Subgroups — C₁₇×#2431 #14525437

Order 2176.

952 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₇×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₇×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₇×#121	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#392	C₂	1	4	yes	maximal, complemented
6	C₁₇×#391	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#390	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₇×#386	C₂	1	4	yes	maximal, complemented
9	C₄×C₂²⋊C₄×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₁₇	—	2	4	no	—
11	C₂×#41×C₁₇	—	2	4	no	—
12	C₂×#41×C₁₇	—	2	4	no	—
13	C₂²×Q₈×C₁₇	C₂²	1	1	yes	characteristic
14	C₂²×D₄×C₁₇	C₂²	1	1	yes	characteristic
15	C₂²×D₄×C₁₇	C₂²	1	2	yes	—
16	C₂²×D₄×C₁₇	C₂²	1	4	yes	—
17	C₂²×D₄×C₁₇	C₂²	1	1	yes	characteristic
18	C₂²×D₄×C₁₇	C₂²	1	2	yes	—
19	C₂³×C₄×C₁₇	C₂²	1	2	yes	—
20	C₂³×C₄×C₁₇	C₂²	1	1	yes	characteristic
21	C₁₇×#127	—	2	4	no	—
22	C₁₇×#124	—	2	4	no	—
23	C₁₇×#121	—	2	4	no	—
24	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₁₇	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₄×C₁₇	C₂²	1	4	yes	—
27	C₂×C₂²⋊C₄×C₁₇	C₂²	1	2	yes	—
28	C₂×C₂²⋊C₄×C₁₇	C₂²	1	4	yes	—
29	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
30	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
31	C₁₇×#95	C₂²	1	2	yes	—
32	C₁₇×#95	C₂²	1	4	yes	—
33	C₂⁴×C₁₇	C₂³	1	2	yes	—
34	C₂⁴×C₁₇	C₂³	1	1	yes	characteristic
35	C₂⁴×C₁₇	C₂³	1	2	yes	—
36	#41×C₁₇	—	2	32	no	—
37	C₂×Q₈×C₁₇	—	2	2	no	—
38	C₂×Q₈×C₁₇	—	2	2	no	—
39	C₂×Q₈×C₁₇	D₄	1	4	yes	complemented
40	C₂×D₄×C₁₇	—	2	4	no	—
41	C₂×D₄×C₁₇	—	2	4	no	—
42	C₂×D₄×C₁₇	—	2	8	no	—
43	C₂×D₄×C₁₇	—	2	4	no	—
44	C₂×D₄×C₁₇	—	2	4	no	—
45	C₂×D₄×C₁₇	—	2	2	no	—
46	C₂×D₄×C₁₇	—	2	2	no	—
47	C₂×D₄×C₁₇	—	2	2	no	—
48	C₂×D₄×C₁₇	—	2	2	no	—
49	C₂×D₄×C₁₇	—	2	2	no	—
50	C₂×D₄×C₁₇	—	2	8	no	—
51	C₂×D₄×C₁₇	—	4	4	no	—
52	C₂×D₄×C₁₇	—	4	4	no	—
53	C₂×D₄×C₁₇	D₄	1	4	yes	complemented
54	C₂²×C₄×C₁₇	—	2	4	no	—
55	C₂²×C₄×C₁₇	—	2	4	no	—
56	C₂²×C₄×C₁₇	—	2	4	no	—
57	C₂²×C₄×C₁₇	—	2	2	no	—
58	C₂²×C₄×C₁₇	—	2	2	no	—
59	C₂²×C₄×C₁₇	D₄	1	4	yes	complemented
60	C₂²×C₄×C₁₇	C₂³	1	2	yes	—
61	C₂²×C₄×C₁₇	C₂³	1	2	yes	—
62	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₇	C₂³	1	2	yes	—
64	C₂²×C₄×C₁₇	C₂³	1	2	yes	—
65	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₁₇	—	2	2	no	—
67	C₂²⋊C₄×C₁₇	—	2	8	no	—
68	C₂²⋊C₄×C₁₇	—	2	4	no	—
69	C₂²⋊C₄×C₁₇	—	2	4	no	—
70	C₂²⋊C₄×C₁₇	—	4	4	no	—
71	C₄²×C₁₇	—	2	2	no	—
