Subgroups — C₂×C₁₇×#2315 #16891793

Order 4352.

5182 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₂²⋊C₄×C₁₇	C₂	1	2	yes	maximal
2	C₂³×C₂²⋊C₄×C₁₇	C₂	1	1	yes	maximal, characteristic
3	C₁₇×#2315	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₁₇×#385	C₂	1	8	yes	maximal, complemented
5	C₂×C₁₇×#380	C₂	1	4	yes	maximal, complemented
6	C₂⁶×C₁₇	C₂²	1	1	yes	characteristic
7	C₂⁴×C₄×C₁₇	C₂²	1	4	yes	—
8	C₂⁴×C₄×C₁₇	C₂²	1	2	yes	—
9	C₂²×C₄⋊C₄×C₁₇	C₂²	1	8	yes	complemented
10	C₂²×C₂²⋊C₄×C₁₇	—	2	16	no	—
11	C₂²×C₂²⋊C₄×C₁₇	C₄	1	16	yes	complemented
12	C₂²×C₂²⋊C₄×C₁₇	C₂²	1	4	yes	—
13	C₂²×C₂²⋊C₄×C₁₇	C₂²	1	4	yes	—
14	C₂²×C₂²⋊C₄×C₁₇	C₂²	1	16	yes	—
15	C₂²×C₂²⋊C₄×C₁₇	C₂²	1	4	yes	—
16	C₂²×C₂²⋊C₄×C₁₇	C₂²	1	8	yes	—
17	C₁₇×#385	C₂²	1	64	yes	complemented
18	C₁₇×#380	C₂²	1	32	yes	complemented
19	C₂×C₁₇×#95	C₂²	1	8	yes	complemented
20	C₂⁵×C₁₇	—	4	8	no	—
21	C₂⁵×C₁₇	C₂×C₄	1	8	yes	—
22	C₂⁵×C₁₇	D₄	1	8	yes	—
23	C₂⁵×C₁₇	Q₈	1	8	yes	—
24	C₂⁵×C₁₇	C₂³	1	1	yes	characteristic
25	C₂⁵×C₁₇	C₂³	1	2	yes	—
26	C₂⁵×C₁₇	C₂³	1	4	yes	—
27	C₂³×C₄×C₁₇	—	2	16	no	—
28	C₂³×C₄×C₁₇	—	4	8	no	—
29	C₂³×C₄×C₁₇	—	4	16	no	—
30	C₂³×C₄×C₁₇	C₂×C₄	1	16	yes	complemented
31	C₂³×C₄×C₁₇	C₂³	1	8	yes	—
32	C₂³×C₄×C₁₇	C₂³	1	4	yes	—
33	C₂³×C₄×C₁₇	C₂³	1	16	yes	—
34	C₂³×C₄×C₁₇	C₂³	1	8	yes	—
35	C₂×C₄⋊C₄×C₁₇	—	4	16	no	—
36	C₂×C₄⋊C₄×C₁₇	C₂³	1	32	yes	complemented
37	C₂×C₂²⋊C₄×C₁₇	—	2	16	no	—
38	C₂×C₂²⋊C₄×C₁₇	—	2	16	no	—
39	C₂×C₂²⋊C₄×C₁₇	—	2	64	no	—
40	C₂×C₂²⋊C₄×C₁₇	C₂×C₄	1	32	yes	complemented
41	C₂×C₂²⋊C₄×C₁₇	C₂×C₄	1	64	yes	complemented
42	C₂×C₂²⋊C₄×C₁₇	C₂³	1	16	yes	—
43	C₂×C₂²⋊C₄×C₁₇	C₂³	1	16	yes	—
44	C₂×C₂²⋊C₄×C₁₇	C₂³	1	16	yes	—
45	C₁₇×#95	C₂³	1	32	yes	complemented
46	C₂⁴×C₁₇	—	2	8	no	—
47	C₂⁴×C₁₇	—	2	16	no	—
48	C₂⁴×C₁₇	—	4	32	no	—
