Subgroups — C₁₇×#7588 #16865332

Order 4352.

478 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₁₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₁₇×#1831	C₂	1	1	yes	maximal, characteristic
3	C₁₇×#1744	C₂	1	1	yes	maximal, characteristic
4	C₁₇×#1271	C₂	1	4	yes	maximal
5	C₂×C₁₇×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₁₇	—	2	1	no	—
7	C₂×C₁₇×C₄⋊C₈	—	2	1	no	—
8	C₂×C₄×C₈×C₁₇	—	2	1	no	—
9	C₁₇×#391	—	2	1	no	—
10	C₁₇×#389	—	2	1	no	—
11	C₁₇×#389	C₂²	1	2	yes	—
12	C₁₇×#384	C₄	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₁₇	—	2	1	no	—
14	C₄×C₄⋊C₄×C₁₇	C₄	1	2	yes	—
15	C₄³×C₁₇	C₂²	1	1	yes	characteristic
16	C₁₇×#336	—	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	—	2	1	no	—
21	C₂²×C₈×C₁₇	—	2	2	no	—
22	C₂²×C₈×C₁₇	—	2	2	no	—
23	C₁₇×C₄⋊Q₈	—	2	2	no	—
24	C₁₇×C₄⋊Q₈	C₈	1	4	yes	complemented
25	C₄×Q₈×C₁₇	—	4	2	no	—
26	C₄×Q₈×C₁₇	—	4	2	no	—
27	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
28	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
30	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
31	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₁₇	—	2	2	no	—
34	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₇	D₄	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	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
41	C₁₇×C₄⋊C₈	—	4	2	no	—
42	C₄×C₈×C₁₇	—	2	4	no	—
43	C₁₇×#95	—	2	2	no	—
44	C₁₇×#95	—	2	2	no	—
45	C₂×Q₈×C₁₇	—	2	4	no	—
46	C₂×Q₈×C₁₇	—	4	2	no	—
47	C₂×Q₈×C₁₇	—	4	2	no	—
48	C₂²×C₄×C₁₇	—	2	2	no	—
49	C₂²×C₄×C₁₇	—	2	2	no	—
50	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₁₇	—	2	4	no	—
58	C₂×C₈×C₁₇	—	4	2	no	—
59	C₂×C₈×C₁₇	—	4	2	no	—
60	C₂×C₈×C₁₇	—	4	2	no	—
61	C₂×C₈×C₁₇	—	4	2	no	—
62	C₂×C₈×C₁₇	—	4	2	no	—
63	C₄⋊C₄×C₁₇	—	2	4	no	—
64	C₄⋊C₄×C₁₇	—	2	2	no	—
65	C₄⋊C₄×C₁₇	—	4	2	no	—
66	C₄⋊C₄×C₁₇	—	4	2	no	—
67	C₄⋊C₄×C₁₇	—	4	2	no	—
68	C₄⋊C₄×C₁₇	—	4	2	no	—
69	C₄⋊C₄×C₁₇	—	4	2	no	—
70	C₄⋊C₄×C₁₇	C₂×C₈	1	4	yes	—
71	C₄²×C₁₇	—	2	2	no	—
72	C₄²×C₁₇	—	2	2	no	—
73	C₄²×C₁₇	—	2	2	no	—
74	C₄²×C₁₇	—	4	1	no	—
75	C₄²×C₁₇	—	4	1	no	—
76	C₄²×C₁₇	—	4	4	no	—
77	C₄²×C₁₇	C₂×C₈	1	2	yes	—
78	C₄²×C₁₇	M₁₆	1	2	yes	—
79	C₄²×C₁₇	D₈	1	2	yes	—
80	C₄²×C₁₇	SD₁₆	1	2	yes	—
81	#7588	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂³×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	Q₈×C₁₇	—	4	8	no	—
84	C₂×C₄×C₁₇	—	2	1	no	—
85	C₂×C₄×C₁₇	—	2	1	no	—
86	C₂×C₄×C₁₇	—	2	1	no	—
87	C₂×C₄×C₁₇	—	2	1	no	—
88	C₂×C₄×C₁₇	—	2	1	no	—
89	C₂×C₄×C₁₇	—	2	1	no	—
90	C₂×C₄×C₁₇	—	2	1	no	—
91	C₂×C₄×C₁₇	—	2	1	no	—
92	C₂×C₄×C₁₇	—	2	1	no	—
93	C₂×C₄×C₁₇	—	2	1	no	—
94	C₂×C₄×C₁₇	—	2	1	no	—
95	C₂×C₄×C₁₇	—	2	1	no	—
96	C₂×C₄×C₁₇	—	2	4	no	—
97	C₂×C₄×C₁₇	—	4	2	no	—
98	C₂×C₄×C₁₇	—	4	2	no	—
