Subgroups — C₁₇×#2933 #14525939

Order 2176.

502 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₇×#131	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×M₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₇×#525	C₂	1	1	yes	maximal, characteristic
4	C₁₇×#436	C₂	1	2	yes	maximal, complemented
5	C₁₇×#435	C₂	1	2	yes	maximal, complemented
6	C₁₇×#435	C₂	1	2	yes	maximal, complemented
7	C₈×D₄×C₁₇	C₂	1	2	yes	maximal, complemented
8	C₁₇×#424	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₇×#408	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₇×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₇×#405	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₁₇	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₁₇	C₂²	1	1	yes	characteristic
14	C₁₇×#131	—	2	4	no	—
15	C₂×M₁₆×C₁₇	—	2	4	no	—
16	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
17	C₂×M₁₆×C₁₇	C₂²	1	2	yes	—
18	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×C₁₇	C₂²	1	2	yes	complemented
20	C₂²×C₈×C₁₇	C₂²	1	2	yes	—
21	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
22	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
23	C₂²×C₈×C₁₇	C₂²	1	2	yes	complemented
24	C₁₇×#123	C₄	1	2	yes	—
25	C₁₇×C₂²⋊Q₈	C₄	1	1	yes	characteristic
26	C₁₇×C₂²⋊Q₈	C₄	1	1	yes	characteristic
27	C₁₇×#121	C₄	1	1	yes	characteristic
28	C₁₇×#121	C₄	1	1	yes	characteristic
29	C₁₇×#120	C₄	1	2	yes	—
30	C₄×D₄×C₁₇	C₂²	1	2	yes	—
31	C₄×D₄×C₁₇	C₂²	1	2	yes	—
32	C₁₇×#117	C₂²	1	1	yes	characteristic
33	C₁₇×C₄⋊C₈	C₂²	1	2	yes	complemented
34	C₁₇×C₄⋊C₈	C₂²	1	2	yes	complemented
35	C₁₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
36	C₁₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
37	C₁₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
38	C₁₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
39	C₁₇×C₂²⋊C₈	C₂²	1	2	yes	complemented
40	C₁₇×C₂²⋊C₈	C₂²	1	2	yes	complemented
41	C₁₇×#97	C₂²	1	2	yes	complemented
42	C₄×C₈×C₁₇	C₂²	1	2	yes	complemented
43	C₂⁴×C₁₇	C₂×C₄	1	1	yes	characteristic
44	#41×C₁₇	—	2	2	no	—
45	#41×C₁₇	—	2	2	no	—
46	C₂×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
47	C₂×D₄×C₁₇	C₂×C₄	1	2	yes	—
48	C₂×D₄×C₁₇	C₂×C₄	1	1	yes	characteristic
49	C₂×D₄×C₁₇	C₂×C₄	1	2	yes	—
50	C₂×D₄×C₁₇	C₂×C₄	1	2	yes	—
51	C₂²×C₄×C₁₇	—	2	2	no	—
52	C₂²×C₄×C₁₇	—	2	2	no	—
53	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₇	C₂³	1	2	yes	—
58	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
60	M₁₆×C₁₇	—	2	2	no	—
61	M₁₆×C₁₇	—	2	4	no	—
62	M₁₆×C₁₇	—	2	4	no	—
63	M₁₆×C₁₇	D₄	1	4	yes	complemented
64	C₂×C₈×C₁₇	—	2	2	no	—
65	C₂×C₈×C₁₇	—	2	2	no	—
66	C₂×C₈×C₁₇	—	2	4	no	—
67	C₂×C₈×C₁₇	—	2	4	no	—
68	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₁₇	C₂³	1	2	yes	—
70	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₁₇	C₂³	1	2	yes	—
74	C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
75	C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
