Subgroups — C₁₇×#2539 #14525545

Order 2176.

900 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₇×#123	C₂	1	2	yes	maximal, complemented
2	C₂×C₁₇×#121	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₇×#120	C₂	1	2	yes	maximal, complemented
4	C₂²×C₂²⋊C₄×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#399	C₂	1	2	yes	maximal, complemented
6	C₁₇×#392	C₂	1	2	yes	maximal, complemented
7	C₁₇×#390	C₂	1	4	yes	maximal, complemented
8	C₄×C₂²⋊C₄×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁵×C₁₇	C₂²	1	1	yes	characteristic
10	C₂²×D₄×C₁₇	C₂²	1	2	yes	—
11	C₂²×D₄×C₁₇	C₂²	1	4	yes	—
12	C₂²×D₄×C₁₇	C₂²	1	1	yes	characteristic, complemented
13	C₂³×C₄×C₁₇	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₁₇	C₂²	1	1	yes	characteristic
15	C₁₇×#123	—	2	8	no	—
16	C₁₇×#121	—	2	4	no	—
17	C₁₇×#120	—	2	8	no	—
18	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic, complemented
19	C₂×C₄⋊C₄×C₁₇	C₂²	1	4	yes	complemented
20	C₂×C₂²⋊C₄×C₁₇	—	2	4	no	—
21	C₂×C₂²⋊C₄×C₁₇	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₁₇	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₁₇	C₂²	1	4	yes	—
24	C₂×C₂²⋊C₄×C₁₇	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₄×C₁₇	C₂²	1	2	yes	complemented
26	C₂×C₂²⋊C₄×C₁₇	C₂²	1	4	yes	complemented
27	C₂×C₄²×C₁₇	C₂²	1	2	yes	complemented
28	C₁₇×#95	C₂²	1	2	yes	complemented
29	C₂⁴×C₁₇	—	2	2	no	—
30	C₂⁴×C₁₇	—	2	4	no	—
31	C₂⁴×C₁₇	—	2	4	no	—
32	C₂⁴×C₁₇	—	4	2	no	—
33	C₂⁴×C₁₇	C₂³	1	2	yes	—
34	C₂⁴×C₁₇	C₂³	1	2	yes	—
35	C₂⁴×C₁₇	C₂³	1	1	yes	characteristic
36	C₂×D₄×C₁₇	—	2	4	no	—
37	C₂×D₄×C₁₇	—	2	4	no	—
38	C₂×D₄×C₁₇	—	2	4	no	—
39	C₂×D₄×C₁₇	—	2	8	no	—
40	C₂×D₄×C₁₇	—	2	8	no	—
41	C₂×D₄×C₁₇	—	2	2	no	—
42	C₂×D₄×C₁₇	—	4	4	no	—
43	C₂×D₄×C₁₇	—	4	2	no	—
44	C₂²×C₄×C₁₇	—	2	4	no	—
45	C₂²×C₄×C₁₇	—	2	2	no	—
46	C₂²×C₄×C₁₇	—	4	2	no	—
47	C₂²×C₄×C₁₇	D₄	1	4	yes	complemented
48	C₂²×C₄×C₁₇	C₂³	1	2	yes	—
49	C₂²×C₄×C₁₇	C₂³	1	2	yes	—
50	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₁₇	C₂³	1	4	yes	—
53	C₄⋊C₄×C₁₇	—	2	8	no	—
54	C₄⋊C₄×C₁₇	—	2	2	no	—
55	C₂²⋊C₄×C₁₇	—	2	8	no	—
56	C₂²⋊C₄×C₁₇	—	2	4	no	—
57	C₂²⋊C₄×C₁₇	—	2	4	no	—
58	C₂²⋊C₄×C₁₇	—	2	8	no	—
59	C₂²⋊C₄×C₁₇	—	2	4	no	—
60	C₂²⋊C₄×C₁₇	D₄	1	8	yes	complemented
61	C₄²×C₁₇	—	4	2	no	—
