Subgroups — Q₈×#10587530 #17978671

Order 4352.

530 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14545200	C₂	1	6	yes	maximal, complemented
2	Q₈×#63216	C₂	1	2	yes	maximal, complemented
3	#14545117	C₂	1	3	yes	maximal
4	C₄×#10587530	C₂	1	3	yes	maximal
5	C₈×Q₈×D₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587530	C₂²	1	3	yes	—
7	C₂×#10587530	C₂²	1	1	yes	characteristic
8	#11763730	C₂²	1	6	yes	—
9	C₄⋊#63216	C₂²	1	6	yes	—
10	#11763698	C₂²	1	6	yes	—
11	C₄×#63216	C₂²	1	6	yes	—
12	C₄×Q₈×D₁₇	C₄	1	1	yes	characteristic
13	Q₈×#3622	C₂²	1	1	yes	characteristic
14	C₈×Dic₃₄	C₄	1	3	yes	—
15	#11762933	C₂²	1	3	yes	—
16	C₄⋊C₈×D₁₇	C₄	1	3	yes	—
17	C₄×C₈×D₁₇	C₂²	1	3	yes	—
18	C₈×Q₈×C₁₇	C₄	1	1	yes	characteristic
19	C₂×Q₈×D₁₇	C₈	1	1	yes	characteristic
20	C₄×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
21	Q₈×Dic₁₇	C₈	1	1	yes	characteristic
22	C₂×C₈×D₁₇	C₂×C₄	1	3	yes	—
23	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×D₁₇	C₈	1	3	yes	—
25	#10587573	C₈	1	3	yes	—
26	C₄²×D₁₇	C₂×C₄	1	3	yes	—
27	C₄×Dic₃₄	C₂×C₄	1	3	yes	—
28	C₂×#63216	C₂³	1	6	yes	—
29	C₂×#63216	C₂³	1	2	yes	—
30	#10587530	—	2	6	no	—
31	#10587530	Q₈	1	4	yes	direct product factor
32	C₁₇×C₄⋊C₈	C₂×C₄	1	3	yes	—
33	C₄×C₈×C₁₇	C₂×C₄	1	3	yes	—
34	#10587485	C₂×C₄	1	6	yes	—
35	C₈×Dic₁₇	C₂×C₄	1	3	yes	—
36	C₈×Dic₁₇	C₂³	1	3	yes	—
37	C₄⋊#3622	C₂×C₄	1	3	yes	—
38	C₄×#3622	C₂³	1	3	yes	—
39	C₂×Q₈×C₁₇	C₂×C₈	1	1	yes	characteristic
40	Q₈×D₁₇	—	2	4	no	—
41	C₂×C₄×D₁₇	C₂×C₈	1	3	yes	—
42	C₂×C₄×D₁₇	C₂×C₈	1	3	yes	—
43	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×Dic₃₄	C₂×C₈	1	3	yes	—
45	C₂×C₈×C₁₇	C₂²×C₄	1	3	yes	—
46	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
47	C₄⋊C₄×C₁₇	C₂×C₈	1	3	yes	—
48	C₄²×C₁₇	C₂²×C₄	1	3	yes	—
49	C₄⋊Dic₁₇	C₂×C₈	1	3	yes	—
50	#63225	C₂×C₈	1	6	yes	—
51	C₄×Dic₁₇	C₂×C₈	1	3	yes	—
52	C₄×Dic₁₇	C₂×C₈	1	3	yes	—
53	C₄×Dic₁₇	C₂²×C₄	1	3	yes	—
54	C₂×#3622	C₂²×C₄	1	3	yes	—
55	C₂×#3622	C₂⁴	1	1	yes	characteristic
56	C₈×D₁₇	—	2	6	no	—
57	C₈×D₁₇	C₂×Q₈	1	2	yes	—
58	C₈×D₁₇	#41	1	2	yes	—
59	#63216	—	2	6	no	—
60	#63216	C₂×Q₈	1	4	yes	complemented
61	Q₈×M₃₂	—	17	1	no	maximal, Sylow
