Subgroups — C₁₇×#7860 #16865604

Order 4352.

666 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₇×#431	C₂	1	2	yes	maximal
2	C₂×C₄×M₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
3	C₁₇×#1275	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₁₇	C₄	1	2	yes	—
5	C₂²×C₄²×C₁₇	C₂²	1	1	yes	characteristic
6	C₁₇×#431	—	2	2	no	—
7	C₁₇×#431	—	2	2	no	—
8	C₁₇×#431	C₄	1	4	yes	—
9	C₁₇×#431	C₂²	1	4	yes	—
10	C₂×C₁₇×C₄⋊C₈	—	2	2	no	—
11	C₂×C₁₇×C₂²⋊C₈	—	2	2	no	—
12	C₄×M₁₆×C₁₇	—	2	2	no	—
13	C₄×M₁₆×C₁₇	—	2	2	no	—
14	C₂×C₁₇×#97	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₇	C₄	1	4	yes	—
16	C₂×C₄×C₈×C₁₇	C₄	1	2	yes	—
17	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₁₇	D₄	1	1	yes	characteristic
19	C₂³×C₄×C₁₇	Q₈	1	1	yes	characteristic
20	C₂×M₁₆×C₁₇	—	2	4	no	—
21	C₂×M₁₆×C₁₇	—	2	4	no	—
22	C₂×M₁₆×C₁₇	C₈	1	8	yes	complemented
23	C₂²×C₈×C₁₇	—	2	4	no	—
24	C₂²×C₈×C₁₇	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₇	D₄	1	2	yes	—
30	C₂×C₄²×C₁₇	D₄	1	2	yes	—
31	C₂×C₄²×C₁₇	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
33	C₁₇×C₄⋊C₈	—	2	4	no	—
34	C₁₇×C₄⋊C₈	—	2	4	no	—
35	C₁₇×C₂²⋊C₈	—	2	4	no	—
36	C₁₇×C₂²⋊C₈	—	2	4	no	—
37	C₁₇×#97	—	2	4	no	—
38	C₄×C₈×C₁₇	—	2	2	no	—
39	C₄×C₈×C₁₇	—	2	2	no	—
40	C₄×C₈×C₁₇	—	2	4	no	—
41	C₄×C₈×C₁₇	C₂×C₄	1	8	yes	—
42	C₂⁴×C₁₇	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₇	—	4	2	no	—
44	C₂²×C₄×C₁₇	—	4	2	no	—
45	C₂²×C₄×C₁₇	C₄²	1	2	yes	—
46	C₂²×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₇	C₂×C₈	1	4	yes	—
55	C₂²×C₄×C₁₇	M₁₆	1	4	yes	—
56	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
59	M₁₆×C₁₇	—	2	16	no	—
60	C₂×C₈×C₁₇	—	2	4	no	—
61	C₂×C₈×C₁₇	—	2	4	no	—
62	C₂×C₈×C₁₇	—	2	8	no	—
63	C₂×C₈×C₁₇	—	4	8	no	—
64	C₂×C₈×C₁₇	C₂×C₈	1	8	yes	complemented
65	C₄²×C₁₇	C₄²	1	1	yes	characteristic
66	C₄²×C₁₇	C₄²	1	1	yes	characteristic
67	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₄²×C₁₇	C₂²⋊C₄	1	2	yes	—
70	C₄²×C₁₇	C₂²⋊C₄	1	2	yes	—
71	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
72	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
73	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
74	C₄²×C₁₇	C₂×D₄	1	2	yes	—
75	#7860	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂³×C₁₇	—	4	1	no	—
77	C₂³×C₁₇	—	4	1	no	—
78	C₂³×C₁₇	#95	1	2	yes	—
79	C₂³×C₁₇	C₄⋊C₈	1	4	yes	—
80	C₂³×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₇	—	2	4	no	—
82	C₂×C₄×C₁₇	—	2	2	no	—
83	C₂×C₄×C₁₇	—	2	2	no	—
84	C₂×C₄×C₁₇	—	2	4	no	—
85	C₂×C₄×C₁₇	—	4	2	no	—
86	C₂×C₄×C₁₇	—	4	2	no	—
87	C₂×C₄×C₁₇	—	4	1	no	—
88	C₂×C₄×C₁₇	—	4	1	no	—
89	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
90	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
91	C₂×C₄×C₁₇	#95	1	2	yes	—
92	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
93	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
94	C₂×C₄×C₁₇	C₄×C₈	1	4	yes	—
