Subgroups — C₁₇×#13201 #16870945

Order 4352.

420 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×#2930	C₂	1	1	yes	maximal, characteristic
2	C₁₇×#2153	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₇×#2117	C₂	1	2	yes	maximal, complemented
4	C₂×C₁₇×#349	C₂	1	2	yes	maximal, complemented
5	C₁₇×#2110	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₇×#131	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
8	C₁₇×#525	C₄	1	1	yes	characteristic
9	C₂×C₁₇×M₃₂	C₂²	1	2	yes	—
10	C₂×C₁₇×M₃₂	C₂²	1	2	yes	—
11	C₁₇×#436	—	2	2	no	—
12	C₁₇×#435	—	2	2	no	—
13	C₁₇×#424	C₄	1	1	yes	characteristic
14	C₁₇×#408	C₄	1	1	yes	characteristic
15	C₁₇×#407	C₄	1	1	yes	characteristic
16	C₁₇×#405	C₂²	1	1	yes	characteristic
17	C₁₇×#364	—	2	2	no	—
18	C₁₇×#350	—	2	4	no	—
19	C₁₇×#349	—	2	4	no	—
20	C₁₇×#347	—	2	2	no	—
21	C₂×#41×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₁₇×#131	—	2	2	no	—
24	C₁₇×#131	—	2	2	no	—
25	C₂×M₁₆×C₁₇	—	2	2	no	—
26	C₂×M₁₆×C₁₇	—	2	1	no	—
27	C₂×M₁₆×C₁₇	—	2	1	no	—
28	C₂×M₁₆×C₁₇	—	2	1	no	—
29	C₂×M₁₆×C₁₇	—	2	2	no	—
30	C₂×M₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
31	C₂×M₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₇	—	2	1	no	—
33	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
35	C₁₇×#123	—	2	1	no	—
36	C₁₇×C₂²⋊Q₈	—	2	1	no	—
37	C₁₇×#121	—	2	1	no	—
38	C₁₇×#120	—	2	1	no	—
39	C₄×D₄×C₁₇	—	2	2	no	—
40	C₁₇×#117	C₂×C₄	1	1	yes	characteristic
41	C₁₇×M₃₂	—	2	2	no	—
42	C₁₇×M₃₂	—	2	2	no	—
43	C₁₇×M₃₂	—	2	2	no	—
44	C₁₇×M₃₂	D₄	1	4	yes	complemented
45	C₂×C₁₆×C₁₇	—	2	2	no	—
46	C₂×C₁₆×C₁₇	—	2	2	no	—
47	C₁₇×C₄⋊C₈	—	2	2	no	—
48	C₁₇×C₂²⋊C₈	—	2	2	no	—
49	C₁₇×C₂²⋊C₈	—	2	2	no	—
50	C₁₇×#97	D₄	1	2	yes	—
51	C₄×C₈×C₁₇	D₄	1	2	yes	—
52	C₂⁴×C₁₇	M₁₆	1	1	yes	characteristic
53	#41×C₁₇	—	2	2	no	—
54	#41×C₁₇	—	2	2	no	—
55	C₂×Q₈×C₁₇	M₁₆	1	1	yes	characteristic
56	C₂×D₄×C₁₇	—	2	1	no	—
57	C₂×D₄×C₁₇	—	2	2	no	—
58	C₂×D₄×C₁₇	M₁₆	1	1	yes	characteristic
59	C₂²×C₄×C₁₇	—	2	1	no	—
60	C₂²×C₄×C₁₇	—	2	1	no	—
61	C₂²×C₄×C₁₇	—	2	2	no	—
62	C₂²×C₄×C₁₇	—	2	1	no	—
63	C₂²×C₄×C₁₇	—	2	2	no	—
64	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
65	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
66	M₁₆×C₁₇	—	2	2	no	—
