Subgroups — C₁₇×#21366 #16879110

Order 4352.

850 conjugacy classes of subgroups.

The subgroup lattice has 292 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	1	yes	maximal, characteristic
2	C₁₇×#2283	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₇×#117	C₂	1	1	yes	maximal, characteristic
4	C₁₇×#1949	C₂	1	4	yes	maximal, complemented
5	C₁₇×#1854	C₂	1	4	yes	maximal
6	C₁₇×#1776	C₂	1	4	yes	maximal, complemented
7	C₂×C₁₇×#117	C₂²	1	2	yes	—
8	C₂²×C₄²×C₁₇	C₂²	1	1	yes	characteristic
9	C₁₇×#432	C₄	1	8	yes	complemented
10	C₂×C₁₇×#107	C₂²	1	4	yes	—
11	C₂×C₁₇×#106	C₂²	1	4	yes	—
12	C₂×C₁₇×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₁₇×C₂²⋊C₈	C₂²	1	2	yes	—
14	C₂×C₁₇×#97	C₂²	1	2	yes	—
15	C₁₇×#383	—	2	1	no	—
16	C₁₇×#382	C₂²	1	2	yes	—
17	C₁₇×#381	—	2	1	no	—
18	C₁₇×#380	C₂²	1	2	yes	—
19	C₄×C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
20	C₄×C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
21	C₄×C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
22	C₄×C₂²⋊C₄×C₁₇	—	2	1	no	—
23	C₄×C₂²⋊C₄×C₁₇	—	2	1	no	—
24	C₄×C₂²⋊C₄×C₁₇	—	2	1	no	—
25	C₁₇×#376	—	2	1	no	—
26	C₁₇×#376	—	2	1	no	—
27	C₄³×C₁₇	—	2	1	no	—
28	C₁₇×#340	C₂²	1	8	yes	—
29	C₂³×C₄×C₁₇	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₁₇	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₁₇	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₁₇	C₂³	1	4	yes	—
33	C₁₇×#117	—	2	8	no	—
34	C₂×C₄⋊C₄×C₁₇	C₂³	1	4	yes	—
35	C₂×C₄⋊C₄×C₁₇	C₂³	1	4	yes	—
36	C₂×C₂²⋊C₄×C₁₇	—	2	2	no	—
37	C₂×C₂²⋊C₄×C₁₇	—	2	2	no	—
38	C₂×C₄²×C₁₇	—	2	2	no	—
39	C₂×C₄²×C₁₇	—	2	2	no	—
40	C₂×C₄²×C₁₇	—	2	2	no	—
41	C₂×C₄²×C₁₇	—	2	2	no	—
42	C₂×C₄²×C₁₇	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₁₇	Q₈	1	4	yes	—
46	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
48	C₁₇×#107	—	2	8	no	—
49	C₁₇×#106	—	2	8	no	—
50	C₁₇×C₄⋊C₈	C₂×C₄	1	8	yes	—
51	C₁₇×C₂²⋊C₈	C₂×C₄	1	8	yes	—
52	C₁₇×#97	C₂×C₄	1	8	yes	—
53	C₁₇×#95	—	2	2	no	—
54	C₁₇×#95	—	2	2	no	—
55	C₁₇×#95	—	2	2	no	—
56	C₁₇×#95	—	2	2	no	—
57	C₁₇×#95	—	2	2	no	—
58	C₁₇×#95	—	2	2	no	—
59	C₂⁴×C₁₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₇	—	2	4	no	—
61	C₂²×C₄×C₁₇	—	2	4	no	—
