Subgroups — C₁₇×#8048 #16865792

Order 4352.

788 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×#2920	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₇×#117	C₂	1	2	yes	maximal
3	C₁₇×#1297	C₂	1	4	yes	maximal
4	C₂×C₁₇×#117	—	2	2	no	—
5	C₂²×C₄²×C₁₇	C₂²	1	1	yes	characteristic
6	C₁₇×#432	—	2	2	no	—
7	C₁₇×#431	—	2	2	no	—
8	C₁₇×#423	C₄	1	2	yes	complemented
9	C₁₇×#407	C₂²	1	2	yes	—
10	C₄×M₁₆×C₁₇	C₄	1	2	yes	complemented
11	C₄×C₄⋊C₄×C₁₇	—	2	2	no	—
12	C₄×C₄⋊C₄×C₁₇	C₄	1	4	yes	complemented
13	C₄×C₂²⋊C₄×C₁₇	—	2	2	no	—
14	C₄×C₂²⋊C₄×C₁₇	C₂²	1	4	yes	—
15	C₁₇×#376	C₄	1	4	yes	complemented
16	C₄³×C₁₇	—	2	2	no	—
17	C₂³×C₄×C₁₇	D₄	1	2	yes	—
18	C₂³×C₄×C₁₇	C₂³	1	1	yes	characteristic
19	C₂×M₁₆×C₁₇	C₂×C₄	1	4	yes	—
20	C₁₇×#117	—	2	16	no	—
21	C₂×C₄⋊C₄×C₁₇	—	2	4	no	—
22	C₂×C₂²⋊C₄×C₁₇	—	2	4	no	—
23	C₂×C₄²×C₁₇	—	2	4	no	—
24	C₂×C₄²×C₁₇	—	2	4	no	—
25	C₂×C₄²×C₁₇	—	2	4	no	—
26	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₇	D₄	1	2	yes	—
31	C₂×C₄²×C₁₇	D₄	1	2	yes	—
32	C₂×C₄²×C₁₇	Q₈	1	2	yes	—
33	C₁₇×C₄⋊C₈	—	2	4	no	—
34	C₁₇×C₂²⋊C₈	—	2	4	no	—
35	C₁₇×#97	—	2	2	no	—
36	C₄×C₈×C₁₇	—	2	2	no	—
37	C₁₇×#95	—	2	4	no	—
38	C₁₇×#95	C₂×C₄	1	8	yes	—
39	C₂⁴×C₁₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₇	—	2	8	no	—
41	C₂²×C₄×C₁₇	—	2	8	no	—
42	C₂²×C₄×C₁₇	—	4	2	no	—
43	C₂²×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₇	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₁₇	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
53	M₁₆×C₁₇	—	4	4	no	—
54	C₂×C₈×C₁₇	—	2	4	no	—
55	C₄⋊C₄×C₁₇	—	2	16	no	—
56	C₂²⋊C₄×C₁₇	—	2	16	no	—
57	C₄²×C₁₇	—	2	16	no	—
58	C₄²×C₁₇	—	4	8	no	—
59	C₄²×C₁₇	—	4	8	no	—
60	C₄²×C₁₇	C₄²	1	1	yes	characteristic
61	C₄²×C₁₇	C₄²	1	2	yes	—
62	C₄²×C₁₇	C₄²	1	1	yes	characteristic
63	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₄²×C₁₇	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
67	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
68	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
69	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
70	#8048	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂³×C₁₇	—	2	4	no	—
72	C₂³×C₁₇	—	4	1	no	—
73	C₂³×C₁₇	C₂×C₂²⋊C₄	1	2	yes	—
74	C₂³×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₇	—	2	4	no	—
76	C₂×C₄×C₁₇	—	2	4	no	—
77	C₂×C₄×C₁₇	—	2	4	no	—
78	C₂×C₄×C₁₇	—	2	4	no	—
79	C₂×C₄×C₁₇	—	2	4	no	—
80	C₂×C₄×C₁₇	—	2	4	no	—
81	C₂×C₄×C₁₇	—	2	2	no	—
