Subgroups — #17060914

Order 4352.

664 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14529721	C₂	1	4	yes	maximal
2	#14529343	C₂	1	8	yes	maximal
3	#14528927	C₂	1	2	yes	maximal
4	C₄×C₁₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#3622	C₂²	1	4	yes	—
6	C₂×C₄×#3622	C₂²	1	4	yes	—
7	#11762564	C₂²	1	8	yes	—
8	#11762564	C₂²	1	8	yes	—
9	#11762564	C₂²	1	4	yes	—
10	C₂×C₁₇×C₄⋊Q₈	C₄	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₁₇	C₂²	1	2	yes	—
12	C₁₇×#391	C₄	1	2	yes	—
13	C₁₇×#389	C₄	1	4	yes	—
14	C₁₇×#384	C₄	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₁₇	C₂²	1	4	yes	—
16	C₄³×C₁₇	C₂²	1	1	yes	characteristic
17	C₂²×Q₈×C₁₇	C₂×C₄	1	2	yes	—
18	C₁₇×C₄⋊Q₈	C₈	1	8	yes	complemented
19	C₄×Q₈×C₁₇	—	4	2	no	—
20	C₄×Q₈×C₁₇	—	4	2	no	—
21	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
22	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	4	yes	—
23	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	4	yes	—
24	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
29	C₂×C₄²×C₁₇	C₂³	1	4	yes	—
30	C₂×C₄²×C₁₇	C₂³	1	2	yes	—
31	C₂²×#3622	C₂³	1	8	yes	—
32	C₁₇×#95	C₂×C₄	1	8	yes	—
33	C₄⋊#3622	—	4	4	no	—
34	C₄×#3622	—	4	4	no	—
35	C₂×Q₈×C₁₇	—	4	2	no	—
36	C₂×Q₈×C₁₇	—	4	2	no	—
37	C₂×Q₈×C₁₇	C₂×C₈	1	8	yes	—
38	C₂²×C₄×C₁₇	C₂²×C₄	1	2	yes	—
39	C₂²×C₄×C₁₇	C₂²×C₄	1	4	yes	—
40	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₇	C₂²×C₄	1	4	yes	—
43	C₂²×C₄×C₁₇	C₂²×C₄	1	2	yes	—
44	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
45	C₄⋊C₄×C₁₇	—	4	4	no	—
46	C₄⋊C₄×C₁₇	—	4	2	no	—
47	C₄⋊C₄×C₁₇	—	4	4	no	—
48	C₄⋊C₄×C₁₇	C₂×C₈	1	16	yes	—
49	C₄²×C₁₇	—	2	4	no	—
50	C₄²×C₁₇	—	2	4	no	—
51	C₄²×C₁₇	—	2	2	no	—
52	C₄²×C₁₇	—	2	4	no	—
53	C₄²×C₁₇	—	4	2	no	—
54	C₄²×C₁₇	—	4	1	no	—
55	C₄²×C₁₇	C₂×C₈	1	4	yes	—
56	C₂×#3622	—	2	8	no	—
57	C₂×#3622	—	2	16	no	—
58	#23363	—	17	1	no	maximal, Sylow
59	C₂³×C₁₇	C₂³×C₄	1	1	yes	characteristic
60	Q₈×C₁₇	—	4	8	no	—
61	C₂×C₄×C₁₇	—	2	1	no	—
62	C₂×C₄×C₁₇	—	2	4	no	—
63	C₂×C₄×C₁₇	—	2	2	no	—
64	C₂×C₄×C₁₇	—	2	2	no	—
65	C₂×C₄×C₁₇	—	2	1	no	—
66	C₂×C₄×C₁₇	—	2	4	no	—
67	C₂×C₄×C₁₇	—	2	1	no	—
68	C₂×C₄×C₁₇	—	2	4	no	—
69	C₂×C₄×C₁₇	—	2	2	no	—
70	C₂×C₄×C₁₇	—	2	2	no	—
71	C₂×C₄×C₁₇	—	2	2	no	—
72	C₂×C₄×C₁₇	—	2	2	no	—
73	C₂×C₄×C₁₇	—	2	8	no	—
74	C₂×C₄×C₁₇	C₂²×C₈	1	4	yes	—
75	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
76	C₂×C₄×C₁₇	C₂²×C₈	1	8	yes	—
77	C₂×C₄×C₁₇	H(5,𝔽₂)	1	2	yes	—
78	C₂×C₄×C₁₇	2₋¹⁺⁴	1	4	yes	—
79	#3622	—	4	8	no	—
80	C₄×C₄⋊Q₈	D₁₇	1	1	yes	characteristic
81	#1984	—	17	4	no	—
82	#1916	—	17	8	no	—
83	#1839	—	17	2	no	—
84	C₂²×C₁₇	#518	1	1	yes	characteristic
85	C₂²×C₁₇	#519	1	2	yes	—
86	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
87	C₂²×C₁₇	#568	1	1	yes	characteristic
88	C₂²×C₁₇	#568	1	2	yes	—
89	C₄×C₁₇	—	2	8	no	—
90	C₄×C₁₇	—	2	2	no	—
91	C₄×C₁₇	—	2	4	no	—
92	C₄×C₁₇	—	4	1	no	—
93	C₄×C₁₇	—	4	4	no	—
94	C₄×C₁₇	—	4	2	no	—
95	C₄×C₁₇	#563	1	4	yes	—
96	C₂×C₄⋊Q₈	Dic₁₇	1	1	yes	characteristic
97	C₂×C₄×Q₈	C₂×D₁₇	1	2	yes	—
98	C₂×C₄⋊C₈	—	17	4	no	—
99	C₂×C₄×C₈	—	17	4	no	—
100	#391	Dic₁₇	1	2	yes	—
