Subgroups — C₂×C₁₇×#2309 #16891785

Order 4352.

7710 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₄×C₁₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₇×#117	C₂	1	2	yes	maximal, complemented
3	C₁₇×#2309	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₁₇×#390	C₂	1	8	yes	maximal, complemented
5	C₂×C₁₇×#379	C₂	1	4	yes	maximal, complemented
6	C₂⁶×C₁₇	C₂²	1	2	yes	—
7	C₂³×D₄×C₁₇	—	2	16	no	—
8	C₂³×D₄×C₁₇	C₄	1	16	yes	complemented
9	C₂³×D₄×C₁₇	C₂²	1	4	yes	—
10	C₂³×D₄×C₁₇	C₂²	1	8	yes	—
11	C₂⁴×C₄×C₁₇	C₂²	1	1	yes	characteristic
12	C₂×C₁₇×#117	—	2	16	no	—
13	C₂×C₁₇×#117	C₂²	1	16	yes	complemented
14	C₂²×C₄⋊C₄×C₁₇	C₂²	1	4	yes	complemented
15	C₂²×C₂²⋊C₄×C₁₇	C₂²	1	4	yes	—
16	C₂²×C₂²⋊C₄×C₁₇	C₂²	1	16	yes	complemented
17	C₂²×C₄²×C₁₇	C₂²	1	4	yes	complemented
18	C₁₇×#390	C₂²	1	64	yes	complemented
19	C₁₇×#379	C₂²	1	32	yes	complemented
20	C₂⁵×C₁₇	—	2	16	no	—
21	C₂⁵×C₁₇	—	4	16	no	—
22	C₂⁵×C₁₇	C₂×C₄	1	16	yes	—
23	C₂⁵×C₁₇	C₂³	1	4	yes	—
24	C₂⁵×C₁₇	C₂³	1	1	yes	characteristic
25	C₂⁵×C₁₇	C₂³	1	8	yes	—
26	C₂²×D₄×C₁₇	—	2	128	no	—
27	C₂²×D₄×C₁₇	—	2	16	no	—
28	C₂²×D₄×C₁₇	—	2	16	no	—
29	C₂²×D₄×C₁₇	—	2	64	no	—
30	C₂²×D₄×C₁₇	C₂×C₄	1	32	yes	—
31	C₂²×D₄×C₁₇	C₂×C₄	1	64	yes	complemented
32	C₂²×D₄×C₁₇	C₂³	1	16	yes	—
33	C₂³×C₄×C₁₇	C₂×C₄	1	8	yes	—
34	C₂³×C₄×C₁₇	D₄	1	16	yes	—
35	C₂³×C₄×C₁₇	C₂³	1	8	yes	—
36	C₂³×C₄×C₁₇	C₂³	1	4	yes	—
37	C₂³×C₄×C₁₇	C₂³	1	2	yes	—
38	C₁₇×#117	—	2	64	no	—
39	C₂×C₄⋊C₄×C₁₇	—	2	16	no	—
40	C₂×C₄⋊C₄×C₁₇	C₂³	1	16	yes	complemented
41	C₂×C₂²⋊C₄×C₁₇	—	2	16	no	—
42	C₂×C₂²⋊C₄×C₁₇	—	4	32	no	—
43	C₂×C₂²⋊C₄×C₁₇	C₂³	1	16	yes	—
44	C₂×C₂²⋊C₄×C₁₇	C₂³	1	64	yes	complemented
45	C₂×C₄²×C₁₇	—	2	16	no	—
46	C₂×C₄²×C₁₇	C₂³	1	16	yes	complemented
47	C₂⁴×C₁₇	—	2	8	no	—
48	C₂⁴×C₁₇	—	2	64	no	—
49	C₂⁴×C₁₇	—	2	32	no	—
50	C₂⁴×C₁₇	—	2	16	no	—
51	C₂⁴×C₁₇	—	2	32	no	—
52	C₂⁴×C₁₇	—	4	64	no	—
53	C₂⁴×C₁₇	—	4	32	no	—
54	C₂⁴×C₁₇	—	4	128	no	—
55	C₂⁴×C₁₇	C₂²×C₄	1	16	yes	—
56	C₂⁴×C₁₇	C₂²×C₄	1	4	yes	—
57	C₂⁴×C₁₇	C₂²×C₄	1	32	yes	—
