Subgroups — C₂×Q₈⋊(C₅₉⋊C₁₆) #28498488

Order 15104.

382 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₅₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊(C₅₉⋊C₁₆)	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₈×Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₅₉⋊C₁₆	—	2	1	no	—
6	C₂²×C₅₉⋊C₁₆	—	2	1	no	—
7	C₄⋊(C₅₉⋊C₁₆)	—	2	1	no	—
8	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	—
9	C₄×C₅₉⋊C₁₆	—	2	1	no	—
10	C₄×C₅₉⋊C₁₆	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₅₉	C₄	1	1	yes	characteristic
12	C₈×Q₈×C₅₉	—	2	1	no	—
13	C₈×Q₈×C₅₉	—	2	1	no	—
14	C₈×Q₈×C₅₉	C₄	1	2	yes	—
15	C₈×Q₈×C₅₉	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₅₉	—	2	1	no	—
17	C₂×C₄⋊C₈×C₅₉	C₄	1	1	yes	characteristic
18	C₂×C₄×C₈×C₅₉	—	2	1	no	—
19	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×C₅₉⋊C₁₆	—	2	4	no	—
21	C₂×C₅₉⋊C₁₆	—	2	2	no	—
22	C₂×C₅₉⋊C₁₆	—	2	2	no	—
23	C₂×C₅₉⋊C₁₆	—	4	2	no	—
24	C₂²×Q₈×C₅₉	C₈	1	1	yes	characteristic
25	C₂²×C₈×C₅₉	—	2	1	no	—
26	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
28	C₄×Q₈×C₅₉	—	2	1	no	—
29	C₄×Q₈×C₅₉	—	2	1	no	—
30	C₄×Q₈×C₅₉	C₈	1	2	yes	—
31	C₄×Q₈×C₅₉	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
33	C₂×C₄⋊C₄×C₅₉	C₈	1	1	yes	characteristic
34	C₂×C₄²×C₅₉	—	2	1	no	—
35	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
36	C₄⋊C₈×C₅₉	—	2	2	no	—
37	C₄⋊C₈×C₅₉	—	2	2	no	—
38	C₄⋊C₈×C₅₉	—	2	1	no	—
39	C₄⋊C₈×C₅₉	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₅₉	—	2	2	no	—
41	C₄×C₈×C₅₉	—	2	2	no	—
42	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
44	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
45	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
46	C₂×Q₈×C₅₉	—	2	1	no	—
47	C₂×Q₈×C₅₉	—	2	2	no	—
48	C₂×Q₈×C₅₉	C₁₆	1	4	yes	complemented
49	C₂×Q₈×C₅₉	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₅₉	—	2	1	no	—
51	C₂²×C₄×C₅₉	—	2	1	no	—
52	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₉	M₁₆	1	1	yes	characteristic
55	C₂×C₈×C₅₉	—	2	2	no	—
56	C₂×C₈×C₅₉	—	2	2	no	—
57	C₂×C₈×C₅₉	—	4	2	no	—
58	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
61	C₂×C₈×C₅₉	Q₁₆	1	2	yes	—
62	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
64	C₄⋊C₄×C₅₉	—	2	2	no	—
65	C₄⋊C₄×C₅₉	—	2	1	no	—
66	C₄⋊C₄×C₅₉	—	2	2	no	—
67	C₄⋊C₄×C₅₉	C₂×C₈	1	2	yes	—
68	C₄²×C₅₉	—	2	2	no	—
69	C₄²×C₅₉	—	2	2	no	—
70	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
72	C₄²×C₅₉	M₁₆	1	1	yes	characteristic
73	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
74	C₅₉⋊C₁₆	—	2	4	no	—
75	C₅₉⋊C₁₆	—	4	2	no	—
76	C₂³×C₅₉	C₂²⋊C₈	1	1	yes	characteristic
77	Q₈×C₅₉	—	2	4	no	—
78	Q₈×C₅₉	—	2	2	no	—
79	Q₈×C₅₉	C₂×C₁₆	1	4	yes	complemented
80	C₂×C₄×C₅₉	—	2	4	no	—
81	C₂×C₄×C₅₉	—	2	2	no	—
82	C₂×C₄×C₅₉	—	2	2	no	—
83	C₂×C₄×C₅₉	—	2	2	no	—
84	C₂×C₄×C₅₉	—	4	2	no	—
85	C₂×C₄×C₅₉	C₂²⋊C₈	1	1	yes	characteristic
86	C₂×C₄×C₅₉	C₂²⋊C₈	1	1	yes	characteristic
87	C₂×C₄×C₅₉	C₂²⋊C₈	1	1	yes	characteristic
