Subgroups — C₂×#21426321 #28743974

Order 15104.

726 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×#16837779	C₂	1	1	yes	maximal, characteristic
2	#21426321	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16837202	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅₉×#500	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄₇₂	—	2	1	no	—
7	C₂²×D₄₇₂	C₂²	1	2	yes	—
8	C₂×#14273724	—	2	1	no	—
9	C₂²×C₅₉⋊C₁₆	C₂²	1	2	yes	—
10	#16837779	—	2	2	no	—
11	#16837779	C₂²	1	4	yes	—
12	#16837202	C₂²	1	16	yes	complemented
13	C₄×C₅₉⋊C₁₆	C₂²	1	4	yes	complemented
14	C₂×C₄⋊Q₈×C₅₉	—	2	1	no	—
15	C₅₉×#500	—	2	2	no	—
16	C₅₉×#500	C₂²	1	4	yes	complemented
17	C₂×#107×C₅₉	—	2	1	no	—
18	C₂×#107×C₅₉	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₂²×D₂₃₆	—	2	2	no	—
21	C₂²×D₂₃₆	—	4	1	no	—
22	C₂²×D₂₃₆	—	4	1	no	—
23	C₂×D₄₇₂	—	2	8	no	—
24	C₂×D₄₇₂	—	2	4	no	—
25	C₂×D₄₇₂	—	2	2	no	—
26	C₂×D₄₇₂	—	4	4	no	—
27	C₂×D₄₇₂	C₂³	1	4	yes	—
28	#14273724	—	2	4	no	—
29	#14273724	—	4	2	no	—
30	C₂×C₅₉⋊C₁₆	—	2	4	no	—
31	C₂×C₅₉⋊C₁₆	C₂³	1	4	yes	—
32	C₂²×Q₈×C₅₉	—	4	1	no	—
33	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
35	C₄⋊Q₈×C₅₉	—	2	4	no	—
36	C₄⋊Q₈×C₅₉	—	4	2	no	—
37	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
38	C₂×C₄⋊C₄×C₅₉	—	4	1	no	—
39	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
40	#107×C₅₉	—	2	4	no	—
41	#107×C₅₉	—	2	2	no	—
42	#107×C₅₉	C₂³	1	4	yes	—
43	C₄×C₈×C₅₉	D₄	1	2	yes	—
44	C₄×C₈×C₅₉	C₂³	1	2	yes	—
45	C₂³×D₅₉	—	4	2	no	—
46	C₂×Q₈×C₅₉	—	4	4	no	—
47	C₂×Q₈×C₅₉	—	8	4	no	—
48	C₂²×C₄×C₅₉	—	4	2	no	—
49	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
52	C₂×D₂₃₆	—	2	4	no	—
53	C₂×D₂₃₆	—	4	4	no	—
54	C₂×D₂₃₆	—	4	4	no	—
55	C₂×D₂₃₆	—	4	4	no	—
56	C₂×D₂₃₆	—	4	4	no	—
57	C₂×D₂₃₆	—	4	2	no	—
58	C₂×D₂₃₆	—	8	4	no	—
59	C₂×D₂₃₆	—	8	4	no	—
60	C₂×C₈×C₅₉	SD₁₆	1	4	yes	—
61	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
63	C₂×C₈×C₅₉	#41	1	4	yes	—
64	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
65	C₄⋊C₄×C₅₉	—	2	4	no	—
66	C₄⋊C₄×C₅₉	—	4	2	no	—
67	C₄⋊C₄×C₅₉	—	4	4	no	—
68	C₄²×C₅₉	D₈	1	2	yes	—
69	C₄²×C₅₉	C₂×D₄	1	2	yes	—
70	D₄₇₂	—	2	8	no	—
71	D₄₇₂	—	4	4	no	—
72	D₄₇₂	—	4	8	no	—
73	C₅₉⋊C₁₆	—	2	4	no	—
74	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
75	C₂²×D₅₉	—	4	4	no	—
76	C₂²×D₅₉	—	8	4	no	—
77	C₂²×D₅₉	—	8	4	no	—
78	C₂²×D₅₉	—	8	4	no	—
79	Q₈×C₅₉	—	8	8	no	—
80	C₂×C₄×C₅₉	—	4	4	no	—
81	C₂×C₄×C₅₉	—	8	4	no	—
82	C₂×C₄×C₅₉	D₁₆	1	4	yes	—
83	C₂×C₄×C₅₉	SD₃₂	1	4	yes	—
84	C₂×C₄×C₅₉	#124	1	2	yes	—
85	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
87	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
88	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