72	C₄²×C₁₇	—	2	2	no	—
73	C₂³×C₁₇	—	2	2	no	—
74	C₂³×C₁₇	—	2	1	no	—
75	C₂³×C₁₇	—	2	2	no	—
76	C₂³×C₁₇	—	2	1	no	—
77	C₂³×C₁₇	—	2	2	no	—
78	C₂³×C₁₇	—	2	4	no	—
79	C₂³×C₁₇	—	2	4	no	—
80	C₂³×C₁₇	—	2	2	no	—
81	C₂³×C₁₇	—	2	2	no	—
82	C₂³×C₁₇	—	2	2	no	—
83	C₂³×C₁₇	—	2	2	no	—
84	C₂³×C₁₇	—	2	4	no	—
85	C₂³×C₁₇	—	2	4	no	—
86	C₂³×C₁₇	C₂×D₄	1	4	yes	—
87	C₂³×C₁₇	C₂×D₄	1	2	yes	—
88	C₂³×C₁₇	C₂⁴	1	1	yes	characteristic
89	Q₈×C₁₇	—	2	8	no	—
90	D₄×C₁₇	—	2	16	no	—
91	D₄×C₁₇	—	2	8	no	—
92	D₄×C₁₇	—	4	16	no	—
93	D₄×C₁₇	—	4	8	no	—
94	D₄×C₁₇	—	4	4	no	—
95	C₂×C₄×C₁₇	—	2	16	no	—
96	C₂×C₄×C₁₇	—	2	2	no	—
97	C₂×C₄×C₁₇	—	2	1	no	—
98	C₂×C₄×C₁₇	—	2	1	no	—
99	C₂×C₄×C₁₇	—	2	2	no	—
100	C₂×C₄×C₁₇	—	2	2	no	—
101	C₂×C₄×C₁₇	—	2	2	no	—
102	C₂×C₄×C₁₇	—	2	8	no	—
103	C₂×C₄×C₁₇	—	2	2	no	—
104	C₂×C₄×C₁₇	—	2	2	no	—
105	C₂×C₄×C₁₇	—	2	2	no	—
106	C₂×C₄×C₁₇	—	2	2	no	—
107	C₂×C₄×C₁₇	—	2	4	no	—
108	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₁₇	C₂×D₄	1	4	yes	—
110	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
111	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
112	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
113	C₂×C₄×C₁₇	#41	1	4	yes	complemented
114	#2431	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂²×C₁₇	—	2	4	no	—
116	C₂²×C₁₇	—	2	2	no	—
117	C₂²×C₁₇	—	2	2	no	—
118	C₂²×C₁₇	—	2	2	no	—
119	C₂²×C₁₇	—	2	8	no	—
120	C₂²×C₁₇	—	4	2	no	—
121	C₂²×C₁₇	—	4	2	no	—
122	C₂²×C₁₇	—	4	2	no	—
123	C₂²×C₁₇	—	4	2	no	—
124	C₂²×C₁₇	—	4	4	no	—
125	C₂²×C₁₇	—	4	2	no	—
126	C₂²×C₁₇	—	4	2	no	—
127	C₂²×C₁₇	—	4	2	no	—
128	C₂²×C₁₇	—	4	4	no	—
129	C₂²×C₁₇	—	4	4	no	—
130	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
131	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
132	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
133	C₂²×C₁₇	C₂×#41	1	2	yes	—
134	C₂²×C₁₇	H(5,𝔽₂)	1	2	yes	—
135	C₄×C₁₇	—	2	2	no	—
136	C₄×C₁₇	—	2	4	no	—
137	C₄×C₁₇	—	2	4	no	—
138	C₄×C₁₇	—	4	2	no	—
139	C₄×C₁₇	—	4	2	no	—
140	C₄×C₁₇	#120	1	4	yes	—
141	C₂²×#41	C₂×C₁₇	1	1	yes	characteristic, complemented
142	C₂×#127	C₂×C₁₇	1	1	yes	characteristic, complemented
143	C₂×#124	C₂×C₁₇	1	1	yes	characteristic, complemented
144	C₂×#121	C₂×C₁₇	1	1	yes	characteristic, complemented
145	#392	C₂×C₁₇	1	4	yes	complemented
146	#391	C₂×C₁₇	1	1	yes	characteristic, complemented
147	#390	C₂×C₁₇	1	1	yes	characteristic, complemented
148	#386	C₂×C₁₇	1	4	yes	complemented
149	C₄×C₂²⋊C₄	C₂×C₁₇	1	1	yes	characteristic, complemented
