Subgroups — C₄×C₁₇×#123 #14525307

Order 2176.

688 conjugacy classes of subgroups.

The subgroup lattice has 290 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	1	yes	maximal, characteristic
2	C₂×C₄×D₄×C₁₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄²×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₇×#388	C₂	1	2	yes	maximal, complemented
5	C₁₇×#387	C₂	1	2	yes	maximal, complemented
6	C₁₇×#386	C₂	1	1	yes	maximal, characteristic
7	C₁₇×#385	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₇×#381	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₄⋊C₄×C₁₇	C₂	1	2	yes	maximal, complemented
10	C₄×C₂²⋊C₄×C₁₇	C₂	1	2	yes	maximal, complemented
11	C₄×C₂²⋊C₄×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄×C₁₇	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₁₇	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₁₇	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₁₇	C₂²	1	1	yes	characteristic
16	C₂³×C₄×C₁₇	C₂²	1	1	yes	characteristic
17	C₁₇×#123	C₄	1	8	yes	direct product factor
18	C₄×D₄×C₁₇	—	2	4	no	—
19	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₁₇	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊C₄×C₁₇	C₂²	1	2	yes	—
28	C₂×C₄²×C₁₇	—	2	4	no	—
29	C₂×C₄²×C₁₇	C₂²	1	2	yes	—
30	C₂×C₄²×C₁₇	C₂²	1	2	yes	—
31	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
32	C₂×C₄²×C₁₇	C₂²	1	2	yes	—
33	C₁₇×#95	C₂²	1	2	yes	—
34	C₁₇×#95	C₂²	1	2	yes	—
35	C₁₇×#95	C₂²	1	2	yes	—
36	C₁₇×#95	C₂²	1	2	yes	—
37	C₁₇×#95	C₂²	1	2	yes	—
38	C₂⁴×C₁₇	C₂³	1	1	yes	characteristic
39	C₂⁴×C₁₇	C₂³	1	1	yes	characteristic
40	C₂×D₄×C₁₇	—	2	2	no	—
41	C₂×D₄×C₁₇	—	2	2	no	—
42	C₂×D₄×C₁₇	C₂×C₄	1	4	yes	—
43	C₂²×C₄×C₁₇	—	2	2	no	—
44	C₂²×C₄×C₁₇	—	2	2	no	—
45	C₂²×C₄×C₁₇	—	2	2	no	—
46	C₂²×C₄×C₁₇	—	2	4	no	—
47	C₂²×C₄×C₁₇	—	2	4	no	—
48	C₂²×C₄×C₁₇	—	2	2	no	—
49	C₂²×C₄×C₁₇	—	2	4	no	—
50	C₂²×C₄×C₁₇	C₂×C₄	1	4	yes	complemented
51	C₂²×C₄×C₁₇	D₄	1	4	yes	—
52	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₇	C₂³	1	2	yes	—
54	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₁₇	C₂³	1	2	yes	—
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	2	yes	—
60	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₁₇	—	2	2	no	—
62	C₄⋊C₄×C₁₇	—	2	4	no	—
63	C₄⋊C₄×C₁₇	C₂×C₄	1	8	yes	complemented
64	C₂²⋊C₄×C₁₇	—	2	2	no	—
65	C₂²⋊C₄×C₁₇	—	2	2	no	—
66	C₂²⋊C₄×C₁₇	—	2	2	no	—
67	C₂²⋊C₄×C₁₇	—	2	4	no	—
68	C₂²⋊C₄×C₁₇	C₂×C₄	1	4	yes	complemented
69	C₂²⋊C₄×C₁₇	C₂×C₄	1	8	yes	complemented
