Subgroups — C₁₇×#8040 #16865784

Order 4352.

726 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₇×#432	C₂	1	2	yes	maximal
2	C₂×C₁₇×#384	C₂	1	1	yes	maximal, characteristic
3	C₁₇×#1296	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄⋊C₄×C₁₇	—	2	2	no	—
5	C₂²×C₄⋊C₄×C₁₇	C₄	1	2	yes	—
6	C₂²×C₄²×C₁₇	C₂²	1	1	yes	characteristic
7	C₁₇×#432	—	2	4	no	—
8	C₁₇×#432	C₄	1	8	yes	complemented
9	C₂×C₁₇×C₄⋊C₈	C₂²	1	4	yes	—
10	C₂×C₁₇×C₂²⋊C₈	—	2	2	no	—
11	C₂×C₁₇×#97	—	2	2	no	—
12	C₁₇×#384	—	2	2	no	—
13	C₁₇×#384	C₄	1	2	yes	—
14	C₁₇×#384	C₂²	1	2	yes	—
15	C₂³×C₄×C₁₇	—	2	2	no	—
16	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂³×C₄×C₁₇	D₄	1	2	yes	—
18	C₂²×C₈×C₁₇	—	2	4	no	—
19	C₂×C₄⋊C₄×C₁₇	—	2	4	no	—
20	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
21	C₂×C₄⋊C₄×C₁₇	—	2	4	no	—
22	C₂×C₄⋊C₄×C₁₇	—	4	8	no	—
23	C₂×C₄⋊C₄×C₁₇	—	4	4	no	—
24	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₇	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₁₇	D₄	1	2	yes	—
28	C₂×C₄²×C₁₇	D₄	1	2	yes	—
29	C₂×C₄²×C₁₇	Q₈	1	2	yes	—
30	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
31	C₁₇×C₄⋊C₈	—	2	4	no	—
32	C₁₇×C₄⋊C₈	C₂×C₄	1	8	yes	complemented
33	C₁₇×C₂²⋊C₈	—	2	8	no	—
34	C₁₇×#97	—	2	8	no	—
35	C₂⁴×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₇	—	2	2	no	—
37	C₂²×C₄×C₁₇	—	2	2	no	—
38	C₂²×C₄×C₁₇	—	2	4	no	—
39	C₂²×C₄×C₁₇	—	2	4	no	—
40	C₂²×C₄×C₁₇	—	2	4	no	—
41	C₂²×C₄×C₁₇	—	4	8	no	—
42	C₂²×C₄×C₁₇	—	4	4	no	—
43	C₂²×C₄×C₁₇	C₄²	1	2	yes	—
44	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₁₇	C₂²⋊C₄	1	4	yes	—
47	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₁₇	C₄⋊C₄	1	4	yes	—
49	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₇	C₂×D₄	1	2	yes	—
51	C₂×C₈×C₁₇	—	2	8	no	—
52	C₂×C₈×C₁₇	—	4	4	no	—
53	C₂×C₈×C₁₇	—	4	4	no	—
54	C₄⋊C₄×C₁₇	—	4	4	no	—
55	C₄⋊C₄×C₁₇	—	4	4	no	—
56	C₄⋊C₄×C₁₇	—	4	8	no	—
57	C₄⋊C₄×C₁₇	—	4	8	no	—
58	C₄²×C₁₇	C₄²	1	2	yes	—
59	C₄²×C₁₇	C₂²⋊C₄	1	4	yes	—
60	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
61	C₄²×C₁₇	C₄⋊C₄	1	4	yes	—
62	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
63	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
65	#8040	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂³×C₁₇	—	2	2	no	—
67	C₂³×C₁₇	—	2	2	no	—
68	C₂³×C₁₇	#95	1	2	yes	—
69	C₂³×C₁₇	#100	1	2	yes	—
70	C₂³×C₁₇	#101	1	2	yes	—
71	C₂³×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₇	—	2	4	no	—
73	C₂×C₄×C₁₇	—	2	4	no	—
74	C₂×C₄×C₁₇	—	2	1	no	—
75	C₂×C₄×C₁₇	—	2	1	no	—