67	M₁₆×C₁₇	—	2	2	no	—
68	M₁₆×C₁₇	—	4	2	no	—
69	M₁₆×C₁₇	—	4	1	no	—
70	M₁₆×C₁₇	—	4	1	no	—
71	M₁₆×C₁₇	—	4	1	no	—
72	M₁₆×C₁₇	—	4	1	no	—
73	C₂×C₈×C₁₇	—	2	1	no	—
74	C₂×C₈×C₁₇	—	2	1	no	—
75	C₂×C₈×C₁₇	—	2	2	no	—
76	C₂×C₈×C₁₇	—	4	2	no	—
77	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
78	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₁₇	#41	1	2	yes	—
82	C₄⋊C₄×C₁₇	—	2	2	no	—
83	C₄⋊C₄×C₁₇	—	2	1	no	—
84	C₂²⋊C₄×C₁₇	—	2	2	no	—
85	C₂²⋊C₄×C₁₇	—	2	2	no	—
86	C₂²⋊C₄×C₁₇	—	2	1	no	—
87	C₄²×C₁₇	C₂²⋊C₄	1	2	yes	—
88	C₁₆×C₁₇	—	2	4	no	—
89	C₁₆×C₁₇	—	4	2	no	—
90	#13201	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₁₇	—	2	1	no	—
92	C₂³×C₁₇	—	2	1	no	—
93	C₂³×C₁₇	—	2	2	no	—
94	C₂³×C₁₇	—	8	1	no	—
95	C₂³×C₁₇	C₂×M₁₆	1	1	yes	characteristic
96	Q₈×C₁₇	—	2	2	no	—
97	D₄×C₁₇	—	2	2	no	—
98	D₄×C₁₇	—	4	2	no	—
99	D₄×C₁₇	—	8	2	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	2	no	—
104	C₂×C₄×C₁₇	—	4	1	no	—
105	C₂×C₄×C₁₇	—	4	1	no	—
106	C₂×C₄×C₁₇	—	4	1	no	—
107	C₂×C₄×C₁₇	—	4	1	no	—
108	C₂×C₄×C₁₇	—	4	2	no	—
109	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
112	C₂×C₄×C₁₇	#131	1	2	yes	—
113	C₈×C₁₇	—	2	2	no	—
114	C₈×C₁₇	—	4	1	no	—
115	C₈×C₁₇	—	4	1	no	—
116	C₈×C₁₇	#121	1	2	yes	—
117	C₈×C₁₇	#124	1	1	yes	characteristic
118	C₈×C₁₇	#127	1	1	yes	characteristic
119	#2930	C₂×C₁₇	1	1	yes	characteristic
120	#2153	C₂×C₁₇	1	1	yes	characteristic, complemented
121	#2117	C₂×C₁₇	1	2	yes	complemented
122	C₂×#349	C₂×C₁₇	1	2	yes	complemented
123	#2110	C₂×C₁₇	1	1	yes	characteristic, complemented
124	C₂²×C₁₇	—	4	1	no	—
125	C₂²×C₁₇	—	4	1	no	—
126	C₂²×C₁₇	—	4	2	no	—
127	C₂²×C₁₇	—	4	1	no	—
128	C₂²×C₁₇	—	4	1	no	—
129	C₂²×C₁₇	—	4	1	no	—
130	C₂²×C₁₇	—	8	2	no	—
131	C₂²×C₁₇	#407	1	1	yes	characteristic
132	C₂²×C₁₇	#435	1	2	yes	—
133	C₄×C₁₇	—	4	2	no	—
134	C₄×C₁₇	—	4	1	no	—
135	C₄×C₁₇	—	4	1	no	—
136	C₄×C₁₇	#390	1	1	yes	characteristic
137	C₄×C₁₇	#408	1	1	yes	characteristic
138	C₄×C₁₇	#435	1	2	yes	—
139	C₂×#131	C₂²×C₁₇	1	1	yes	characteristic
140	C₂²×M₁₆	C₂²×C₁₇	1	1	yes	characteristic
141	#525	C₄×C₁₇	1	1	yes	characteristic
142	C₂×M₃₂	C₂²×C₁₇	1	2	yes	—
143	C₂×M₃₂	C₂²×C₁₇	1	2	yes	—
144	#436	—	2	2	no	—
145	#435	—	2	2	no	—
