Subgroups — Q₈×#10587489 #17053164

Order 4352.

838 conjugacy classes of subgroups.

The subgroup lattice has 212 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	#14529543	C₂	1	6	yes	maximal
3	#14528944	C₂	1	3	yes	maximal
4	C₄×#10587489	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×#10587489	C₂²	1	3	yes	—
8	C₂×#10587489	C₂²	1	6	yes	—
9	C₂×#10587489	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	#10587489	—	2	6	no	—
35	#10587489	—	4	6	no	—
36	#10587489	Q₈	1	4	yes	direct product factor
37	C₂×Q₈×C₁₇	C₄⋊C₄	1	4	yes	—
38	C₂×Q₈×C₁₇	D₈	1	4	yes	—
39	C₂×Q₈×C₁₇	Q₁₆	1	4	yes	—
40	C₂²×C₄×C₁₇	C₂×D₄	1	3	yes	—
41	C₂²×C₄×C₁₇	C₂×D₄	1	3	yes	—
42	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
43	C₂²×Dic₁₇	—	2	2	no	—
44	C₂²×Dic₁₇	—	2	6	no	—
45	C₂×C₈×C₁₇	—	2	3	no	—
46	C₂×C₈×C₁₇	—	2	3	no	—
47	C₂×C₈×C₁₇	C₂²×C₄	1	6	yes	—
48	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
49	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
50	C₂×C₈×C₁₇	#41	1	2	yes	—
51	C₄⋊C₄×C₁₇	C₄⋊C₄	1	12	yes	—
52	C₄²×C₁₇	C₂²×C₄	1	6	yes	—
53	C₄²×C₁₇	C₂×Q₈	1	6	yes	—
54	C₄⋊Dic₁₇	—	2	2	no	—
55	C₄⋊Dic₁₇	—	2	6	no	—
56	C₄⋊Dic₁₇	—	2	6	no	—
57	C₄⋊Dic₁₇	—	4	12	no	—
58	C₄⋊Dic₁₇	—	8	6	no	—
59	C₄⋊Dic₁₇	C₂×Q₈	1	4	yes	—
60	C₄×Dic₁₇	—	4	12	no	—
61	Q₈×#107	—	17	1	no	maximal, Sylow
62	C₂³×C₁₇	C₂²×D₄	1	1	yes	characteristic
63	Q₈×C₁₇	#107	1	16	yes	complemented
64	C₂×C₄×C₁₇	—	2	3	no	—
65	C₂×C₄×C₁₇	—	2	3	no	—
66	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	6	yes	—
67	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	6	yes	—
68	C₂×C₄×C₁₇	C₂×D₈	1	6	yes	—
69	C₂×C₄×C₁₇	C₂×Q₁₆	1	6	yes	—
70	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
74	C₂×C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
75	C₂×C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
76	C₂×Dic₁₇	—	2	4	no	—
77	C₂×Dic₁₇	—	4	2	no	—
78	C₂×Dic₁₇	—	4	2	no	—
79	C₂×Dic₁₇	—	4	6	no	—
80	C₂×Dic₁₇	—	4	6	no	—
81	C₂×Dic₁₇	—	4	6	no	—
82	C₈×C₁₇	—	2	6	no	—
83	C₈×C₁₇	C₄×Q₈	1	4	yes	complemented
84	C₂×C₈×Q₈	D₁₇	1	1	yes	characteristic
85	Q₈×C₄⋊C₄	—	17	2	no	—
86	#1947	—	17	6	no	—
87	#1845	—	17	3	no	—
88	C₄×#107	—	17	3	no	—
89	C₂²×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₁₇	D₄×Q₈	1	1	yes	characteristic
