Subgroups — Q₈×#10587488 #17053151

Order 4352.

838 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₄⋊Dic₁₇	C₂	1	2	yes	maximal
2	#14529548	C₂	1	6	yes	maximal
3	#14528941	C₂	1	3	yes	maximal
4	C₄×#10587488	C₂	1	3	yes	maximal
5	C₂×C₈×Q₈×C₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₈×Dic₁₇	—	2	2	no	—
7	C₂×#10587488	C₂²	1	3	yes	—
8	C₂×#10587488	C₂²	1	6	yes	—
9	C₂×#10587488	C₂²	1	1	yes	characteristic
10	#11762708	—	2	6	no	—
11	#11762679	C₂²	1	6	yes	—
12	C₄×C₄⋊Dic₁₇	C₂²	1	6	yes	—
13	#11762572	C₂²	1	6	yes	—
14	C₂×C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
15	C₈×Q₈×C₁₇	C₄	1	8	yes	complemented
16	C₂×C₁₇×C₄⋊C₈	C₂²	1	3	yes	—
17	C₂×C₄×C₈×C₁₇	C₂²	1	3	yes	—
18	C₂²×Q₈×C₁₇	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₇	C₂³	1	3	yes	—
20	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
21	C₄×Q₈×C₁₇	C₂×C₄	1	4	yes	—
22	C₄×Q₈×C₁₇	Q₈	1	4	yes	—
23	C₂×C₄⋊C₄×C₁₇	D₄	1	3	yes	—
24	C₂×C₄²×C₁₇	C₂³	1	3	yes	—
25	Q₈×Dic₁₇	—	4	8	no	—
26	C₂×C₄⋊Dic₁₇	—	2	6	no	—
27	C₂×C₄⋊Dic₁₇	—	2	6	no	—
28	C₂×C₄⋊Dic₁₇	C₂³	1	6	yes	—
29	C₂×C₄⋊Dic₁₇	C₂³	1	2	yes	—
30	C₂×C₄×Dic₁₇	—	2	6	no	—
31	C₁₇×C₄⋊C₈	C₂×C₄	1	12	yes	—
32	C₄×C₈×C₁₇	C₂×C₄	1	12	yes	—
33	#10587502	—	2	6	no	—
34	#10587488	—	2	6	no	—
35	#10587488	—	4	6	no	—
36	#10587488	Q₈	1	4	yes	direct product factor
37	C₂×Q₈×C₁₇	C₄⋊C₄	1	4	yes	—
38	C₂×Q₈×C₁₇	SD₁₆	1	8	yes	—
39	C₂²×C₄×C₁₇	C₂×D₄	1	3	yes	—
40	C₂²×C₄×C₁₇	C₂×D₄	1	3	yes	—
41	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
42	C₂²×Dic₁₇	—	2	2	no	—
43	C₂²×Dic₁₇	—	2	6	no	—
44	C₂×C₈×C₁₇	—	2	3	no	—
45	C₂×C₈×C₁₇	—	2	3	no	—
46	C₂×C₈×C₁₇	C₂²×C₄	1	6	yes	—
47	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
48	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
49	C₂×C₈×C₁₇	#41	1	2	yes	—
50	C₄⋊C₄×C₁₇	C₄⋊C₄	1	12	yes	—
51	C₄²×C₁₇	C₂²×C₄	1	6	yes	—
52	C₄²×C₁₇	C₂×Q₈	1	6	yes	—
53	C₄⋊Dic₁₇	—	2	2	no	—
54	C₄⋊Dic₁₇	—	2	6	no	—
55	C₄⋊Dic₁₇	—	2	6	no	—
56	C₄⋊Dic₁₇	—	4	12	no	—
57	C₄⋊Dic₁₇	—	8	6	no	—
58	C₄⋊Dic₁₇	C₂×Q₈	1	4	yes	—
59	C₄×Dic₁₇	—	4	12	no	—
60	Q₈×#106	—	17	1	no	maximal, Sylow
61	C₂³×C₁₇	C₂²×D₄	1	1	yes	characteristic
62	Q₈×C₁₇	#106	1	16	yes	complemented
63	C₂×C₄×C₁₇	—	2	6	no	—
64	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	6	yes	—
65	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	6	yes	—
66	C₂×C₄×C₁₇	C₂×SD₁₆	1	12	yes	—
67	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