95	C₂×C₄×C₁₇	#97	1	4	yes	—
96	C₂×C₄×C₁₇	C₂²⋊C₈	1	8	yes	—
97	C₂×C₄×C₁₇	C₄⋊C₈	1	4	yes	—
98	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
99	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	2	yes	—
100	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
103	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
104	C₈×C₁₇	—	2	8	no	—
105	C₈×C₁₇	—	4	8	no	—
106	C₂×#431	C₂×C₁₇	1	2	yes	—
107	C₂×C₄×M₁₆	C₂×C₁₇	1	1	yes	characteristic
108	#1275	C₂×C₁₇	1	4	yes	complemented
109	C₂²×C₁₇	—	2	2	no	—
110	C₂²×C₁₇	—	2	1	no	—
111	C₂²×C₁₇	—	2	2	no	—
112	C₂²×C₁₇	—	2	1	no	—
113	C₂²×C₁₇	—	4	4	no	—
114	C₂²×C₁₇	#336	1	4	yes	—
115	C₂²×C₁₇	C₂×#95	1	1	yes	characteristic
116	C₂²×C₁₇	C₂×C₄⋊C₈	1	2	yes	—
117	C₄×C₁₇	—	2	2	no	—
118	C₄×C₁₇	—	2	1	no	—
119	C₄×C₁₇	—	2	1	no	—
120	C₄×C₁₇	—	2	2	no	—
121	C₄×C₁₇	#336	1	4	yes	—
122	C₄×C₁₇	H(3,ℤ/4)	1	2	yes	—
123	C₄×C₁₇	#338	1	2	yes	—
124	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
125	C₄×C₁₇	C₂×#97	1	1	yes	characteristic
126	C₄×C₁₇	C₂×C₂²⋊C₈	1	2	yes	—
127	C₂²×M₁₆	C₄×C₁₇	1	2	yes	—
128	C₂²×C₄²	C₂²×C₁₇	1	1	yes	characteristic
129	#431	—	2	2	no	—
130	#431	—	2	2	no	—
131	#431	C₄×C₁₇	1	4	yes	—
132	#431	C₂²×C₁₇	1	4	yes	—
133	C₂×C₄⋊C₈	—	2	2	no	—
134	C₂×C₂²⋊C₈	—	2	2	no	—
135	C₄×M₁₆	—	2	2	no	—
136	C₄×M₁₆	—	2	2	no	—
137	C₂×#97	C₂²×C₁₇	1	2	yes	—
138	C₂×C₄×C₈	C₄×C₁₇	1	2	yes	—
139	C₂×C₄×C₈	C₄×C₁₇	1	4	yes	—
140	C₂×C₁₇	—	2	4	no	—
141	C₂×C₁₇	—	2	2	no	—
142	C₂×C₁₇	C₂×#336	1	1	yes	characteristic
143	C₂×C₁₇	C₂×H(3,ℤ/4)	1	1	yes	characteristic
144	C₂×C₁₇	C₂×#338	1	1	yes	characteristic
145	C₂³×C₄	C₂×C₄×C₁₇	1	1	yes	characteristic
146	C₂³×C₄	D₄×C₁₇	1	1	yes	characteristic
147	C₂³×C₄	Q₈×C₁₇	1	1	yes	characteristic
148	C₂×M₁₆	—	2	4	no	—
149	C₂×M₁₆	—	2	4	no	—
150	C₂×M₁₆	C₈×C₁₇	1	8	yes	complemented
151	C₂²×C₈	—	2	4	no	—
152	C₂²×C₈	C₂×C₄×C₁₇	1	4	yes	—
153	C₂×C₄²	C₂×C₄×C₁₇	1	1	yes	characteristic
154	C₂×C₄²	C₂×C₄×C₁₇	1	2	yes	—
155	C₂×C₄²	C₂×C₄×C₁₇	1	2	yes	—
156	C₂×C₄²	D₄×C₁₇	1	1	yes	characteristic
157	C₂×C₄²	D₄×C₁₇	1	2	yes	—
158	C₂×C₄²	D₄×C₁₇	1	2	yes	—
159	C₂×C₄²	Q₈×C₁₇	1	1	yes	characteristic
160	C₂×C₄²	C₂³×C₁₇	1	1	yes	characteristic
161	C₄⋊C₈	—	2	4	no	—
162	C₄⋊C₈	—	2	4	no	—
163	C₂²⋊C₈	—	2	4	no	—
164	C₂²⋊C₈	—	2	4	no	—
165	#97	—	2	4	no	—
166	C₄×C₈	—	2	2	no	—
167	C₄×C₈	—	2	2	no	—
168	C₄×C₈	—	2	4	no	—
169	C₄×C₈	C₂×C₄×C₁₇	1	8	yes	—
170	C₁₇	#7860	1	1	yes	characteristic, direct product factor, Sylow
171	C₂⁴	C₄⋊C₄×C₁₇	1	1	yes	characteristic
172	C₂²×C₄	—	4	2	no	—
173	C₂²×C₄	—	4	2	no	—
174	C₂²×C₄	C₄²×C₁₇	1	2	yes	—
175	C₂²×C₄	C₂²⋊C₄×C₁₇	1	2	yes	—
176	C₂²×C₄	C₂²⋊C₄×C₁₇	1	2	yes	—
177	C₂²×C₄	C₂²⋊C₄×C₁₇	1	2	yes	—
178	C₂²×C₄	C₄⋊C₄×C₁₇	1	1	yes	characteristic
179	C₂²×C₄	C₄⋊C₄×C₁₇	1	1	yes	characteristic
180	C₂²×C₄	C₄⋊C₄×C₁₇	1	2	yes	—
181	C₂²×C₄	C₄⋊C₄×C₁₇	1	2	yes	—
182	C₂²×C₄	C₄⋊C₄×C₁₇	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₈×C₁₇	1	4	yes	—