62	C₂²×D₁₇	C₂²×C₈	1	1	yes	characteristic
63	Q₈×C₁₇	M₃₂	1	4	yes	complemented
64	C₂×C₄×C₁₇	C₂²×C₈	1	3	yes	—
65	C₂×C₄×C₁₇	C₂²×C₈	1	3	yes	—
66	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
67	C₂×Dic₁₇	C₂²×C₈	1	3	yes	—
68	C₂×Dic₁₇	C₂²×C₈	1	3	yes	—
69	C₂×Dic₁₇	C₂²×C₈	1	1	yes	characteristic
70	C₄×D₁₇	—	2	6	no	—
71	C₄×D₁₇	—	4	3	no	—
72	C₄×D₁₇	C₄×Q₈	1	2	yes	—
73	C₄×D₁₇	#131	1	2	yes	—
74	Dic₃₄	—	2	6	no	—
75	C₈×C₁₇	—	2	3	no	—
76	C₈×C₁₇	C₄×Q₈	1	2	yes	—
77	#3622	—	2	3	no	—
78	#3622	C₂²×Q₈	1	1	yes	characteristic
79	#3622	C₂×#41	1	1	yes	characteristic
80	C₂×C₈×Q₈	—	17	1	no	—
81	#2183	—	17	6	no	—
82	Q₈×C₁₆	—	17	2	no	—
83	#2150	—	17	3	no	—
84	C₄×M₃₂	—	17	3	no	—
85	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
86	C₂×D₁₇	—	2	1	no	—
87	C₂×D₁₇	—	2	1	no	—
88	C₂×D₁₇	C₈×Q₈	1	2	yes	—
89	C₄×C₁₇	—	2	3	no	—
90	C₄×C₁₇	C₂×M₃₂	1	6	yes	—
91	C₄×C₁₇	C₂×C₄×Q₈	1	1	yes	characteristic
92	C₄×C₁₇	C₂×#131	1	1	yes	characteristic
93	Dic₁₇	—	2	3	no	—
94	Dic₁₇	—	2	3	no	—
95	Dic₁₇	C₈×Q₈	1	2	yes	—
96	C₂×C₄×Q₈	—	17	1	no	—
97	C₂×M₃₂	—	17	1	no	—
98	C₂×M₃₂	—	17	3	no	—
99	C₈×Q₈	—	17	1	no	—
100	C₈×Q₈	—	17	3	no	—
101	C₈×Q₈	—	17	3	no	—
102	C₈×Q₈	#595	1	1	yes	characteristic
103	C₂×C₄⋊C₈	—	17	3	no	—
104	C₂×C₄×C₈	—	17	3	no	—
105	C₄⋊C₁₆	—	17	6	no	—
106	C₄⋊C₁₆	—	17	6	no	—
107	#346	—	17	6	no	—
108	C₄×C₁₆	—	17	6	no	—
109	C₂×C₁₇	C₂×C₈×Q₈	1	1	yes	characteristic
110	C₂×C₁₇	C₂²×M₃₂	1	1	yes	characteristic
111	C₂×C₁₇	#3407	1	1	yes	characteristic
112	D₁₇	—	2	2	no	—
113	C₂²×Q₈	—	17	1	no	—
114	C₂²×C₈	—	17	1	no	—
115	C₂²×C₈	—	17	3	no	—
116	C₄×Q₈	—	17	1	no	—
117	C₄×Q₈	—	17	3	no	—
118	C₄×Q₈	—	17	3	no	—
119	C₄×Q₈	C₂×#595	1	1	yes	characteristic
120	C₂×C₄⋊C₄	—	17	3	no	—
121	C₂×C₄²	—	17	3	no	—
122	M₃₂	—	17	4	no	—
123	M₃₂	—	34	6	no	—
124	C₂×C₁₆	—	17	2	no	—
125	C₂×C₁₆	—	17	6	no	—
126	C₄⋊C₈	—	17	6	no	—
127	C₄⋊C₈	—	17	3	no	—
128	C₄⋊C₈	C₂×#595	1	3	yes	—
129	C₄×C₈	—	17	3	no	—
130	C₄×C₈	—	17	3	no	—
131	C₄×C₈	—	17	3	no	—
132	C₄×C₈	C₂×#595	1	3	yes	—
133	C₁₇	Q₈×M₃₂	1	1	yes	characteristic, complemented, Sylow
134	C₂×Q₈	—	17	3	no	—
135	C₂×Q₈	—	34	4	no	—
