Subgroups — Q₈×#14273646 #28605264

Order 15104.

970 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	Q₈×#11745664	C₂	1	2	yes	maximal
2	#21425086	C₂	1	6	yes	maximal
3	#21424092	C₂	1	3	yes	maximal
4	C₄×#14273646	C₂	1	3	yes	maximal
5	C₂×Q₈²×C₅₉	C₂	1	1	yes	maximal, characteristic
6	#16837533	—	2	6	no	—
7	Q₈×C₅₉⋊C₈	—	2	2	no	—
8	#16837426	C₂²	1	6	yes	—
9	C₂×#14273646	C₂²	1	3	yes	—
10	C₂×#14273646	C₂²	1	1	yes	characteristic
11	C₄×#11745664	C₂²	1	6	yes	—
12	#16837171	C₂²	1	6	yes	—
13	#16837171	C₂²	1	6	yes	—
14	Q₈²×C₅₉	—	2	2	no	—
15	Q₈²×C₅₉	—	2	6	no	—
16	C₂×C₄⋊Q₈×C₅₉	C₄	1	6	yes	—
17	C₂×C₄⋊Q₈×C₅₉	C₂²	1	3	yes	—
18	C₂×C₄×Q₈×C₅₉	C₄	1	2	yes	—
19	C₂×C₄×Q₈×C₅₉	C₂²	1	3	yes	—
20	C₂×C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×#11745664	C₂³	1	6	yes	—
22	C₂×#11745664	C₂³	1	2	yes	—
23	C₄⋊(C₅₉⋊C₈)	—	2	6	no	—
24	C₄⋊(C₅₉⋊C₈)	—	2	6	no	—
25	#14273646	—	2	6	no	—
26	#14273646	Q₈	1	4	yes	direct product factor
27	#14273638	—	2	6	no	—
28	C₄×C₅₉⋊C₈	—	2	6	no	—
29	C₂²×Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×Q₈×C₅₉	C₂×C₄	1	6	yes	—
31	C₂²×Q₈×C₅₉	C₂³	1	1	yes	characteristic
32	C₄⋊Q₈×C₅₉	—	2	12	no	—
33	C₄⋊Q₈×C₅₉	—	2	12	no	—
34	C₄⋊Q₈×C₅₉	—	2	6	no	—
35	C₄⋊Q₈×C₅₉	—	2	6	no	—
36	C₄×Q₈×C₅₉	—	2	6	no	—
37	C₄×Q₈×C₅₉	—	2	2	no	—
38	C₄×Q₈×C₅₉	—	2	6	no	—
39	C₄×Q₈×C₅₉	—	2	6	no	—
40	C₄×Q₈×C₅₉	D₄	1	2	yes	—
41	C₄×Q₈×C₅₉	D₄	1	6	yes	—
42	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	6	yes	—
43	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	6	yes	—
44	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	3	yes	—
45	C₂×C₄⋊C₄×C₅₉	C₂³	1	3	yes	—
46	C₂×C₄²×C₅₉	C₂×C₄	1	6	yes	—
47	C₂×C₄²×C₅₉	C₂³	1	3	yes	—
48	C₂×Q₈×C₅₉	—	2	4	no	—
49	C₂×Q₈×C₅₉	—	2	12	no	—
50	C₂×Q₈×C₅₉	—	2	2	no	—
51	C₂×Q₈×C₅₉	—	2	2	no	—
52	C₂×Q₈×C₅₉	—	8	6	no	—
53	C₂×Q₈×C₅₉	C₂²⋊C₄	1	3	yes	—
54	C₂×Q₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×Q₈×C₅₉	C₂×Q₈	1	2	yes	—
56	C₂×Q₈×C₅₉	#41	1	2	yes	—
57	C₂²×C₄×C₅₉	C₂²×C₄	1	3	yes	—
58	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₅₉	C₂²×C₄	1	6	yes	—
60	C₂²×C₄×C₅₉	C₂²×C₄	1	3	yes	—
61	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
62	C₄⋊C₄×C₅₉	—	2	12	no	—
63	C₄⋊C₄×C₅₉	—	2	6	no	—
64	C₄⋊C₄×C₅₉	—	2	6	no	—
65	C₄⋊C₄×C₅₉	—	2	6	no	—
66	C₄⋊C₄×C₅₉	C₂²⋊C₄	1	6	yes	—
67	C₄⋊C₄×C₅₉	C₂²⋊C₄	1	6	yes	—
68	C₄²×C₅₉	—	2	6	no	—
69	C₄²×C₅₉	—	2	6	no	—
70	C₄²×C₅₉	C₂×D₄	1	6	yes	—
71	C₄²×C₅₉	C₂×D₄	1	6	yes	—
72	#11745664	—	2	2	no	—
73	#11745664	—	2	6	no	—
74	#11745664	—	2	6	no	—
75	#11745664	C₂×Q₈	1	4	yes	—
76	C₂×C₅₉⋊C₈	—	2	2	no	—
77	C₂×C₅₉⋊C₈	—	2	6	no	—
78	C₂³×C₅₉	C₂³×C₄	1	1	yes	characteristic
79	Q₈×C₅₉	—	2	4	no	—
80	Q₈×C₅₉	—	2	4	no	—
81	Q₈×C₅₉	—	2	6	no	—
82	Q₈×C₅₉	—	8	12	no	—
