Subgroups — C₁₇×#10652 #16868396

Order 4352.

350 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×#1948	C₂	1	2	yes	maximal
2	C₁₇×#1939	C₂	1	1	yes	maximal, characteristic
3	C₁₇×#1845	C₂	1	1	yes	maximal, characteristic
4	C₁₇×#1296	C₂	1	2	yes	maximal
5	C₁₇×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₇×#107	—	2	1	no	—
7	C₂×C₁₇×#107	—	2	1	no	—
8	C₂×C₁₇×#106	—	2	2	no	—
9	C₂×C₁₇×C₄⋊C₈	—	2	1	no	—
10	C₂×C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₁₇×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
13	C₁₇×#384	C₂²	1	2	yes	—
14	C₄×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
15	C₁₇×#340	—	2	2	no	—
16	C₁₇×#336	—	2	1	no	—
17	C₂²×C₈×C₁₇	—	2	2	no	—
18	C₂²×C₈×C₁₇	—	2	1	no	—
19	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
22	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
23	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
24	C₂×C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₇	—	2	1	no	—
27	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
28	C₁₇×#107	—	4	2	no	—
29	C₁₇×#107	—	4	2	no	—
30	C₁₇×#106	—	4	4	no	—
31	C₁₇×C₄⋊C₈	—	2	2	no	—
32	C₁₇×C₄⋊C₈	—	2	2	no	—
33	C₁₇×C₄⋊C₈	—	4	2	no	—
34	C₁₇×C₄⋊C₈	Q₈	1	4	yes	—
35	C₄×C₈×C₁₇	—	2	2	no	—
36	C₁₇×#95	—	2	1	no	—
37	C₂²×C₄×C₁₇	—	2	1	no	—
38	C₂²×C₄×C₁₇	—	2	1	no	—
39	C₂²×C₄×C₁₇	—	4	2	no	—
40	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
43	C₂×C₈×C₁₇	—	2	1	no	—
44	C₂×C₈×C₁₇	—	2	1	no	—
45	C₂×C₈×C₁₇	—	2	1	no	—
46	C₂×C₈×C₁₇	—	2	1	no	—
47	C₂×C₈×C₁₇	—	2	4	no	—
48	C₂×C₈×C₁₇	—	4	2	no	—
49	C₂×C₈×C₁₇	—	4	2	no	—
50	C₂×C₈×C₁₇	—	4	1	no	—
51	C₂×C₈×C₁₇	—	4	1	no	—
52	C₂×C₈×C₁₇	—	4	1	no	—
53	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
54	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
55	C₄⋊C₄×C₁₇	—	2	1	no	—
56	C₄⋊C₄×C₁₇	—	2	2	no	—
57	C₄⋊C₄×C₁₇	—	4	2	no	—
58	C₄⋊C₄×C₁₇	—	4	2	no	—
59	C₄⋊C₄×C₁₇	—	4	2	no	—
60	C₄⋊C₄×C₁₇	—	4	2	no	—
61	C₄⋊C₄×C₁₇	—	8	2	no	—
62	C₄⋊C₄×C₁₇	Q₁₆	1	2	yes	—
63	C₄²×C₁₇	—	4	1	no	—
64	C₄²×C₁₇	—	4	1	no	—
65	C₄²×C₁₇	C₂×Q₈	1	2	yes	—
66	C₄²×C₁₇	#41	1	2	yes	—
67	#10652	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂³×C₁₇	#121	1	1	yes	characteristic
69	C₂×C₄×C₁₇	—	2	1	no	—
70	C₂×C₄×C₁₇	—	2	1	no	—
71	C₂×C₄×C₁₇	—	2	2	no	—
72	C₂×C₄×C₁₇	—	4	1	no	—
73	C₂×C₄×C₁₇	—	4	1	no	—
74	C₂×C₄×C₁₇	—	4	1	no	—
75	C₂×C₄×C₁₇	—	4	1	no	—
76	C₂×C₄×C₁₇	—	4	1	no	—
77	C₂×C₄×C₁₇	—	8	2	no	—
78	C₂×C₄×C₁₇	—	8	2	no	—
79	C₂×C₄×C₁₇	—	8	2	no	—
80	C₂×C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
81	C₂×C₄×C₁₇	#123	1	1	yes	characteristic
82	C₂×C₄×C₁₇	#125	1	2	yes	—
83	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
84	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