62	C₂²×C₄×C₁₇	—	2	4	no	—
63	C₂²×C₄×C₁₇	—	4	2	no	—
64	C₂²×C₄×C₁₇	—	4	2	no	—
65	C₂²×C₄×C₁₇	C₄⋊C₄	1	4	yes	—
66	C₂²×C₄×C₁₇	C₄⋊C₄	1	4	yes	—
67	C₂²×C₄×C₁₇	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₇	C₂×Q₈	1	2	yes	—
74	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
75	C₂×C₈×C₁₇	—	2	4	no	—
76	C₂×C₈×C₁₇	—	2	4	no	—
77	C₂×C₈×C₁₇	C₂²×C₄	1	8	yes	—
78	C₄⋊C₄×C₁₇	—	2	4	no	—
79	C₄⋊C₄×C₁₇	—	2	4	no	—
80	C₄⋊C₄×C₁₇	—	2	4	no	—
81	C₄⋊C₄×C₁₇	#41	1	8	yes	—
82	C₂²⋊C₄×C₁₇	—	2	8	no	—
83	C₂²⋊C₄×C₁₇	—	4	4	no	—
84	C₄²×C₁₇	—	2	8	no	—
85	C₄²×C₁₇	—	4	4	no	—
86	C₄²×C₁₇	—	4	4	no	—
87	C₄²×C₁₇	—	4	4	no	—
88	C₄²×C₁₇	C₄⋊C₄	1	4	yes	—
89	C₄²×C₁₇	C₄⋊C₄	1	4	yes	—
90	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
91	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
92	C₄²×C₁₇	C₂×Q₈	1	2	yes	—
93	C₄²×C₁₇	C₂×Q₈	1	2	yes	—
94	#21366	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂³×C₁₇	—	2	2	no	—
96	C₂³×C₁₇	—	2	2	no	—
97	C₂³×C₁₇	—	2	2	no	—
98	C₂³×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
99	C₂³×C₁₇	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₇	—	2	2	no	—
101	C₂×C₄×C₁₇	—	2	2	no	—
102	C₂×C₄×C₁₇	—	2	2	no	—
103	C₂×C₄×C₁₇	—	2	2	no	—
104	C₂×C₄×C₁₇	—	2	2	no	—
105	C₂×C₄×C₁₇	—	2	4	no	—
106	C₂×C₄×C₁₇	—	2	4	no	—
107	C₂×C₄×C₁₇	—	2	8	no	—
108	C₂×C₄×C₁₇	—	4	4	no	—
109	C₂×C₄×C₁₇	—	4	4	no	—
110	C₂×C₄×C₁₇	—	4	4	no	—
111	C₂×C₄×C₁₇	—	4	4	no	—
112	C₂×C₄×C₁₇	—	4	4	no	—
113	C₂×C₄×C₁₇	—	4	4	no	—
114	C₂×C₄×C₁₇	—	4	4	no	—
115	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
122	C₂×C₄×C₁₇	C₂×#41	1	4	yes	—
123	C₈×C₁₇	—	2	8	no	—
124	C₂×#432	C₂×C₁₇	1	1	yes	characteristic
125	#2283	C₂×C₁₇	1	1	yes	characteristic
126	C₄×#117	C₂×C₁₇	1	1	yes	characteristic
127	#1949	C₂×C₁₇	1	4	yes	complemented
128	#1854	C₂×C₁₇	1	4	yes	—
129	#1776	C₂×C₁₇	1	4	yes	complemented
130	C₂²×C₁₇	—	2	2	no	—
131	C₂²×C₁₇	—	2	2	no	—
132	C₂²×C₁₇	—	2	2	no	—
133	C₂²×C₁₇	—	4	4	no	—
134	C₂²×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
135	C₂²×C₁₇	#525	1	2	yes	—
136	C₂²×C₁₇	#575	1	2	yes	—
137	C₂²×C₁₇	#575	1	2	yes	—
138	C₄×C₁₇	—	2	2	no	—
139	C₄×C₁₇	—	2	2	no	—
140	C₄×C₁₇	—	2	2	no	—
141	C₄×C₁₇	—	2	2	no	—
142	C₄×C₁₇	—	2	2	no	—
