Subgroups — C₁₇×#7727 #16865471

Order 4352.

494 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×#2857	C₂	1	1	yes	maximal, characteristic
2	C₁₇×#1817	C₂	1	1	yes	maximal, characteristic
3	C₁₇×#1797	C₂	1	1	yes	maximal, characteristic
4	C₁₇×#1272	C₂	1	4	yes	maximal, complemented
5	C₂×C₁₇×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₂×C₁₇×C₂²⋊Q₈	C₄	1	2	yes	—
7	C₂²×C₄²×C₁₇	C₂²	1	1	yes	characteristic
8	C₂×C₁₇×C₄⋊C₈	—	2	1	no	—
9	C₂×C₁₇×C₂²⋊C₈	—	2	1	no	—
10	C₂×C₁₇×C₂²⋊C₈	—	2	1	no	—
11	C₁₇×#391	C₂²	1	2	yes	—
12	C₁₇×#389	—	2	1	no	—
13	C₁₇×#389	—	2	1	no	—
14	C₁₇×#384	C₄	1	1	yes	characteristic
15	C₁₇×#380	—	2	1	no	—
16	C₁₇×#380	—	2	1	no	—
17	C₁₇×#336	—	2	1	no	—
18	C₁₇×#336	C₂²	1	2	yes	—
19	C₁₇×#336	C₂²	1	2	yes	—
20	C₂²×Q₈×C₁₇	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₇	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₇	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₇	—	2	2	no	—
25	C₂²×C₈×C₁₇	—	2	2	no	—
26	C₁₇×C₄⋊Q₈	—	4	2	no	—
27	C₁₇×C₂²⋊Q₈	—	4	4	no	—
28	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
29	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
30	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
31	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
32	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
33	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
34	C₂×C₂²⋊C₄×C₁₇	—	2	2	no	—
35	C₂×C₄²×C₁₇	—	4	2	no	—
36	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
40	C₁₇×C₄⋊C₈	—	4	2	no	—
41	C₁₇×C₂²⋊C₈	—	4	2	no	—
42	C₁₇×C₂²⋊C₈	—	4	2	no	—
43	C₁₇×#95	—	2	2	no	—
44	C₁₇×#95	—	2	2	no	—
45	C₂⁴×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×Q₈×C₁₇	—	2	4	no	—
47	C₂×Q₈×C₁₇	—	4	2	no	—
48	C₂×Q₈×C₁₇	—	4	2	no	—
49	C₂²×C₄×C₁₇	—	2	2	no	—
50	C₂²×C₄×C₁₇	—	2	2	no	—
51	C₂²×C₄×C₁₇	—	2	2	no	—
52	C₂²×C₄×C₁₇	—	2	2	no	—
53	C₂²×C₄×C₁₇	—	2	2	no	—
54	C₂²×C₄×C₁₇	—	2	2	no	—
55	C₂²×C₄×C₁₇	—	4	1	no	—
56	C₂²×C₄×C₁₇	—	4	1	no	—
57	C₂²×C₄×C₁₇	—	4	1	no	—
58	C₂²×C₄×C₁₇	—	4	1	no	—
59	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₇	SD₁₆	1	2	yes	—
63	C₂²×C₄×C₁₇	Q₁₆	1	2	yes	—
64	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₁₇	—	4	2	no	—
68	C₂×C₈×C₁₇	—	4	2	no	—
69	C₂×C₈×C₁₇	—	4	2	no	—
70	C₂×C₈×C₁₇	—	4	2	no	—
71	C₂×C₈×C₁₇	—	4	2	no	—
72	C₂×C₈×C₁₇	—	4	2	no	—
73	C₄⋊C₄×C₁₇	—	4	2	no	—
74	C₄⋊C₄×C₁₇	—	4	2	no	—
75	C₄⋊C₄×C₁₇	—	4	2	no	—
76	C₄⋊C₄×C₁₇	—	4	2	no	—
77	C₄⋊C₄×C₁₇	—	4	2	no	—
78	C₄⋊C₄×C₁₇	—	4	4	no	—
79	C₂²⋊C₄×C₁₇	—	4	4	no	—
80	C₄²×C₁₇	—	2	2	no	—
81	C₄²×C₁₇	—	2	2	no	—
82	C₄²×C₁₇	—	4	1	no	—
83	C₄²×C₁₇	—	4	1	no	—
84	#7727	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂³×C₁₇	—	2	1	no	—
86	C₂³×C₁₇	—	2	1	no	—
87	C₂³×C₁₇	—	2	1	no	—
88	C₂³×C₁₇	—	2	1	no	—
89	C₂³×C₁₇	—	2	1	no	—
90	C₂³×C₁₇	—	2	1	no	—
91	C₂³×C₁₇	#103	1	2	yes	—
92	C₂³×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	Q₈×C₁₇	—	8	4	no	—
94	C₂×C₄×C₁₇	—	2	1	no	—
95	C₂×C₄×C₁₇	—	2	1	no	—