70	C₄²×C₁₇	—	2	4	no	—
71	C₄²×C₁₇	—	2	4	no	—
72	C₄²×C₁₇	—	2	2	no	—
73	C₄²×C₁₇	—	2	4	no	—
74	C₂³×C₁₇	—	2	2	no	—
75	C₂³×C₁₇	—	2	2	no	—
76	C₂³×C₁₇	—	2	1	no	—
77	C₂³×C₁₇	—	2	2	no	—
78	C₂³×C₁₇	—	2	2	no	—
79	C₂³×C₁₇	—	2	1	no	—
80	C₂³×C₁₇	C₂²×C₄	1	2	yes	—
81	C₂³×C₁₇	C₂²×C₄	1	2	yes	—
82	C₂³×C₁₇	C₂×D₄	1	2	yes	—
83	C₂³×C₁₇	#41	1	2	yes	—
84	C₂³×C₁₇	C₂⁴	1	1	yes	characteristic
85	D₄×C₁₇	—	2	8	no	—
86	C₂×C₄×C₁₇	—	2	8	no	—
87	C₂×C₄×C₁₇	—	2	8	no	—
88	C₂×C₄×C₁₇	—	2	2	no	—
89	C₂×C₄×C₁₇	—	2	2	no	—
90	C₂×C₄×C₁₇	—	2	1	no	—
91	C₂×C₄×C₁₇	—	2	2	no	—
92	C₂×C₄×C₁₇	—	2	2	no	—
93	C₂×C₄×C₁₇	—	2	2	no	—
94	C₂×C₄×C₁₇	—	2	2	no	—
95	C₂×C₄×C₁₇	—	2	2	no	—
96	C₂×C₄×C₁₇	—	2	1	no	—
97	C₂×C₄×C₁₇	—	2	8	no	—
98	C₂×C₄×C₁₇	—	2	1	no	—
99	C₂×C₄×C₁₇	—	2	1	no	—
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	—	4	4	no	—
106	C₂×C₄×C₁₇	C₂²×C₄	1	4	yes	—
107	C₂×C₄×C₁₇	C₂²×C₄	1	4	yes	—
108	C₂×C₄×C₁₇	C₂²×C₄	1	2	yes	—
109	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
111	C₂×C₄×C₁₇	#41	1	4	yes	—
112	C₂×C₄×C₁₇	#41	1	4	yes	—
113	C₄×#123	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂²×C₁₇	—	2	4	no	—
115	C₂²×C₁₇	—	2	2	no	—
116	C₂²×C₁₇	—	2	2	no	—
117	C₂²×C₁₇	—	2	4	no	—
118	C₂²×C₁₇	—	2	2	no	—
119	C₂²×C₁₇	—	2	4	no	—
120	C₂²×C₁₇	—	4	1	no	—
121	C₂²×C₁₇	—	4	1	no	—
122	C₂²×C₁₇	—	4	2	no	—
123	C₂²×C₁₇	C₄×D₄	1	4	yes	—
124	C₂²×C₁₇	C₂³×C₄	1	1	yes	characteristic
125	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
126	C₂²×C₁₇	C₂×#41	1	1	yes	characteristic
127	C₂²×C₁₇	C₂×#41	1	2	yes	—
128	C₂²×C₁₇	C₂×#41	1	2	yes	—
129	C₄×C₁₇	—	2	4	no	—
130	C₄×C₁₇	—	2	2	no	—
131	C₄×C₁₇	—	2	4	no	—
132	C₄×C₁₇	—	2	2	no	—
133	C₄×C₁₇	—	2	4	no	—
134	C₄×C₁₇	—	4	1	no	—
135	C₄×C₁₇	—	4	1	no	—
136	C₄×C₁₇	—	4	2	no	—
137	C₄×C₁₇	#123	1	4	yes	direct product factor
138	C₂×#123	C₂×C₁₇	1	1	yes	characteristic
139	C₂×C₄×D₄	C₂×C₁₇	1	1	yes	characteristic
140	C₂²×C₄²	C₂×C₁₇	1	1	yes	characteristic, complemented
141	#388	C₂×C₁₇	1	2	yes	complemented
142	#387	C₂×C₁₇	1	2	yes	complemented
143	#386	C₂×C₁₇	1	1	yes	characteristic
144	#385	C₂×C₁₇	1	1	yes	characteristic, complemented
145	#381	C₂×C₁₇	1	1	yes	characteristic, complemented
146	C₄×C₄⋊C₄	C₂×C₁₇	1	2	yes	complemented
147	C₄×C₂²⋊C₄	C₂×C₁₇	1	2	yes	complemented
148	C₄×C₂²⋊C₄	C₂×C₁₇	1	1	yes	characteristic, complemented
149	C₂×C₁₇	—	2	4	no	—
150	C₂×C₁₇	—	4	2	no	—
151	C₂×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
152	C₂×C₁₇	C₄×#41	1	2	yes	—
153	C₂×C₁₇	C₂×#123	1	1	yes	characteristic