58	C₂⁴×C₁₇	C₂×D₄	1	8	yes	—
59	C₂⁴×C₁₇	C₂⁴	1	8	yes	—
60	C₂⁴×C₁₇	C₂⁴	1	2	yes	—
61	C₂⁴×C₁₇	C₂⁴	1	1	yes	characteristic
62	C₂×D₄×C₁₇	—	2	128	no	—
63	C₂×D₄×C₁₇	—	2	256	no	—
64	C₂×D₄×C₁₇	—	2	32	no	—
65	C₂×D₄×C₁₇	—	2	32	no	—
66	C₂×D₄×C₁₇	C₂²×C₄	1	64	yes	—
67	C₂²×C₄×C₁₇	—	2	16	no	—
68	C₂²×C₄×C₁₇	—	2	32	no	—
69	C₂²×C₄×C₁₇	C₂²⋊C₄	1	64	yes	complemented
70	C₂²×C₄×C₁₇	C₂²×C₄	1	8	yes	—
71	C₂²×C₄×C₁₇	C₂²×C₄	1	16	yes	—
72	C₂²×C₄×C₁₇	C₂×D₄	1	8	yes	—
73	C₂²×C₄×C₁₇	C₂×D₄	1	8	yes	—
74	C₂²×C₄×C₁₇	C₂×D₄	1	32	yes	—
75	C₂²×C₄×C₁₇	C₂⁴	1	16	yes	—
76	C₂²×C₄×C₁₇	C₂⁴	1	4	yes	—
77	C₄⋊C₄×C₁₇	—	2	32	no	—
78	C₂²⋊C₄×C₁₇	—	2	32	no	—
79	C₂²⋊C₄×C₁₇	—	4	64	no	—
80	C₄²×C₁₇	—	2	32	no	—
81	C₂×#2309	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂³×C₁₇	—	2	16	no	—
83	C₂³×C₁₇	—	2	32	no	—
84	C₂³×C₁₇	—	2	8	no	—
85	C₂³×C₁₇	—	2	64	no	—
86	C₂³×C₁₇	—	2	32	no	—
87	C₂³×C₁₇	—	2	64	no	—
88	C₂³×C₁₇	—	2	16	no	—
89	C₂³×C₁₇	—	2	32	no	—
90	C₂³×C₁₇	—	4	64	no	—
91	C₂³×C₁₇	—	4	256	no	—
92	C₂³×C₁₇	—	4	128	no	—
93	C₂³×C₁₇	—	4	64	no	—
94	C₂³×C₁₇	C₂×C₂²⋊C₄	1	16	yes	—
95	C₂³×C₁₇	C₂³×C₄	1	16	yes	—
96	C₂³×C₁₇	C₂³×C₄	1	4	yes	—
97	C₂³×C₁₇	C₂³×C₄	1	2	yes	—
98	C₂³×C₁₇	C₂²×D₄	1	8	yes	—
99	C₂³×C₁₇	C₂²×D₄	1	4	yes	—
100	C₂³×C₁₇	H(5,𝔽₂)	1	8	yes	—
101	C₂³×C₁₇	C₂⁵	1	1	yes	characteristic
102	D₄×C₁₇	—	2	128	no	—
103	C₂×C₄×C₁₇	—	2	16	no	—
104	C₂×C₄×C₁₇	—	2	32	no	—
105	C₂×C₄×C₁₇	—	4	32	no	—
106	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	32	yes	—
107	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	64	yes	complemented
108	C₂×C₄×C₁₇	C₂³×C₄	1	8	yes	—
109	C₂×C₄×C₁₇	C₂²×D₄	1	8	yes	—
110	C₂×C₄×C₁₇	C₂²×D₄	1	8	yes	—
111	C₂⁴×D₄	C₂×C₁₇	1	1	yes	characteristic
112	C₂²×#117	C₂×C₁₇	1	2	yes	complemented
113	#2309	C₂×C₁₇	1	16	yes	direct product factor
114	C₂×#390	C₂×C₁₇	1	8	yes	complemented
115	C₂×#379	C₂×C₁₇	1	4	yes	complemented
116	C₂²×C₁₇	—	2	32	no	—
117	C₂²×C₁₇	—	2	8	no	—
118	C₂²×C₁₇	—	2	16	no	—
119	C₂²×C₁₇	—	2	16	no	—
120	C₂²×C₁₇	—	2	32	no	—
121	C₂²×C₁₇	—	4	128	no	—
