Subgroups — Q₈×#10587665 #17983189

Order 4352.

582 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	#14546008	C₂	1	6	yes	maximal, complemented
2	Q₈×#63241	C₂	1	2	yes	maximal, complemented
3	#14545970	C₂	1	3	yes	maximal
4	C₄×#10587665	C₂	1	3	yes	maximal
5	Q₈×#63242	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587665	C₂²	1	3	yes	—
7	C₂×#10587665	C₂²	1	1	yes	characteristic
8	#11763912	C₂²	1	6	yes	—
9	C₄⋊#63241	C₂²	1	6	yes	—
10	C₄×#63241	C₂²	1	6	yes	—
11	C₄×#63241	C₂²	1	6	yes	—
12	#11763858	C₄	1	3	yes	—
13	#11763858	C₂²	1	3	yes	—
14	Q₈×#3624	C₄	1	1	yes	characteristic
15	Q₈×#3624	C₂²	1	1	yes	characteristic
16	#11763810	C₄	1	3	yes	—
17	C₄×#63242	C₂²	1	3	yes	—
18	C₄×Q₈×D₁₇	C₄	1	1	yes	characteristic
19	C₂×Q₈×D₁₇	C₈	1	1	yes	characteristic
20	C₂×#63242	C₂×C₄	1	3	yes	—
21	C₂×#63242	C₂³	1	1	yes	characteristic
22	C₂×#63241	C₂³	1	6	yes	—
23	C₂×#63241	C₂³	1	2	yes	—
24	#10587665	—	2	6	no	—
25	#10587665	Q₈	1	4	yes	direct product factor
26	C₄×Q₈×C₁₇	C₈	1	1	yes	characteristic
27	Q₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
28	C₄⋊C₄×D₁₇	C₈	1	3	yes	—
29	#10587573	C₂×C₄	1	3	yes	—
30	C₄²×D₁₇	C₂×C₄	1	3	yes	—
31	C₄×Dic₃₄	C₈	1	3	yes	—
32	#10587544	C₂×C₄	1	6	yes	—
33	C₄⋊#3624	C₂×C₄	1	3	yes	—
34	C₄⋊#3624	C₂×C₄	1	3	yes	—
35	C₄×#3624	C₂×C₄	1	3	yes	—
36	C₄×#3624	C₂×C₄	1	3	yes	—
37	C₄×#3624	C₂³	1	3	yes	—
38	C₄×#3624	C₂³	1	3	yes	—
39	C₂×Q₈×C₁₇	C₂×C₈	1	1	yes	characteristic
40	Q₈×D₁₇	C₁₆	1	8	yes	complemented
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×#3624	C₂²×C₄	1	3	yes	—
46	C₂×#3624	C₂²×C₄	1	3	yes	—
47	C₂×#3624	C₂²×C₄	1	1	yes	characteristic
48	C₂×#3624	C₂⁴	1	1	yes	characteristic
49	#63242	—	2	6	no	—
50	#63242	C₂×Q₈	1	2	yes	—
51	#63242	#41	1	2	yes	—
52	#63241	—	2	6	no	—
53	#63241	C₂×Q₈	1	4	yes	complemented
54	C₄⋊C₄×C₁₇	C₂×C₈	1	3	yes	—
55	C₄²×C₁₇	C₂×C₈	1	3	yes	—
56	C₄⋊Dic₁₇	C₂×C₈	1	3	yes	—
57	#63225	C₂×C₈	1	6	yes	—
58	C₄×Dic₁₇	C₂×C₈	1	3	yes	—
59	C₄×Dic₁₇	C₂²×C₄	1	3	yes	—
60	C₄×Dic₁₇	C₂²×C₄	1	3	yes	—
61	C₂×Q₈×C₁₆	—	17	1	no	maximal, Sylow
62	C₂²×D₁₇	C₂²×C₈	1	1	yes	characteristic