88	C₂×C₄×C₅₉	#103	1	2	yes	—
89	C₂×C₄×C₅₉	#104	1	2	yes	—
90	C₂×C₄×C₅₉	C₂×C₁₆	1	2	yes	—
91	C₂×C₄×C₅₉	M₃₂	1	2	yes	—
92	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
95	C₈×C₅₉	—	2	1	no	—
96	C₈×C₅₉	—	2	1	no	—
97	C₈×C₅₉	—	4	2	no	—
98	C₈×C₅₉	#103	1	2	yes	—
99	C₈×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
100	C₈×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
101	C₂×Q₈⋊C₁₆	—	59	1	no	maximal, Sylow
102	C₂²×C₅₉	Q₈⋊C₈	1	2	yes	—
103	C₂²×C₅₉	C₂²⋊C₁₆	1	2	yes	—
104	C₂²×C₅₉	#350	1	2	yes	—
105	C₂²×C₅₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
106	C₄×C₅₉	—	2	4	no	—
107	C₄×C₅₉	—	2	1	no	—
108	C₄×C₅₉	—	2	1	no	—
109	C₄×C₅₉	—	4	2	no	—
110	C₄×C₅₉	Q₈⋊C₈	1	2	yes	—
111	C₄×C₅₉	C₂²⋊C₁₆	1	2	yes	—
112	C₄×C₅₉	C₂×#103	1	1	yes	characteristic
113	C₄×C₅₉	C₂×#104	1	1	yes	characteristic
114	C₄×C₅₉	C₂²×C₁₆	1	1	yes	characteristic
115	C₄×C₅₉	C₂×M₃₂	1	1	yes	characteristic
116	C₂×C₈×Q₈	D₅₉	1	1	yes	characteristic
117	C₂×C₄⋊C₁₆	—	59	1	no	—
118	C₂×C₄×C₁₆	—	59	1	no	—
119	Q₈⋊C₁₆	—	59	4	no	—
120	C₂×C₅₉	Q₈⋊C₁₆	1	4	yes	complemented
121	C₂×C₅₉	C₂×Q₈⋊C₈	1	1	yes	characteristic
122	C₂×C₅₉	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
123	C₂×C₅₉	C₂×#350	1	1	yes	characteristic
124	C₂×C₄×Q₈	Dic₅₉	1	1	yes	characteristic
125	C₂²×C₁₆	—	118	1	no	—
126	C₂²×C₁₆	—	118	1	no	—
127	C₈×Q₈	—	2	1	no	—
128	C₈×Q₈	—	2	1	no	—
129	C₈×Q₈	Dic₅₉	1	2	yes	—
130	C₈×Q₈	C₂×D₅₉	1	2	yes	—
131	C₂×C₄⋊C₈	—	2	1	no	—
132	C₂×C₄⋊C₈	Dic₅₉	1	1	yes	characteristic
133	C₂×C₄×C₈	—	2	1	no	—
134	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
135	C₄⋊C₁₆	—	59	2	no	—
136	C₄⋊C₁₆	—	118	1	no	—
137	C₄×C₁₆	—	59	2	no	—
138	C₄×C₁₆	—	118	1	no	—
139	C₅₉	C₂×Q₈⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
140	C₂²×Q₈	C₅₉⋊C₈	1	1	yes	characteristic
141	C₂²×C₈	—	2	1	no	—
142	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
143	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
144	C₄×Q₈	—	2	1	no	—
145	C₄×Q₈	—	2	1	no	—
146	C₄×Q₈	C₅₉⋊C₈	1	2	yes	—
147	C₄×Q₈	C₂×Dic₅₉	1	2	yes	—
148	C₂×C₄⋊C₄	—	2	1	no	—
149	C₂×C₄⋊C₄	C₅₉⋊C₈	1	1	yes	characteristic
150	C₂×C₄²	—	2	1	no	—
151	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
152	C₂×C₁₆	—	118	4	no	—
153	C₂×C₁₆	—	118	2	no	—
154	C₂×C₁₆	—	118	2	no	—
155	C₂×C₁₆	—	236	2	no	—
156	C₄⋊C₈	—	2	2	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	2	1	no	—
159	C₄⋊C₈	C₂×Dic₅₉	1	2	yes	—
160	C₄×C₈	—	2	2	no	—
161	C₄×C₈	—	2	2	no	—
162	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
163	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
164	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
165	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
166	C₂×Q₈	—	2	1	no	—
167	C₂×Q₈	—	2	2	no	—
168	C₂×Q₈	C₅₉⋊C₁₆	1	4	yes	complemented
169	C₂×Q₈	C₂×C₅₉⋊C₈	1	2	yes	—
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	C₂×C₅₉⋊C₈	1	1	yes	characteristic
173	C₂²×C₄	#11745664	1	1	yes	characteristic
174	C₂²×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	4	2	no	—