49	C₂⁴×C₁₇	—	4	32	no	—
50	C₂⁴×C₁₇	—	4	32	no	—
51	C₂⁴×C₁₇	—	4	16	no	—
52	C₂⁴×C₁₇	C₄⋊C₄	1	64	yes	complemented
53	C₂⁴×C₁₇	C₂²×C₄	1	4	yes	—
54	C₂⁴×C₁₇	C₂²×C₄	1	8	yes	—
55	C₂⁴×C₁₇	C₂²×C₄	1	16	yes	—
56	C₂⁴×C₁₇	C₂×D₄	1	8	yes	—
57	C₂⁴×C₁₇	C₂×D₄	1	4	yes	—
58	C₂⁴×C₁₇	C₂×D₄	1	16	yes	—
59	C₂⁴×C₁₇	C₂×Q₈	1	4	yes	—
60	C₂⁴×C₁₇	C₂×Q₈	1	8	yes	—
61	C₂⁴×C₁₇	C₂×Q₈	1	16	yes	—
62	C₂⁴×C₁₇	C₂⁴	1	4	yes	—
63	C₂⁴×C₁₇	C₂⁴	1	2	yes	—
64	C₂⁴×C₁₇	C₂⁴	1	1	yes	characteristic
65	C₂²×C₄×C₁₇	—	2	16	no	—
66	C₂²×C₄×C₁₇	—	2	32	no	—
67	C₂²×C₄×C₁₇	—	2	16	no	—
68	C₂²×C₄×C₁₇	—	2	32	no	—
69	C₂²×C₄×C₁₇	—	4	32	no	—
70	C₂²×C₄×C₁₇	—	4	16	no	—
71	C₂²×C₄×C₁₇	—	4	32	no	—
72	C₂²×C₄×C₁₇	—	4	64	no	—
73	C₂²×C₄×C₁₇	C₂²×C₄	1	16	yes	—
74	C₂²×C₄×C₁₇	C₂²×C₄	1	32	yes	complemented
75	C₂²×C₄×C₁₇	C₂⁴	1	16	yes	—
76	C₂²×C₄×C₁₇	C₂⁴	1	8	yes	—
77	C₄⋊C₄×C₁₇	—	4	32	no	—
78	C₂²⋊C₄×C₁₇	—	2	32	no	—
79	C₂²⋊C₄×C₁₇	—	2	32	no	—
80	C₂²⋊C₄×C₁₇	C₂²×C₄	1	64	yes	complemented
81	C₂×#2315	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	32	no	—
85	C₂³×C₁₇	—	2	16	no	—
86	C₂³×C₁₇	—	2	16	no	—
87	C₂³×C₁₇	—	2	32	no	—
88	C₂³×C₁₇	—	2	8	no	—
89	C₂³×C₁₇	—	2	16	no	—
90	C₂³×C₁₇	—	4	128	no	—
91	C₂³×C₁₇	—	4	32	no	—
92	C₂³×C₁₇	—	4	32	no	—
93	C₂³×C₁₇	—	4	32	no	—
94	C₂³×C₁₇	C₂×C₄⋊C₄	1	16	yes	—
95	C₂³×C₁₇	C₂×C₄⋊C₄	1	32	yes	—
96	C₂³×C₁₇	C₂×C₄⋊C₄	1	64	yes	complemented
97	C₂³×C₁₇	C₂³×C₄	1	8	yes	—
98	C₂³×C₁₇	C₂³×C₄	1	4	yes	—
99	C₂³×C₁₇	C₂³×C₄	1	2	yes	—
100	C₂³×C₁₇	C₂²×D₄	1	4	yes	—
101	C₂³×C₁₇	C₂²×D₄	1	8	yes	—
102	C₂³×C₁₇	C₂²×D₄	1	2	yes	—
103	C₂³×C₁₇	C₂²×Q₈	1	8	yes	—
104	C₂³×C₁₇	C₂²×Q₈	1	4	yes	—
105	C₂³×C₁₇	C₂²×Q₈	1	2	yes	—
106	C₂³×C₁₇	H(5,𝔽₂)	1	8	yes	—
107	C₂³×C₁₇	C₂⁵	1	1	yes	characteristic
108	C₂×C₄×C₁₇	—	2	32	no	—
109	C₂×C₄×C₁₇	—	2	16	no	—
110	C₂×C₄×C₁₇	—	2	16	no	—