63	Q₈×C₁₇	C₂×C₁₆	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	C₂²×C₈	1	1	yes	characteristic
67	C₂×Dic₁₇	C₂²×C₈	1	3	yes	—
68	C₂×Dic₁₇	C₂²×C₈	1	3	yes	—
69	C₂×Dic₁₇	C₂³×C₄	1	1	yes	characteristic
70	C₄×D₁₇	—	2	6	no	—
71	C₄×D₁₇	C₂×C₁₆	1	12	yes	—
72	C₄×D₁₇	C₄×Q₈	1	2	yes	—
73	C₄×D₁₇	#131	1	2	yes	—
74	Dic₃₄	C₂×C₁₆	1	12	yes	complemented
75	#3624	—	2	3	no	—
76	#3624	—	2	3	no	—
77	#3624	C₄×Q₈	1	2	yes	—
78	#3624	C₂²×Q₈	1	1	yes	characteristic
79	#3624	C₂×#41	1	1	yes	characteristic
80	C₂×C₈×Q₈	—	17	1	no	—
81	Q₈×C₁₆	—	17	2	no	—
82	Q₈×C₁₆	—	17	6	no	—
83	C₂×C₄⋊C₁₆	—	17	3	no	—
84	C₂×C₄×C₁₆	—	17	3	no	—
85	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
86	C₂×D₁₇	C₈×Q₈	1	2	yes	—
87	C₂×D₁₇	C₂²×C₁₆	1	2	yes	—
88	C₂×D₁₇	#504	1	2	yes	—
89	C₄×C₁₇	—	2	3	no	—
90	C₄×C₁₇	C₈×Q₈	1	2	yes	—
91	C₄×C₁₇	C₂²×C₁₆	1	6	yes	—
92	Dic₁₇	—	2	3	no	—
93	Dic₁₇	C₂²×C₁₆	1	6	yes	—
94	Dic₁₇	C₂×C₄×Q₈	1	1	yes	characteristic
95	Dic₁₇	C₂×#131	1	1	yes	characteristic
96	C₂×C₄×Q₈	—	17	1	no	—
97	C₂²×C₁₆	—	17	1	no	—
98	C₂²×C₁₆	—	17	3	no	—
99	C₈×Q₈	—	17	1	no	—
100	C₈×Q₈	—	17	3	no	—
101	C₈×Q₈	—	17	1	no	—
102	C₈×Q₈	—	17	3	no	—
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	C₄×C₁₆	—	17	6	no	—
108	C₄×C₁₆	—	17	6	no	—
109	C₂×C₁₇	C₂×C₈×Q₈	1	1	yes	characteristic
110	C₂×C₁₇	C₂³×C₁₆	1	1	yes	characteristic
111	C₂×C₁₇	C₂×#504	1	1	yes	characteristic
112	D₁₇	Q₈×C₁₆	1	4	yes	complemented
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	#3633	1	1	yes	characteristic
120	C₂×C₄⋊C₄	—	17	3	no	—
121	C₂×C₄²	—	17	3	no	—
122	C₂×C₁₆	—	17	4	no	—
123	C₂×C₁₆	—	17	2	no	—
124	C₂×C₁₆	—	17	6	no	—
125	C₂×C₁₆	—	34	6	no	—
126	C₄⋊C₈	—	17	3	no	—
127	C₄⋊C₈	—	17	3	no	—
128	C₄⋊C₈	—	17	6	no	—
129	C₄×C₈	—	17	3	no	—
130	C₄×C₈	—	17	3	no	—
131	C₄×C₈	—	17	3	no	—
132	C₄×C₈	—	17	3	no	—
133	C₁₇	C₂×Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
134	C₂×Q₈	—	17	8	no	—
135	C₂×Q₈	—	17	3	no	—
136	C₂×Q₈	C₂×#3633	1	1	yes	characteristic
137	C₂²×C₄	—	17	3	no	—
138	C₂²×C₄	—	17	1	no	—
139	C₂²×C₄	—	17	3	no	—
140	C₂×C₈	—	17	2	no	—