82	C₂×C₄×C₁₇	—	2	16	no	—
83	C₂×C₄×C₁₇	—	4	2	no	—
84	C₂×C₄×C₁₇	—	4	1	no	—
85	C₂×C₄×C₁₇	—	4	8	no	—
86	C₂×C₄×C₁₇	—	4	8	no	—
87	C₂×C₄×C₁₇	#95	1	2	yes	—
88	C₂×C₄×C₁₇	#95	1	2	yes	—
89	C₂×C₄×C₁₇	#95	1	2	yes	—
90	C₂×C₄×C₁₇	#95	1	2	yes	—
91	C₂×C₄×C₁₇	#104	1	8	yes	—
92	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
93	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
95	C₈×C₁₇	—	4	4	no	—
96	#2920	C₂×C₁₇	1	1	yes	characteristic
97	C₄×#117	C₂×C₁₇	1	2	yes	—
98	#1297	C₂×C₁₇	1	4	yes	—
99	C₂²×C₁₇	—	2	2	no	—
100	C₂²×C₁₇	—	2	2	no	—
101	C₂²×C₁₇	—	2	2	no	—
102	C₂²×C₁₇	—	2	1	no	—
103	C₂²×C₁₇	—	4	4	no	—
104	C₂²×C₁₇	C₂×#95	1	1	yes	characteristic
105	C₂²×C₁₇	C₂×#104	1	4	yes	—
106	C₄×C₁₇	—	2	2	no	—
107	C₄×C₁₇	—	2	2	no	—
108	C₄×C₁₇	—	2	1	no	—
109	C₄×C₁₇	—	2	2	no	—
110	C₄×C₁₇	—	4	16	no	—
111	C₄×C₁₇	#339	1	4	yes	—
112	C₄×C₁₇	#339	1	4	yes	—
113	C₄×C₁₇	#410	1	2	yes	—
114	C₂×#117	—	2	2	no	—
115	C₂²×C₄²	C₂²×C₁₇	1	1	yes	characteristic
116	#432	—	2	2	no	—
117	#431	—	2	2	no	—
118	#423	C₄×C₁₇	1	2	yes	complemented
119	#407	C₂²×C₁₇	1	2	yes	—
120	C₄×M₁₆	C₄×C₁₇	1	2	yes	complemented
121	C₄×C₄⋊C₄	—	2	2	no	—
122	C₄×C₄⋊C₄	C₄×C₁₇	1	4	yes	complemented
123	C₄×C₂²⋊C₄	—	2	2	no	—
124	C₄×C₂²⋊C₄	C₂²×C₁₇	1	4	yes	—
125	#376	C₄×C₁₇	1	4	yes	complemented
126	C₄³	—	2	2	no	—
127	C₂×C₁₇	—	2	4	no	—
128	C₂×C₁₇	—	4	1	no	—
129	C₂×C₁₇	C₂×#339	1	2	yes	—
130	C₂×C₁₇	#1740	1	1	yes	characteristic
131	C₂³×C₄	D₄×C₁₇	1	2	yes	—
132	C₂³×C₄	C₂³×C₁₇	1	1	yes	characteristic
133	C₂×M₁₆	C₂×C₄×C₁₇	1	4	yes	—
134	#117	—	2	16	no	—
135	C₂×C₄⋊C₄	—	2	4	no	—
136	C₂×C₂²⋊C₄	—	2	4	no	—
137	C₂×C₄²	—	2	4	no	—
138	C₂×C₄²	—	2	4	no	—
139	C₂×C₄²	—	2	4	no	—
140	C₂×C₄²	C₂×C₄×C₁₇	1	1	yes	characteristic
141	C₂×C₄²	C₂×C₄×C₁₇	1	1	yes	characteristic
142	C₂×C₄²	C₂×C₄×C₁₇	1	2	yes	—
143	C₂×C₄²	C₂×C₄×C₁₇	1	2	yes	—
144	C₂×C₄²	D₄×C₁₇	1	2	yes	—
145	C₂×C₄²	D₄×C₁₇	1	2	yes	—
146	C₂×C₄²	Q₈×C₁₇	1	2	yes	—
147	C₄⋊C₈	—	2	4	no	—
148	C₂²⋊C₈	—	2	4	no	—
149	#97	—	2	2	no	—
150	C₄×C₈	—	2	2	no	—
151	#95	—	2	4	no	—
152	#95	C₂×C₄×C₁₇	1	8	yes	—
153	C₁₇	#8048	1	1	yes	characteristic, direct product factor, Sylow
154	C₂⁴	C₂×D₄×C₁₇	1	1	yes	characteristic
155	C₂²×C₄	—	2	8	no	—
156	C₂²×C₄	—	2	8	no	—
157	C₂²×C₄	—	4	2	no	—
158	C₂²×C₄	C₂²⋊C₄×C₁₇	1	2	yes	—
159	C₂²×C₄	C₂²⋊C₄×C₁₇	1	2	yes	—
160	C₂²×C₄	C₂²⋊C₄×C₁₇	1	2	yes	—
161	C₂²×C₄	C₂²⋊C₄×C₁₇	1	2	yes	—
162	C₂²×C₄	C₄⋊C₄×C₁₇	1	2	yes	—
163	C₂²×C₄	C₄⋊C₄×C₁₇	1	2	yes	—