154	C₂×C₁₇	#525	1	1	yes	characteristic
155	C₂×C₁₇	#529	1	2	yes	—
156	C₂²×D₄	C₂²×C₁₇	1	1	yes	characteristic
157	C₂³×C₄	C₂²×C₁₇	1	1	yes	characteristic
158	C₂³×C₄	C₂²×C₁₇	1	1	yes	characteristic
159	C₂³×C₄	C₂²×C₁₇	1	1	yes	characteristic
160	C₂³×C₄	C₂²×C₁₇	1	1	yes	characteristic
161	#123	C₄×C₁₇	1	8	yes	direct product factor
162	C₄×D₄	—	2	4	no	—
163	C₂×C₄⋊C₄	C₂²×C₁₇	1	1	yes	characteristic
164	C₂×C₄⋊C₄	C₂²×C₁₇	1	2	yes	—
165	C₂×C₄⋊C₄	C₂²×C₁₇	1	2	yes	—
166	C₂×C₂²⋊C₄	C₂²×C₁₇	1	1	yes	characteristic
167	C₂×C₂²⋊C₄	C₂²×C₁₇	1	1	yes	characteristic
168	C₂×C₂²⋊C₄	C₂²×C₁₇	1	1	yes	characteristic
169	C₂×C₂²⋊C₄	C₂²×C₁₇	1	2	yes	—
170	C₂×C₂²⋊C₄	C₂²×C₁₇	1	1	yes	characteristic
171	C₂×C₂²⋊C₄	C₂²×C₁₇	1	2	yes	—
172	C₂×C₄²	—	2	4	no	—
173	C₂×C₄²	C₂²×C₁₇	1	1	yes	characteristic
174	C₂×C₄²	C₂²×C₁₇	1	2	yes	—
175	C₂×C₄²	C₂²×C₁₇	1	2	yes	—
176	C₂×C₄²	C₂²×C₁₇	1	2	yes	—
177	#95	C₂²×C₁₇	1	2	yes	—
178	#95	C₂²×C₁₇	1	2	yes	—
179	#95	C₂²×C₁₇	1	2	yes	—
180	#95	C₂²×C₁₇	1	2	yes	—
181	#95	C₂²×C₁₇	1	2	yes	—
182	C₁₇	C₄×#123	1	1	yes	characteristic, direct product factor, Sylow
183	C₂⁴	C₂³×C₁₇	1	1	yes	characteristic
184	C₂⁴	C₂³×C₁₇	1	1	yes	characteristic
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	C₂×C₄×C₁₇	1	4	yes	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	C₂×C₄×C₁₇	1	4	yes	complemented
196	C₂²×C₄	D₄×C₁₇	1	4	yes	—
197	C₂²×C₄	C₂³×C₁₇	1	2	yes	—
198	C₂²×C₄	C₂³×C₁₇	1	1	yes	characteristic
199	C₂²×C₄	C₂³×C₁₇	1	1	yes	characteristic
200	C₂²×C₄	C₂³×C₁₇	1	2	yes	—
201	C₂²×C₄	C₂³×C₁₇	1	2	yes	—
202	C₂²×C₄	C₂³×C₁₇	1	1	yes	characteristic
203	C₂²×C₄	C₂³×C₁₇	1	1	yes	characteristic
204	C₂²×C₄	C₂³×C₁₇	1	2	yes	—
205	C₂²×C₄	C₂³×C₁₇	1	1	yes	characteristic
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	C₂×C₄×C₁₇	1	8	yes	complemented
209	C₂²⋊C₄	—	2	2	no	—
210	C₂²⋊C₄	—	2	2	no	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₂²⋊C₄	—	2	4	no	—
213	C₂²⋊C₄	C₂×C₄×C₁₇	1	4	yes	complemented
214	C₂²⋊C₄	C₂×C₄×C₁₇	1	8	yes	complemented
215	C₄²	—	2	4	no	—
216	C₄²	—	2	4	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	2	4	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	2	1	no	—
222	C₂³	—	2	2	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	2	1	no	—
225	C₂³	C₂²×C₄×C₁₇	1	2	yes	—
226	C₂³	C₂²×C₄×C₁₇	1	2	yes	—
227	C₂³	C₂×D₄×C₁₇	1	2	yes	—
228	C₂³	#41×C₁₇	1	2	yes	—
229	C₂³	C₂⁴×C₁₇	1	1	yes	characteristic
230	D₄	—	2	8	no	—
231	C₂×C₄	—	2	8	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	2	8	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