76	C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
77	C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
78	C₂²⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
79	C₂²⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
80	C₂²⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
81	C₂²⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
82	C₂²⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
83	C₄²×C₁₇	C₂³	1	2	yes	—
84	C₂³×C₁₇	—	2	2	no	—
85	C₂³×C₁₇	—	2	2	no	—
86	C₂³×C₁₇	—	4	1	no	—
87	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
88	C₂³×C₁₇	C₂²×C₄	1	2	yes	—
89	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
90	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
91	Q₈×C₁₇	—	2	2	no	—
92	D₄×C₁₇	—	2	4	no	—
93	D₄×C₁₇	—	2	4	no	—
94	D₄×C₁₇	—	2	2	no	—
95	D₄×C₁₇	—	4	2	no	—
96	C₂×C₄×C₁₇	—	2	2	no	—
97	C₂×C₄×C₁₇	—	2	2	no	—
98	C₂×C₄×C₁₇	—	2	2	no	—
99	C₂×C₄×C₁₇	—	2	1	no	—
100	C₂×C₄×C₁₇	—	2	2	no	—
101	C₂×C₄×C₁₇	—	2	1	no	—
102	C₂×C₄×C₁₇	—	2	1	no	—
103	C₂×C₄×C₁₇	—	2	1	no	—
104	C₂×C₄×C₁₇	—	2	2	no	—
105	C₂×C₄×C₁₇	C₂²×C₄	1	2	yes	—
106	C₂×C₄×C₁₇	C₂²×C₄	1	2	yes	—
107	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₇	C₂²×C₄	1	2	yes	—
110	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
112	C₂×C₄×C₁₇	#41	1	2	yes	—
113	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
114	C₈×C₁₇	—	2	1	no	—
115	C₈×C₁₇	—	2	1	no	—
116	C₈×C₁₇	—	2	2	no	—
117	C₈×C₁₇	—	2	1	no	—
118	C₈×C₁₇	—	2	1	no	—
119	C₈×C₁₇	C₂×D₄	1	4	yes	—
120	#2933	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂²×C₁₇	—	2	1	no	—
122	C₂²×C₁₇	—	2	1	no	—
123	C₂²×C₁₇	—	2	1	no	—
124	C₂²×C₁₇	—	2	1	no	—
125	C₂²×C₁₇	—	2	1	no	—
126	C₂²×C₁₇	—	2	2	no	—
127	C₂²×C₁₇	—	2	2	no	—
128	C₂²×C₁₇	—	2	1	no	—
129	C₂²×C₁₇	—	2	1	no	—
130	C₂²×C₁₇	—	2	2	no	—
131	C₂²×C₁₇	—	4	4	no	—
132	C₂²×C₁₇	C₄×D₄	1	2	yes	—
133	C₂²×C₁₇	#131	1	2	yes	complemented
134	C₂²×C₁₇	C₂³×C₄	1	1	yes	characteristic
135	C₄×C₁₇	—	2	1	no	—
136	C₄×C₁₇	—	2	1	no	—
137	C₄×C₁₇	—	2	2	no	—
138	C₄×C₁₇	—	2	2	no	—
139	C₄×C₁₇	—	2	2	no	—
140	C₄×C₁₇	C₄×D₄	1	2	yes	—
141	C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
142	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
143	C₂×#131	C₂×C₁₇	1	1	yes	characteristic, complemented
144	C₂²×M₁₆	C₂×C₁₇	1	1	yes	characteristic, complemented
145	#525	C₂×C₁₇	1	1	yes	characteristic
146	#436	C₂×C₁₇	1	2	yes	complemented
147	#435	C₂×C₁₇	1	2	yes	complemented
148	#435	C₂×C₁₇	1	2	yes	complemented
149	C₈×D₄	C₂×C₁₇	1	2	yes	complemented
150	#424	C₂×C₁₇	1	1	yes	characteristic, complemented
151	#408	C₂×C₁₇	1	1	yes	characteristic, complemented
152	C₂×C₂²⋊C₈	C₂×C₁₇	1	1	yes	characteristic, complemented
153	#405	C₂×C₁₇	1	1	yes	characteristic, complemented
154	C₂×C₁₇	—	2	2	no	—
155	C₂×C₁₇	—	2	2	no	—
156	C₂×C₁₇	—	4	2	no	—
157	C₂×C₁₇	—	4	1	no	—
158	C₂×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
159	C₂×C₁₇	C₂×#131	1	1	yes	characteristic