89	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
90	D₂₃₆	—	4	4	no	—
91	D₂₃₆	—	4	4	no	—
92	D₂₃₆	—	8	4	no	—
93	D₂₃₆	—	8	8	no	—
94	D₂₃₆	—	8	8	no	—
95	C₈×C₅₉	#124	1	2	yes	—
96	C₈×C₅₉	C₂×SD₁₆	1	4	yes	—
97	C₈×C₅₉	C₂×#41	1	2	yes	—
98	C₂×#2240	—	59	1	no	maximal, Sylow
99	C₂²×C₅₉	#486	1	2	yes	—
100	C₂²×C₅₉	C₂×D₁₆	1	2	yes	—
101	C₂²×C₅₉	C₂×SD₃₂	1	2	yes	—
102	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
103	C₂×D₅₉	—	8	4	no	—
104	C₂×D₅₉	—	8	4	no	—
105	C₂×D₅₉	—	8	4	no	—
106	C₂×D₅₉	—	16	8	no	—
107	C₄×C₅₉	—	8	4	no	—
108	C₄×C₅₉	#486	1	2	yes	—
109	C₄×C₅₉	C₂×D₁₆	1	4	yes	—
110	C₄×C₅₉	C₂×SD₃₂	1	4	yes	—
111	C₄×C₅₉	C₂×#124	1	1	yes	characteristic
112	C₄×C₅₉	C₂²×SD₁₆	1	1	yes	characteristic
113	C₂×#500	D₅₉	1	1	yes	characteristic, complemented
114	C₂×#493	—	59	1	no	—
115	#2240	—	59	8	no	—
116	C₂×#357	—	59	4	no	—
117	C₂×C₄×C₁₆	—	59	1	no	—
118	C₂×C₅₉	#2240	1	4	yes	complemented
119	C₂×C₅₉	C₂×#486	1	1	yes	characteristic
120	C₂×C₅₉	C₂²×D₁₆	1	1	yes	characteristic
121	C₂×C₅₉	C₂²×SD₃₂	1	1	yes	characteristic
122	D₅₉	—	16	4	no	—
123	C₂²×D₈	—	59	2	no	—
124	C₂²×D₈	—	118	1	no	—
125	C₂×C₄⋊Q₈	—	2	1	no	—
126	C₂×#127	—	118	1	no	—
127	C₂²×C₁₆	—	59	2	no	—
128	#500	—	2	2	no	—
129	#500	C₂×D₅₉	1	4	yes	complemented
130	#493	—	59	4	no	—
131	#493	—	118	2	no	—
132	C₂×#107	—	2	1	no	—
133	C₂×#107	C₂×D₅₉	1	2	yes	—
134	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
135	#357	—	59	16	no	—
136	C₄×C₁₆	—	59	4	no	—
137	C₅₉	C₂×#2240	1	1	yes	characteristic, complemented, Sylow
138	C₂²×Q₈	—	4	1	no	—
139	C₂²×D₄	—	118	2	no	—
140	C₂²×D₄	—	236	1	no	—
141	C₂²×D₄	—	236	1	no	—
142	C₂×D₈	—	59	4	no	—
143	C₂×D₈	—	118	8	no	—
144	C₂×D₈	—	118	2	no	—
145	C₂×D₈	—	118	4	no	—
146	C₂×D₈	—	236	4	no	—
147	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
148	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
149	C₄⋊Q₈	—	2	4	no	—
150	C₄⋊Q₈	—	4	2	no	—
151	#127	—	118	4	no	—
152	#127	—	236	2	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	—	4	1	no	—
155	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
156	C₂×C₁₆	—	59	4	no	—
157	C₂×C₁₆	—	118	4	no	—
158	#107	—	2	4	no	—
159	#107	—	2	2	no	—
160	#107	C₂²×D₅₉	1	4	yes	—
161	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
162	C₄×C₈	C₂²×D₅₉	1	2	yes	—
163	C₂⁴	—	236	2	no	—
164	C₂×Q₈	—	4	4	no	—
165	C₂×Q₈	—	8	4	no	—
166	C₂×D₄	—	118	4	no	—
167	C₂×D₄	—	472	4	no	—
168	C₂×D₄	—	472	4	no	—
169	C₂×D₄	—	236	4	no	—
170	C₂×D₄	—	236	4	no	—
171	C₂×D₄	—	236	4	no	—
172	C₂×D₄	—	236	2	no	—
173	C₂×D₄	—	236	4	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	#11745669	1	1	yes	characteristic
