Subgroups — Q₈×#10587490 #17053217

Order 4352.

658 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#63223	C₂	1	2	yes	maximal
2	#14529560	C₂	1	6	yes	maximal
3	#14528956	C₂	1	3	yes	maximal
4	C₄×#10587490	C₂	1	3	yes	maximal
5	C₂×C₈×Q₈×C₁₇	C₂	1	1	yes	maximal, characteristic
6	#11762940	—	2	6	no	—
7	Q₈×#3622	—	2	2	no	—
8	C₂×#10587490	C₂²	1	3	yes	—
9	C₂×#10587490	C₂²	1	6	yes	—
10	C₂×#10587490	C₂²	1	1	yes	characteristic
11	#11762833	C₂²	1	6	yes	—
12	C₄×#63223	C₂²	1	6	yes	—
13	#11762574	C₂²	1	6	yes	—
14	C₂×C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
15	C₈×Q₈×C₁₇	C₄	1	6	yes	—
16	C₈×Q₈×C₁₇	C₄	1	2	yes	—
17	C₂×C₁₇×C₄⋊C₈	C₂²	1	3	yes	—
18	C₂×C₄×C₈×C₁₇	C₂²	1	3	yes	—
19	C₂²×Q₈×C₁₇	Q₈	1	1	yes	characteristic
20	C₂²×C₈×C₁₇	C₂³	1	3	yes	—
21	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
22	C₄×Q₈×C₁₇	C₂×C₄	1	3	yes	—
23	C₄×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₄×Q₈×C₁₇	D₄	1	1	yes	characteristic
25	C₄×Q₈×C₁₇	D₄	1	3	yes	—
26	C₂×C₄⋊C₄×C₁₇	Q₈	1	3	yes	—
27	C₂×C₄²×C₁₇	C₂³	1	3	yes	—
28	C₂×#63223	C₂³	1	6	yes	—
29	C₂×#63223	C₂³	1	2	yes	—
30	C₁₇×C₄⋊C₈	C₂×C₄	1	6	yes	—
31	C₁₇×C₄⋊C₈	C₂×C₄	1	6	yes	—
32	C₄×C₈×C₁₇	C₂×C₄	1	6	yes	—
33	C₄×C₈×C₁₇	C₂×C₄	1	6	yes	—
34	#10587490	—	2	6	no	—
35	#10587490	—	4	6	no	—
36	#10587490	Q₈	1	4	yes	direct product factor
37	C₄⋊#3622	—	2	6	no	—
38	C₄⋊#3622	—	2	6	no	—
39	#10587474	—	2	6	no	—
40	C₄×#3622	—	2	6	no	—
41	C₂×Q₈×C₁₇	—	2	4	no	—
42	C₂×Q₈×C₁₇	C₄⋊C₄	1	3	yes	—
43	C₂×Q₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₇	C₂×Q₈	1	3	yes	—
45	C₂²×C₄×C₁₇	C₂×Q₈	1	3	yes	—
46	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
47	C₂×C₈×C₁₇	—	2	3	no	—
48	C₂×C₈×C₁₇	—	2	3	no	—
49	C₂×C₈×C₁₇	C₂²×C₄	1	6	yes	—
50	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
51	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
52	C₂×C₈×C₁₇	#41	1	2	yes	—
53	C₄⋊C₄×C₁₇	C₄⋊C₄	1	6	yes	—
54	C₄⋊C₄×C₁₇	C₄⋊C₄	1	3	yes	—
55	C₄⋊C₄×C₁₇	C₄⋊C₄	1	3	yes	—
56	C₄²×C₁₇	C₂²×C₄	1	3	yes	—
57	C₄²×C₁₇	C₂²×C₄	1	3	yes	—
58	C₄²×C₁₇	C₂×D₄	1	3	yes	—
59	C₄²×C₁₇	C₂×D₄	1	3	yes	—
60	#63223	—	2	2	no	—
61	#63223	—	2	6	no	—
62	#63223	—	2	6	no	—
63	#63223	C₂×Q₈	1	4	yes	—
64	C₂×#3622	—	2	2	no	—
65	C₂×#3622	—	2	6	no	—
66	Q₈×#108	—	17	1	no	maximal, Sylow
67	C₂³×C₁₇	C₂²×Q₈	1	1	yes	characteristic
68	Q₈×C₁₇	—	2	6	no	—
69	Q₈×C₁₇	#108	1	4	yes	complemented
70	C₂×C₄×C₁₇	—	2	6	no	—
71	C₂×C₄×C₁₇	—	4	3	no	—
72	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	3	yes	—