111	C₂×C₄×C₁₇	—	2	32	no	—
112	C₂×C₄×C₁₇	—	4	32	no	—
113	C₂×C₄×C₁₇	—	4	64	no	—
114	C₂×C₄×C₁₇	—	4	32	no	—
115	C₂×C₄×C₁₇	C₂³×C₄	1	16	yes	—
116	C₂³×C₂²⋊C₄	C₂×C₁₇	1	2	yes	—
117	C₂³×C₂²⋊C₄	C₂×C₁₇	1	1	yes	characteristic
118	#2315	C₂×C₁₇	1	16	yes	direct product factor
119	C₂×#385	C₂×C₁₇	1	8	yes	complemented
120	C₂×#380	C₂×C₁₇	1	4	yes	complemented
121	C₂²×C₁₇	—	2	32	no	—
122	C₂²×C₁₇	—	2	64	no	—
123	C₂²×C₁₇	—	2	8	no	—
124	C₂²×C₁₇	—	2	16	no	—
125	C₂²×C₁₇	—	2	16	no	—
126	C₂²×C₁₇	—	2	8	no	—
127	C₂²×C₁₇	—	2	8	no	—
128	C₂²×C₁₇	—	2	16	no	—
129	C₂²×C₁₇	—	4	64	no	—
130	C₂²×C₁₇	C₂²×C₄⋊C₄	1	16	yes	—
131	C₂²×C₁₇	C₂²×C₄⋊C₄	1	8	yes	—
132	C₂²×C₁₇	C₂²×C₄⋊C₄	1	4	yes	—
133	C₂²×C₁₇	#518	1	8	yes	—
134	C₂²×C₁₇	#534	1	8	yes	—
135	C₂²×C₁₇	#543	1	8	yes	—
136	C₂²×C₁₇	C₂⁴×C₄	1	1	yes	characteristic
137	C₂²×C₁₇	C₂³×D₄	1	1	yes	characteristic
138	C₂²×C₁₇	C₂³×Q₈	1	1	yes	characteristic
139	C₂²×C₁₇	C₂×H(5,𝔽₂)	1	4	yes	—
140	C₄×C₁₇	—	2	16	no	—
141	C₄×C₁₇	—	4	16	no	—
142	C₂⁶	C₂²×C₁₇	1	1	yes	characteristic
143	C₂⁴×C₄	C₂²×C₁₇	1	4	yes	—
144	C₂⁴×C₄	C₂²×C₁₇	1	2	yes	—
145	C₂²×C₄⋊C₄	C₂²×C₁₇	1	8	yes	complemented
146	C₂²×C₂²⋊C₄	—	2	16	no	—
147	C₂²×C₂²⋊C₄	C₄×C₁₇	1	16	yes	complemented
148	C₂²×C₂²⋊C₄	C₂²×C₁₇	1	16	yes	—
149	C₂²×C₂²⋊C₄	C₂²×C₁₇	1	4	yes	—
150	C₂²×C₂²⋊C₄	C₂²×C₁₇	1	4	yes	—
151	C₂²×C₂²⋊C₄	C₂²×C₁₇	1	8	yes	—
152	C₂²×C₂²⋊C₄	C₂²×C₁₇	1	4	yes	—
153	#385	C₂²×C₁₇	1	64	yes	complemented
154	#380	C₂²×C₁₇	1	32	yes	complemented
155	C₂×#95	C₂²×C₁₇	1	8	yes	complemented
156	C₂×C₁₇	—	2	8	no	—
157	C₂×C₁₇	—	2	16	no	—
158	C₂×C₁₇	#2315	1	8	yes	direct product factor
159	C₂×C₁₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₁₇	C₂×#518	1	2	yes	—
161	C₂×C₁₇	C₂×#534	1	2	yes	—
162	C₂×C₁₇	C₂×#543	1	2	yes	—