83	Q₈×C₅₉	#101	1	4	yes	complemented
84	C₂×C₄×C₅₉	—	2	2	no	—
85	C₂×C₄×C₅₉	—	2	2	no	—
86	C₂×C₄×C₅₉	—	2	6	no	—
87	C₂×C₄×C₅₉	—	2	6	no	—
88	C₂×C₄×C₅₉	—	4	3	no	—
89	C₂×C₄×C₅₉	—	4	6	no	—
90	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	3	yes	—
91	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	6	yes	—
92	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	3	yes	—
93	C₂×C₄×C₅₉	C₄×Q₈	1	4	yes	—
94	C₂×C₄×C₅₉	C₂²×D₄	1	2	yes	—
95	C₂×C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
96	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
97	C₂×C₄×C₅₉	2₋¹⁺⁴	1	2	yes	—
98	C₅₉⋊C₈	—	2	4	no	—
99	C₅₉⋊C₈	—	4	6	no	—
100	Q₈×#101	—	59	1	no	maximal, Sylow
101	C₂²×C₅₉	—	2	1	no	—
102	C₂²×C₅₉	—	2	1	no	—
103	C₂²×C₅₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
104	C₂²×C₅₉	C₂×C₄×Q₈	1	1	yes	characteristic
105	C₂²×C₅₉	#519	1	1	yes	characteristic
106	C₄×C₅₉	—	2	4	no	—
107	C₄×C₅₉	—	2	3	no	—
108	C₄×C₅₉	—	2	6	no	—
109	C₄×C₅₉	—	4	6	no	—
110	C₄×C₅₉	C₂×#101	1	6	yes	—
111	C₄×C₅₉	#548	1	2	yes	—
112	C₄×C₅₉	D₄×Q₈	1	2	yes	—
113	C₂×Q₈²	D₅₉	1	1	yes	characteristic
114	Q₈×M₁₆	—	59	2	no	—
115	#1997	—	59	6	no	—
116	#1801	—	59	3	no	—
117	C₄×#101	—	59	3	no	—
118	C₂×C₅₉	—	2	1	no	—
119	C₂×C₅₉	—	2	1	no	—
120	C₂×C₅₉	Q₈×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₅₉	C₂²×#101	1	1	yes	characteristic
122	C₂×C₅₉	#2888	1	1	yes	characteristic
123	Q₈²	—	2	2	no	—
124	Q₈²	—	2	6	no	—
125	C₂×C₄⋊Q₈	Dic₅₉	1	6	yes	—
126	C₂×C₄⋊Q₈	C₂×D₅₉	1	3	yes	—
127	C₂×C₄×Q₈	Dic₅₉	1	2	yes	—
128	C₂×C₄×Q₈	C₂×D₅₉	1	3	yes	—
129	C₂×C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
130	#446	—	118	6	no	—
131	C₈×Q₈	—	118	2	no	—
132	#423	—	59	6	no	—
133	C₂×#101	—	59	1	no	—
134	C₂×#101	—	59	3	no	—
135	C₄×M₁₆	—	59	6	no	—
136	#343	—	59	6	no	—
137	#343	—	59	6	no	—
138	C₅₉	Q₈×#101	1	1	yes	characteristic, complemented, Sylow
139	C₂²×Q₈	C₂×Dic₅₉	1	1	yes	characteristic
140	C₂²×Q₈	C₂×Dic₅₉	1	6	yes	—
141	C₂²×Q₈	C₂²×D₅₉	1	1	yes	characteristic
142	C₂×M₁₆	—	59	2	no	—
143	C₂×M₁₆	—	59	6	no	—
144	C₄⋊Q₈	—	2	12	no	—
145	C₄⋊Q₈	—	2	12	no	—
146	C₄⋊Q₈	—	2	6	no	—
147	C₄⋊Q₈	—	2	6	no	—
148	C₄×Q₈	—	2	6	no	—
149	C₄×Q₈	—	2	2	no	—
150	C₄×Q₈	—	2	6	no	—
151	C₄×Q₈	—	2	6	no	—
152	C₄×Q₈	C₂²⋊D₅₉	1	2	yes	—
153	C₄×Q₈	C₂²⋊D₅₉	1	6	yes	—
154	C₂×C₄⋊C₄	C₂×Dic₅₉	1	6	yes	—
155	C₂×C₄⋊C₄	C₂×Dic₅₉	1	6	yes	—
156	C₂×C₄⋊C₄	C₂×Dic₅₉	1	3	yes	—
157	C₂×C₄⋊C₄	C₂²×D₅₉	1	3	yes	—
158	C₂×C₄²	C₂×Dic₅₉	1	6	yes	—
159	C₂×C₄²	C₂²×D₅₉	1	3	yes	—
160	C₄⋊C₈	—	118	6	no	—
161	C₄⋊C₈	—	118	6	no	—
162	#101	—	59	4	no	—
163	#101	—	118	6	no	—
164	#97	—	118	6	no	—
165	C₄×C₈	—	118	6	no	—
166	C₂×Q₈	—	2	4	no	—
167	C₂×Q₈	—	2	12	no	—
168	C₂×Q₈	—	2	2	no	—
169	C₂×Q₈	—	2	2	no	—
170	C₂×Q₈	—	8	6	no	—