143	C₄×C₁₇	—	4	8	no	—
144	C₄×C₁₇	—	8	4	no	—
145	C₄×C₁₇	C₄×#41	1	2	yes	—
146	C₄×C₁₇	#533	1	2	yes	—
147	C₂×#117	C₂²×C₁₇	1	2	yes	—
148	C₂²×C₄²	C₂²×C₁₇	1	1	yes	characteristic
149	#432	C₄×C₁₇	1	8	yes	complemented
150	C₂×#107	C₂²×C₁₇	1	4	yes	—
151	C₂×#106	C₂²×C₁₇	1	4	yes	—
152	C₂×C₄⋊C₈	C₂²×C₁₇	1	2	yes	—
153	C₂×C₂²⋊C₈	C₂²×C₁₇	1	2	yes	—
154	C₂×#97	C₂²×C₁₇	1	2	yes	—
155	#383	—	2	1	no	—
156	#382	C₂²×C₁₇	1	2	yes	—
157	#381	—	2	1	no	—
158	#380	C₂²×C₁₇	1	2	yes	—
159	C₄×C₄⋊C₄	C₂²×C₁₇	1	2	yes	—
160	C₄×C₄⋊C₄	C₂²×C₁₇	1	2	yes	—
161	C₄×C₄⋊C₄	C₂²×C₁₇	1	2	yes	—
162	C₄×C₂²⋊C₄	—	2	1	no	—
163	C₄×C₂²⋊C₄	—	2	1	no	—
164	C₄×C₂²⋊C₄	—	2	1	no	—
165	#376	—	2	1	no	—
166	#376	—	2	1	no	—
167	C₄³	—	2	1	no	—
168	#340	C₂²×C₁₇	1	8	yes	—
169	C₂×C₁₇	—	4	2	no	—
170	C₂×C₁₇	#2296	1	1	yes	characteristic
171	C₂×C₁₇	#2911	1	1	yes	characteristic
172	C₂×C₁₇	#2911	1	1	yes	characteristic
173	C₂×C₁₇	#3111	1	4	yes	—
174	C₂³×C₄	D₄×C₁₇	1	1	yes	characteristic
175	C₂³×C₄	D₄×C₁₇	1	1	yes	characteristic
176	C₂³×C₄	C₂³×C₁₇	1	1	yes	characteristic
177	C₂²×C₈	C₂³×C₁₇	1	4	yes	—
178	#117	—	2	8	no	—
179	C₂×C₄⋊C₄	C₂³×C₁₇	1	4	yes	—
180	C₂×C₄⋊C₄	C₂³×C₁₇	1	4	yes	—
181	C₂×C₂²⋊C₄	—	2	2	no	—
182	C₂×C₂²⋊C₄	—	2	2	no	—
183	C₂×C₄²	—	2	2	no	—
184	C₂×C₄²	—	2	2	no	—
185	C₂×C₄²	—	2	2	no	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	C₂×C₄×C₁₇	1	4	yes	—
188	C₂×C₄²	D₄×C₁₇	1	1	yes	characteristic
189	C₂×C₄²	D₄×C₁₇	1	1	yes	characteristic
190	C₂×C₄²	Q₈×C₁₇	1	4	yes	—
191	C₂×C₄²	C₂³×C₁₇	1	1	yes	characteristic
192	C₂×C₄²	C₂³×C₁₇	1	1	yes	characteristic
193	#107	—	2	8	no	—
194	#106	—	2	8	no	—
195	C₄⋊C₈	C₂×C₄×C₁₇	1	8	yes	—
196	C₂²⋊C₈	C₂×C₄×C₁₇	1	8	yes	—
197	#97	C₂×C₄×C₁₇	1	8	yes	—
198	#95	—	2	2	no	—
199	#95	—	2	2	no	—
200	#95	—	2	2	no	—
201	#95	—	2	2	no	—
202	#95	—	2	2	no	—
203	#95	—	2	2	no	—
204	C₁₇	#21366	1	1	yes	characteristic, direct product factor, Sylow
205	C₂⁴	C₂×D₄×C₁₇	1	1	yes	characteristic
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	4	2	no	—
210	C₂²×C₄	—	4	2	no	—
211	C₂²×C₄	C₄⋊C₄×C₁₇	1	4	yes	—
212	C₂²×C₄	C₄⋊C₄×C₁₇	1	4	yes	—
213	C₂²×C₄	C₂²×C₄×C₁₇	1	2	yes	—
214	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
219	C₂²×C₄	C₂×Q₈×C₁₇	1	2	yes	—