122	C₂²×C₁₇	—	4	64	no	—
123	C₂²×C₁₇	—	4	32	no	—
124	C₂²×C₁₇	C₂²×C₂²⋊C₄	1	8	yes	—
125	C₂²×C₁₇	C₂²×C₂²⋊C₄	1	4	yes	—
126	C₂²×C₁₇	#518	1	8	yes	—
127	C₂²×C₁₇	#535	1	16	yes	—
128	C₂²×C₁₇	C₂⁴×C₄	1	1	yes	characteristic
129	C₂²×C₁₇	C₂³×D₄	1	2	yes	—
130	C₂²×C₁₇	C₂×H(5,𝔽₂)	1	4	yes	—
131	C₄×C₁₇	—	4	16	no	—
132	C₄×C₁₇	C₂²×C₂²⋊C₄	1	16	yes	—
133	C₂⁶	C₂²×C₁₇	1	2	yes	—
134	C₂³×D₄	—	2	16	no	—
135	C₂³×D₄	C₄×C₁₇	1	16	yes	complemented
136	C₂³×D₄	C₂²×C₁₇	1	4	yes	—
137	C₂³×D₄	C₂²×C₁₇	1	8	yes	—
138	C₂⁴×C₄	C₂²×C₁₇	1	1	yes	characteristic
139	C₂×#117	—	2	16	no	—
140	C₂×#117	C₂²×C₁₇	1	16	yes	complemented
141	C₂²×C₄⋊C₄	C₂²×C₁₇	1	4	yes	complemented
142	C₂²×C₂²⋊C₄	C₂²×C₁₇	1	4	yes	—
143	C₂²×C₂²⋊C₄	C₂²×C₁₇	1	16	yes	complemented
144	C₂²×C₄²	C₂²×C₁₇	1	4	yes	complemented
145	#390	C₂²×C₁₇	1	64	yes	complemented
146	#379	C₂²×C₁₇	1	32	yes	complemented
147	C₂×C₁₇	—	2	8	no	—
148	C₂×C₁₇	—	4	16	no	—
149	C₂×C₁₇	#2309	1	8	yes	direct product factor
150	C₂×C₁₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic
151	C₂×C₁₇	C₂×#518	1	2	yes	—
152	C₂×C₁₇	C₂×#535	1	4	yes	—
153	C₂⁵	—	2	16	no	—
154	C₂⁵	—	4	16	no	—
155	C₂⁵	C₂×C₄×C₁₇	1	16	yes	—
156	C₂⁵	C₂³×C₁₇	1	4	yes	—
157	C₂⁵	C₂³×C₁₇	1	1	yes	characteristic
158	C₂⁵	C₂³×C₁₇	1	8	yes	—
159	C₂²×D₄	—	2	128	no	—
160	C₂²×D₄	—	2	16	no	—
161	C₂²×D₄	—	2	16	no	—
162	C₂²×D₄	—	2	64	no	—
163	C₂²×D₄	C₂×C₄×C₁₇	1	32	yes	—
164	C₂²×D₄	C₂×C₄×C₁₇	1	64	yes	complemented
165	C₂²×D₄	C₂³×C₁₇	1	16	yes	—
166	C₂³×C₄	C₂×C₄×C₁₇	1	8	yes	—
167	C₂³×C₄	D₄×C₁₇	1	16	yes	—
168	C₂³×C₄	C₂³×C₁₇	1	8	yes	—
169	C₂³×C₄	C₂³×C₁₇	1	2	yes	—
170	C₂³×C₄	C₂³×C₁₇	1	4	yes	—
171	#117	—	2	64	no	—
172	C₂×C₄⋊C₄	—	2	16	no	—
173	C₂×C₄⋊C₄	C₂³×C₁₇	1	16	yes	complemented
174	C₂×C₂²⋊C₄	—	2	16	no	—
175	C₂×C₂²⋊C₄	—	4	32	no	—
176	C₂×C₂²⋊C₄	C₂³×C₁₇	1	16	yes	—
177	C₂×C₂²⋊C₄	C₂³×C₁₇	1	64	yes	complemented
178	C₂×C₄²	—	2	16	no	—
179	C₂×C₄²	C₂³×C₁₇	1	16	yes	complemented
180	C₁₇	C₂×#2309	1	1	yes	characteristic, direct product factor, Sylow
181	C₂⁴	—	2	8	no	—
182	C₂⁴	—	2	32	no	—