62	C₂³×C₁₇	—	2	2	no	—
63	C₂³×C₁₇	—	2	2	no	—
64	C₂³×C₁₇	—	2	8	no	—
65	C₂³×C₁₇	—	2	2	no	—
66	C₂³×C₁₇	—	2	4	no	—
67	C₂³×C₁₇	—	2	2	no	—
68	C₂³×C₁₇	—	2	4	no	—
69	C₂³×C₁₇	—	2	2	no	—
70	C₂³×C₁₇	—	2	2	no	—
71	C₂³×C₁₇	—	2	2	no	—
72	C₂³×C₁₇	—	2	4	no	—
73	C₂³×C₁₇	—	2	4	no	—
74	C₂³×C₁₇	—	4	16	no	—
75	C₂³×C₁₇	—	4	4	no	—
76	C₂³×C₁₇	—	4	4	no	—
77	C₂³×C₁₇	C₂×D₄	1	4	yes	—
78	C₂³×C₁₇	C₂×D₄	1	2	yes	—
79	C₂³×C₁₇	#41	1	4	yes	—
80	C₂³×C₁₇	C₂⁴	1	1	yes	characteristic
81	D₄×C₁₇	—	4	16	no	—
82	D₄×C₁₇	—	4	4	no	—
83	D₄×C₁₇	—	4	8	no	—
84	C₂×C₄×C₁₇	—	2	4	no	—
85	C₂×C₄×C₁₇	—	2	2	no	—
86	C₂×C₄×C₁₇	—	2	4	no	—
87	C₂×C₄×C₁₇	—	2	2	no	—
88	C₂×C₄×C₁₇	—	2	2	no	—
89	C₂×C₄×C₁₇	—	2	2	no	—
90	C₂×C₄×C₁₇	—	2	4	no	—
91	C₂×C₄×C₁₇	—	2	2	no	—
92	C₂×C₄×C₁₇	—	2	2	no	—
93	C₂×C₄×C₁₇	—	4	4	no	—
94	C₂×C₄×C₁₇	—	4	4	no	—
95	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₁₇	C₂×D₄	1	8	yes	—
97	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
98	#2539	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂²×C₁₇	—	2	2	no	—
100	C₂²×C₁₇	—	2	2	no	—
101	C₂²×C₁₇	—	2	8	no	—
102	C₂²×C₁₇	—	2	4	no	—
103	C₂²×C₁₇	—	2	8	no	—
104	C₂²×C₁₇	—	4	16	no	—
105	C₂²×C₁₇	—	4	2	no	—
106	C₂²×C₁₇	—	4	2	no	—
107	C₂²×C₁₇	—	4	4	no	—
108	C₂²×C₁₇	—	4	2	no	—
109	C₂²×C₁₇	—	4	2	no	—
110	C₂²×C₁₇	—	4	2	no	—
111	C₂²×C₁₇	—	4	4	no	—
112	C₂²×C₁₇	—	4	4	no	—
113	C₂²×C₁₇	#127	1	4	yes	—
114	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
115	C₂²×C₁₇	C₂²×D₄	1	2	yes	—
116	C₂²×C₁₇	C₂×#41	1	2	yes	—
117	C₂²×C₁₇	H(5,𝔽₂)	1	2	yes	—
118	C₄×C₁₇	—	2	2	no	—
119	C₄×C₁₇	—	2	2	no	—
120	C₄×C₁₇	—	2	8	no	—
121	C₄×C₁₇	—	4	2	no	—
122	C₄×C₁₇	—	4	2	no	—
123	C₂×#123	C₂×C₁₇	1	2	yes	complemented
124	C₂×#121	C₂×C₁₇	1	1	yes	characteristic, complemented
125	C₂×#120	C₂×C₁₇	1	2	yes	complemented
126	C₂²×C₂²⋊C₄	C₂×C₁₇	1	1	yes	characteristic, complemented
127	#399	C₂×C₁₇	1	2	yes	complemented
128	#392	C₂×C₁₇	1	2	yes	complemented
129	#390	C₂×C₁₇	1	4	yes	complemented
130	C₄×C₂²⋊C₄	C₂×C₁₇	1	1	yes	characteristic, complemented
131	C₂×C₁₇	—	2	4	no	—
132	C₂×C₁₇	—	4	4	no	—
133	C₂×C₁₇	—	8	2	no	—
134	C₂×C₁₇	#525	1	1	yes	characteristic
135	C₂×C₁₇	C₂×#127	1	1	yes	characteristic
136	C₂×C₁₇	#534	1	1	yes	characteristic
137	C₂×C₁₇	#546	1	4	yes	—
138	C₂⁵	C₂²×C₁₇	1	1	yes	characteristic
