Subgroups — C₂×#21426245 #28743315

Order 15104.

574 conjugacy classes of subgroups.

The subgroup lattice has 244 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	#21426245	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16837227	C₂	1	2	yes	maximal
4	C₂×C₄⋊(C₅₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
5	C₂×#16837205	C₂	1	2	yes	maximal
6	C₂×C₅₉×#500	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₃₆	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₁₈	—	2	1	no	—
9	C₂²×C₅₉⋊C₁₆	C₂²	1	2	yes	—
10	C₄×Dic₂₃₆	—	2	2	no	—
11	C₄×Dic₂₃₆	C₂²	1	4	yes	—
12	C₂×#14273661	C₂²	1	1	yes	characteristic
13	C₂×#14273652	—	2	1	no	—
14	#16837227	C₂²	1	8	yes	—
15	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	4	yes	—
16	#16837205	C₂²	1	8	yes	—
17	C₂×C₄⋊Q₈×C₅₉	—	2	1	no	—
18	C₅₉×#500	—	2	2	no	—
19	C₅₉×#500	C₂²	1	4	yes	—
20	C₂×#107×C₅₉	—	2	1	no	—
21	C₂×#107×C₅₉	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₂²×Dic₁₁₈	—	2	1	no	—
24	C₂×Dic₂₃₆	—	2	1	no	—
25	C₂×Dic₂₃₆	—	2	2	no	—
26	C₂×Dic₂₃₆	Q₈	1	4	yes	complemented
27	C₂×Dic₂₃₆	C₂³	1	2	yes	—
28	C₄×Dic₁₁₈	—	2	4	no	—
29	C₄×Dic₁₁₈	—	4	2	no	—
30	C₂×C₄⋊Dic₅₉	—	2	1	no	—
31	C₂×Dic₅₉⋊C₄	—	4	1	no	—
32	C₂×C₄×Dic₅₉	—	4	1	no	—
33	C₂×C₅₉⋊C₁₆	—	2	4	no	—
34	C₂×C₅₉⋊C₁₆	C₂³	1	4	yes	—
35	#14273661	—	2	1	no	—
36	#14273661	C₂³	1	2	yes	—
37	#14273652	—	2	4	no	—
38	C₂²×Q₈×C₅₉	—	4	1	no	—
39	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
41	C₄⋊Q₈×C₅₉	—	2	4	no	—
42	C₄⋊Q₈×C₅₉	—	4	2	no	—
43	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
44	C₂×C₄⋊C₄×C₅₉	—	4	1	no	—
45	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
46	#107×C₅₉	—	2	4	no	—
47	#107×C₅₉	—	2	2	no	—
48	#107×C₅₉	C₂³	1	4	yes	—
49	C₄×C₈×C₅₉	D₄	1	2	yes	—
50	C₄×C₈×C₅₉	C₂³	1	2	yes	—
51	C₂×Q₈×C₅₉	—	4	4	no	—
52	C₂×Q₈×C₅₉	—	8	4	no	—
53	C₂²×C₄×C₅₉	—	4	2	no	—
54	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₅₉	—	4	1	no	—
58	C₂²×Dic₅₉	—	4	1	no	—
59	C₂×Dic₁₁₈	—	2	4	no	—
60	C₂×Dic₁₁₈	—	2	2	no	—
61	C₂×Dic₁₁₈	—	4	1	no	—
62	C₂×Dic₁₁₈	—	4	2	no	—
63	C₂×C₈×C₅₉	—	2	2	no	—
64	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
65	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
67	C₂×C₈×C₅₉	#41	1	2	yes	—
68	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
69	C₄⋊C₄×C₅₉	—	2	4	no	—
70	C₄⋊C₄×C₅₉	—	4	2	no	—
71	C₄⋊C₄×C₅₉	—	4	4	no	—
72	C₄²×C₅₉	D₈	1	2	yes	—
73	C₄²×C₅₉	C₂×D₄	1	2	yes	—
74	C₄⋊Dic₅₉	—	2	2	no	—
75	C₄⋊Dic₅₉	—	4	1	no	—
76	Dic₅₉⋊C₄	—	4	4	no	—
77	C₄×Dic₅₉	—	4	4	no	—
78	Dic₂₃₆	—	2	4	no	—
79	Dic₂₃₆	—	2	2	no	—
80	Dic₂₃₆	C₂×Q₈	1	4	yes	complemented
81	C₅₉⋊C₁₆	—	2	4	no	—
82	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
83	Q₈×C₅₉	—	8	8	no	—
84	C₂×C₄×C₅₉	—	2	2	no	—
85	C₂×C₄×C₅₉	—	4	4	no	—