183	C₂⁴	—	2	16	no	—
184	C₂⁴	—	2	32	no	—
185	C₂⁴	—	2	64	no	—
186	C₂⁴	—	4	64	no	—
187	C₂⁴	—	4	32	no	—
188	C₂⁴	—	4	128	no	—
189	C₂⁴	C₂²×C₄×C₁₇	1	16	yes	—
190	C₂⁴	C₂²×C₄×C₁₇	1	4	yes	—
191	C₂⁴	C₂²×C₄×C₁₇	1	32	yes	—
192	C₂⁴	C₂×D₄×C₁₇	1	8	yes	—
193	C₂⁴	C₂⁴×C₁₇	1	8	yes	—
194	C₂⁴	C₂⁴×C₁₇	1	2	yes	—
195	C₂⁴	C₂⁴×C₁₇	1	1	yes	characteristic
196	C₂×D₄	—	2	32	no	—
197	C₂×D₄	—	2	32	no	—
198	C₂×D₄	—	2	128	no	—
199	C₂×D₄	—	2	256	no	—
200	C₂×D₄	C₂²×C₄×C₁₇	1	64	yes	—
201	C₂²×C₄	—	2	16	no	—
202	C₂²×C₄	—	2	32	no	—
203	C₂²×C₄	C₂²⋊C₄×C₁₇	1	64	yes	complemented
204	C₂²×C₄	C₂²×C₄×C₁₇	1	8	yes	—
205	C₂²×C₄	C₂²×C₄×C₁₇	1	16	yes	—
206	C₂²×C₄	C₂×D₄×C₁₇	1	8	yes	—
207	C₂²×C₄	C₂×D₄×C₁₇	1	8	yes	—
208	C₂²×C₄	C₂×D₄×C₁₇	1	32	yes	—
209	C₂²×C₄	C₂⁴×C₁₇	1	16	yes	—
210	C₂²×C₄	C₂⁴×C₁₇	1	4	yes	—
211	C₄⋊C₄	—	2	32	no	—
212	C₂²⋊C₄	—	2	32	no	—
213	C₂²⋊C₄	—	4	64	no	—
214	C₄²	—	2	32	no	—
215	C₂³	—	2	16	no	—
216	C₂³	—	2	32	no	—
217	C₂³	—	2	8	no	—
218	C₂³	—	2	64	no	—
219	C₂³	—	2	32	no	—
220	C₂³	—	2	64	no	—
221	C₂³	—	2	16	no	—
222	C₂³	—	2	32	no	—
223	C₂³	—	4	64	no	—
224	C₂³	—	4	256	no	—
225	C₂³	—	4	128	no	—
226	C₂³	—	4	64	no	—
227	C₂³	C₂×C₂²⋊C₄×C₁₇	1	16	yes	—
228	C₂³	C₂³×C₄×C₁₇	1	16	yes	—
229	C₂³	C₂³×C₄×C₁₇	1	4	yes	—
230	C₂³	C₂³×C₄×C₁₇	1	2	yes	—
231	C₂³	C₂²×D₄×C₁₇	1	8	yes	—
232	C₂³	C₂²×D₄×C₁₇	1	4	yes	—
233	C₂³	C₁₇×H(5,𝔽₂)	1	8	yes	—
234	C₂³	C₂⁵×C₁₇	1	1	yes	characteristic
235	D₄	—	2	128	no	—
236	C₂×C₄	—	2	16	no	—
237	C₂×C₄	—	2	32	no	—
238	C₂×C₄	—	4	32	no	—
239	C₂×C₄	C₂×C₂²⋊C₄×C₁₇	1	32	yes	—
240	C₂×C₄	C₂×C₂²⋊C₄×C₁₇	1	64	yes	complemented
241	C₂×C₄	C₂³×C₄×C₁₇	1	8	yes	—
242	C₂×C₄	C₂²×D₄×C₁₇	1	8	yes	—
243	C₂×C₄	C₂²×D₄×C₁₇	1	8	yes	—
244	C₂²	—	2	32	no	—
245	C₂²	—	2	8	no	—
246	C₂²	—	2	16	no	—
247	C₂²	—	2	16	no	—
248	C₂²	—	2	32	no	—
249	C₂²	—	4	128	no	—
250	C₂²	—	4	64	no	—
251	C₂²	—	4	32	no	—
252	C₂²	C₂²×C₂²⋊C₄×C₁₇	1	8	yes	—
253	C₂²	C₂²×C₂²⋊C₄×C₁₇	1	4	yes	—
254	C₂²	C₁₇×#518	1	8	yes	—
255	C₂²	C₁₇×#535	1	16	yes	—
256	C₂²	C₂⁴×C₄×C₁₇	1	1	yes	characteristic