73	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	3	yes	—
74	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	3	yes	—
75	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	3	yes	—
76	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
78	C₂×C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
80	C₂×C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
81	C₂×C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
82	C₈×C₁₇	—	2	6	no	—
83	C₈×C₁₇	C₄×Q₈	1	4	yes	—
84	#3622	—	2	4	no	—
85	#3622	—	4	6	no	—
86	Q₈×M₁₆	—	17	2	no	—
87	C₂×C₈×Q₈	D₁₇	1	1	yes	characteristic
88	#1950	—	17	6	no	—
89	#1848	—	17	3	no	—
90	C₄×#108	—	17	3	no	—
91	C₂²×C₁₇	—	2	1	no	—
92	C₂²×C₁₇	—	2	1	no	—
93	C₂²×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
94	C₂²×C₁₇	#557	1	1	yes	characteristic
95	C₂²×C₁₇	Q₈²	1	1	yes	characteristic
96	C₄×C₁₇	—	2	3	no	—
97	C₄×C₁₇	—	2	3	no	—
98	C₄×C₁₇	—	2	3	no	—
99	C₄×C₁₇	C₂×#108	1	6	yes	—
100	C₄×C₁₇	C₂×C₄×Q₈	1	1	yes	characteristic
101	C₄×C₁₇	#520	1	1	yes	characteristic
102	C₄×C₁₇	D₄×Q₈	1	1	yes	characteristic
103	C₄×C₁₇	#550	1	1	yes	characteristic
104	C₂×C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
105	#446	—	34	6	no	—
106	C₈×Q₈	—	34	2	no	—
107	C₈×Q₈	Dic₁₇	1	6	yes	—
108	C₈×Q₈	Dic₁₇	1	2	yes	—
109	C₂×#108	—	17	1	no	—
110	C₂×#108	—	17	3	no	—
111	C₂×#108	—	17	6	no	—
112	#423	—	17	6	no	—
113	C₂×C₄⋊C₈	C₂×D₁₇	1	3	yes	—
114	C₄×M₁₆	—	17	6	no	—
115	C₂×C₄×C₈	C₂×D₁₇	1	3	yes	—
116	#341	—	17	6	no	—
117	C₂×C₁₇	—	2	1	no	—
118	C₂×C₁₇	—	2	1	no	—
119	C₂×C₁₇	Q₈×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₁₇	C₂²×#108	1	1	yes	characteristic
121	C₂×C₁₇	#2916	1	1	yes	characteristic
122	C₂²×Q₈	Dic₃₄	1	1	yes	characteristic
123	C₂×M₁₆	—	17	2	no	—
124	C₂×M₁₆	—	17	6	no	—
125	C₂²×C₈	C₂²×D₁₇	1	3	yes	—
126	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
127	C₄×Q₈	D₆₈	1	1	yes	characteristic
128	C₄×Q₈	D₆₈	1	3	yes	—
129	C₄×Q₈	C₂×Dic₁₇	1	3	yes	—
130	C₄×Q₈	C₂×Dic₁₇	1	1	yes	characteristic
131	C₂×C₄⋊C₄	Dic₃₄	1	3	yes	—
132	C₂×C₄²	C₂²×D₁₇	1	3	yes	—
133	#108	—	17	4	no	—
134	#108	—	34	6	no	—
135	#108	—	68	6	no	—
136	C₄⋊C₈	—	34	6	no	—
137	C₄⋊C₈	—	34	6	no	—
138	C₄⋊C₈	C₂×Dic₁₇	1	6	yes	—
139	C₄⋊C₈	C₂×Dic₁₇	1	6	yes	—
140	#97	—	34	6	no	—
141	C₄×C₈	—	34	6	no	—
142	C₄×C₈	C₂×Dic₁₇	1	6	yes	—
143	C₄×C₈	C₂×Dic₁₇	1	6	yes	—
144	C₁₇	Q₈×#108	1	1	yes	characteristic, complemented, Sylow
145	C₂×Q₈	—	2	4	no	—
146	C₂×Q₈	C₄⋊Dic₁₇	1	3	yes	—
147	C₂×Q₈	C₄⋊Dic₁₇	1	1	yes	characteristic