139	C₂²×D₄	C₂²×C₁₇	1	2	yes	—
140	C₂²×D₄	C₂²×C₁₇	1	4	yes	—
141	C₂²×D₄	C₂²×C₁₇	1	1	yes	characteristic, complemented
142	C₂³×C₄	C₂²×C₁₇	1	1	yes	characteristic
143	C₂³×C₄	C₂²×C₁₇	1	1	yes	characteristic
144	#123	—	2	8	no	—
145	#121	—	2	4	no	—
146	#120	—	2	8	no	—
147	C₂×C₄⋊C₄	C₂²×C₁₇	1	1	yes	characteristic, complemented
148	C₂×C₄⋊C₄	C₂²×C₁₇	1	4	yes	complemented
149	C₂×C₂²⋊C₄	—	2	4	no	—
150	C₂×C₂²⋊C₄	C₂²×C₁₇	1	2	yes	—
151	C₂×C₂²⋊C₄	C₂²×C₁₇	1	2	yes	—
152	C₂×C₂²⋊C₄	C₂²×C₁₇	1	4	yes	—
153	C₂×C₂²⋊C₄	C₂²×C₁₇	1	2	yes	—
154	C₂×C₂²⋊C₄	C₂²×C₁₇	1	2	yes	complemented
155	C₂×C₂²⋊C₄	C₂²×C₁₇	1	4	yes	complemented
156	C₂×C₄²	C₂²×C₁₇	1	2	yes	complemented
157	#95	C₂²×C₁₇	1	2	yes	complemented
158	C₁₇	#2539	1	1	yes	characteristic, direct product factor, Sylow
159	C₂⁴	—	2	2	no	—
160	C₂⁴	—	2	4	no	—
161	C₂⁴	—	2	4	no	—
162	C₂⁴	—	4	2	no	—
163	C₂⁴	C₂³×C₁₇	1	2	yes	—
164	C₂⁴	C₂³×C₁₇	1	2	yes	—
165	C₂⁴	C₂³×C₁₇	1	1	yes	characteristic
166	C₂×D₄	—	2	4	no	—
167	C₂×D₄	—	2	4	no	—
168	C₂×D₄	—	2	4	no	—
169	C₂×D₄	—	2	8	no	—
170	C₂×D₄	—	2	8	no	—
171	C₂×D₄	—	2	2	no	—
172	C₂×D₄	—	4	4	no	—
173	C₂×D₄	—	4	2	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	2	4	no	—
176	C₂²×C₄	—	4	2	no	—
177	C₂²×C₄	D₄×C₁₇	1	4	yes	complemented
178	C₂²×C₄	C₂³×C₁₇	1	2	yes	—
179	C₂²×C₄	C₂³×C₁₇	1	2	yes	—
180	C₂²×C₄	C₂³×C₁₇	1	1	yes	characteristic
181	C₂²×C₄	C₂³×C₁₇	1	4	yes	—
182	C₂²×C₄	C₂³×C₁₇	1	1	yes	characteristic
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	2	8	no	—
185	C₂²⋊C₄	—	2	4	no	—
186	C₂²⋊C₄	—	2	4	no	—
187	C₂²⋊C₄	—	2	8	no	—
188	C₂²⋊C₄	—	2	4	no	—
189	C₂²⋊C₄	—	2	8	no	—
190	C₂²⋊C₄	D₄×C₁₇	1	8	yes	complemented
191	C₄²	—	4	2	no	—
192	C₂³	—	2	2	no	—
193	C₂³	—	2	2	no	—
194	C₂³	—	2	8	no	—
195	C₂³	—	2	2	no	—
196	C₂³	—	2	4	no	—
197	C₂³	—	2	2	no	—
198	C₂³	—	2	4	no	—
199	C₂³	—	2	2	no	—
200	C₂³	—	2	2	no	—
201	C₂³	—	2	2	no	—
202	C₂³	—	2	4	no	—
203	C₂³	—	2	4	no	—
204	C₂³	—	4	16	no	—
205	C₂³	—	4	4	no	—
206	C₂³	—	4	4	no	—
207	C₂³	C₂×D₄×C₁₇	1	4	yes	—
208	C₂³	C₂×D₄×C₁₇	1	2	yes	—
209	C₂³	#41×C₁₇	1	4	yes	—
210	C₂³	C₂⁴×C₁₇	1	1	yes	characteristic
211	D₄	—	4	16	no	—
212	D₄	—	4	4	no	—
213	D₄	—	4	8	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	C₂×D₄×C₁₇	1	2	yes	—
226	C₂×C₄	C₂×D₄×C₁₇	1	8	yes	—