184	C₂²×C₄	M₁₆×C₁₇	1	4	yes	—
185	C₂²×C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
186	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
187	C₂²×C₄	C₂×Q₈×C₁₇	1	1	yes	characteristic
188	M₁₆	—	2	16	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	2	8	no	—
192	C₂×C₈	—	4	8	no	—
193	C₂×C₈	C₂×C₈×C₁₇	1	8	yes	complemented
194	C₄²	C₄²×C₁₇	1	1	yes	characteristic
195	C₄²	C₄²×C₁₇	1	1	yes	characteristic
196	C₄²	C₂²⋊C₄×C₁₇	1	1	yes	characteristic
197	C₄²	C₂²⋊C₄×C₁₇	1	1	yes	characteristic
198	C₄²	C₂²⋊C₄×C₁₇	1	2	yes	—
199	C₄²	C₂²⋊C₄×C₁₇	1	2	yes	—
200	C₄²	C₄⋊C₄×C₁₇	1	2	yes	—
201	C₄²	C₄⋊C₄×C₁₇	1	2	yes	—
202	C₄²	C₂²×C₄×C₁₇	1	2	yes	—
203	C₄²	C₂×D₄×C₁₇	1	2	yes	—
204	C₂³	—	4	1	no	—
205	C₂³	—	4	1	no	—
206	C₂³	C₁₇×#95	1	2	yes	—
207	C₂³	C₁₇×C₄⋊C₈	1	4	yes	—
208	C₂³	C₂×C₄⋊C₄×C₁₇	1	1	yes	characteristic
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	1	no	—
216	C₂×C₄	—	4	1	no	—
217	C₂×C₄	C₁₇×#95	1	1	yes	characteristic
218	C₂×C₄	C₁₇×#95	1	1	yes	characteristic
219	C₂×C₄	C₁₇×#95	1	2	yes	—
220	C₂×C₄	C₁₇×#95	1	1	yes	characteristic
221	C₂×C₄	C₁₇×#95	1	1	yes	characteristic
222	C₂×C₄	C₄×C₈×C₁₇	1	4	yes	—
223	C₂×C₄	C₁₇×#97	1	4	yes	—
224	C₂×C₄	C₁₇×C₂²⋊C₈	1	8	yes	—
225	C₂×C₄	C₁₇×C₄⋊C₈	1	4	yes	—
226	C₂×C₄	C₂×C₄²×C₁₇	1	1	yes	characteristic
227	C₂×C₄	C₂×C₂²⋊C₄×C₁₇	1	2	yes	—
228	C₂×C₄	C₂×C₂²⋊C₄×C₁₇	1	1	yes	characteristic
229	C₂×C₄	C₂×C₄⋊C₄×C₁₇	1	2	yes	—
230	C₂×C₄	C₂²×C₈×C₁₇	1	2	yes	—
231	C₂×C₄	C₂×M₁₆×C₁₇	1	2	yes	—
232	C₈	—	2	8	no	—
233	C₈	—	4	8	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	4	4	no	—
239	C₂²	C₁₇×#336	1	4	yes	—
240	C₂²	C₂×C₁₇×#95	1	1	yes	characteristic
241	C₂²	C₂×C₁₇×C₄⋊C₈	1	2	yes	—
242	C₄	—	2	2	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	2	2	no	—
246	C₄	C₁₇×#336	1	4	yes	—
247	C₄	C₁₇×H(3,ℤ/4)	1	2	yes	—
248	C₄	C₁₇×#338	1	2	yes	—
249	C₄	C₂×C₄×C₈×C₁₇	1	1	yes	characteristic
250	C₄	C₂×C₁₇×#97	1	1	yes	characteristic
251	C₄	C₂×C₁₇×C₂²⋊C₈	1	2	yes	—
252	C₂	—	2	4	no	—
253	C₂	—	2	2	no	—
254	C₂	C₂×C₁₇×#336	1	1	yes	characteristic
255	C₂	C₂×C₁₇×H(3,ℤ/4)	1	1	yes	characteristic
256	C₂	C₂×C₁₇×#338	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₁₇	C₄²
commutator	C₄	C₂×C₄×C₈×C₁₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₇
socle	C₂²×C₁₇	C₂×#95
prime radical	C₂⁴×C₁₇	C₄⋊C₄
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄²×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₇×#1275	C₂	semidirect
22. C₂×M₁₆×C₁₇	C₈	semidirect
64. C₂×C₈×C₁₇	C₂×C₈	semidirect
75. #7860	C₁₇	direct (N × H)
108. #1275	C₂×C₁₇	semidirect
150. C₂×M₁₆	C₈×C₁₇	semidirect
170. C₁₇	#7860	direct (N × H)
193. C₂×C₈	C₂×C₈×C₁₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₇×#431	2
C₂×C₄×M₁₆×C₁₇	2
C₁₇×#1275	2
#7860	17

Subgroups — C17×#7860 #16865604

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#7860 #16865604