257	C₂²	C₂³×D₄×C₁₇	1	2	yes	—
258	C₂²	C₂×C₁₇×H(5,𝔽₂)	1	4	yes	—
259	C₄	—	4	16	no	—
260	C₄	C₂²×C₂²⋊C₄×C₁₇	1	16	yes	—
261	C₂	—	2	8	no	—
262	C₂	—	4	16	no	—
263	C₂	C₁₇×#2309	1	8	yes	direct product factor
264	C₂	C₂³×C₂²⋊C₄×C₁₇	1	1	yes	characteristic
265	C₂	C₂×C₁₇×#518	1	2	yes	—
266	C₂	C₂×C₁₇×#535	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₁₇	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₁₇
Frattini, cosocle	C₂³	C₂⁵×C₁₇
prime radical	C₂⁴×D₄×C₁₇	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₁₇×#117	C₂	semidirect
3. C₁₇×#2309	C₂	direct (N × H)
4. C₂×C₁₇×#390	C₂	semidirect
5. C₂×C₁₇×#379	C₂	semidirect
8. C₂³×D₄×C₁₇	C₄	semidirect
13. C₂×C₁₇×#117	C₂²	semidirect
14. C₂²×C₄⋊C₄×C₁₇	C₂²	semidirect
16. C₂²×C₂²⋊C₄×C₁₇	C₂²	semidirect
17. C₂²×C₄²×C₁₇	C₂²	semidirect
18. C₁₇×#390	C₂²	semidirect
19. C₁₇×#379	C₂²	semidirect
31. C₂²×D₄×C₁₇	C₂×C₄	semidirect
40. C₂×C₄⋊C₄×C₁₇	C₂³	semidirect
44. C₂×C₂²⋊C₄×C₁₇	C₂³	semidirect
46. C₂×C₄²×C₁₇	C₂³	semidirect
69. C₂²×C₄×C₁₇	C₂²⋊C₄	semidirect
81. C₂×#2309	C₁₇	direct (N × H)
107. C₂×C₄×C₁₇	C₂×C₂²⋊C₄	semidirect
112. C₂²×#117	C₂×C₁₇	semidirect
113. #2309	C₂×C₁₇	direct (N × H)
114. C₂×#390	C₂×C₁₇	semidirect
115. C₂×#379	C₂×C₁₇	semidirect
135. C₂³×D₄	C₄×C₁₇	semidirect
140. C₂×#117	C₂²×C₁₇	semidirect
141. C₂²×C₄⋊C₄	C₂²×C₁₇	semidirect
143. C₂²×C₂²⋊C₄	C₂²×C₁₇	semidirect
144. C₂²×C₄²	C₂²×C₁₇	semidirect
145. #390	C₂²×C₁₇	semidirect
146. #379	C₂²×C₁₇	semidirect
149. C₂×C₁₇	#2309	direct (N × H)
164. C₂²×D₄	C₂×C₄×C₁₇	semidirect
173. C₂×C₄⋊C₄	C₂³×C₁₇	semidirect
177. C₂×C₂²⋊C₄	C₂³×C₁₇	semidirect
179. C₂×C₄²	C₂³×C₁₇	semidirect
180. C₁₇	C₂×#2309	direct (N × H)
203. C₂²×C₄	C₂²⋊C₄×C₁₇	semidirect
240. C₂×C₄	C₂×C₂²⋊C₄×C₁₇	semidirect
263. C₂	C₁₇×#2309	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×D₄×C₁₇	2
C₂²×C₁₇×#117	2
C₁₇×#2309	2
C₂×C₁₇×#390	2
C₂×C₁₇×#379	2
C₂×#2309	17

Subgroups — C2×C17×#2309 #16891785

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₇×#2309 #16891785