Subgroups — C₁₇×#9703 #16867447

Order 4352.

538 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×#2868	C₂	1	1	yes	maximal, characteristic
2	C₁₇×#1831	C₂	1	1	yes	maximal, characteristic
3	C₁₇×#1767	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₇×#332	C₂	1	4	yes	maximal
5	C₂×C₁₇×C₄⋊Q₈	—	2	2	no	—
6	C₂×C₁₇×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₂×C₁₇×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₁₇×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₁₇×#97	—	2	1	no	—
10	C₁₇×#384	—	2	1	no	—
11	C₁₇×#384	—	2	1	no	—
12	C₁₇×#384	C₄	1	4	yes	—
13	C₄³×C₁₇	C₂²	1	1	yes	characteristic
14	C₁₇×#336	—	2	1	no	—
15	C₁₇×#336	—	2	2	no	—
16	C₁₇×#332	—	2	8	no	—
17	C₂²×Q₈×C₁₇	—	2	2	no	—
18	C₂²×C₈×C₁₇	—	2	2	no	—
19	C₂²×C₈×C₁₇	—	2	2	no	—
20	C₁₇×C₄⋊Q₈	—	2	4	no	—
21	C₁₇×C₄⋊Q₈	—	2	4	no	—
22	C₁₇×C₄⋊Q₈	—	4	8	no	—
23	C₂×C₄⋊C₄×C₁₇	—	2	4	no	—
24	C₂×C₄⋊C₄×C₁₇	—	2	4	no	—
25	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
26	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₁₇	D₄	1	2	yes	—
29	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
32	C₁₇×C₄⋊C₈	—	2	4	no	—
33	C₁₇×C₄⋊C₈	—	2	2	no	—
34	C₁₇×C₄⋊C₈	D₄	1	4	yes	—
35	C₁₇×#97	—	4	2	no	—
36	C₂×Q₈×C₁₇	—	4	2	no	—
37	C₂×Q₈×C₁₇	—	4	2	no	—
38	C₂×Q₈×C₁₇	—	4	8	no	—
39	C₂²×C₄×C₁₇	—	2	4	no	—
40	C₂²×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₁₇	—	2	4	no	—
46	C₂×C₈×C₁₇	—	4	2	no	—
47	C₂×C₈×C₁₇	—	4	2	no	—
48	C₂×C₈×C₁₇	—	4	2	no	—
49	C₂×C₈×C₁₇	—	4	4	no	—
50	C₄⋊C₄×C₁₇	—	2	8	no	—
51	C₄⋊C₄×C₁₇	—	4	8	no	—
52	C₄⋊C₄×C₁₇	—	4	8	no	—
53	C₄⋊C₄×C₁₇	—	4	4	no	—
54	C₄²×C₁₇	—	2	4	no	—
55	C₄²×C₁₇	—	2	4	no	—
56	C₄²×C₁₇	—	2	2	no	—
57	C₄²×C₁₇	SD₁₆	1	2	yes	—
58	C₄²×C₁₇	Q₁₆	1	2	yes	—
59	C₄²×C₁₇	C₂×D₄	1	2	yes	—
60	C₄²×C₁₇	#41	1	2	yes	—
61	#9703	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂³×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	Q₈×C₁₇	—	8	4	no	—
64	Q₈×C₁₇	—	8	4	no	—
65	C₂×C₄×C₁₇	—	2	2	no	—
66	C₂×C₄×C₁₇	—	2	2	no	—
67	C₂×C₄×C₁₇	—	2	2	no	—
68	C₂×C₄×C₁₇	—	2	2	no	—
69	C₂×C₄×C₁₇	—	2	2	no	—
70	C₂×C₄×C₁₇	—	4	4	no	—
71	C₂×C₄×C₁₇	—	4	8	no	—
72	C₂×C₄×C₁₇	#101	1	4	yes	—
73	C₂×C₄×C₁₇	#103	1	4	yes	—
74	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
75	C₂×C₄×C₁₇	#121	1	2	yes	—
76	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
77	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
78	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
80	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	4	yes	—
81	C₂×C₄×C₁₇	#137	1	2	yes	—
82	C₈×C₁₇	—	4	4	no	—
83	C₈×C₁₇	—	8	2	no	—
84	#2868	C₂×C₁₇	1	1	yes	characteristic
85	#1831	C₂×C₁₇	1	1	yes	characteristic
86	#1767	C₂×C₁₇	1	1	yes	characteristic
87	C₂×#332	C₂×C₁₇	1	4	yes	—
88	C₂²×C₁₇	#390	1	1	yes	characteristic
89	C₂²×C₁₇	C₂×#101	1	2	yes	—
90	C₂²×C₁₇	C₂×#103	1	1	yes	characteristic
91	C₂²×C₁₇	#418	1	2	yes	—
92	C₂²×C₁₇	#419	1	1	yes	characteristic
93	C₄×C₁₇	—	2	4	no	—
94	C₄×C₁₇	—	2	4	no	—
95	C₄×C₁₇	—	2	2	no	—
96	C₄×C₁₇	—	8	4	no	—
97	C₄×C₁₇	#461	1	2	yes	—
98	C₄×C₁₇	#462	1	2	yes	—
99	C₄×C₁₇	#490	1	2	yes	—
100	C₄×C₁₇	#496	1	2	yes	—
101	C₂×C₄⋊Q₈	—	2	2	no	—
102	C₂×C₄⋊Q₈	C₂²×C₁₇	1	2	yes	—
103	C₂×C₄⋊C₈	C₂²×C₁₇	1	2	yes	—
104	C₂×C₄⋊C₈	C₂²×C₁₇	1	2	yes	—
105	C₂×#97	—	2	1	no	—
106	#384	—	2	1	no	—
107	#384	—	2	1	no	—