163	C₂⁵	—	4	8	no	—
164	C₂⁵	C₂×C₄×C₁₇	1	8	yes	—
165	C₂⁵	D₄×C₁₇	1	8	yes	—
166	C₂⁵	Q₈×C₁₇	1	8	yes	—
167	C₂⁵	C₂³×C₁₇	1	2	yes	—
168	C₂⁵	C₂³×C₁₇	1	1	yes	characteristic
169	C₂⁵	C₂³×C₁₇	1	4	yes	—
170	C₂³×C₄	—	2	16	no	—
171	C₂³×C₄	—	4	8	no	—
172	C₂³×C₄	—	4	16	no	—
173	C₂³×C₄	C₂×C₄×C₁₇	1	16	yes	complemented
174	C₂³×C₄	C₂³×C₁₇	1	8	yes	—
175	C₂³×C₄	C₂³×C₁₇	1	4	yes	—
176	C₂³×C₄	C₂³×C₁₇	1	16	yes	—
177	C₂³×C₄	C₂³×C₁₇	1	8	yes	—
178	C₂×C₄⋊C₄	—	4	16	no	—
179	C₂×C₄⋊C₄	C₂³×C₁₇	1	32	yes	complemented
180	C₂×C₂²⋊C₄	—	2	16	no	—
181	C₂×C₂²⋊C₄	—	2	16	no	—
182	C₂×C₂²⋊C₄	—	2	64	no	—
183	C₂×C₂²⋊C₄	C₂×C₄×C₁₇	1	32	yes	complemented
184	C₂×C₂²⋊C₄	C₂×C₄×C₁₇	1	64	yes	complemented
185	C₂×C₂²⋊C₄	C₂³×C₁₇	1	16	yes	—
186	C₂×C₂²⋊C₄	C₂³×C₁₇	1	16	yes	—
187	C₂×C₂²⋊C₄	C₂³×C₁₇	1	16	yes	—
188	#95	C₂³×C₁₇	1	32	yes	complemented
189	C₁₇	C₂×#2315	1	1	yes	characteristic, direct product factor, Sylow
190	C₂⁴	—	2	8	no	—
191	C₂⁴	—	2	16	no	—
192	C₂⁴	—	4	32	no	—
193	C₂⁴	—	4	32	no	—
194	C₂⁴	—	4	32	no	—
195	C₂⁴	—	4	16	no	—
196	C₂⁴	C₄⋊C₄×C₁₇	1	64	yes	complemented
197	C₂⁴	C₂²×C₄×C₁₇	1	8	yes	—
198	C₂⁴	C₂²×C₄×C₁₇	1	4	yes	—
199	C₂⁴	C₂²×C₄×C₁₇	1	16	yes	—
200	C₂⁴	C₂×D₄×C₁₇	1	4	yes	—
201	C₂⁴	C₂×D₄×C₁₇	1	8	yes	—
202	C₂⁴	C₂×D₄×C₁₇	1	16	yes	—
203	C₂⁴	C₂×Q₈×C₁₇	1	8	yes	—
204	C₂⁴	C₂×Q₈×C₁₇	1	4	yes	—
205	C₂⁴	C₂×Q₈×C₁₇	1	16	yes	—
206	C₂⁴	C₂⁴×C₁₇	1	4	yes	—
207	C₂⁴	C₂⁴×C₁₇	1	2	yes	—
208	C₂⁴	C₂⁴×C₁₇	1	1	yes	characteristic
209	C₂²×C₄	—	2	16	no	—
210	C₂²×C₄	—	2	32	no	—
211	C₂²×C₄	—	2	16	no	—
212	C₂²×C₄	—	2	32	no	—
213	C₂²×C₄	—	4	32	no	—
214	C₂²×C₄	—	4	16	no	—
215	C₂²×C₄	—	4	32	no	—
216	C₂²×C₄	—	4	64	no	—
217	C₂²×C₄	C₂²×C₄×C₁₇	1	16	yes	—
218	C₂²×C₄	C₂²×C₄×C₁₇	1	32	yes	complemented
219	C₂²×C₄	C₂⁴×C₁₇	1	16	yes	—
220	C₂²×C₄	C₂⁴×C₁₇	1	8	yes	—