176	C₂²×C₄	#11745669	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
178	D₈	—	118	8	no	—
179	D₈	—	236	4	no	—
180	D₈	—	236	8	no	—
181	C₂×C₈	#11745671	1	4	yes	—
182	C₂×C₈	#11745689	1	4	yes	—
183	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
184	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
185	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	—	4	4	no	—
189	C₄²	#11745669	1	2	yes	—
190	C₄²	C₂×C₂²⋊D₅₉	1	2	yes	—
191	C₁₆	—	118	4	no	—
192	C₂³	—	472	4	no	—
193	C₂³	—	472	4	no	—
194	C₂³	—	472	4	no	—
195	C₂³	—	236	4	no	—
196	C₂³	C₂×#11745669	1	1	yes	characteristic
197	Q₈	—	8	8	no	—
198	D₄	—	472	4	no	—
199	D₄	—	472	8	no	—
200	D₄	—	472	8	no	—
201	D₄	—	236	4	no	—
202	D₄	—	236	4	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	—	8	4	no	—
205	C₂×C₄	#14273668	1	4	yes	—
206	C₂×C₄	#14273670	1	4	yes	—
207	C₂×C₄	#14273716	1	2	yes	—
208	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
209	C₂×C₄	C₂×#11745669	1	2	yes	—
210	C₂×C₄	C₂×#11745669	1	2	yes	—
211	C₂×C₄	C₂×#11745671	1	2	yes	—
212	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
213	C₈	#14273716	1	2	yes	—
214	C₈	C₂×#11745671	1	4	yes	—
215	C₈	C₂×#11745689	1	2	yes	—
216	C₂²	—	944	8	no	—
217	C₂²	—	472	4	no	—
218	C₂²	—	472	4	no	—
219	C₂²	—	472	4	no	—
220	C₂²	#16837750	1	2	yes	—
221	C₂²	C₂×#14273668	1	2	yes	—
222	C₂²	C₂×#14273670	1	2	yes	—
223	C₂²	C₂²×#11745669	1	1	yes	characteristic
224	C₄	—	8	4	no	—
225	C₄	#16837750	1	2	yes	—
226	C₄	C₂×#14273668	1	4	yes	—
227	C₄	C₂×#14273670	1	4	yes	—
228	C₄	C₂×#14273716	1	1	yes	characteristic
229	C₄	C₂²×#11745671	1	1	yes	characteristic
230	C₂	—	944	4	no	—
231	C₂	#21426321	1	4	yes	direct product factor
232	C₂	C₂×#16837750	1	1	yes	characteristic
233	C₂	C₂²×#14273668	1	1	yes	characteristic
234	C₂	C₂²×#14273670	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11745669
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂×C₅₉×#500	C₂
socle	C₂³×C₅₉	C₂×D₈
prime radical	C₂×#16837779	C₂
hypercenter	C₂×#500	D₅₉
hypocenter	C₅₉	C₂×#2240
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
2. #21426321	C₂	direct (N × H)
3. C₂×#16837202	C₂	semidirect
4. C₂×C₄×C₅₉⋊C₁₆	C₂	semidirect
5. C₂×C₅₉×#500	C₂	semidirect
12. #16837202	C₂²	semidirect
13. C₄×C₅₉⋊C₁₆	C₂²	semidirect
16. C₅₉×#500	C₂²	semidirect
113. C₂×#500	D₅₉	semidirect
118. C₂×C₅₉	#2240	semidirect
129. #500	C₂×D₅₉	semidirect
137. C₅₉	C₂×#2240	semidirect
231. C₂	#21426321	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16837779	2
#21426321	2
C₂×#16837202	2
C₂×C₄×C₅₉⋊C₁₆	2
C₂×C₅₉×#500	2
C₂×#2240	59

Subgroups — C2×#21426321 #28743974

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21426321 #28743974