Subgroups — #17053329

Order 4352.

642 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	Q₈×#63223	C₂	1	2	yes	maximal
2	#14529560	C₂	1	6	yes	maximal
3	#14528922	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	6	yes	—
9	#11762833	C₂²	1	6	yes	—
10	C₄×#63223	C₂²	1	6	yes	—
11	#11762574	C₂²	1	6	yes	—
12	#11762574	C₂²	1	3	yes	—
13	#11762574	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
21	C₂²×C₈×C₁₇	C₂³	1	3	yes	—
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	12	no	—
35	C₄⋊#3622	—	2	6	no	—
36	C₄⋊#3622	—	2	6	no	—
37	#10587474	—	2	6	no	—
38	C₄×#3622	—	2	6	no	—
39	C₂×Q₈×C₁₇	—	2	4	no	—
40	C₂×Q₈×C₁₇	C₄⋊C₄	1	3	yes	—
41	C₂×Q₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₇	C₂×Q₈	1	3	yes	—
43	C₂²×C₄×C₁₇	C₂×Q₈	1	3	yes	—
44	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
45	C₂×C₈×C₁₇	—	2	6	no	—
46	C₂×C₈×C₁₇	—	2	2	no	—
47	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
48	C₂×C₈×C₁₇	C₂²×C₄	1	6	yes	—
49	C₄⋊C₄×C₁₇	C₄⋊C₄	1	6	yes	—
50	C₄⋊C₄×C₁₇	C₄⋊C₄	1	3	yes	—
51	C₄⋊C₄×C₁₇	C₄⋊C₄	1	3	yes	—
52	C₄²×C₁₇	C₂²×C₄	1	3	yes	—
53	C₄²×C₁₇	C₂²×C₄	1	3	yes	—
54	C₄²×C₁₇	C₂×D₄	1	3	yes	—
55	C₄²×C₁₇	C₂×D₄	1	3	yes	—
56	#63223	—	2	2	no	—
57	#63223	—	2	6	no	—
58	#63223	—	2	6	no	—
59	#63223	#41	1	4	yes	—
60	C₂×#3622	—	2	2	no	—
61	C₂×#3622	—	2	6	no	—
62	#22746	—	17	1	no	maximal, Sylow
63	C₂³×C₁₇	C₂²×Q₈	1	1	yes	characteristic
64	Q₈×C₁₇	—	2	6	no	—
65	Q₈×C₁₇	#108	1	4	yes	complemented
66	C₂×C₄×C₁₇	—	2	6	no	—
67	C₂×C₄×C₁₇	—	4	3	no	—
68	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	3	yes	—
69	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	3	yes	—
70	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	3	yes	—
71	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	3	yes	—
72	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
74	C₂×C₄×C₁₇	C₂×#41	1	2	yes	—
75	C₂×C₄×C₁₇	2₋¹⁺⁴	1	2	yes	—
76	C₈×C₁₇	—	2	6	no	—
77	C₈×C₁₇	—	2	2	no	—
78	#3622	—	2	4	no	—
79	#3622	—	4	6	no	—
80	Q₈×M₁₆	—	17	2	no	—
81	C₂×C₈×Q₈	D₁₇	1	1	yes	characteristic
82	#1950	—	17	6	no	—
83	#1837	—	17	3	no	—
84	C₄×#108	—	17	3	no	—
85	C₂²×C₁₇	—	2	1	no	—
86	C₂²×C₁₇	—	2	1	no	—
87	C₂²×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₁₇	#557	1	2	yes	—
89	C₄×C₁₇	—	2	3	no	—
90	C₄×C₁₇	—	2	3	no	—
91	C₄×C₁₇	—	2	3	no	—
92	C₄×C₁₇	C₂×#108	1	6	yes	—
93	C₄×C₁₇	C₄×#41	1	1	yes	characteristic
94	C₄×C₁₇	#519	1	1	yes	characteristic
95	C₄×C₁₇	#548	1	2	yes	—
96	C₂×C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
97	#446	—	34	6	no	—
98	C₈×Q₈	—	34	2	no	—
99	C₈×Q₈	Dic₁₇	1	6	yes	—
100	C₈×Q₈	Dic₁₇	1	2	yes	—
101	C₂×#108	—	17	6	no	—
102	#423	—	17	6	no	—
103	C₂×C₄⋊C₈	C₂×D₁₇	1	3	yes	—
104	C₄×M₁₆	—	17	6	no	—
105	C₂×C₄×C₈	C₂×D₁₇	1	3	yes	—
106	#341	—	17	3	no	—