221	C₄⋊C₄	—	4	32	no	—
222	C₂²⋊C₄	—	2	32	no	—
223	C₂²⋊C₄	—	2	32	no	—
224	C₂²⋊C₄	C₂²×C₄×C₁₇	1	64	yes	complemented
225	C₂³	—	2	16	no	—
226	C₂³	—	2	32	no	—
227	C₂³	—	2	32	no	—
228	C₂³	—	2	16	no	—
229	C₂³	—	2	16	no	—
230	C₂³	—	2	32	no	—
231	C₂³	—	2	8	no	—
232	C₂³	—	2	16	no	—
233	C₂³	—	4	128	no	—
234	C₂³	—	4	32	no	—
235	C₂³	—	4	32	no	—
236	C₂³	—	4	32	no	—
237	C₂³	C₂×C₄⋊C₄×C₁₇	1	32	yes	—
238	C₂³	C₂×C₄⋊C₄×C₁₇	1	16	yes	—
239	C₂³	C₂×C₄⋊C₄×C₁₇	1	64	yes	complemented
240	C₂³	C₂³×C₄×C₁₇	1	8	yes	—
241	C₂³	C₂³×C₄×C₁₇	1	4	yes	—
242	C₂³	C₂³×C₄×C₁₇	1	2	yes	—
243	C₂³	C₂²×D₄×C₁₇	1	4	yes	—
244	C₂³	C₂²×D₄×C₁₇	1	8	yes	—
245	C₂³	C₂²×D₄×C₁₇	1	2	yes	—
246	C₂³	C₂²×Q₈×C₁₇	1	8	yes	—
247	C₂³	C₂²×Q₈×C₁₇	1	4	yes	—
248	C₂³	C₂²×Q₈×C₁₇	1	2	yes	—
249	C₂³	C₁₇×H(5,𝔽₂)	1	8	yes	—
250	C₂³	C₂⁵×C₁₇	1	1	yes	characteristic
251	C₂×C₄	—	2	32	no	—
252	C₂×C₄	—	2	16	no	—
253	C₂×C₄	—	2	16	no	—
254	C₂×C₄	—	2	32	no	—
255	C₂×C₄	—	4	32	no	—
256	C₂×C₄	—	4	64	no	—
257	C₂×C₄	—	4	32	no	—
258	C₂×C₄	C₂³×C₄×C₁₇	1	16	yes	—
259	C₂²	—	2	32	no	—
260	C₂²	—	2	64	no	—
261	C₂²	—	2	8	no	—
262	C₂²	—	2	16	no	—
263	C₂²	—	2	16	no	—
264	C₂²	—	2	8	no	—
265	C₂²	—	2	8	no	—
266	C₂²	—	2	16	no	—
267	C₂²	—	4	64	no	—
268	C₂²	C₂²×C₄⋊C₄×C₁₇	1	16	yes	—
269	C₂²	C₂²×C₄⋊C₄×C₁₇	1	8	yes	—
270	C₂²	C₂²×C₄⋊C₄×C₁₇	1	4	yes	—
271	C₂²	C₁₇×#518	1	8	yes	—
272	C₂²	C₁₇×#534	1	8	yes	—
273	C₂²	C₁₇×#543	1	8	yes	—
274	C₂²	C₂⁴×C₄×C₁₇	1	1	yes	characteristic
275	C₂²	C₂³×D₄×C₁₇	1	1	yes	characteristic
276	C₂²	C₂³×Q₈×C₁₇	1	1	yes	characteristic
277	C₂²	C₂×C₁₇×H(5,𝔽₂)	1	4	yes	—
278	C₄	—	2	16	no	—
279	C₄	—	4	16	no	—
280	C₂	—	2	8	no	—
281	C₂	—	2	16	no	—
282	C₂	C₁₇×#2315	1	8	yes	direct product factor
283	C₂	C₂³×C₄⋊C₄×C₁₇	1	1	yes	characteristic
284	C₂	C₂×C₁₇×#518	1	2	yes	—