101	#389	Dic₁₇	1	4	yes	—
102	#384	Dic₁₇	1	1	yes	characteristic
103	C₄×C₄⋊C₄	C₂×D₁₇	1	4	yes	—
104	C₄³	C₂×D₁₇	1	1	yes	characteristic
105	#336	—	17	8	no	—
106	#336	—	17	4	no	—
107	#336	—	17	8	no	—
108	C₂×C₁₇	#2381	1	1	yes	characteristic
109	C₂×C₁₇	#2972	1	1	yes	characteristic
110	C₂×C₁₇	#2980	1	1	yes	characteristic
111	C₂×C₁₇	#2982	1	2	yes	—
112	C₂×C₁₇	#2987	1	2	yes	—
113	C₂²×Q₈	C₂×Dic₁₇	1	2	yes	—
114	C₂²×C₈	—	17	8	no	—
115	C₄⋊Q₈	#3622	1	8	yes	complemented
116	C₄×Q₈	—	4	2	no	—
117	C₄×Q₈	—	4	2	no	—
118	C₂×C₄⋊C₄	C₂×Dic₁₇	1	2	yes	—
119	C₂×C₄⋊C₄	C₂×Dic₁₇	1	4	yes	—
120	C₂×C₄⋊C₄	C₂×Dic₁₇	1	4	yes	—
121	C₂×C₄⋊C₄	C₂×Dic₁₇	1	4	yes	—
122	C₂×C₄²	C₂×Dic₁₇	1	2	yes	—
123	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
124	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
125	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
126	C₂×C₄²	C₂²×D₁₇	1	4	yes	—
127	C₂×C₄²	C₂²×D₁₇	1	2	yes	—
128	C₄⋊C₈	—	68	4	no	—
129	C₄×C₈	—	68	4	no	—
130	#95	C₂×Dic₁₇	1	8	yes	—
131	C₁₇	#23363	1	1	yes	characteristic, complemented, Sylow
132	C₂×Q₈	—	4	2	no	—
133	C₂×Q₈	—	4	2	no	—
134	C₂×Q₈	C₂×#3622	1	8	yes	—
135	C₂²×C₄	C₂²×Dic₁₇	1	2	yes	—
136	C₂²×C₄	C₂²×Dic₁₇	1	4	yes	—
137	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
138	C₂²×C₄	C₂²×Dic₁₇	1	4	yes	—
139	C₂²×C₄	C₂²×Dic₁₇	1	2	yes	—
140	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
141	C₂²×C₄	C₂³×D₁₇	1	1	yes	characteristic
142	C₂×C₈	—	34	8	no	—
143	C₂×C₈	—	34	16	no	—
144	C₄⋊C₄	—	4	4	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	4	4	no	—
147	C₄⋊C₄	C₂×#3622	1	16	yes	—
148	C₄²	—	2	4	no	—
149	C₄²	—	2	4	no	—
150	C₄²	—	2	2	no	—
151	C₄²	—	2	4	no	—
152	C₄²	—	4	2	no	—
153	C₄²	—	4	1	no	—
154	C₄²	C₂×#3622	1	4	yes	—
155	C₂³	C₂³×Dic₁₇	1	1	yes	characteristic
156	Q₈	—	4	8	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	8	no	—
170	C₂×C₄	C₂²×#3622	1	4	yes	—
171	C₂×C₄	C₂²×#3622	1	2	yes	—
172	C₂×C₄	C₂²×#3622	1	8	yes	—
173	C₂×C₄	#10587691	1	2	yes	—
174	C₂×C₄	#10587694	1	4	yes	—
175	C₈	—	68	8	no	—
176	C₂²	#11763284	1	1	yes	characteristic
177	C₂²	#11763287	1	2	yes	—
178	C₂²	C₂³×#3622	1	1	yes	characteristic
179	C₂²	#11763576	1	1	yes	characteristic
180	C₂²	#11763576	1	2	yes	—
181	C₄	—	2	2	no	—
182	C₄	—	2	8	no	—
183	C₄	—	2	4	no	—
184	C₄	—	4	1	no	—
185	C₄	—	4	4	no	—
186	C₄	—	4	2	no	—
187	C₄	#11763558	1	4	yes	—
188	C₂	#14531965	1	1	yes	characteristic
189	C₂	#14537808	1	1	yes	characteristic
190	C₂	#14537855	1	1	yes	characteristic
191	C₂	#14537865	1	2	yes	—
192	C₂	#14537890	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₇
commutator	C₂²×C₁₇	C₂³×C₈
Fitting	C₄×C₁₇×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₁₇	C₂³×C₄
cosocle	C₂²×C₄×C₁₇	C₂⁴
hypercenter	C₄×C₄⋊Q₈	D₁₇
hypocenter	C₁₇	#23363
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
18. C₁₇×C₄⋊Q₈	C₈	semidirect
115. C₄⋊Q₈	#3622	semidirect
131. C₁₇	#23363	semidirect

Maximal subgroups

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

subgroup	index
#14529721	2
#14529343	2
#14528927	2
C₄×C₁₇×C₄⋊Q₈	2
#23363	17