76	C₂×C₄×C₁₇	—	2	1	no	—
77	C₂×C₄×C₁₇	—	2	1	no	—
78	C₂×C₄×C₁₇	—	2	2	no	—
79	C₂×C₄×C₁₇	—	2	2	no	—
80	C₂×C₄×C₁₇	—	2	8	no	—
81	C₂×C₄×C₁₇	—	2	8	no	—
82	C₂×C₄×C₁₇	—	4	4	no	—
83	C₂×C₄×C₁₇	—	4	8	no	—
84	C₂×C₄×C₁₇	—	8	8	no	—
85	C₂×C₄×C₁₇	#95	1	4	yes	—
86	C₂×C₄×C₁₇	#95	1	2	yes	—
87	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
88	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	2	yes	—
89	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
91	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	4	yes	—
92	C₂×C₄×C₁₇	#137	1	4	yes	—
93	C₈×C₁₇	—	4	8	no	—
94	C₂×#432	C₂×C₁₇	1	2	yes	—
95	C₂×#384	C₂×C₁₇	1	1	yes	characteristic
96	#1296	C₂×C₁₇	1	4	yes	complemented
97	C₂²×C₁₇	—	2	2	no	—
98	C₂²×C₁₇	—	2	4	no	—
99	C₂²×C₁₇	—	2	2	no	—
100	C₂²×C₁₇	—	2	4	no	—
101	C₂²×C₁₇	#343	1	4	yes	—
102	C₂²×C₁₇	C₂×#95	1	1	yes	characteristic
103	C₂²×C₁₇	C₂×#100	1	1	yes	characteristic
104	C₂²×C₁₇	C₂×#101	1	1	yes	characteristic
105	C₂²×C₁₇	#418	1	2	yes	—
106	C₂²×C₁₇	#419	1	2	yes	—
107	C₄×C₁₇	—	2	4	no	—
108	C₄×C₁₇	—	2	2	no	—
109	C₄×C₁₇	—	2	1	no	—
110	C₄×C₁₇	—	2	1	no	—
111	C₄×C₁₇	—	8	4	no	—
112	C₄×C₁₇	#417	1	4	yes	—
113	C₄×C₁₇	#428	1	4	yes	—
114	C₂²×C₄⋊C₄	—	2	2	no	—
115	C₂²×C₄⋊C₄	C₄×C₁₇	1	2	yes	—
116	C₂²×C₄²	C₂²×C₁₇	1	1	yes	characteristic
117	#432	—	2	4	no	—
118	#432	C₄×C₁₇	1	8	yes	complemented
119	C₂×C₄⋊C₈	C₂²×C₁₇	1	4	yes	—
120	C₂×C₂²⋊C₈	—	2	2	no	—
121	C₂×#97	—	2	2	no	—
122	#384	—	2	2	no	—
123	#384	C₄×C₁₇	1	2	yes	—
124	#384	C₂²×C₁₇	1	2	yes	—
125	C₂×C₁₇	—	2	4	no	—
126	C₂×C₁₇	#1555	1	1	yes	characteristic
127	C₂×C₁₇	#1556	1	2	yes	—
128	C₂×C₁₇	#1558	1	1	yes	characteristic
129	C₂×C₁₇	#1736	1	2	yes	—
130	C₂×C₁₇	C₂×#343	1	1	yes	characteristic
131	C₂³×C₄	—	2	2	no	—
132	C₂³×C₄	C₂×C₄×C₁₇	1	1	yes	characteristic
133	C₂³×C₄	D₄×C₁₇	1	2	yes	—
134	C₂²×C₈	—	2	4	no	—
135	C₂×C₄⋊C₄	—	2	4	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	2	4	no	—
138	C₂×C₄⋊C₄	—	4	4	no	—
139	C₂×C₄⋊C₄	—	4	8	no	—
140	C₂×C₄⋊C₄	C₂×C₄×C₁₇	1	4	yes	—
141	C₂×C₄²	C₂×C₄×C₁₇	1	1	yes	characteristic
142	C₂×C₄²	C₂×C₄×C₁₇	1	4	yes	—
143	C₂×C₄²	D₄×C₁₇	1	2	yes	—
144	C₂×C₄²	D₄×C₁₇	1	2	yes	—
145	C₂×C₄²	Q₈×C₁₇	1	2	yes	—
146	C₂×C₄²	C₂³×C₁₇	1	1	yes	characteristic
147	C₄⋊C₈	—	2	4	no	—
148	C₄⋊C₈	C₂×C₄×C₁₇	1	8	yes	complemented
149	C₂²⋊C₈	—	2	8	no	—
150	#97	—	2	8	no	—
151	C₁₇	#8040	1	1	yes	characteristic, direct product factor, Sylow
152	C₂⁴	C₂²⋊C₄×C₁₇	1	1	yes	characteristic
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	—	2	4	no	—
157	C₂²×C₄	—	2	4	no	—