285	C₂	C₂×C₁₇×#534	1	2	yes	—
286	C₂	C₂×C₁₇×#543	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₁₇	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₁₇
Frattini, cosocle	C₂³	C₂⁵×C₁₇
prime radical	C₂⁶×C₁₇	C₂²
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
3. C₁₇×#2315	C₂	direct (N × H)
4. C₂×C₁₇×#385	C₂	semidirect
5. C₂×C₁₇×#380	C₂	semidirect
9. C₂²×C₄⋊C₄×C₁₇	C₂²	semidirect
11. C₂²×C₂²⋊C₄×C₁₇	C₄	semidirect
17. C₁₇×#385	C₂²	semidirect
18. C₁₇×#380	C₂²	semidirect
19. C₂×C₁₇×#95	C₂²	semidirect
30. C₂³×C₄×C₁₇	C₂×C₄	semidirect
36. C₂×C₄⋊C₄×C₁₇	C₂³	semidirect
40. C₂×C₂²⋊C₄×C₁₇	C₂×C₄	semidirect
41. C₂×C₂²⋊C₄×C₁₇	C₂×C₄	semidirect
45. C₁₇×#95	C₂³	semidirect
52. C₂⁴×C₁₇	C₄⋊C₄	semidirect
74. C₂²×C₄×C₁₇	C₂²×C₄	semidirect
80. C₂²⋊C₄×C₁₇	C₂²×C₄	semidirect
81. C₂×#2315	C₁₇	direct (N × H)
96. C₂³×C₁₇	C₂×C₄⋊C₄	semidirect
118. #2315	C₂×C₁₇	direct (N × H)
119. C₂×#385	C₂×C₁₇	semidirect
120. C₂×#380	C₂×C₁₇	semidirect
145. C₂²×C₄⋊C₄	C₂²×C₁₇	semidirect
147. C₂²×C₂²⋊C₄	C₄×C₁₇	semidirect
153. #385	C₂²×C₁₇	semidirect
154. #380	C₂²×C₁₇	semidirect
155. C₂×#95	C₂²×C₁₇	semidirect
158. C₂×C₁₇	#2315	direct (N × H)
173. C₂³×C₄	C₂×C₄×C₁₇	semidirect
179. C₂×C₄⋊C₄	C₂³×C₁₇	semidirect
183. C₂×C₂²⋊C₄	C₂×C₄×C₁₇	semidirect
184. C₂×C₂²⋊C₄	C₂×C₄×C₁₇	semidirect
188. #95	C₂³×C₁₇	semidirect
189. C₁₇	C₂×#2315	direct (N × H)
196. C₂⁴	C₄⋊C₄×C₁₇	semidirect
218. C₂²×C₄	C₂²×C₄×C₁₇	semidirect
224. C₂²⋊C₄	C₂²×C₄×C₁₇	semidirect
239. C₂³	C₂×C₄⋊C₄×C₁₇	semidirect
282. C₂	C₁₇×#2315	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₂²⋊C₄×C₁₇	2
C₁₇×#2315	2
C₂×C₁₇×#385	2
C₂×C₁₇×#380	2
C₂×#2315	17

Subgroups — C2×C17×#2315 #16891793

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₇×#2315 #16891793