136	C₂×Q₈	#63242	1	1	yes	characteristic
137	C₂²×C₄	—	17	1	no	—
138	C₂²×C₄	—	17	3	no	—
139	C₂²×C₄	—	17	3	no	—
140	C₂×C₈	—	17	1	no	—
141	C₂×C₈	—	17	2	no	—
142	C₂×C₈	—	17	2	no	—
143	C₂×C₈	—	17	3	no	—
144	C₂×C₈	—	34	6	no	—
145	C₂×C₈	C₂²×#595	1	3	yes	—
146	C₂×C₈	C₂²×#595	1	1	yes	characteristic
147	C₄⋊C₄	—	17	6	no	—
148	C₄⋊C₄	—	17	3	no	—
149	C₄⋊C₄	#63242	1	3	yes	—
150	C₄²	—	17	3	no	—
151	C₄²	—	17	3	no	—
152	C₄²	—	17	3	no	—
153	C₄²	C₂²×#595	1	3	yes	—
154	C₁₆	—	17	4	no	—
155	C₁₆	—	34	6	no	—
156	C₂³	—	17	1	no	—
157	Q₈	—	34	6	no	—
158	Q₈	#10587530	1	4	yes	direct product factor
159	C₂×C₄	—	17	1	no	—
160	C₂×C₄	—	17	2	no	—
161	C₂×C₄	—	17	2	no	—
162	C₂×C₄	—	17	3	no	—
163	C₂×C₄	—	17	3	no	—
164	C₂×C₄	—	34	6	no	—
165	C₂×C₄	—	68	3	no	—
166	C₂×C₄	C₂×#63242	1	3	yes	—
167	C₂×C₄	C₂×#63242	1	3	yes	—
168	C₂×C₄	C₂³×#595	1	1	yes	characteristic
169	C₈	—	2	3	no	—
170	C₈	—	17	1	no	—
171	C₈	—	17	1	no	—
172	C₈	—	34	3	no	—
173	C₈	Q₈×#595	1	2	yes	—
174	C₂²	—	17	2	no	—
175	C₂²	—	34	1	no	—
176	C₂²	—	34	1	no	—
177	C₂²	C₂²×#63242	1	1	yes	characteristic
178	C₄	—	2	3	no	—
179	C₄	—	17	2	no	—
180	C₄	—	34	3	no	—
181	C₄	—	34	3	no	—
182	C₄	C₂×#10587530	1	6	yes	—
183	C₄	C₂×Q₈×#595	1	1	yes	characteristic
184	C₄	D₁₇⋊#131	1	1	yes	characteristic
185	C₂	—	34	2	no	—
186	C₂	Q₈×#63242	1	1	yes	characteristic
187	C₂	C₂²×#10587530	1	1	yes	characteristic
188	C₂	#14545752	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#595
commutator, socle	C₂²×C₁₇	C₂³×C₈
Fitting	C₈×Q₈×C₁₇	C₄
prime radical	C₂²×D₁₇	C₂²×C₈
cosocle	C₂×#3622	C₂⁴
hypercenter	C₈×Q₈	#595
hypocenter	C₁₇	Q₈×M₃₂
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. #14545200	C₂	semidirect
2. Q₈×#63216	C₂	semidirect
31. #10587530	Q₈	direct (N × H)
60. #63216	C₂×Q₈	semidirect
63. Q₈×C₁₇	M₃₂	semidirect
133. C₁₇	Q₈×M₃₂	semidirect
158. Q₈	#10587530	direct (N × H)

Maximal subgroups

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

subgroup	index
#14545200	2
Q₈×#63216	2
#14545117	2
C₄×#10587530	2
C₈×Q₈×D₁₇	2
Q₈×M₃₂	17

Subgroups — Q8×#10587530 #17978671

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#10587530 #17978671