—
193	Q₈	—	472	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	236	2	no	—
197	C₂×C₄	—	236	2	no	—
198	C₂×C₄	#14273649	1	1	yes	characteristic
199	C₂×C₄	#14273649	1	1	yes	characteristic
200	C₂×C₄	#14273649	1	1	yes	characteristic
201	C₂×C₄	#14273649	1	1	yes	characteristic
202	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
203	C₂×C₄	#14273707	1	2	yes	—
204	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
205	C₂×C₄	#14273760	1	2	yes	—
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	236	4	no	—
209	C₈	#14273692	1	2	yes	—
210	C₈	#14273714	1	1	yes	characteristic
211	C₈	#14273725	1	1	yes	characteristic
212	C₂²	—	2	1	no	—
213	C₂²	—	2	1	no	—
214	C₂²	C₂×#14273649	1	1	yes	characteristic
215	C₂²	#16837688	1	2	yes	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	472	2	no	—
220	C₄	—	472	2	no	—
221	C₄	#16837210	1	2	yes	—
222	C₄	#16837307	1	1	yes	characteristic
223	C₄	#16837412	1	1	yes	characteristic
224	C₄	#16837683	1	2	yes	—
225	C₂	—	2	2	no	—
226	C₂	#21425047	1	1	yes	characteristic
227	C₂	C₂×#16837210	1	1	yes	characteristic
228	C₂	#21425825	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄₇₂
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×C₄⋊C₁₆	C₂
socle	C₂²×C₅₉	C₂×#102
prime radical	C₂³×C₅₉	C₂×SD₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₅₉
hypocenter	C₅₉	#13274
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
30. #14273663	Q₈	semidirect
134. C₅₉	#13274	semidirect

Maximal subgroups

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

subgroup	index
#21434610	2
C₄×#14273663	2
#21422192	2
C₂×C₅₉×C₄⋊C₁₆	2
#13274	59