91	C₂²×C₁₇	#550	1	1	yes	characteristic
92	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
93	C₂²×C₁₇	C₂²×Q₁₆	1	1	yes	characteristic
94	C₂²×C₁₇	#576	1	1	yes	characteristic
95	C₂²×C₁₇	#578	1	1	yes	characteristic
96	C₄×C₁₇	—	2	6	no	—
97	C₄×C₁₇	C₂×#107	1	12	yes	—
98	C₄×C₁₇	C₂×C₄×Q₈	1	1	yes	characteristic
99	C₄×C₁₇	#520	1	1	yes	characteristic
100	C₄×C₁₇	#557	1	1	yes	characteristic
101	C₄×C₁₇	Q₈²	1	1	yes	characteristic
102	Dic₁₇	—	4	4	no	—
103	Dic₁₇	—	8	6	no	—
104	C₂×C₄×Q₈	—	34	2	no	—
105	C₂×C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
106	C₈×Q₈	Dic₁₇	1	8	yes	complemented
107	C₂×#107	—	17	1	no	—
108	C₂×#107	—	17	3	no	—
109	C₂×#107	—	17	6	no	—
110	C₂×C₄⋊C₈	C₂×D₁₇	1	3	yes	—
111	C₂×C₄×C₈	C₂×D₁₇	1	3	yes	—
112	#389	—	34	6	no	—
113	#384	—	17	6	no	—
114	C₄×C₄⋊C₄	—	17	6	no	—
115	#340	—	17	6	no	—
116	C₂×C₁₇	Q₈×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₁₇	C₂²×#107	1	1	yes	characteristic
118	C₂×C₁₇	#2913	1	1	yes	characteristic
119	C₂×C₁₇	Q₈×D₈	1	1	yes	characteristic
120	C₂×C₁₇	Q₈×Q₁₆	1	1	yes	characteristic
121	C₂×C₁₇	#3391	1	1	yes	characteristic
122	C₂×C₁₇	#3395	1	1	yes	characteristic
123	C₂²×Q₈	D₆₈	1	1	yes	characteristic
124	C₂²×C₈	C₂²×D₁₇	1	3	yes	—
125	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
126	C₄×Q₈	—	68	8	no	—
127	C₄×Q₈	Dic₃₄	1	4	yes	—
128	C₄×Q₈	C₂×Dic₁₇	1	4	yes	—
129	C₂×C₄⋊C₄	—	17	2	no	—
130	C₂×C₄⋊C₄	—	17	6	no	—
131	C₂×C₄⋊C₄	—	34	6	no	—
132	C₂×C₄⋊C₄	—	34	6	no	—
133	C₂×C₄⋊C₄	D₆₈	1	3	yes	—
134	C₂×C₄²	—	34	6	no	—
135	C₂×C₄²	C₂²×D₁₇	1	3	yes	—
136	#107	—	17	4	no	—
137	#107	—	34	6	no	—
138	#107	—	68	6	no	—
139	C₄⋊C₈	C₂×Dic₁₇	1	12	yes	—
140	C₄×C₈	C₂×Dic₁₇	1	12	yes	—
141	#95	—	34	6	no	—
142	C₁₇	Q₈×#107	1	1	yes	characteristic, complemented, Sylow
143	C₂×Q₈	D₁₃₆	1	4	yes	—
144	C₂×Q₈	Dic₆₈	1	4	yes	—
145	C₂×Q₈	C₄⋊Dic₁₇	1	4	yes	—
146	C₂²×C₄	—	34	2	no	—
147	C₂²×C₄	—	34	6	no	—
148	C₂²×C₄	C₂×D₆₈	1	3	yes	—
149	C₂²×C₄	C₂×D₆₈	1	3	yes	—
150	C₂²×C₄	C₂³×D₁₇	1	1	yes	characteristic
151	C₂×C₈	—	2	3	no	—
152	C₂×C₈	—	2	3	no	—
153	C₂×C₈	Q₈×D₁₇	1	2	yes	—
154	C₂×C₈	#63256	1	2	yes	—
155	C₂×C₈	C₂²×Dic₁₇	1	6	yes	—
156	C₂×C₈	C₂²×Dic₁₇	1	2	yes	—
157	C₄⋊C₄	—	17	4	no	—
158	C₄⋊C₄	—	34	2	no	—
159	C₄⋊C₄	—	34	6	no	—
160	C₄⋊C₄	—	34	6	no	—