96	C₂×C₄×C₁₇	—	2	1	no	—
97	C₂×C₄×C₁₇	—	2	1	no	—
98	C₂×C₄×C₁₇	—	2	1	no	—
99	C₂×C₄×C₁₇	—	2	1	no	—
100	C₂×C₄×C₁₇	—	2	1	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	1	no	—
105	C₂×C₄×C₁₇	—	2	1	no	—
106	C₂×C₄×C₁₇	—	4	2	no	—
107	C₂×C₄×C₁₇	—	4	2	no	—
108	C₂×C₄×C₁₇	—	4	4	no	—
109	C₂×C₄×C₁₇	—	4	4	no	—
110	C₂×C₄×C₁₇	—	4	2	no	—
111	C₂×C₄×C₁₇	—	4	2	no	—
112	C₂×C₄×C₁₇	—	4	2	no	—
113	C₂×C₄×C₁₇	—	4	2	no	—
114	C₂×C₄×C₁₇	—	4	2	no	—
115	C₂×C₄×C₁₇	—	4	2	no	—
116	C₂×C₄×C₁₇	#99	1	2	yes	—
117	C₂×C₄×C₁₇	#100	1	2	yes	—
118	C₂×C₄×C₁₇	#103	1	2	yes	—
119	C₂×C₄×C₁₇	#131	1	2	yes	—
120	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
121	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
122	C₂×C₄×C₁₇	#137	1	2	yes	—
123	C₈×C₁₇	—	8	2	no	—
124	C₈×C₁₇	—	8	2	no	—
125	#2857	C₂×C₁₇	1	1	yes	characteristic
126	#1817	C₂×C₁₇	1	1	yes	characteristic
127	#1797	C₂×C₁₇	1	1	yes	characteristic
128	#1272	C₂×C₁₇	1	4	yes	complemented
129	C₂²×C₁₇	—	2	2	no	—
130	C₂²×C₁₇	—	2	2	no	—
131	C₂²×C₁₇	—	2	2	no	—
132	C₂²×C₁₇	—	4	1	no	—
133	C₂²×C₁₇	—	4	1	no	—
134	C₂²×C₁₇	—	4	1	no	—
135	C₂²×C₁₇	—	4	1	no	—
136	C₂²×C₁₇	#408	1	1	yes	characteristic
137	C₂²×C₁₇	C₂×#99	1	1	yes	characteristic
138	C₂²×C₁₇	C₂×#100	1	1	yes	characteristic
139	C₂²×C₁₇	C₂×#103	1	1	yes	characteristic
140	C₂²×C₁₇	#419	1	1	yes	characteristic
141	C₂²×C₁₇	#421	1	1	yes	characteristic
142	C₂²×C₁₇	#421	1	1	yes	characteristic
143	C₄×C₁₇	—	2	2	no	—
144	C₄×C₁₇	—	2	2	no	—
145	C₄×C₁₇	—	2	2	no	—
146	C₄×C₁₇	—	2	2	no	—
147	C₄×C₁₇	—	2	1	no	—
148	C₄×C₁₇	—	2	2	no	—
149	C₄×C₁₇	—	2	1	no	—
150	C₄×C₁₇	—	8	2	no	—
151	C₄×C₁₇	—	8	2	no	—
152	C₂×C₄⋊Q₈	C₄×C₁₇	1	1	yes	characteristic
153	C₂×C₂²⋊Q₈	C₄×C₁₇	1	2	yes	—
154	C₂²×C₄²	C₂²×C₁₇	1	1	yes	characteristic
155	C₂×C₄⋊C₈	—	2	1	no	—
156	C₂×C₂²⋊C₈	—	2	1	no	—
157	C₂×C₂²⋊C₈	—	2	1	no	—
158	#391	C₂²×C₁₇	1	2	yes	—
159	#389	—	2	1	no	—
160	#389	—	2	1	no	—
161	#384	C₄×C₁₇	1	1	yes	characteristic
162	#380	—	2	1	no	—
163	#380	—	2	1	no	—
164	#336	—	2	1	no	—
165	#336	C₂²×C₁₇	1	2	yes	—
166	#336	C₂²×C₁₇	1	2	yes	—
167	C₂×C₁₇	—	4	2	no	—
168	C₂×C₁₇	#1471	1	1	yes	characteristic
169	C₂×C₁₇	#1481	1	1	yes	characteristic
170	C₂×C₁₇	#1497	1	1	yes	characteristic
171	C₂×C₁₇	#1511	1	1	yes	characteristic
172	C₂×C₁₇	#1520	1	1	yes	characteristic
173	C₂×C₁₇	#1536	1	1	yes	characteristic
174	C₂×C₁₇	#1548	1	1	yes	characteristic
175	C₂²×Q₈	C₂×C₄×C₁₇	1	2	yes	—
176	C₂³×C₄	C₂×C₄×C₁₇	1	1	yes	characteristic
177	C₂³×C₄	D₄×C₁₇	1	1	yes	characteristic
178	C₂³×C₄	D₄×C₁₇	1	1	yes	characteristic
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₄⋊Q₈	—	4	2	no	—
182	C₂²⋊Q₈	—	4	4	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	2	1	no	—
185	C₂×C₄⋊C₄	—	2	1	no	—
186	C₂×C₄⋊C₄	—	2	1	no	—
187	C₂×C₄⋊C₄	—	2	1	no	—
188	C₂×C₄⋊C₄	C₂×C₄×C₁₇	1	2	yes	—
189	C₂×C₂²⋊C₄	—	2	2	no	—
190	C₂×C₄²	—	4	2	no	—
191	C₂×C₄²	C₂×C₄×C₁₇	1	1	yes	characteristic
192	C₂×C₄²	D₄×C₁₇	1	1	yes	characteristic