141	C₂×C₈	—	17	2	no	—
142	C₂×C₈	—	17	1	no	—
143	C₂×C₈	—	17	1	no	—
144	C₂×C₈	—	17	3	no	—
145	C₂×C₈	—	17	3	no	—
146	C₂×C₈	—	34	6	no	—
147	C₄⋊C₄	—	17	6	no	—
148	C₄⋊C₄	—	17	3	no	—
149	C₄⋊C₄	C₂×#3633	1	3	yes	—
150	C₄²	—	17	3	no	—
151	C₄²	—	17	3	no	—
152	C₄²	—	17	3	no	—
153	C₄²	C₂×#3633	1	3	yes	—
154	C₁₆	—	17	4	no	—
155	C₁₆	—	34	6	no	—
156	C₂³	—	17	1	no	—
157	Q₈	—	17	12	no	—
158	Q₈	#10587665	1	4	yes	direct product factor
159	C₂×C₄	—	17	1	no	—
160	C₂×C₄	—	17	3	no	—
161	C₂×C₄	—	17	2	no	—
162	C₂×C₄	—	17	2	no	—
163	C₂×C₄	—	17	12	no	—
164	C₂×C₄	—	17	3	no	—
165	C₂×C₄	—	34	6	no	—
166	C₂×C₄	C₂²×#3633	1	3	yes	—
167	C₂×C₄	C₂²×#3633	1	3	yes	—
168	C₂×C₄	C₂²×#3633	1	1	yes	characteristic
169	C₈	—	17	1	no	—
170	C₈	—	17	1	no	—
171	C₈	—	17	2	no	—
172	C₈	—	34	3	no	—
173	C₈	—	34	3	no	—
174	C₂²	—	17	2	no	—
175	C₂²	—	17	2	no	—
176	C₂²	—	17	2	no	—
177	C₂²	C₂³×#3633	1	1	yes	characteristic
178	C₄	—	2	3	no	—
179	C₄	—	17	1	no	—
180	C₄	—	17	1	no	—
181	C₄	—	17	6	no	—
182	C₄	—	34	3	no	—
183	C₄	Q₈×#3633	1	2	yes	—
184	C₄	C₂×#10587665	1	6	yes	—
185	C₂	—	17	4	no	—
186	C₂	C₂×Q₈×#3633	1	1	yes	characteristic
187	C₂	C₂²×#10587665	1	1	yes	characteristic
188	C₂	#14546035	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×#3633
commutator	C₂×C₁₇	C₂³×C₁₆
Frattini	C₂²	C₂³×#3633
Fitting	C₄×Q₈×C₁₇	C₈
socle	C₂²×C₁₇	C₂³×C₈
prime radical	C₂²×D₁₇	C₂²×C₈
cosocle	C₂×#3624	C₂⁴
hypercenter	C₄×Q₈	#3633
hypocenter	C₁₇	C₂×Q₈×C₁₆
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #14546008	C₂	semidirect
2. Q₈×#63241	C₂	semidirect
25. #10587665	Q₈	direct (N × H)
40. Q₈×D₁₇	C₁₆	semidirect
53. #63241	C₂×Q₈	semidirect
63. Q₈×C₁₇	C₂×C₁₆	semidirect
74. Dic₃₄	C₂×C₁₆	semidirect
112. D₁₇	Q₈×C₁₆	semidirect
133. C₁₇	C₂×Q₈×C₁₆	semidirect
158. Q₈	#10587665	direct (N × H)

Maximal subgroups

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

subgroup	index
#14546008	2
Q₈×#63241	2
#14545970	2
C₄×#10587665	2
Q₈×#63242	2
C₂×Q₈×C₁₆	17

Subgroups — Q8×#10587665 #17983189

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#10587665 #17983189