107	#341	—	17	6	no	—
108	#341	—	17	1	no	—
109	C₂×C₁₇	—	2	2	no	—
110	C₂×C₁₇	#2351	1	1	yes	characteristic
111	C₂×C₁₇	C₂²×#108	1	1	yes	characteristic
112	C₂×C₁₇	#2916	1	1	yes	characteristic
113	C₂²×Q₈	Dic₃₄	1	1	yes	characteristic
114	C₂×M₁₆	—	17	2	no	—
115	C₂×M₁₆	—	17	6	no	—
116	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
117	C₂²×C₈	C₂²×D₁₇	1	3	yes	—
118	C₄×Q₈	D₆₈	1	1	yes	characteristic
119	C₄×Q₈	D₆₈	1	3	yes	—
120	C₄×Q₈	C₂×Dic₁₇	1	3	yes	—
121	C₄×Q₈	C₂×Dic₁₇	1	1	yes	characteristic
122	C₂×C₄⋊C₄	Dic₃₄	1	3	yes	—
123	C₂×C₄²	C₂²×D₁₇	1	3	yes	—
124	#108	—	34	12	no	—
125	C₄⋊C₈	—	34	6	no	—
126	C₄⋊C₈	—	34	6	no	—
127	C₄⋊C₈	C₂×Dic₁₇	1	6	yes	—
128	C₄⋊C₈	C₂×Dic₁₇	1	6	yes	—
129	#97	—	34	6	no	—
130	C₄×C₈	—	34	6	no	—
131	C₄×C₈	C₂×Dic₁₇	1	6	yes	—
132	C₄×C₈	C₂×Dic₁₇	1	6	yes	—
133	C₁₇	#22746	1	1	yes	characteristic, complemented, Sylow
134	C₂×Q₈	—	2	4	no	—
135	C₂×Q₈	C₄⋊Dic₁₇	1	3	yes	—
136	C₂×Q₈	C₄⋊Dic₁₇	1	1	yes	characteristic
137	C₂²×C₄	C₂×Dic₃₄	1	3	yes	—
138	C₂²×C₄	C₂×Dic₃₄	1	3	yes	—
139	C₂²×C₄	C₂³×D₁₇	1	1	yes	characteristic
140	M₁₆	—	17	4	no	—
141	M₁₆	—	34	2	no	—
142	M₁₆	—	34	6	no	—
143	M₁₆	—	34	6	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	6	no	—
146	C₂×C₈	—	34	2	no	—
147	C₂×C₈	—	34	6	no	—
148	C₂×C₈	C₂²×Dic₁₇	1	2	yes	—
149	C₂×C₈	C₂²×Dic₁₇	1	6	yes	—
150	C₄⋊C₄	C₄⋊Dic₁₇	1	6	yes	—
151	C₄⋊C₄	C₄⋊Dic₁₇	1	3	yes	—
152	C₄⋊C₄	C₄⋊Dic₁₇	1	3	yes	—
153	C₄²	C₂×D₆₈	1	3	yes	—
154	C₄²	C₂×D₆₈	1	3	yes	—
155	C₄²	C₂²×Dic₁₇	1	3	yes	—
156	C₄²	C₂²×Dic₁₇	1	3	yes	—
157	C₂³	C₂²×Dic₃₄	1	1	yes	characteristic
158	Q₈	—	2	6	no	—
159	Q₈	#10587490	1	4	yes	complemented
160	C₂×C₄	—	2	6	no	—
161	C₂×C₄	—	4	3	no	—
162	C₂×C₄	C₂×C₄⋊Dic₁₇	1	3	yes	—
163	C₂×C₄	C₂×C₄⋊Dic₁₇	1	3	yes	—
164	C₂×C₄	C₂×C₄⋊Dic₁₇	1	3	yes	—
165	C₂×C₄	C₂×C₄⋊Dic₁₇	1	3	yes	—
166	C₂×C₄	C₂²×D₆₈	1	1	yes	characteristic
167	C₂×C₄	#10587694	1	2	yes	—
168	C₂×C₄	#41×D₁₇	1	2	yes	—
169	C₂×C₄	C₂³×Dic₁₇	1	1	yes	characteristic
170	C₈	—	2	6	no	—
171	C₈	—	2	2	no	—
172	C₈	—	34	4	no	—
173	C₈	—	68	6	no	—
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	C₂²×C₄⋊Dic₁₇	1	1	yes	characteristic
177	C₂²	#11763535	1	2	yes	—
178	C₄	—	2	3	no	—
179	C₄	—	2	3	no	—
180	C₄	—	2	3	no	—
181	C₄	C₂×#10587490	1	6	yes	—
182	C₄	#41×Dic₁₇	1	1	yes	characteristic
183	C₄	#11763287	1	1	yes	characteristic
184	C₄	#11763471	1	2	yes	—
185	C₂	—	2	2	no	—
186	C₂	#14531705	1	1	yes	characteristic
187	C₂	C₂²×#10587490	1	1	yes	characteristic
188	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	#22746
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
65. Q₈×C₁₇	#108	semidirect
133. C₁₇	#22746	semidirect
159. Q₈	#10587490	semidirect

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
#14528922	2
C₄×#10587490	2
C₂×C₈×Q₈×C₁₇	2
#22746	17