160	C₂×C₁₇	#568	1	1	yes	characteristic
161	C₂×#41	C₂²×C₁₇	1	1	yes	characteristic
162	C₂³×C₄	C₂²×C₁₇	1	1	yes	characteristic
163	#131	—	2	4	no	—
164	C₂×M₁₆	—	2	4	no	—
165	C₂×M₁₆	C₂²×C₁₇	1	1	yes	characteristic
166	C₂×M₁₆	C₂²×C₁₇	1	2	yes	—
167	C₂×M₁₆	C₂²×C₁₇	1	1	yes	characteristic
168	C₂×M₁₆	C₂²×C₁₇	1	2	yes	complemented
169	C₂²×C₈	C₂²×C₁₇	1	2	yes	—
170	C₂²×C₈	C₂²×C₁₇	1	1	yes	characteristic
171	C₂²×C₈	C₂²×C₁₇	1	1	yes	characteristic
172	C₂²×C₈	C₂²×C₁₇	1	2	yes	complemented
173	#123	C₄×C₁₇	1	2	yes	—
174	C₂²⋊Q₈	C₄×C₁₇	1	1	yes	characteristic
175	C₂²⋊Q₈	C₄×C₁₇	1	1	yes	characteristic
176	#121	C₄×C₁₇	1	1	yes	characteristic
177	#121	C₄×C₁₇	1	1	yes	characteristic
178	#120	C₄×C₁₇	1	2	yes	—
179	C₄×D₄	C₂²×C₁₇	1	2	yes	—
180	C₄×D₄	C₂²×C₁₇	1	2	yes	—
181	#117	C₂²×C₁₇	1	1	yes	characteristic
182	C₄⋊C₈	C₂²×C₁₇	1	2	yes	complemented
183	C₄⋊C₈	C₂²×C₁₇	1	2	yes	complemented
184	C₂²⋊C₈	C₂²×C₁₇	1	1	yes	characteristic
185	C₂²⋊C₈	C₂²×C₁₇	1	1	yes	characteristic
186	C₂²⋊C₈	C₂²×C₁₇	1	1	yes	characteristic
187	C₂²⋊C₈	C₂²×C₁₇	1	1	yes	characteristic
188	C₂²⋊C₈	C₂²×C₁₇	1	2	yes	complemented
189	C₂²⋊C₈	C₂²×C₁₇	1	2	yes	complemented
190	#97	C₂²×C₁₇	1	2	yes	complemented
191	C₄×C₈	C₂²×C₁₇	1	2	yes	complemented
192	C₁₇	#2933	1	1	yes	characteristic, direct product factor, Sylow
193	C₂⁴	C₂×C₄×C₁₇	1	1	yes	characteristic
194	#41	—	2	2	no	—
195	#41	—	2	2	no	—
196	C₂×Q₈	C₂×C₄×C₁₇	1	1	yes	characteristic
197	C₂×D₄	C₂×C₄×C₁₇	1	2	yes	—
198	C₂×D₄	C₂×C₄×C₁₇	1	2	yes	—
199	C₂×D₄	C₂×C₄×C₁₇	1	1	yes	characteristic
200	C₂×D₄	C₂×C₄×C₁₇	1	2	yes	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	C₂×C₄×C₁₇	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₄×C₁₇	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₄×C₁₇	1	1	yes	characteristic
206	C₂²×C₄	C₂³×C₁₇	1	1	yes	characteristic
207	C₂²×C₄	C₂³×C₁₇	1	1	yes	characteristic
208	C₂²×C₄	C₂³×C₁₇	1	2	yes	—
209	C₂²×C₄	C₂³×C₁₇	1	1	yes	characteristic
210	M₁₆	—	2	2	no	—
211	M₁₆	—	2	4	no	—
212	M₁₆	—	2	4	no	—
213	M₁₆	D₄×C₁₇	1	4	yes	complemented
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	C₂³×C₁₇	1	1	yes	characteristic
219	C₂×C₈	C₂³×C₁₇	1	1	yes	characteristic
220	C₂×C₈	C₂³×C₁₇	1	1	yes	characteristic
221	C₂×C₈	C₂³×C₁₇	1	1	yes	characteristic
222	C₂×C₈	C₂³×C₁₇	1	2	yes	—
223	C₂×C₈	C₂³×C₁₇	1	2	yes	—
224	C₄⋊C₄	C₂×C₄×C₁₇	1	1	yes	characteristic
225	C₄⋊C₄	C₂×C₄×C₁₇	1	1	yes	characteristic
226	C₄⋊C₄	C₂×C₄×C₁₇	1	2	yes	—
227	C₄⋊C₄	C₂×C₄×C₁₇	1	2	yes	—
228	C₂²⋊C₄	C₂×C₄×C₁₇	1	2	yes	—
229	C₂²⋊C₄	C₂×C₄×C₁₇	1	2	yes	—
230	C₂²⋊C₄	C₂×C₄×C₁₇	1	2	yes	—
231	C₂²⋊C₄	C₂×C₄×C₁₇	1	2	yes	—
232	C₂²⋊C₄	C₂×C₄×C₁₇	1	2	yes	—
233	C₄²	C₂³×C₁₇	1	2	yes	—
234	C₂³	—	2	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	4	1	no	—
237	C₂³	C₂²×C₄×C₁₇	1	1	yes	characteristic
238	C₂³	C₂²×C₄×C₁₇	1	2	yes	—
239	C₂³	C₂²×C₄×C₁₇	1	1	yes	characteristic
240	C₂³	C₂²×C₄×C₁₇	1	1	yes	characteristic
241	Q₈	—	2	2	no	—
242	D₄	—	2	4	no	—