220	C₂²×C₄	C₂⁴×C₁₇	1	1	yes	characteristic
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	C₂²×C₄×C₁₇	1	8	yes	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	#41×C₁₇	1	8	yes	—
228	C₂²⋊C₄	—	2	8	no	—
229	C₂²⋊C₄	—	4	4	no	—
230	C₄²	—	2	8	no	—
231	C₄²	—	4	4	no	—
232	C₄²	—	4	4	no	—
233	C₄²	—	4	4	no	—
234	C₄²	C₄⋊C₄×C₁₇	1	4	yes	—
235	C₄²	C₄⋊C₄×C₁₇	1	4	yes	—
236	C₄²	C₂²×C₄×C₁₇	1	2	yes	—
237	C₄²	C₂²×C₄×C₁₇	1	2	yes	—
238	C₄²	C₂×Q₈×C₁₇	1	2	yes	—
239	C₄²	C₂×Q₈×C₁₇	1	2	yes	—
240	C₂³	—	2	2	no	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	C₂×C₄⋊C₄×C₁₇	1	2	yes	—
244	C₂³	C₂²×D₄×C₁₇	1	1	yes	characteristic
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	8	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	C₂×C₄⋊C₄×C₁₇	1	2	yes	—
261	C₂×C₄	C₂×C₄⋊C₄×C₁₇	1	2	yes	—
262	C₂×C₄	C₂×C₄⋊C₄×C₁₇	1	2	yes	—
263	C₂×C₄	C₂×C₄⋊C₄×C₁₇	1	2	yes	—
264	C₂×C₄	C₂×C₄⋊C₄×C₁₇	1	2	yes	—
265	C₂×C₄	C₂³×C₄×C₁₇	1	1	yes	characteristic
266	C₂×C₄	C₂²×Q₈×C₁₇	1	1	yes	characteristic
267	C₂×C₄	C₂×#41×C₁₇	1	4	yes	—
268	C₈	—	2	8	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	4	4	no	—
273	C₂²	C₂²×C₄⋊C₄×C₁₇	1	1	yes	characteristic
274	C₂²	C₁₇×#525	1	2	yes	—
275	C₂²	C₁₇×#575	1	2	yes	—
276	C₂²	C₁₇×#575	1	2	yes	—
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	4	8	no	—
283	C₄	—	8	4	no	—
284	C₄	C₄×#41×C₁₇	1	2	yes	—
285	C₄	C₁₇×#533	1	2	yes	—
286	C₂	—	4	2	no	—
287	C₂	C₁₇×#2296	1	1	yes	characteristic
288	C₂	C₁₇×#2911	1	1	yes	characteristic
289	C₂	C₁₇×#2911	1	1	yes	characteristic
290	C₂	C₁₇×#3111	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₇
prime radical	C₂⁴×C₁₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ 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
4. C₁₇×#1949	C₂	semidirect
6. C₁₇×#1776	C₂	semidirect
9. C₁₇×#432	C₄	semidirect
94. #21366	C₁₇	direct (N × H)
127. #1949	C₂×C₁₇	semidirect
129. #1776	C₂×C₁₇	semidirect
149. #432	C₄×C₁₇	semidirect
204. C₁₇	#21366	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
C₁₇×#2283	2
C₄×C₁₇×#117	2
C₁₇×#1949	2
C₁₇×#1854	2
C₁₇×#1776	2
#21366	17

Subgroups — C17×#21366 #16879110

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#21366 #16879110