171	C₂×Q₈	C₂²⋊Dic₅₉	1	3	yes	—
172	C₂×Q₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
173	C₂×Q₈	Q₈×D₅₉	1	2	yes	—
174	C₂×Q₈	#11745689	1	2	yes	—
175	C₂²×C₄	C₂²×Dic₅₉	1	3	yes	—
176	C₂²×C₄	C₂²×Dic₅₉	1	2	yes	—
177	C₂²×C₄	C₂²×Dic₅₉	1	6	yes	—
178	C₂²×C₄	C₂²×Dic₅₉	1	3	yes	—
179	C₂²×C₄	C₂³×D₅₉	1	1	yes	characteristic
180	M₁₆	—	59	4	no	—
181	M₁₆	—	118	2	no	—
182	M₁₆	—	118	6	no	—
183	M₁₆	—	118	6	no	—
184	C₂×C₈	—	118	2	no	—
185	C₂×C₈	—	118	6	no	—
186	C₄⋊C₄	—	2	12	no	—
187	C₄⋊C₄	—	2	6	no	—
188	C₄⋊C₄	—	2	6	no	—
189	C₄⋊C₄	—	2	6	no	—
190	C₄⋊C₄	C₂²⋊Dic₅₉	1	6	yes	—
191	C₄⋊C₄	C₂²⋊Dic₅₉	1	6	yes	—
192	C₄²	—	2	6	no	—
193	C₄²	—	2	6	no	—
194	C₄²	C₂×C₂²⋊D₅₉	1	6	yes	—
195	C₄²	C₂×C₂²⋊D₅₉	1	6	yes	—
196	C₂³	C₂³×Dic₅₉	1	1	yes	characteristic
197	Q₈	—	2	6	no	—
198	Q₈	—	2	4	no	—
199	Q₈	—	2	4	no	—
200	Q₈	—	8	12	no	—
201	Q₈	#14273646	1	4	yes	direct product factor
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	6	no	—
205	C₂×C₄	—	2	6	no	—
206	C₂×C₄	—	4	3	no	—
207	C₂×C₄	—	4	6	no	—
208	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	3	yes	—
209	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	6	yes	—
210	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	3	yes	—
211	C₂×C₄	Q₈×Dic₅₉	1	4	yes	—
212	C₂×C₄	C₂²×C₂²⋊D₅₉	1	2	yes	—
213	C₂×C₄	C₂×Q₈×D₅₉	1	1	yes	characteristic
214	C₂×C₄	C₂×#11745689	1	1	yes	characteristic
215	C₂×C₄	#14273775	1	2	yes	—
216	C₈	—	118	4	no	—
217	C₈	—	236	6	no	—
218	C₂²	—	2	1	no	—
219	C₂²	—	2	1	no	—
220	C₂²	C₂²×C₂²⋊Dic₅₉	1	1	yes	characteristic
221	C₂²	C₂×Q₈×Dic₅₉	1	1	yes	characteristic
222	C₂²	#16837880	1	1	yes	characteristic
223	C₄	—	2	4	no	—
224	C₄	—	2	3	no	—
225	C₄	—	2	6	no	—
226	C₄	—	4	6	no	—
227	C₄	C₂×#14273646	1	6	yes	—
228	C₄	#16838063	1	2	yes	—
229	C₄	Q₈×C₂²⋊D₅₉	1	2	yes	—
230	C₂	—	2	1	no	—
231	C₂	—	2	1	no	—
232	C₂	Q₈×C₂²⋊Dic₅₉	1	1	yes	characteristic
233	C₂	C₂²×#14273646	1	1	yes	characteristic
234	C₂	#21432575	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊Dic₅₉
commutator, prime radical	C₂³×C₅₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₉
Fitting	C₂×Q₈²×C₅₉	C₂
socle	C₂²×C₅₉	C₂²×C₂²⋊C₄
cosocle	C₂²×C₄×C₅₉	C₂⁴
hypercenter	C₂×Q₈²	D₅₉
hypocenter	C₅₉	Q₈×#101
LCS	G ▷ C₂³×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ C₂×Q₈²

Complements

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

N	H ≅ G/N	type
26. #14273646	Q₈	direct (N × H)
83. Q₈×C₅₉	#101	semidirect
138. C₅₉	Q₈×#101	semidirect
201. Q₈	#14273646	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×#11745664	2
#21425086	2
#21424092	2
C₄×#14273646	2
C₂×Q₈²×C₅₉	2
Q₈×#101	59

Subgroups — Q8×#14273646 #28605264

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#14273646 #28605264