227	C₂×C₄	C₂×D₄×C₁₇	1	2	yes	—
228	C₂²	—	2	2	no	—
229	C₂²	—	2	2	no	—
230	C₂²	—	2	8	no	—
231	C₂²	—	2	4	no	—
232	C₂²	—	2	8	no	—
233	C₂²	—	4	16	no	—
234	C₂²	—	4	2	no	—
235	C₂²	—	4	2	no	—
236	C₂²	—	4	4	no	—
237	C₂²	—	4	2	no	—
238	C₂²	—	4	2	no	—
239	C₂²	—	4	2	no	—
240	C₂²	—	4	4	no	—
241	C₂²	—	4	4	no	—
242	C₂²	C₁₇×#127	1	4	yes	—
243	C₂²	C₂²×D₄×C₁₇	1	1	yes	characteristic
244	C₂²	C₂²×D₄×C₁₇	1	2	yes	—
245	C₂²	C₂×#41×C₁₇	1	2	yes	—
246	C₂²	C₁₇×H(5,𝔽₂)	1	2	yes	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	2	8	no	—
250	C₄	—	4	2	no	—
251	C₄	—	4	2	no	—
252	C₂	—	2	4	no	—
253	C₂	—	4	4	no	—
254	C₂	—	8	2	no	—
255	C₂	C₁₇×#525	1	1	yes	characteristic
256	C₂	C₂×C₁₇×#127	1	1	yes	characteristic
257	C₂	C₁₇×#534	1	1	yes	characteristic
258	C₂	C₁₇×#546	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₇
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₇×#123	C₂	semidirect
2. C₂×C₁₇×#121	C₂	semidirect
3. C₂×C₁₇×#120	C₂	semidirect
4. C₂²×C₂²⋊C₄×C₁₇	C₂	semidirect
5. C₁₇×#399	C₂	semidirect
6. C₁₇×#392	C₂	semidirect
7. C₁₇×#390	C₂	semidirect
8. C₄×C₂²⋊C₄×C₁₇	C₂	semidirect
12. C₂²×D₄×C₁₇	C₂²	semidirect
18. C₂×C₄⋊C₄×C₁₇	C₂²	semidirect
19. C₂×C₄⋊C₄×C₁₇	C₂²	semidirect
25. C₂×C₂²⋊C₄×C₁₇	C₂²	semidirect
26. C₂×C₂²⋊C₄×C₁₇	C₂²	semidirect
27. C₂×C₄²×C₁₇	C₂²	semidirect
28. C₁₇×#95	C₂²	semidirect
47. C₂²×C₄×C₁₇	D₄	semidirect
60. C₂²⋊C₄×C₁₇	D₄	semidirect
98. #2539	C₁₇	direct (N × H)
123. C₂×#123	C₂×C₁₇	semidirect
124. C₂×#121	C₂×C₁₇	semidirect
125. C₂×#120	C₂×C₁₇	semidirect
126. C₂²×C₂²⋊C₄	C₂×C₁₇	semidirect
127. #399	C₂×C₁₇	semidirect
128. #392	C₂×C₁₇	semidirect
129. #390	C₂×C₁₇	semidirect
130. C₄×C₂²⋊C₄	C₂×C₁₇	semidirect
141. C₂²×D₄	C₂²×C₁₇	semidirect
147. C₂×C₄⋊C₄	C₂²×C₁₇	semidirect
148. C₂×C₄⋊C₄	C₂²×C₁₇	semidirect
154. C₂×C₂²⋊C₄	C₂²×C₁₇	semidirect
155. C₂×C₂²⋊C₄	C₂²×C₁₇	semidirect
156. C₂×C₄²	C₂²×C₁₇	semidirect
157. #95	C₂²×C₁₇	semidirect
158. C₁₇	#2539	direct (N × H)
177. C₂²×C₄	D₄×C₁₇	semidirect
190. C₂²⋊C₄	D₄×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×#123	2
C₂×C₁₇×#121	2
C₂×C₁₇×#120	2
C₂²×C₂²⋊C₄×C₁₇	2
C₁₇×#399	2
C₁₇×#392	2
C₁₇×#390	2
C₄×C₂²⋊C₄×C₁₇	2
#2539	17

Subgroups — C17×#2539 #14525545

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#2539 #14525545