99	C₂×C₄×C₁₇	—	4	2	no	—
100	C₂×C₄×C₁₇	—	4	2	no	—
101	C₂×C₄×C₁₇	—	4	2	no	—
102	C₂×C₄×C₁₇	C₂²⋊C₈	1	2	yes	—
103	C₂×C₄×C₁₇	C₂²⋊C₈	1	2	yes	—
104	C₂×C₄×C₁₇	#101	1	2	yes	—
105	C₂×C₄×C₁₇	#102	1	2	yes	—
106	C₂×C₄×C₁₇	#102	1	2	yes	—
107	C₂×C₄×C₁₇	#104	1	2	yes	—
108	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
110	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
112	C₂×C₄×C₁₇	#137	1	2	yes	—
113	C₈×C₁₇	—	4	4	no	—
114	C₈×C₁₇	—	8	2	no	—
115	C₄×C₄⋊Q₈	C₂×C₁₇	1	1	yes	characteristic
116	#1831	C₂×C₁₇	1	1	yes	characteristic
117	#1744	C₂×C₁₇	1	1	yes	characteristic
118	#1271	C₂×C₁₇	1	4	yes	—
119	C₂²×C₁₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
120	C₂²×C₁₇	#410	1	1	yes	characteristic
121	C₂²×C₁₇	C₂×#101	1	1	yes	characteristic
122	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
123	C₂²×C₁₇	#419	1	1	yes	characteristic
124	C₂²×C₁₇	C₂×#104	1	1	yes	characteristic
125	C₂²×C₁₇	#421	1	1	yes	characteristic
126	C₄×C₁₇	—	2	2	no	—
127	C₄×C₁₇	—	2	1	no	—
128	C₄×C₁₇	—	2	1	no	—
129	C₄×C₁₇	—	2	2	no	—
130	C₄×C₁₇	—	2	2	no	—
131	C₄×C₁₇	—	2	1	no	—
132	C₄×C₁₇	—	2	1	no	—
133	C₄×C₁₇	—	2	2	no	—
134	C₄×C₁₇	—	4	4	no	—
135	C₄×C₁₇	—	8	2	no	—
136	C₄×C₁₇	D₄⋊C₈	1	4	yes	—
137	C₂×C₄⋊Q₈	C₄×C₁₇	1	1	yes	characteristic
138	C₂×C₄×Q₈	—	2	1	no	—
139	C₂×C₄⋊C₈	—	2	1	no	—
140	C₂×C₄×C₈	—	2	1	no	—
141	#391	—	2	1	no	—
142	#389	—	2	1	no	—
143	#389	C₂²×C₁₇	1	2	yes	—
144	#384	C₄×C₁₇	1	1	yes	characteristic
145	C₄×C₄⋊C₄	—	2	1	no	—
146	C₄×C₄⋊C₄	C₄×C₁₇	1	2	yes	—
147	C₄³	C₂²×C₁₇	1	1	yes	characteristic
148	#336	—	2	1	no	—
149	#336	—	2	1	no	—
150	#336	C₂²×C₁₇	1	2	yes	—
151	#336	C₂²×C₁₇	1	2	yes	—
152	C₂×C₁₇	#1461	1	1	yes	characteristic
153	C₂×C₁₇	C₂×D₄⋊C₈	1	1	yes	characteristic
154	C₂×C₁₇	#1479	1	1	yes	characteristic
155	C₂×C₁₇	#1506	1	1	yes	characteristic
156	C₂×C₁₇	#1517	1	1	yes	characteristic
157	C₂×C₁₇	#1531	1	1	yes	characteristic
158	C₂×C₁₇	#1546	1	1	yes	characteristic
159	C₂²×Q₈	—	2	1	no	—
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	—	2	2	no	—
162	C₄⋊Q₈	—	2	2	no	—
163	C₄⋊Q₈	C₈×C₁₇	1	4	yes	complemented
164	C₄×Q₈	—	4	2	no	—
165	C₄×Q₈	—	4	2	no	—
166	C₂×C₄⋊C₄	—	2	1	no	—
167	C₂×C₄⋊C₄	—	2	1	no	—
168	C₂×C₄⋊C₄	—	2	1	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	C₂×C₄×C₁₇	1	2	yes	—
171	C₂×C₄⋊C₄	C₂×C₄×C₁₇	1	2	yes	—
172	C₂×C₄²	—	2	2	no	—
173	C₂×C₄²	C₂×C₄×C₁₇	1	1	yes	characteristic
174	C₂×C₄²	C₂×C₄×C₁₇	1	1	yes	characteristic
175	C₂×C₄²	D₄×C₁₇	1	1	yes	characteristic
176	C₂×C₄²	D₄×C₁₇	1	1	yes	characteristic
177	C₂×C₄²	D₄×C₁₇	1	1	yes	characteristic
178	C₂×C₄²	D₄×C₁₇	1	1	yes	characteristic
179	C₂×C₄²	C₂³×C₁₇	1	1	yes	characteristic
180	C₄⋊C₈	—	4	2	no	—
181	C₄×C₈	—	2	4	no	—
182	#95	—	2	2	no	—