158	C₂²×C₄	—	4	8	no	—
159	C₂²×C₄	—	4	4	no	—
160	C₂²×C₄	C₄²×C₁₇	1	2	yes	—
161	C₂²×C₄	C₂²⋊C₄×C₁₇	1	2	yes	—
162	C₂²×C₄	C₂²⋊C₄×C₁₇	1	1	yes	characteristic
163	C₂²×C₄	C₂²⋊C₄×C₁₇	1	4	yes	—
164	C₂²×C₄	C₄⋊C₄×C₁₇	1	4	yes	—
165	C₂²×C₄	C₄⋊C₄×C₁₇	1	2	yes	—
166	C₂²×C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
167	C₂²×C₄	C₂×D₄×C₁₇	1	2	yes	—
168	C₂×C₈	—	2	8	no	—
169	C₂×C₈	—	4	4	no	—
170	C₂×C₈	—	4	4	no	—
171	C₄⋊C₄	—	4	4	no	—
172	C₄⋊C₄	—	4	4	no	—
173	C₄⋊C₄	—	4	8	no	—
174	C₄⋊C₄	—	4	8	no	—
175	C₄²	C₄²×C₁₇	1	2	yes	—
176	C₄²	C₂²⋊C₄×C₁₇	1	4	yes	—
177	C₄²	C₄⋊C₄×C₁₇	1	2	yes	—
178	C₄²	C₄⋊C₄×C₁₇	1	4	yes	—
179	C₄²	C₂²×C₄×C₁₇	1	2	yes	—
180	C₄²	C₂×D₄×C₁₇	1	1	yes	characteristic
181	C₄²	C₂×Q₈×C₁₇	1	1	yes	characteristic
182	C₂³	—	2	2	no	—
183	C₂³	—	2	2	no	—
184	C₂³	C₁₇×#95	1	2	yes	—
185	C₂³	C₁₇×#100	1	2	yes	—
186	C₂³	C₁₇×#101	1	2	yes	—
187	C₂³	C₂×C₂²⋊C₄×C₁₇	1	1	yes	characteristic
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	8	no	—
197	C₂×C₄	—	2	8	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	—	4	8	no	—
200	C₂×C₄	—	8	8	no	—
201	C₂×C₄	C₁₇×#95	1	4	yes	—
202	C₂×C₄	C₁₇×#95	1	2	yes	—
203	C₂×C₄	C₂×C₄²×C₁₇	1	1	yes	characteristic
204	C₂×C₄	C₂×C₂²⋊C₄×C₁₇	1	2	yes	—
205	C₂×C₄	C₂×C₄⋊C₄×C₁₇	1	1	yes	characteristic
206	C₂×C₄	C₂×C₄⋊C₄×C₁₇	1	2	yes	—
207	C₂×C₄	C₁₇×AGL(1,ℤ/8)	1	4	yes	—
208	C₂×C₄	C₁₇×#137	1	4	yes	—
209	C₈	—	4	8	no	—
210	C₂²	—	2	4	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	4	no	—
213	C₂²	—	2	2	no	—
214	C₂²	C₁₇×#343	1	4	yes	—
215	C₂²	C₂×C₁₇×#95	1	1	yes	characteristic
216	C₂²	C₂×C₁₇×#100	1	1	yes	characteristic
217	C₂²	C₂×C₁₇×#101	1	1	yes	characteristic
218	C₂²	C₁₇×#418	1	2	yes	—
219	C₂²	C₁₇×#419	1	2	yes	—
220	C₄	—	2	4	no	—
221	C₄	—	2	2	no	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	8	4	no	—
225	C₄	C₁₇×#417	1	4	yes	—
226	C₄	C₁₇×#428	1	4	yes	—
227	C₂	—	2	4	no	—
228	C₂	C₁₇×#1555	1	1	yes	characteristic
229	C₂	C₁₇×#1556	1	2	yes	—
230	C₂	C₁₇×#1558	1	1	yes	characteristic
231	C₂	C₁₇×#1736	1	2	yes	—
232	C₂	C₂×C₁₇×#343	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₁₇×#1296	C₂	semidirect
8. C₁₇×#432	C₄	semidirect
32. C₁₇×C₄⋊C₈	C₂×C₄	semidirect
65. #8040	C₁₇	direct (N × H)
96. #1296	C₂×C₁₇	semidirect
118. #432	C₄×C₁₇	semidirect
148. C₄⋊C₈	C₂×C₄×C₁₇	semidirect
151. C₁₇	#8040	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₇×#432	2
C₂×C₁₇×#384	2
C₁₇×#1296	2
#8040	17

Subgroups — C17×#8040 #16865784

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#8040 #16865784