86	C₂×C₄×C₅₉	—	8	4	no	—
87	C₂×C₄×C₅₉	Q₃₂	1	4	yes	—
88	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
89	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
90	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
92	C₂×C₄×C₅₉	#137	1	2	yes	—
93	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
94	C₂×Dic₅₉	—	4	2	no	—
95	C₂×Dic₅₉	—	4	4	no	—
96	C₂×Dic₅₉	—	4	2	no	—
97	C₂×Dic₅₉	—	8	2	no	—
98	Dic₁₁₈	—	2	4	no	—
99	Dic₁₁₈	—	4	2	no	—
100	Dic₁₁₈	—	4	4	no	—
101	C₈×C₅₉	—	2	2	no	—
102	C₈×C₅₉	C₂²⋊Q₈	1	2	yes	—
103	C₈×C₅₉	C₂²×Q₈	1	1	yes	characteristic
104	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
105	C₂×#2227	—	59	1	no	maximal, Sylow
106	C₂²×C₅₉	#474	1	2	yes	—
107	C₂²×C₅₉	C₂×Q₃₂	1	2	yes	—
108	C₂²×C₅₉	#509	1	2	yes	—
109	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
110	C₄×C₅₉	—	2	2	no	—
111	C₄×C₅₉	—	8	4	no	—
112	C₄×C₅₉	#474	1	2	yes	—
113	C₄×C₅₉	C₂×Q₃₂	1	4	yes	—
114	C₄×C₅₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
115	C₄×C₅₉	C₂×#137	1	1	yes	characteristic
116	Dic₅₉	—	4	4	no	—
117	Dic₅₉	—	8	2	no	—
118	C₂×#500	D₅₉	1	1	yes	characteristic
119	C₂×C₄×Q₁₆	—	59	1	no	—
120	#2227	—	59	8	no	—
121	C₂×#366	—	59	2	no	—
122	C₂×C₄⋊C₁₆	—	59	1	no	—
123	C₂×#358	—	59	2	no	—
124	C₂×C₅₉	#2227	1	4	yes	complemented
125	C₂×C₅₉	C₂×#474	1	1	yes	characteristic
126	C₂×C₅₉	C₂²×Q₃₂	1	1	yes	characteristic
127	C₂×C₅₉	C₂×#509	1	1	yes	characteristic
128	C₂²×Q₁₆	—	59	1	no	—
129	C₂×C₄⋊Q₈	—	2	1	no	—
130	C₂×C₄×Q₈	—	118	1	no	—
131	C₂²×C₁₆	—	59	2	no	—
132	#500	—	2	2	no	—
133	#500	C₂×D₅₉	1	4	yes	—
134	C₄×Q₁₆	—	59	4	no	—
135	C₄×Q₁₆	—	118	2	no	—
136	C₂×#107	—	2	1	no	—
137	C₂×#107	—	59	1	no	—
138	C₂×#107	C₂×D₅₉	1	2	yes	—
139	C₂×#103	—	118	1	no	—
140	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
141	#366	—	59	8	no	—
142	C₄⋊C₁₆	—	59	4	no	—
143	#358	—	59	8	no	—
144	C₅₉	C₂×#2227	1	1	yes	characteristic, complemented, Sylow
145	C₂²×Q₈	—	4	1	no	—
146	C₂²×Q₈	—	118	1	no	—
147	C₂×Q₁₆	—	59	4	no	—
148	C₂×Q₁₆	—	59	2	no	—
149	C₂×Q₁₆	—	118	2	no	—
150	C₂×Q₁₆	—	118	1	no	—
151	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
152	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
153	C₄⋊Q₈	—	2	4	no	—
154	C₄⋊Q₈	—	4	2	no	—
155	C₄×Q₈	—	118	4	no	—
156	C₄×Q₈	—	236	2	no	—
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	—	4	1	no	—
159	C₂×C₄⋊C₄	—	118	1	no	—
160	C₂×C₄⋊C₄	—	236	1	no	—
161	C₂×C₄²	—	236	1	no	—
162	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
163	C₂×C₁₆	—	59	4	no	—
164	C₂×C₁₆	—	118	4	no	—
165	#107	—	2	4	no	—
166	#107	—	2	2	no	—
167	#107	—	59	2	no	—
168	#107	—	118	1	no	—
169	#107	C₂²×D₅₉	1	4	yes	—
170	#103	—	118	4	no	—
171	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