85	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
86	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
87	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
88	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
89	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
90	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
91	C₈×C₁₇	—	2	2	no	—
92	C₈×C₁₇	—	2	2	no	—
93	C₈×C₁₇	—	4	4	no	—
94	C₈×C₁₇	—	8	1	no	—
95	#1948	C₂×C₁₇	1	2	yes	—
96	#1939	C₂×C₁₇	1	1	yes	characteristic
97	#1845	C₂×C₁₇	1	1	yes	characteristic
98	#1296	C₂×C₁₇	1	2	yes	—
99	#1277	C₂×C₁₇	1	1	yes	characteristic
100	C₂²×C₁₇	#398	1	1	yes	characteristic
101	C₂²×C₁₇	#460	1	1	yes	characteristic
102	C₂²×C₁₇	#461	1	1	yes	characteristic
103	C₂²×C₁₇	#467	1	1	yes	characteristic
104	C₂²×C₁₇	#469	1	1	yes	characteristic
105	C₂²×C₁₇	#472	1	1	yes	characteristic
106	C₂²×C₁₇	#473	1	1	yes	characteristic
107	C₄×C₁₇	—	2	2	no	—
108	C₄×C₁₇	—	4	2	no	—
109	C₄×C₁₇	—	8	1	no	—
110	C₄×C₁₇	—	16	2	no	—
111	C₄×C₁₇	#477	1	2	yes	—
112	C₄×C₁₇	#478	1	2	yes	—
113	C₄×C₁₇	#481	1	1	yes	characteristic
114	C₄×C₁₇	#483	1	1	yes	characteristic
115	C₄×C₁₇	#498	1	1	yes	characteristic
116	C₄×C₁₇	#501	1	1	yes	characteristic
117	C₂×#107	—	2	1	no	—
118	C₂×#107	—	2	1	no	—
119	C₂×#106	—	2	2	no	—
120	C₂×C₄⋊C₈	—	2	1	no	—
121	C₂×C₄⋊C₈	C₂²×C₁₇	1	1	yes	characteristic
122	C₂×C₄⋊C₈	C₂²×C₁₇	1	2	yes	—
123	C₂×C₄×C₈	C₂²×C₁₇	1	1	yes	characteristic
124	#384	C₂²×C₁₇	1	2	yes	—
125	C₄×C₄⋊C₄	C₂²×C₁₇	1	1	yes	characteristic
126	#340	—	2	2	no	—
127	#336	—	2	1	no	—
128	C₂×C₁₇	#1670	1	1	yes	characteristic
129	C₂×C₁₇	#1675	1	1	yes	characteristic
130	C₂×C₁₇	#1688	1	1	yes	characteristic
131	C₂×C₁₇	#1693	1	1	yes	characteristic
132	C₂×C₁₇	#2084	1	1	yes	characteristic
133	C₂×C₁₇	#2087	1	1	yes	characteristic
134	C₂×C₁₇	#2089	1	1	yes	characteristic
135	C₂²×C₈	—	2	2	no	—
136	C₂²×C₈	—	2	1	no	—
137	C₂²×C₈	D₄×C₁₇	1	1	yes	characteristic
138	C₂²×C₈	D₄×C₁₇	1	1	yes	characteristic
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	—	2	2	no	—
141	C₂×C₄⋊C₄	—	2	2	no	—
142	C₂×C₄⋊C₄	D₄×C₁₇	1	1	yes	characteristic
143	C₂×C₄⋊C₄	D₄×C₁₇	1	1	yes	characteristic
144	C₂×C₄²	—	2	1	no	—
145	C₂×C₄²	C₂³×C₁₇	1	1	yes	characteristic
146	#107	—	4	2	no	—
147	#107	—	4	2	no	—
148	#106	—	4	4	no	—
149	C₄⋊C₈	—	2	2	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	4	2	no	—
152	C₄⋊C₈	Q₈×C₁₇	1	4	yes	—
153	C₄×C₈	—	2	2	no	—
154	#95	—	2	1	no	—
155	C₁₇	#10652	1	1	yes	characteristic, direct product factor, Sylow
156	C₂²×C₄	—	2	1	no	—
157	C₂²×C₄	—	2	1	no	—
158	C₂²×C₄	—	4	2	no	—
159	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
160	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