183	#95	—	2	2	no	—
184	C₁₇	#7588	1	1	yes	characteristic, direct product factor, Sylow
185	C₂×Q₈	—	2	4	no	—
186	C₂×Q₈	—	4	2	no	—
187	C₂×Q₈	—	4	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	C₂²⋊C₄×C₁₇	1	1	yes	characteristic
191	C₂²×C₄	C₂²⋊C₄×C₁₇	1	1	yes	characteristic
192	C₂²×C₄	C₂²⋊C₄×C₁₇	1	1	yes	characteristic
193	C₂²×C₄	C₂²⋊C₄×C₁₇	1	1	yes	characteristic
194	C₂²×C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
196	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	4	2	no	—
202	C₂×C₈	—	4	2	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	C₂×C₈×C₁₇	1	4	yes	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₄²	—	4	1	no	—
215	C₄²	—	4	1	no	—
216	C₄²	—	4	4	no	—
217	C₄²	C₂×C₈×C₁₇	1	2	yes	—
218	C₄²	M₁₆×C₁₇	1	2	yes	—
219	C₄²	D₈×C₁₇	1	2	yes	—
220	C₄²	SD₁₆×C₁₇	1	2	yes	—
221	C₂³	C₂×C₂²⋊C₄×C₁₇	1	1	yes	characteristic
222	Q₈	—	4	8	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	C₁₇×C₂²⋊C₈	1	2	yes	—
242	C₂×C₄	C₁₇×C₂²⋊C₈	1	2	yes	—
243	C₂×C₄	C₁₇×#101	1	2	yes	—
244	C₂×C₄	C₁₇×#102	1	2	yes	—
245	C₂×C₄	C₁₇×#102	1	2	yes	—
246	C₂×C₄	C₁₇×#104	1	2	yes	—
247	C₂×C₄	C₂²×C₈×C₁₇	1	1	yes	characteristic
248	C₂×C₄	C₂×M₁₆×C₁₇	1	1	yes	characteristic
249	C₂×C₄	C₂×D₈×C₁₇	1	1	yes	characteristic
250	C₂×C₄	C₂×SD₁₆×C₁₇	1	1	yes	characteristic
251	C₂×C₄	C₁₇×#137	1	2	yes	—
252	C₈	—	4	4	no	—
253	C₈	—	8	2	no	—
254	C₂²	C₂×C₁₇×C₂²⋊C₈	1	1	yes	characteristic
255	C₂²	C₁₇×#410	1	1	yes	characteristic
256	C₂²	C₂×C₁₇×#101	1	1	yes	characteristic
257	C₂²	C₂×C₁₇×#102	1	1	yes	characteristic
258	C₂²	C₁₇×#419	1	1	yes	characteristic
259	C₂²	C₂×C₁₇×#104	1	1	yes	characteristic
260	C₂²	C₁₇×#421	1	1	yes	characteristic
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	2	2	no	—
268	C₄	—	2	2	no	—
269	C₄	—	4	4	no	—
270	C₄	—	8	2	no	—
271	C₄	C₁₇×D₄⋊C₈	1	4	yes	—
272	C₂	C₁₇×#1461	1	1	yes	characteristic
273	C₂	C₂×C₁₇×D₄⋊C₈	1	1	yes	characteristic
274	C₂	C₁₇×#1479	1	1	yes	characteristic
275	C₂	C₁₇×#1506	1	1	yes	characteristic
276	C₂	C₁₇×#1517	1	1	yes	characteristic
277	C₂	C₁₇×#1531	1	1	yes	characteristic
278	C₂	C₁₇×#1546	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₇	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₁₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₇
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₇ ◁ C₄³×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
24. C₁₇×C₄⋊Q₈	C₈	semidirect
81. #7588	C₁₇	direct (N × H)
163. C₄⋊Q₈	C₈×C₁₇	semidirect
184. C₁₇	#7588	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₁₇×C₄⋊Q₈	2
C₁₇×#1831	2
C₁₇×#1744	2
C₁₇×#1271	2
#7588	17

Subgroups — C17×#7588 #16865332

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#7588 #16865332