148	C₂²×C₄	C₂×Dic₃₄	1	3	yes	—
149	C₂²×C₄	C₂×Dic₃₄	1	3	yes	—
150	C₂²×C₄	C₂³×D₁₇	1	1	yes	characteristic
151	M₁₆	—	17	4	no	—
152	M₁₆	—	34	2	no	—
153	M₁₆	—	34	6	no	—
154	M₁₆	—	34	6	no	—
155	C₂×C₈	—	2	3	no	—
156	C₂×C₈	—	2	3	no	—
157	C₂×C₈	—	34	2	no	—
158	C₂×C₈	—	34	6	no	—
159	C₂×C₈	Q₈×D₁₇	1	2	yes	—
160	C₂×C₈	#63256	1	2	yes	—
161	C₂×C₈	C₂²×Dic₁₇	1	6	yes	—
162	C₂×C₈	C₂²×Dic₁₇	1	2	yes	—
163	C₄⋊C₄	C₄⋊Dic₁₇	1	6	yes	—
164	C₄⋊C₄	C₄⋊Dic₁₇	1	3	yes	—
165	C₄⋊C₄	C₄⋊Dic₁₇	1	3	yes	—
166	C₄²	C₂×D₆₈	1	3	yes	—
167	C₄²	C₂×D₆₈	1	3	yes	—
168	C₄²	C₂²×Dic₁₇	1	3	yes	—
169	C₄²	C₂²×Dic₁₇	1	3	yes	—
170	C₂³	C₂²×Dic₃₄	1	1	yes	characteristic
171	Q₈	—	2	6	no	—
172	Q₈	#10587490	1	4	yes	direct product factor
173	C₂×C₄	—	2	6	no	—
174	C₂×C₄	—	4	3	no	—
175	C₂×C₄	C₂×C₄⋊Dic₁₇	1	3	yes	—
176	C₂×C₄	C₂×C₄⋊Dic₁₇	1	3	yes	—
177	C₂×C₄	C₂×C₄⋊Dic₁₇	1	3	yes	—
178	C₂×C₄	C₂×C₄⋊Dic₁₇	1	3	yes	—
179	C₂×C₄	C₂²×D₆₈	1	1	yes	characteristic
180	C₂×C₄	C₂×Q₈×D₁₇	1	1	yes	characteristic
181	C₂×C₄	C₂×#63256	1	1	yes	characteristic
182	C₂×C₄	#10587696	1	1	yes	characteristic
183	C₂×C₄	#10587697	1	1	yes	characteristic
184	C₂×C₄	C₂³×Dic₁₇	1	1	yes	characteristic
185	C₈	—	2	6	no	—
186	C₈	—	34	4	no	—
187	C₈	—	68	6	no	—
188	C₈	Q₈×Dic₁₇	1	4	yes	—
189	C₂²	—	2	1	no	—
190	C₂²	—	2	1	no	—
191	C₂²	C₂²×C₄⋊Dic₁₇	1	1	yes	characteristic
192	C₂²	#11763538	1	1	yes	characteristic
193	C₂²	Q₈×Dic₃₄	1	1	yes	characteristic
194	C₄	—	2	3	no	—
195	C₄	—	2	3	no	—
196	C₄	—	2	3	no	—
197	C₄	C₂×#10587490	1	6	yes	—
198	C₄	C₂×Q₈×Dic₁₇	1	1	yes	characteristic
199	C₄	#11763292	1	1	yes	characteristic
200	C₄	Q₈×D₆₈	1	1	yes	characteristic
201	C₄	#11763481	1	1	yes	characteristic
202	C₂	—	2	1	no	—
203	C₂	—	2	1	no	—
204	C₂	Q₈×C₄⋊Dic₁₇	1	1	yes	characteristic
205	C₂	C₂²×#10587490	1	1	yes	characteristic
206	C₂	#14537395	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₆₈
commutator	C₂×C₄×C₁₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₇
Fitting	C₂×C₈×Q₈×C₁₇	C₂
socle	C₂²×C₁₇	C₂²×C₄⋊C₄
prime radical	C₂³×C₁₇	C₂²×Q₈
cosocle	C₂²×C₄×C₁₇	C₂⁴
hypercenter	C₂×C₈×Q₈	D₁₇
hypocenter	C₁₇	Q₈×#108
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
36. #10587490	Q₈	direct (N × H)
69. Q₈×C₁₇	#108	semidirect
144. C₁₇	Q₈×#108	semidirect
172. Q₈	#10587490	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×#63223	2
#14529560	2
#14528956	2
C₄×#10587490	2
C₂×C₈×Q₈×C₁₇	2
Q₈×#108	17

Subgroups — Q8×#10587490 #17053217

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#10587490 #17053217