71	C₂×C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
72	C₂×C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
73	C₂×Dic₁₇	—	2	4	no	—
74	C₂×Dic₁₇	—	4	2	no	—
75	C₂×Dic₁₇	—	4	2	no	—
76	C₂×Dic₁₇	—	4	6	no	—
77	C₂×Dic₁₇	—	4	6	no	—
78	C₂×Dic₁₇	—	4	6	no	—
79	C₈×C₁₇	—	2	6	no	—
80	C₈×C₁₇	C₄×Q₈	1	4	yes	complemented
81	C₂×C₈×Q₈	D₁₇	1	1	yes	characteristic
82	Q₈×C₄⋊C₄	—	17	2	no	—
83	#1948	—	17	6	no	—
84	#1844	—	17	3	no	—
85	C₄×#106	—	17	3	no	—
86	C₂²×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₁₇	D₄×Q₈	1	1	yes	characteristic
88	C₂²×C₁₇	#550	1	1	yes	characteristic
89	C₂²×C₁₇	C₂²×SD₁₆	1	2	yes	—
90	C₂²×C₁₇	#577	1	2	yes	—
91	C₄×C₁₇	—	2	6	no	—
92	C₄×C₁₇	C₂×#106	1	12	yes	—
93	C₄×C₁₇	C₂×C₄×Q₈	1	1	yes	characteristic
94	C₄×C₁₇	#520	1	1	yes	characteristic
95	C₄×C₁₇	#557	1	1	yes	characteristic
96	C₄×C₁₇	Q₈²	1	1	yes	characteristic
97	Dic₁₇	—	4	4	no	—
98	Dic₁₇	—	8	6	no	—
99	C₂×C₄×Q₈	—	34	2	no	—
100	C₂×C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
101	C₈×Q₈	Dic₁₇	1	8	yes	complemented
102	C₂×#106	—	17	1	no	—
103	C₂×#106	—	17	3	no	—
104	C₂×#106	—	17	6	no	—
105	C₂×C₄⋊C₈	C₂×D₁₇	1	3	yes	—
106	C₂×C₄×C₈	C₂×D₁₇	1	3	yes	—
107	#389	—	34	6	no	—
108	#384	—	17	6	no	—
109	C₄×C₄⋊C₄	—	17	6	no	—
110	#340	—	17	6	no	—
111	C₂×C₁₇	Q₈×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₁₇	C₂²×#106	1	1	yes	characteristic
113	C₂×C₁₇	#2912	1	1	yes	characteristic
114	C₂×C₁₇	Q₈×SD₁₆	1	2	yes	—
115	C₂×C₁₇	#3392	1	2	yes	—
116	C₂²×Q₈	D₆₈	1	1	yes	characteristic
117	C₂²×C₈	C₂²×D₁₇	1	3	yes	—
118	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
119	C₄×Q₈	—	68	8	no	—
120	C₄×Q₈	Dic₃₄	1	4	yes	—
121	C₄×Q₈	C₂×Dic₁₇	1	4	yes	—
122	C₂×C₄⋊C₄	—	17	2	no	—
123	C₂×C₄⋊C₄	—	17	6	no	—
124	C₂×C₄⋊C₄	—	34	6	no	—
125	C₂×C₄⋊C₄	—	34	6	no	—
126	C₂×C₄⋊C₄	D₆₈	1	3	yes	—
127	C₂×C₄²	—	34	6	no	—
128	C₂×C₄²	C₂²×D₁₇	1	3	yes	—
129	#106	—	17	4	no	—
130	#106	—	34	6	no	—
131	#106	—	68	6	no	—
132	C₄⋊C₈	C₂×Dic₁₇	1	12	yes	—
133	C₄×C₈	C₂×Dic₁₇	1	12	yes	—
134	#95	—	34	6	no	—
135	C₁₇	Q₈×#106	1	1	yes	characteristic, complemented, Sylow
136	C₂×Q₈	#63219	1	8	yes	—
137	C₂×Q₈	C₄⋊Dic₁₇	1	4	yes	—
138	C₂²×C₄	—	34	2	no	—
139	C₂²×C₄	—	34	6	no	—
140	C₂²×C₄	C₂×D₆₈	1	3	yes	—
141	C₂²×C₄	C₂×D₆₈	1	3	yes	—
142	C₂²×C₄	C₂³×D₁₇	1	1	yes	characteristic
143	C₂×C₈	—	2	3	no	—
144	C₂×C₈	—	2	3	no	—
145	C₂×C₈	Q₈×D₁₇	1	2	yes	—
146	C₂×C₈	#63256	1	2	yes	—