161	C₂²×C₄	#41×C₁₇	1	1	yes	characteristic
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	4	no	—
167	C₂×C₈	—	4	2	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	—	4	1	no	—
170	C₂×C₈	—	4	1	no	—
171	C₂×C₈	—	4	1	no	—
172	C₂×C₈	SD₁₆×C₁₇	1	2	yes	—
173	C₂×C₈	SD₁₆×C₁₇	1	2	yes	—
174	C₄⋊C₄	—	2	1	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	8	2	no	—
181	C₄⋊C₄	Q₁₆×C₁₇	1	2	yes	—
182	C₄²	—	4	1	no	—
183	C₄²	—	4	1	no	—
184	C₄²	C₂×Q₈×C₁₇	1	2	yes	—
185	C₄²	#41×C₁₇	1	2	yes	—
186	C₂³	C₁₇×#121	1	1	yes	characteristic
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	4	1	no	—
191	C₂×C₄	—	4	1	no	—
192	C₂×C₄	—	4	1	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	8	2	no	—
196	C₂×C₄	—	8	2	no	—
197	C₂×C₄	—	8	2	no	—
198	C₂×C₄	C₁₇×C₂²⋊Q₈	1	2	yes	—
199	C₂×C₄	C₁₇×#123	1	1	yes	characteristic
200	C₂×C₄	C₁₇×#125	1	2	yes	—
201	C₂×C₄	C₁₇×C₄⋊Q₈	1	1	yes	characteristic
202	C₂×C₄	C₂×SD₁₆×C₁₇	1	1	yes	characteristic
203	C₂×C₄	C₂×SD₁₆×C₁₇	1	1	yes	characteristic
204	C₂×C₄	C₂×Q₁₆×C₁₇	1	1	yes	characteristic
205	C₂×C₄	C₁₇×#135	1	1	yes	characteristic
206	C₂×C₄	C₁₇×AGL(1,ℤ/8)	1	1	yes	characteristic
207	C₂×C₄	C₁₇×AGL(1,ℤ/8)	1	2	yes	—
208	C₂×C₄	C₁₇×#137	1	1	yes	characteristic
209	C₈	—	2	2	no	—
210	C₈	—	2	2	no	—
211	C₈	—	4	4	no	—
212	C₈	—	8	1	no	—
213	C₂²	C₁₇×#398	1	1	yes	characteristic
214	C₂²	C₁₇×#460	1	1	yes	characteristic
215	C₂²	C₁₇×#461	1	1	yes	characteristic
216	C₂²	C₁₇×#467	1	1	yes	characteristic
217	C₂²	C₁₇×#469	1	1	yes	characteristic
218	C₂²	C₁₇×#472	1	1	yes	characteristic
219	C₂²	C₁₇×#473	1	1	yes	characteristic
220	C₄	—	2	2	no	—
221	C₄	—	4	2	no	—
222	C₄	—	8	1	no	—
223	C₄	—	16	2	no	—
224	C₄	C₁₇×#477	1	2	yes	—
225	C₄	C₁₇×#478	1	2	yes	—
226	C₄	C₁₇×#481	1	1	yes	characteristic
227	C₄	C₁₇×#483	1	1	yes	characteristic
228	C₄	C₁₇×#498	1	1	yes	characteristic
229	C₄	C₁₇×#501	1	1	yes	characteristic
230	C₂	C₁₇×#1670	1	1	yes	characteristic
231	C₂	C₁₇×#1675	1	1	yes	characteristic
232	C₂	C₁₇×#1688	1	1	yes	characteristic
233	C₂	C₁₇×#1693	1	1	yes	characteristic
234	C₂	C₁₇×#2084	1	1	yes	characteristic
235	C₂	C₁₇×#2087	1	1	yes	characteristic
236	C₂	C₁₇×#2089	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₇	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₇
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₇ ◁ C₂×C₄²×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
67. #10652	C₁₇	direct (N × H)
155. C₁₇	#10652	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×#1948	2
C₁₇×#1939	2
C₁₇×#1845	2
C₁₇×#1296	2
C₁₇×#1277	2
#10652	17

Subgroups — C17×#10652 #16868396

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#10652 #16868396