243	D₄	—	2	4	no	—
244	D₄	—	2	2	no	—
245	D₄	—	4	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	C₂²×C₄×C₁₇	1	2	yes	—
256	C₂×C₄	C₂²×C₄×C₁₇	1	2	yes	—
257	C₂×C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
258	C₂×C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₄×C₁₇	1	2	yes	—
260	C₂×C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
261	C₂×C₄	C₂×D₄×C₁₇	1	2	yes	—
262	C₂×C₄	#41×C₁₇	1	2	yes	—
263	C₂×C₄	C₂⁴×C₁₇	1	1	yes	characteristic
264	C₈	—	2	1	no	—
265	C₈	—	2	1	no	—
266	C₈	—	2	2	no	—
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	C₂×D₄×C₁₇	1	4	yes	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	4	4	no	—
281	C₂²	C₄×D₄×C₁₇	1	2	yes	—
282	C₂²	C₁₇×#131	1	2	yes	complemented
283	C₂²	C₂³×C₄×C₁₇	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	2	no	—
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	C₄×D₄×C₁₇	1	2	yes	—
290	C₄	C₂²×D₄×C₁₇	1	1	yes	characteristic
291	C₄	C₂×#41×C₁₇	1	1	yes	characteristic
292	C₂	—	2	2	no	—
293	C₂	—	2	2	no	—
294	C₂	—	4	1	no	—
295	C₂	—	4	2	no	—
296	C₂	C₂×C₄×D₄×C₁₇	1	1	yes	characteristic
297	C₂	C₂×C₁₇×#131	1	1	yes	characteristic
298	C₂	C₁₇×#568	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₇	C₂⁴
commutator	C₂²	C₂³×C₄×C₁₇
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₇
socle	C₂²×C₁₇	C₂³×C₄
prime radical	C₁₇×#525	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₇×#131	C₂	semidirect
2. C₂²×M₁₆×C₁₇	C₂	semidirect
4. C₁₇×#436	C₂	semidirect
5. C₁₇×#435	C₂	semidirect
6. C₁₇×#435	C₂	semidirect
7. C₈×D₄×C₁₇	C₂	semidirect
8. C₁₇×#424	C₂	semidirect
9. C₁₇×#408	C₂	semidirect
10. C₂×C₁₇×C₂²⋊C₈	C₂	semidirect
11. C₁₇×#405	C₂	semidirect
19. C₂×M₁₆×C₁₇	C₂²	semidirect
23. C₂²×C₈×C₁₇	C₂²	semidirect
33. C₁₇×C₄⋊C₈	C₂²	semidirect
34. C₁₇×C₄⋊C₈	C₂²	semidirect
39. C₁₇×C₂²⋊C₈	C₂²	semidirect
40. C₁₇×C₂²⋊C₈	C₂²	semidirect
41. C₁₇×#97	C₂²	semidirect
42. C₄×C₈×C₁₇	C₂²	semidirect
63. M₁₆×C₁₇	D₄	semidirect
120. #2933	C₁₇	direct (N × H)
133. C₂²×C₁₇	#131	semidirect
143. C₂×#131	C₂×C₁₇	semidirect
144. C₂²×M₁₆	C₂×C₁₇	semidirect
146. #436	C₂×C₁₇	semidirect
147. #435	C₂×C₁₇	semidirect
148. #435	C₂×C₁₇	semidirect
149. C₈×D₄	C₂×C₁₇	semidirect
150. #424	C₂×C₁₇	semidirect
151. #408	C₂×C₁₇	semidirect
152. C₂×C₂²⋊C₈	C₂×C₁₇	semidirect
153. #405	C₂×C₁₇	semidirect
168. C₂×M₁₆	C₂²×C₁₇	semidirect
172. C₂²×C₈	C₂²×C₁₇	semidirect
182. C₄⋊C₈	C₂²×C₁₇	semidirect
183. C₄⋊C₈	C₂²×C₁₇	semidirect
188. C₂²⋊C₈	C₂²×C₁₇	semidirect
189. C₂²⋊C₈	C₂²×C₁₇	semidirect
190. #97	C₂²×C₁₇	semidirect
191. C₄×C₈	C₂²×C₁₇	semidirect
192. C₁₇	#2933	direct (N × H)
213. M₁₆	D₄×C₁₇	semidirect
282. C₂²	C₁₇×#131	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₇×#131	2
C₂²×M₁₆×C₁₇	2
C₁₇×#525	2
C₁₇×#436	2
C₁₇×#435	2
C₈×D₄×C₁₇	2
C₁₇×#424	2
C₁₇×#408	2
C₂×C₁₇×C₂²⋊C₈	2
C₁₇×#405	2
#2933	17

Subgroups — C17×#2933 #14525939

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#2933 #14525939