164	C₂²×C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
165	C₂²×C₄	C₂²×C₄×C₁₇	1	2	yes	—
166	C₂²×C₄	C₂×D₄×C₁₇	1	2	yes	—
167	C₂²×C₄	C₂×Q₈×C₁₇	1	1	yes	characteristic
168	M₁₆	—	4	4	no	—
169	C₂×C₈	—	2	4	no	—
170	C₄⋊C₄	—	2	16	no	—
171	C₂²⋊C₄	—	2	16	no	—
172	C₄²	—	2	16	no	—
173	C₄²	—	4	8	no	—
174	C₄²	—	4	8	no	—
175	C₄²	C₄²×C₁₇	1	1	yes	characteristic
176	C₄²	C₄²×C₁₇	1	2	yes	—
177	C₄²	C₄²×C₁₇	1	1	yes	characteristic
178	C₄²	C₂²⋊C₄×C₁₇	1	1	yes	characteristic
179	C₄²	C₂²⋊C₄×C₁₇	1	2	yes	—
180	C₄²	C₂²⋊C₄×C₁₇	1	1	yes	characteristic
181	C₄²	C₄⋊C₄×C₁₇	1	2	yes	—
182	C₄²	C₄⋊C₄×C₁₇	1	2	yes	—
183	C₄²	C₄⋊C₄×C₁₇	1	2	yes	—
184	C₄²	C₄⋊C₄×C₁₇	1	2	yes	—
185	C₂³	—	2	4	no	—
186	C₂³	—	4	1	no	—
187	C₂³	C₂×C₂²⋊C₄×C₁₇	1	2	yes	—
188	C₂³	C₂×C₄⋊C₄×C₁₇	1	1	yes	characteristic
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	16	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	1	no	—
199	C₂×C₄	—	4	8	no	—
200	C₂×C₄	—	4	8	no	—
201	C₂×C₄	C₁₇×#95	1	2	yes	—
202	C₂×C₄	C₁₇×#95	1	2	yes	—
203	C₂×C₄	C₁₇×#95	1	2	yes	—
204	C₂×C₄	C₁₇×#95	1	2	yes	—
205	C₂×C₄	C₁₇×#104	1	8	yes	—
206	C₂×C₄	C₂×C₄²×C₁₇	1	1	yes	characteristic
207	C₂×C₄	C₂×C₂²⋊C₄×C₁₇	1	1	yes	characteristic
208	C₂×C₄	C₂×C₄⋊C₄×C₁₇	1	2	yes	—
209	C₈	—	4	4	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	2	1	no	—
214	C₂²	—	4	4	no	—
215	C₂²	C₂×C₁₇×#95	1	1	yes	characteristic
216	C₂²	C₂×C₁₇×#104	1	4	yes	—
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	2	no	—
221	C₄	—	4	16	no	—
222	C₄	C₁₇×#339	1	4	yes	—
223	C₄	C₁₇×#339	1	4	yes	—
224	C₄	C₁₇×#410	1	2	yes	—
225	C₂	—	2	4	no	—
226	C₂	—	4	1	no	—
227	C₂	C₂×C₁₇×#339	1	2	yes	—
228	C₂	C₁₇×#1740	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₁₇	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₁₇
Frattini, cosocle	C₂³×C₄	C₂³×C₁₇
socle	C₂²×C₁₇	C₂×#95
prime radical	C₂⁴×C₁₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄²×C₁₇ ◁ C₂²×C₄²×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₁₇×#423	C₄	semidirect
10. C₄×M₁₆×C₁₇	C₄	semidirect
12. C₄×C₄⋊C₄×C₁₇	C₄	semidirect
15. C₁₇×#376	C₄	semidirect
70. #8048	C₁₇	direct (N × H)
118. #423	C₄×C₁₇	semidirect
120. C₄×M₁₆	C₄×C₁₇	semidirect
122. C₄×C₄⋊C₄	C₄×C₁₇	semidirect
125. #376	C₄×C₁₇	semidirect
153. C₁₇	#8048	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×#2920	2
C₄×C₁₇×#117	2
C₁₇×#1297	2
#8048	17

Subgroups — C17×#8048 #16865792

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#8048 #16865792