172	C₄×C₈	C₂²×D₅₉	1	2	yes	—
173	C₂×Q₈	—	4	4	no	—
174	C₂×Q₈	—	8	4	no	—
175	C₂×Q₈	—	118	4	no	—
176	C₂×Q₈	—	118	2	no	—
177	C₂×Q₈	—	236	1	no	—
178	C₂×Q₈	—	236	2	no	—
179	C₂²×C₄	—	4	2	no	—
180	C₂²×C₄	—	236	1	no	—
181	C₂²×C₄	—	236	1	no	—
182	C₂²×C₄	#11745669	1	1	yes	characteristic
183	C₂²×C₄	#11745669	1	1	yes	characteristic
184	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
185	Q₁₆	—	59	4	no	—
186	Q₁₆	—	118	4	no	—
187	Q₁₆	—	118	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	Q₈×D₅₉	1	2	yes	—
190	C₂×C₈	#11745689	1	2	yes	—
191	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
192	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
193	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	4	no	—
197	C₄⋊C₄	—	118	2	no	—
198	C₄⋊C₄	—	236	1	no	—
199	C₄⋊C₄	—	236	4	no	—
200	C₄²	—	236	4	no	—
201	C₄²	#11745669	1	2	yes	—
202	C₄²	C₂×C₂²⋊D₅₉	1	2	yes	—
203	C₁₆	—	118	4	no	—
204	C₂³	C₂×#11745669	1	1	yes	characteristic
205	Q₈	—	8	8	no	—
206	Q₈	—	118	4	no	—
207	Q₈	—	236	4	no	—
208	Q₈	—	236	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	8	4	no	—
212	C₂×C₄	—	472	2	no	—
213	C₂×C₄	—	236	2	no	—
214	C₂×C₄	—	236	2	no	—
215	C₂×C₄	—	236	4	no	—
216	C₂×C₄	#14273673	1	4	yes	—
217	C₂×C₄	#14273703	1	2	yes	—
218	C₂×C₄	C₂×#11745669	1	2	yes	—
219	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
220	C₂×C₄	C₂×#11745669	1	2	yes	—
221	C₂×C₄	#14273758	1	2	yes	—
222	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	#14273703	1	2	yes	—
225	C₈	C₂×Q₈×D₅₉	1	1	yes	characteristic
226	C₈	C₂×#11745689	1	1	yes	characteristic
227	C₂²	#16837679	1	2	yes	—
228	C₂²	C₂×#14273673	1	2	yes	—
229	C₂²	#16837838	1	2	yes	—
230	C₂²	C₂²×#11745669	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	8	4	no	—
233	C₄	—	472	2	no	—
234	C₄	—	236	4	no	—
235	C₄	#16837679	1	2	yes	—
236	C₄	C₂×#14273673	1	4	yes	—
237	C₄	C₂×#14273703	1	1	yes	characteristic
238	C₄	C₂×#14273758	1	1	yes	characteristic
239	C₂	#21426245	1	4	yes	direct product factor
240	C₂	C₂×#16837679	1	1	yes	characteristic
241	C₂	C₂²×#14273673	1	1	yes	characteristic
242	C₂	C₂×#16837838	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, prime radical	C₂³×C₅₉	C₂×D₈
hypercenter	C₂×#500	D₅₉
hypocenter	C₅₉	C₂×#2227
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. #21426245	C₂	direct (N × H)
26. C₂×Dic₂₃₆	Q₈	semidirect
80. Dic₂₃₆	C₂×Q₈	semidirect
124. C₂×C₅₉	#2227	semidirect
144. C₅₉	C₂×#2227	semidirect
239. C₂	#21426245	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×C₄×Dic₂₃₆	2
#21426245	2
C₂×#16837227	2
C₂×C₄⋊(C₅₉⋊C₁₆)	2
C₂×#16837205	2
C₂×C₅₉×#500	2
C₂×#2227	59

Subgroups — C2×#21426245 #28743315

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21426245 #28743315