161	C₄⋊C₄	—	68	12	no	—
162	C₄⋊C₄	—	136	6	no	—
163	C₄⋊C₄	C₄⋊Dic₁₇	1	12	yes	—
164	C₄²	—	68	12	no	—
165	C₄²	C₂×Dic₃₄	1	6	yes	—
166	C₄²	C₂²×Dic₁₇	1	6	yes	—
167	C₂³	C₂²×D₆₈	1	1	yes	characteristic
168	Q₈	#10587489	1	16	yes	direct product factor
169	C₂×C₄	—	2	3	no	—
170	C₂×C₄	—	2	3	no	—
171	C₂×C₄	—	34	4	no	—
172	C₂×C₄	—	68	2	no	—
173	C₂×C₄	—	68	2	no	—
174	C₂×C₄	—	68	6	no	—
175	C₂×C₄	—	68	6	no	—
176	C₂×C₄	—	68	6	no	—
177	C₂×C₄	C₂×D₁₃₆	1	6	yes	—
178	C₂×C₄	C₂×Dic₆₈	1	6	yes	—
179	C₂×C₄	C₂×C₄⋊Dic₁₇	1	6	yes	—
180	C₂×C₄	C₂×C₄⋊Dic₁₇	1	6	yes	—
181	C₂×C₄	C₂²×Dic₃₄	1	1	yes	characteristic
182	C₂×C₄	C₂×Q₈×D₁₇	1	1	yes	characteristic
183	C₂×C₄	C₂×#63256	1	1	yes	characteristic
184	C₂×C₄	#10587696	1	1	yes	characteristic
185	C₂×C₄	#10587697	1	1	yes	characteristic
186	C₂×C₄	C₂³×Dic₁₇	1	1	yes	characteristic
187	C₈	—	2	6	no	—
188	C₈	Q₈×Dic₁₇	1	4	yes	complemented
189	C₂²	C₂²×C₄⋊Dic₁₇	1	1	yes	characteristic
190	C₂²	Q₈×D₆₈	1	1	yes	characteristic
191	C₂²	#11763481	1	1	yes	characteristic
192	C₂²	C₂²×D₁₃₆	1	1	yes	characteristic
193	C₂²	C₂²×Dic₆₈	1	1	yes	characteristic
194	C₂²	#11763614	1	1	yes	characteristic
195	C₂²	#11763626	1	1	yes	characteristic
196	C₄	—	2	6	no	—
197	C₄	—	68	4	no	—
198	C₄	—	136	6	no	—
199	C₄	C₂×#10587489	1	12	yes	—
200	C₄	C₂×Q₈×Dic₁₇	1	1	yes	characteristic
201	C₄	#11763292	1	1	yes	characteristic
202	C₄	#11763538	1	1	yes	characteristic
203	C₄	Q₈×Dic₃₄	1	1	yes	characteristic
204	C₂	Q₈×C₄⋊Dic₁₇	1	1	yes	characteristic
205	C₂	C₂²×#10587489	1	1	yes	characteristic
206	C₂	#14537384	1	1	yes	characteristic
207	C₂	Q₈×D₁₃₆	1	1	yes	characteristic
208	C₂	Q₈×Dic₆₈	1	1	yes	characteristic
209	C₂	#14541759	1	1	yes	characteristic
210	C₂	#14541790	1	1	yes	characteristic

	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×#107
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. #10587489	Q₈	direct (N × H)
63. Q₈×C₁₇	#107	semidirect
83. C₈×C₁₇	C₄×Q₈	semidirect
106. C₈×Q₈	Dic₁₇	semidirect
142. C₁₇	Q₈×#107	semidirect
168. Q₈	#10587489	direct (N × H)
188. 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
#14529543	2
#14528944	2
C₄×#10587489	2
C₂×C₈×Q₈×C₁₇	2
Q₈×#107	17

Subgroups — Q8×#10587489 #17053164

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#10587489 #17053164