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	—	4	4	no	—
251	C₂×C₄	C₂²×C₄×C₁₇	1	4	yes	—
252	C₂×C₄	C₂²×C₄×C₁₇	1	4	yes	—
253	C₂×C₄	C₂²×C₄×C₁₇	1	2	yes	—
254	C₂×C₄	C₂×D₄×C₁₇	1	2	yes	—
255	C₂×C₄	C₂×D₄×C₁₇	1	2	yes	—
256	C₂×C₄	#41×C₁₇	1	4	yes	—
257	C₂×C₄	#41×C₁₇	1	4	yes	—
258	C₂²	—	2	4	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	4	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	4	no	—
264	C₂²	—	4	1	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	2	no	—
267	C₂²	C₄×D₄×C₁₇	1	4	yes	—
268	C₂²	C₂³×C₄×C₁₇	1	1	yes	characteristic
269	C₂²	C₂²×D₄×C₁₇	1	1	yes	characteristic
270	C₂²	C₂×#41×C₁₇	1	1	yes	characteristic
271	C₂²	C₂×#41×C₁₇	1	2	yes	—
272	C₂²	C₂×#41×C₁₇	1	2	yes	—
273	C₄	—	2	4	no	—
274	C₄	—	2	2	no	—
275	C₄	—	2	4	no	—
276	C₄	—	2	2	no	—
277	C₄	—	2	4	no	—
278	C₄	—	4	1	no	—
279	C₄	—	4	1	no	—
280	C₄	—	4	2	no	—
281	C₄	C₁₇×#123	1	4	yes	direct product factor
282	C₂	—	2	4	no	—
283	C₂	—	4	2	no	—
284	C₂	C₂×C₄×D₄×C₁₇	1	1	yes	characteristic
285	C₂	C₄×#41×C₁₇	1	2	yes	—
286	C₂	C₂×C₁₇×#123	1	1	yes	characteristic
287	C₂	C₁₇×#525	1	1	yes	characteristic
288	C₂	C₁₇×#529	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₂²×C₄²×C₁₇	C₂	semidirect
4. C₁₇×#388	C₂	semidirect
5. C₁₇×#387	C₂	semidirect
7. C₁₇×#385	C₂	semidirect
8. C₁₇×#381	C₂	semidirect
9. C₄×C₄⋊C₄×C₁₇	C₂	semidirect
10. C₄×C₂²⋊C₄×C₁₇	C₂	semidirect
11. C₄×C₂²⋊C₄×C₁₇	C₂	semidirect
17. C₁₇×#123	C₄	direct (N × H)
50. C₂²×C₄×C₁₇	C₂×C₄	semidirect
63. C₄⋊C₄×C₁₇	C₂×C₄	semidirect
68. C₂²⋊C₄×C₁₇	C₂×C₄	semidirect
69. C₂²⋊C₄×C₁₇	C₂×C₄	semidirect
113. C₄×#123	C₁₇	direct (N × H)
137. C₄×C₁₇	#123	direct (N × H)
140. C₂²×C₄²	C₂×C₁₇	semidirect
141. #388	C₂×C₁₇	semidirect
142. #387	C₂×C₁₇	semidirect
144. #385	C₂×C₁₇	semidirect
145. #381	C₂×C₁₇	semidirect
146. C₄×C₄⋊C₄	C₂×C₁₇	semidirect
147. C₄×C₂²⋊C₄	C₂×C₁₇	semidirect
148. C₄×C₂²⋊C₄	C₂×C₁₇	semidirect
161. #123	C₄×C₁₇	direct (N × H)
182. C₁₇	C₄×#123	direct (N × H)
195. C₂²×C₄	C₂×C₄×C₁₇	semidirect
208. C₄⋊C₄	C₂×C₄×C₁₇	semidirect
213. C₂²⋊C₄	C₂×C₄×C₁₇	semidirect
214. C₂²⋊C₄	C₂×C₄×C₁₇	semidirect
281. C₄	C₁₇×#123	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×#123	2
C₂×C₄×D₄×C₁₇	2
C₂²×C₄²×C₁₇	2
C₁₇×#388	2
C₁₇×#387	2
C₁₇×#386	2
C₁₇×#385	2
C₁₇×#381	2
C₄×C₄⋊C₄×C₁₇	2
C₄×C₂²⋊C₄×C₁₇	2
C₄×#123	17

Subgroups — C4×C17×#123 #14525307

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₁₇×#123 #14525307