147	C₂×C₈	C₂²×Dic₁₇	1	6	yes	—
148	C₂×C₈	C₂²×Dic₁₇	1	2	yes	—
149	C₄⋊C₄	—	17	4	no	—
150	C₄⋊C₄	—	34	2	no	—
151	C₄⋊C₄	—	34	6	no	—
152	C₄⋊C₄	—	34	6	no	—
153	C₄⋊C₄	—	68	12	no	—
154	C₄⋊C₄	—	136	6	no	—
155	C₄⋊C₄	C₄⋊Dic₁₇	1	12	yes	—
156	C₄²	—	68	12	no	—
157	C₄²	C₂×Dic₃₄	1	6	yes	—
158	C₄²	C₂²×Dic₁₇	1	6	yes	—
159	C₂³	C₂²×D₆₈	1	1	yes	characteristic
160	Q₈	#10587488	1	16	yes	direct product factor
161	C₂×C₄	—	2	6	no	—
162	C₂×C₄	—	34	4	no	—
163	C₂×C₄	—	68	2	no	—
164	C₂×C₄	—	68	2	no	—
165	C₂×C₄	—	68	6	no	—
166	C₂×C₄	—	68	6	no	—
167	C₂×C₄	—	68	6	no	—
168	C₂×C₄	C₂×#63219	1	12	yes	—
169	C₂×C₄	C₂×C₄⋊Dic₁₇	1	6	yes	—
170	C₂×C₄	C₂×C₄⋊Dic₁₇	1	6	yes	—
171	C₂×C₄	C₂²×Dic₃₄	1	1	yes	characteristic
172	C₂×C₄	C₂×Q₈×D₁₇	1	1	yes	characteristic
173	C₂×C₄	C₂×#63256	1	1	yes	characteristic
174	C₂×C₄	#10587696	1	1	yes	characteristic
175	C₂×C₄	#10587697	1	1	yes	characteristic
176	C₂×C₄	C₂³×Dic₁₇	1	1	yes	characteristic
177	C₈	—	2	6	no	—
178	C₈	Q₈×Dic₁₇	1	4	yes	complemented
179	C₂²	C₂²×C₄⋊Dic₁₇	1	1	yes	characteristic
180	C₂²	Q₈×D₆₈	1	1	yes	characteristic
181	C₂²	#11763481	1	1	yes	characteristic
182	C₂²	C₂²×#63219	1	2	yes	—
183	C₂²	#11763621	1	2	yes	—
184	C₄	—	2	6	no	—
185	C₄	—	68	4	no	—
186	C₄	—	136	6	no	—
187	C₄	C₂×#10587488	1	12	yes	—
188	C₄	C₂×Q₈×Dic₁₇	1	1	yes	characteristic
189	C₄	#11763292	1	1	yes	characteristic
190	C₄	#11763538	1	1	yes	characteristic
191	C₄	Q₈×Dic₃₄	1	1	yes	characteristic
192	C₂	Q₈×C₄⋊Dic₁₇	1	1	yes	characteristic
193	C₂	C₂²×#10587488	1	1	yes	characteristic
194	C₂	#14537379	1	1	yes	characteristic
195	C₂	Q₈×#63219	1	2	yes	—
196	C₂	#14541766	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₆₈
commutator	C₂×C₄×C₁₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₇
Fitting	C₂×C₈×Q₈×C₁₇	C₂
socle, prime radical	C₂³×C₁₇	C₂²×D₄
cosocle	C₂²×C₄×C₁₇	C₂⁴
hypercenter	C₂×C₈×Q₈	D₁₇
hypocenter	C₁₇	Q₈×#106
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
15. C₈×Q₈×C₁₇	C₄	semidirect
36. #10587488	Q₈	direct (N × H)
62. Q₈×C₁₇	#106	semidirect
80. C₈×C₁₇	C₄×Q₈	semidirect
101. C₈×Q₈	Dic₁₇	semidirect
135. C₁₇	Q₈×#106	semidirect
160. Q₈	#10587488	direct (N × H)
178. C₈	Q₈×Dic₁₇	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₄⋊Dic₁₇	2
#14529548	2
#14528941	2
C₄×#10587488	2
C₂×C₈×Q₈×C₁₇	2
Q₈×#106	17

Subgroups — Q8×#10587488 #17053151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#10587488 #17053151