193	C₂×C₄²	D₄×C₁₇	1	1	yes	characteristic
194	C₂×C₄²	C₂³×C₁₇	1	1	yes	characteristic
195	C₄⋊C₈	—	4	2	no	—
196	C₂²⋊C₈	—	4	2	no	—
197	C₂²⋊C₈	—	4	2	no	—
198	#95	—	2	2	no	—
199	#95	—	2	2	no	—
200	C₁₇	#7727	1	1	yes	characteristic, direct product factor, Sylow
201	C₂⁴	C₂²⋊C₄×C₁₇	1	1	yes	characteristic
202	C₂×Q₈	—	2	4	no	—
203	C₂×Q₈	—	4	2	no	—
204	C₂×Q₈	—	4	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	4	1	no	—
212	C₂²×C₄	—	4	1	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	C₂²⋊C₄×C₁₇	1	1	yes	characteristic
216	C₂²×C₄	C₂²⋊C₄×C₁₇	1	1	yes	characteristic
217	C₂²×C₄	C₂²⋊C₄×C₁₇	1	1	yes	characteristic
218	C₂²×C₄	SD₁₆×C₁₇	1	2	yes	—
219	C₂²×C₄	Q₁₆×C₁₇	1	2	yes	—
220	C₂²×C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	4	2	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₄⋊C₄	—	4	2	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	—	4	4	no	—
235	C₂²⋊C₄	—	4	4	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	2	2	no	—
238	C₄²	—	4	1	no	—
239	C₄²	—	4	1	no	—
240	C₂³	—	2	1	no	—
241	C₂³	—	2	1	no	—
242	C₂³	—	2	1	no	—
243	C₂³	—	2	1	no	—
244	C₂³	—	2	1	no	—
245	C₂³	—	2	1	no	—
246	C₂³	C₁₇×#103	1	2	yes	—
247	C₂³	C₂×C₂²⋊C₄×C₁₇	1	1	yes	characteristic
248	Q₈	—	8	4	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	C₁₇×#99	1	2	yes	—
272	C₂×C₄	C₁₇×#100	1	2	yes	—
273	C₂×C₄	C₁₇×#103	1	2	yes	—
274	C₂×C₄	C₁₇×#131	1	2	yes	—
275	C₂×C₄	C₂×SD₁₆×C₁₇	1	1	yes	characteristic
276	C₂×C₄	C₂×Q₁₆×C₁₇	1	1	yes	characteristic
277	C₂×C₄	C₁₇×#137	1	2	yes	—
278	C₈	—	8	2	no	—
279	C₈	—	8	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	4	1	no	—
286	C₂²	—	4	1	no	—
287	C₂²	C₁₇×#408	1	1	yes	characteristic
288	C₂²	C₂×C₁₇×#99	1	1	yes	characteristic
289	C₂²	C₂×C₁₇×#100	1	1	yes	characteristic
290	C₂²	C₂×C₁₇×#103	1	1	yes	characteristic
291	C₂²	C₁₇×#419	1	1	yes	characteristic
292	C₂²	C₁₇×#421	1	1	yes	characteristic
293	C₂²	C₁₇×#421	1	1	yes	characteristic
294	C₄	—	2	2	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	2	no	—
301	C₄	—	8	2	no	—
302	C₄	—	8	2	no	—
303	C₂	—	4	2	no	—
304	C₂	C₁₇×#1471	1	1	yes	characteristic
305	C₂	C₁₇×#1481	1	1	yes	characteristic
306	C₂	C₁₇×#1497	1	1	yes	characteristic
307	C₂	C₁₇×#1511	1	1	yes	characteristic
308	C₂	C₁₇×#1520	1	1	yes	characteristic
309	C₂	C₁₇×#1536	1	1	yes	characteristic
310	C₂	C₁₇×#1548	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₇	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₁₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₇
prime radical	C₂⁴×C₁₇	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ 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×#1272	C₂	semidirect
84. #7727	C₁₇	direct (N × H)
128. #1272	C₂×C₁₇	semidirect
200. C₁₇	#7727	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₇×#2857	2
C₁₇×#1817	2
C₁₇×#1797	2
C₁₇×#1272	2
#7727	17

Subgroups — C17×#7727 #16865471

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#7727 #16865471