178	C₂×C₈	#11745671	1	2	yes	—
179	C₂×C₈	#11745672	1	2	yes	—
180	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
181	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
182	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
183	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	2	1	no	—
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	C₂×C₅₉⋊C₈	1	2	yes	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	2	no	—
190	C₄²	C₂×C₅₉⋊C₈	1	1	yes	characteristic
191	C₄²	#11745664	1	1	yes	characteristic
192	C₄²	C₂²⋊Dic₅₉	1	1	yes	characteristic
193	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
194	C₁₆	—	118	4	no	—
195	C₁₆	—	236	2	no	—
196	C₂³	C₂²⋊(C₅₉⋊C₈)	1	1	yes	characteristic
197	Q₈	—	2	4	no	—
198	Q₈	—	2	2	no	—
199	Q₈	C₂×C₅₉⋊C₁₆	1	4	yes	complemented
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	C₂²⋊(C₅₉⋊C₈)	1	1	yes	characteristic
206	C₂×C₄	C₂²⋊(C₅₉⋊C₈)	1	1	yes	characteristic
207	C₂×C₄	C₂²⋊(C₅₉⋊C₈)	1	1	yes	characteristic
208	C₂×C₄	#14273651	1	2	yes	—
209	C₂×C₄	#14273654	1	2	yes	—
210	C₂×C₄	C₂×C₅₉⋊C₁₆	1	2	yes	—
211	C₂×C₄	#14273667	1	2	yes	—
212	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
213	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
214	C₂×C₄	C₂×#11745664	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	—	2	1	no	—
217	C₈	—	4	2	no	—
218	C₈	#14273651	1	2	yes	—
219	C₈	C₂×#11745671	1	1	yes	characteristic
220	C₈	C₂×#11745672	1	1	yes	characteristic
221	C₂²	Q₈⋊(C₅₉⋊C₈)	1	2	yes	—
222	C₂²	C₂²⋊(C₅₉⋊C₁₆)	1	2	yes	—
223	C₂²	#16837186	1	2	yes	—
224	C₂²	C₂×C₂²⋊(C₅₉⋊C₈)	1	1	yes	characteristic
225	C₄	—	2	4	no	—
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	4	2	no	—
229	C₄	Q₈⋊(C₅₉⋊C₈)	1	2	yes	—
230	C₄	C₂²⋊(C₅₉⋊C₁₆)	1	2	yes	—
231	C₄	C₂×#14273651	1	1	yes	characteristic
232	C₄	C₂×#14273654	1	1	yes	characteristic
233	C₄	C₂²×C₅₉⋊C₁₆	1	1	yes	characteristic
234	C₄	C₂×#14273667	1	1	yes	characteristic
235	C₂	Q₈⋊(C₅₉⋊C₁₆)	1	4	yes	direct product factor
236	C₂	C₂×Q₈⋊(C₅₉⋊C₈)	1	1	yes	characteristic
237	C₂	C₂×C₂²⋊(C₅₉⋊C₁₆)	1	1	yes	characteristic
238	C₂	C₂×#16837186	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²⋊D₅₉
commutator	C₄×C₅₉	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₂×C₈×Q₈×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	C₂²⋊C₈
cosocle	C₄×C₈×C₅₉	C₂³
hypercenter	C₂×C₈×Q₈	D₅₉
hypocenter	C₅₉	C₂×Q₈⋊C₁₆
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
3. Q₈⋊(C₅₉⋊C₁₆)	C₂	direct (N × H)
48. C₂×Q₈×C₅₉	C₁₆	semidirect
79. Q₈×C₅₉	C₂×C₁₆	semidirect
120. C₂×C₅₉	Q₈⋊C₁₆	semidirect
139. C₅₉	C₂×Q₈⋊C₁₆	semidirect
168. C₂×Q₈	C₅₉⋊C₁₆	semidirect
199. Q₈	C₂×C₅₉⋊C₁₆	semidirect
235. C₂	Q₈⋊(C₅₉⋊C₁₆)	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⋊(C₅₉⋊C₁₆)	2
C₂×C₄×C₅₉⋊C₁₆	2
Q₈⋊(C₅₉⋊C₁₆)	2
C₂×C₈×Q₈×C₅₉	2
C₂×Q₈⋊C₁₆	59

Subgroups — C2×Q8⋊(C59⋊C16) #28498488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈⋊(C₅₉⋊C₁₆) #28498488