150	C₂×C₁₇	—	4	2	no	—
151	C₂×C₁₇	—	4	4	no	—
152	C₂×C₁₇	—	8	2	no	—
153	C₂×C₁₇	C₂×#120	1	1	yes	characteristic
154	C₂×C₁₇	#532	1	1	yes	characteristic
155	C₂×C₁₇	#535	1	1	yes	characteristic
156	C₂×C₁₇	#546	1	2	yes	—
157	C₂×C₁₇	#550	1	2	yes	—
158	C₂×#41	—	2	4	no	—
159	C₂×#41	—	2	4	no	—
160	C₂×#41	—	2	4	no	—
161	C₂²×Q₈	C₂²×C₁₇	1	1	yes	characteristic
162	C₂²×D₄	C₂²×C₁₇	1	1	yes	characteristic
163	C₂²×D₄	C₂²×C₁₇	1	2	yes	—
164	C₂²×D₄	C₂²×C₁₇	1	4	yes	—
165	C₂²×D₄	C₂²×C₁₇	1	1	yes	characteristic
166	C₂²×D₄	C₂²×C₁₇	1	2	yes	—
167	C₂³×C₄	C₂²×C₁₇	1	2	yes	—
168	C₂³×C₄	C₂²×C₁₇	1	1	yes	characteristic
169	#127	—	2	4	no	—
170	#124	—	2	4	no	—
171	#121	—	2	4	no	—
172	C₂×C₄⋊C₄	C₂²×C₁₇	1	1	yes	characteristic
173	C₂×C₂²⋊C₄	C₂²×C₁₇	1	2	yes	—
174	C₂×C₂²⋊C₄	C₂²×C₁₇	1	4	yes	—
175	C₂×C₂²⋊C₄	C₂²×C₁₇	1	2	yes	—
176	C₂×C₂²⋊C₄	C₂²×C₁₇	1	4	yes	—
177	C₂×C₄²	C₂²×C₁₇	1	1	yes	characteristic
178	C₂×C₄²	C₂²×C₁₇	1	1	yes	characteristic
179	#95	C₂²×C₁₇	1	2	yes	—
180	#95	C₂²×C₁₇	1	4	yes	—
181	C₁₇	#2431	1	1	yes	characteristic, direct product factor, Sylow
182	C₂⁴	C₂³×C₁₇	1	2	yes	—
183	C₂⁴	C₂³×C₁₇	1	1	yes	characteristic
184	C₂⁴	C₂³×C₁₇	1	2	yes	—
185	#41	—	2	32	no	—
186	C₂×Q₈	—	2	2	no	—
187	C₂×Q₈	—	2	2	no	—
188	C₂×Q₈	D₄×C₁₇	1	4	yes	complemented
189	C₂×D₄	—	2	4	no	—
190	C₂×D₄	—	2	4	no	—
191	C₂×D₄	—	2	4	no	—
192	C₂×D₄	—	2	8	no	—
193	C₂×D₄	—	2	4	no	—
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	—	2	2	no	—
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	—	2	8	no	—
200	C₂×D₄	—	4	4	no	—
201	C₂×D₄	—	4	4	no	—
202	C₂×D₄	D₄×C₁₇	1	4	yes	complemented
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	D₄×C₁₇	1	4	yes	complemented
209	C₂²×C₄	C₂³×C₁₇	1	2	yes	—
210	C₂²×C₄	C₂³×C₁₇	1	2	yes	—
211	C₂²×C₄	C₂³×C₁₇	1	1	yes	characteristic
212	C₂²×C₄	C₂³×C₁₇	1	2	yes	—
213	C₂²×C₄	C₂³×C₁₇	1	2	yes	—
214	C₂²×C₄	C₂³×C₁₇	1	1	yes	characteristic
215	C₄⋊C₄	—	2	2	no	—
216	C₂²⋊C₄	—	2	4	no	—
217	C₂²⋊C₄	—	2	8	no	—
218	C₂²⋊C₄	—	2	4	no	—
219	C₂²⋊C₄	—	4	4	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	2	no	—
222	C₂³	—	2	2	no	—
223	C₂³	—	2	1	no	—
224	C₂³	—	2	2	no	—
225	C₂³	—	2	1	no	—
226	C₂³	—	2	2	no	—
227	C₂³	—	2	4	no	—
228	C₂³	—	2	4	no	—
229	C₂³	—	2	2	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	2	no	—
232	C₂³	—	2	2	no	—
233	C₂³	—	2	4	no	—
234	C₂³	—	2	4	no	—
235	C₂³	C₂×D₄×C₁₇	1	4	yes	—
236	C₂³	C₂×D₄×C₁₇	1	2	yes	—
237	C₂³	C₂⁴×C₁₇	1	1	yes	characteristic
238	Q₈	—	2	8	no	—