108	#384	C₄×C₁₇	1	4	yes	—
109	C₄³	C₂²×C₁₇	1	1	yes	characteristic
110	#336	—	2	2	no	—
111	#336	—	2	1	no	—
112	#332	—	2	8	no	—
113	C₂×C₁₇	#1549	1	2	yes	—
114	C₂×C₁₇	#1556	1	2	yes	—
115	C₂×C₁₇	#1961	1	1	yes	characteristic
116	C₂×C₁₇	#1966	1	1	yes	characteristic
117	C₂×C₁₇	#1981	1	1	yes	characteristic
118	C₂²×Q₈	—	2	2	no	—
119	C₂²×C₈	—	2	2	no	—
120	C₂²×C₈	—	2	2	no	—
121	C₄⋊Q₈	—	2	4	no	—
122	C₄⋊Q₈	—	2	4	no	—
123	C₄⋊Q₈	—	4	8	no	—
124	C₂×C₄⋊C₄	—	2	4	no	—
125	C₂×C₄⋊C₄	—	2	4	no	—
126	C₂×C₄⋊C₄	—	2	2	no	—
127	C₂×C₄⋊C₄	C₂×C₄×C₁₇	1	4	yes	—
128	C₂×C₄²	C₂×C₄×C₁₇	1	2	yes	—
129	C₂×C₄²	D₄×C₁₇	1	2	yes	—
130	C₂×C₄²	D₄×C₁₇	1	1	yes	characteristic
131	C₂×C₄²	D₄×C₁₇	1	1	yes	characteristic
132	C₂×C₄²	C₂³×C₁₇	1	1	yes	characteristic
133	C₄⋊C₈	—	2	4	no	—
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	D₄×C₁₇	1	4	yes	—
136	#97	—	4	2	no	—
137	C₁₇	#9703	1	1	yes	characteristic, direct product factor, Sylow
138	C₂×Q₈	—	4	2	no	—
139	C₂×Q₈	—	4	2	no	—
140	C₂×Q₈	—	4	8	no	—
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	C₂²⋊C₄×C₁₇	1	2	yes	—
143	C₂²×C₄	C₂²⋊C₄×C₁₇	1	2	yes	—
144	C₂²×C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
145	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
146	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
147	C₂×C₈	—	2	4	no	—
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	—	4	4	no	—
152	C₄⋊C₄	—	2	8	no	—
153	C₄⋊C₄	—	4	8	no	—
154	C₄⋊C₄	—	4	8	no	—
155	C₄⋊C₄	—	4	4	no	—
156	C₄²	—	2	4	no	—
157	C₄²	—	2	2	no	—
158	C₄²	—	2	4	no	—
159	C₄²	SD₁₆×C₁₇	1	2	yes	—
160	C₄²	Q₁₆×C₁₇	1	2	yes	—
161	C₄²	C₂×D₄×C₁₇	1	2	yes	—
162	C₄²	#41×C₁₇	1	2	yes	—
163	C₂³	C₂×C₂²⋊C₄×C₁₇	1	1	yes	characteristic
164	Q₈	—	8	4	no	—
165	Q₈	—	8	4	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	4	8	no	—
173	C₂×C₄	C₁₇×#101	1	4	yes	—
174	C₂×C₄	C₁₇×#103	1	4	yes	—
175	C₂×C₄	C₄×D₄×C₁₇	1	2	yes	—
176	C₂×C₄	C₁₇×#121	1	2	yes	—
177	C₂×C₄	C₁₇×#124	1	1	yes	characteristic
178	C₂×C₄	C₁₇×#127	1	1	yes	characteristic
179	C₂×C₄	C₂×SD₁₆×C₁₇	1	1	yes	characteristic
180	C₂×C₄	C₂×Q₁₆×C₁₇	1	1	yes	characteristic
181	C₂×C₄	C₁₇×AGL(1,ℤ/8)	1	4	yes	—
182	C₂×C₄	C₁₇×#137	1	2	yes	—
183	C₈	—	4	4	no	—
184	C₈	—	8	2	no	—
185	C₂²	C₁₇×#390	1	1	yes	characteristic
186	C₂²	C₂×C₁₇×#101	1	2	yes	—
187	C₂²	C₂×C₁₇×#103	1	1	yes	characteristic
188	C₂²	C₁₇×#418	1	2	yes	—
189	C₂²	C₁₇×#419	1	1	yes	characteristic
190	C₄	—	2	2	no	—
191	C₄	—	2	4	no	—
192	C₄	—	2	4	no	—
193	C₄	—	8	4	no	—
194	C₄	C₁₇×#461	1	2	yes	—
195	C₄	C₁₇×#462	1	2	yes	—
196	C₄	C₁₇×#490	1	2	yes	—
197	C₄	C₁₇×#496	1	2	yes	—
198	C₂	C₁₇×#1549	1	2	yes	—
199	C₂	C₁₇×#1556	1	2	yes	—
200	C₂	C₁₇×#1961	1	1	yes	characteristic
201	C₂	C₁₇×#1966	1	1	yes	characteristic
202	C₂	C₁₇×#1981	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₇	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₁₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₇
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₇ ◁ C₄³×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
61. #9703	C₁₇	direct (N × H)
137. C₁₇	#9703	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₇×#2868	2
C₁₇×#1831	2
C₁₇×#1767	2
C₂×C₁₇×#332	2
#9703	17

Subgroups — C17×#9703 #16867447

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#9703 #16867447