239	D₄	—	2	16	no	—
240	D₄	—	2	8	no	—
241	D₄	—	4	16	no	—
242	D₄	—	4	8	no	—
243	D₄	—	4	4	no	—
244	C₂×C₄	—	2	16	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	8	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	C₂×D₄×C₁₇	1	2	yes	—
258	C₂×C₄	C₂×D₄×C₁₇	1	4	yes	—
259	C₂×C₄	C₂×D₄×C₁₇	1	2	yes	—
260	C₂×C₄	C₂×D₄×C₁₇	1	2	yes	—
261	C₂×C₄	C₂×D₄×C₁₇	1	2	yes	—
262	C₂×C₄	#41×C₁₇	1	4	yes	complemented
263	C₂²	—	2	4	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	8	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	4	2	no	—
271	C₂²	—	4	2	no	—
272	C₂²	—	4	4	no	—
273	C₂²	—	4	2	no	—
274	C₂²	—	4	2	no	—
275	C₂²	—	4	2	no	—
276	C₂²	—	4	4	no	—
277	C₂²	—	4	4	no	—
278	C₂²	C₂²×D₄×C₁₇	1	1	yes	characteristic
279	C₂²	C₂²×D₄×C₁₇	1	1	yes	characteristic
280	C₂²	C₂²×D₄×C₁₇	1	1	yes	characteristic
281	C₂²	C₂×#41×C₁₇	1	2	yes	—
282	C₂²	C₁₇×H(5,𝔽₂)	1	2	yes	—
283	C₄	—	2	2	no	—
284	C₄	—	2	4	no	—
285	C₄	—	2	4	no	—
286	C₄	—	4	2	no	—
287	C₄	—	4	2	no	—
288	C₄	C₁₇×#120	1	4	yes	—
289	C₂	—	4	2	no	—
290	C₂	—	4	4	no	—
291	C₂	—	8	2	no	—
292	C₂	C₂×C₁₇×#120	1	1	yes	characteristic
293	C₂	C₁₇×#532	1	1	yes	characteristic
294	C₂	C₁₇×#535	1	1	yes	characteristic
295	C₂	C₁₇×#546	1	2	yes	—
296	C₂	C₁₇×#550	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₇
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#41×C₁₇	C₂	semidirect
2. C₂×C₁₇×#127	C₂	semidirect
3. C₂×C₁₇×#124	C₂	semidirect
4. C₂×C₁₇×#121	C₂	semidirect
5. C₁₇×#392	C₂	semidirect
6. C₁₇×#391	C₂	semidirect
7. C₁₇×#390	C₂	semidirect
8. C₁₇×#386	C₂	semidirect
9. C₄×C₂²⋊C₄×C₁₇	C₂	semidirect
39. C₂×Q₈×C₁₇	D₄	semidirect
53. C₂×D₄×C₁₇	D₄	semidirect
59. C₂²×C₄×C₁₇	D₄	semidirect
113. C₂×C₄×C₁₇	#41	semidirect
114. #2431	C₁₇	direct (N × H)
141. C₂²×#41	C₂×C₁₇	semidirect
142. C₂×#127	C₂×C₁₇	semidirect
143. C₂×#124	C₂×C₁₇	semidirect
144. C₂×#121	C₂×C₁₇	semidirect
145. #392	C₂×C₁₇	semidirect
146. #391	C₂×C₁₇	semidirect
147. #390	C₂×C₁₇	semidirect
148. #386	C₂×C₁₇	semidirect
149. C₄×C₂²⋊C₄	C₂×C₁₇	semidirect
181. C₁₇	#2431	direct (N × H)
188. C₂×Q₈	D₄×C₁₇	semidirect
202. C₂×D₄	D₄×C₁₇	semidirect
208. C₂²×C₄	D₄×C₁₇	semidirect
262. C₂×C₄	#41×C₁₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#41×C₁₇	2
C₂×C₁₇×#127	2
C₂×C₁₇×#124	2
C₂×C₁₇×#121	2
C₁₇×#392	2
C₁₇×#391	2
C₁₇×#390	2
C₁₇×#386	2
C₄×C₂²⋊C₄×C₁₇	2
#2431	17

Subgroups — C17×#2431 #14525437

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#2431 #14525437