146	#424	C₄×C₁₇	1	1	yes	characteristic
147	#408	C₄×C₁₇	1	1	yes	characteristic
148	#407	C₄×C₁₇	1	1	yes	characteristic
149	#405	C₂²×C₁₇	1	1	yes	characteristic
150	#364	—	2	2	no	—
151	#350	—	2	4	no	—
152	#349	—	2	4	no	—
153	#347	—	2	2	no	—
154	C₂×C₁₇	—	2	1	no	—
155	C₂×C₁₇	—	2	2	no	—
156	C₂×C₁₇	—	8	1	no	—
157	C₂×C₁₇	—	8	1	no	—
158	C₂×C₁₇	#1955	1	1	yes	characteristic
159	C₂×#41	C₂×C₄×C₁₇	1	1	yes	characteristic
160	C₂³×C₄	C₂×C₄×C₁₇	1	1	yes	characteristic
161	#131	—	2	2	no	—
162	#131	—	2	2	no	—
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	2	1	no	—
165	C₂×M₁₆	—	2	1	no	—
166	C₂×M₁₆	—	2	1	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	C₂×C₄×C₁₇	1	1	yes	characteristic
169	C₂×M₁₆	C₂×C₄×C₁₇	1	1	yes	characteristic
170	C₂²×C₈	—	2	1	no	—
171	C₂²×C₈	C₂×C₄×C₁₇	1	1	yes	characteristic
172	C₂²×C₈	C₂³×C₁₇	1	1	yes	characteristic
173	#123	—	2	1	no	—
174	C₂²⋊Q₈	—	2	1	no	—
175	#121	—	2	1	no	—
176	#120	—	2	1	no	—
177	C₄×D₄	—	2	2	no	—
178	#117	C₂×C₄×C₁₇	1	1	yes	characteristic
179	M₃₂	—	2	2	no	—
180	M₃₂	—	2	2	no	—
181	M₃₂	—	2	2	no	—
182	M₃₂	D₄×C₁₇	1	4	yes	complemented
183	C₂×C₁₆	—	2	2	no	—
184	C₂×C₁₆	—	2	2	no	—
185	C₄⋊C₈	—	2	2	no	—
186	C₂²⋊C₈	—	2	2	no	—
187	C₂²⋊C₈	—	2	2	no	—
188	#97	D₄×C₁₇	1	2	yes	—
189	C₄×C₈	D₄×C₁₇	1	2	yes	—
190	C₁₇	#13201	1	1	yes	characteristic, direct product factor, Sylow
191	C₂⁴	M₁₆×C₁₇	1	1	yes	characteristic
192	#41	—	2	2	no	—
193	#41	—	2	2	no	—
194	C₂×Q₈	M₁₆×C₁₇	1	1	yes	characteristic
195	C₂×D₄	—	2	1	no	—
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	M₁₆×C₁₇	1	1	yes	characteristic
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	M₁₆×C₁₇	1	1	yes	characteristic
204	C₂²×C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
205	M₁₆	—	2	2	no	—
206	M₁₆	—	2	2	no	—
207	M₁₆	—	4	1	no	—
208	M₁₆	—	4	1	no	—
209	M₁₆	—	4	1	no	—
210	M₁₆	—	4	1	no	—
211	M₁₆	—	4	2	no	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	C₂²⋊C₄×C₁₇	1	2	yes	—
217	C₂×C₈	C₂×D₄×C₁₇	1	1	yes	characteristic
218	C₂×C₈	C₂×D₄×C₁₇	1	2	yes	—
219	C₂×C₈	C₂×D₄×C₁₇	1	1	yes	characteristic
220	C₂×C₈	#41×C₁₇	1	2	yes	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₂²⋊C₄	—	2	2	no	—
224	C₂²⋊C₄	—	2	2	no	—
225	C₂²⋊C₄	—	2	1	no	—
226	C₄²	C₂²⋊C₄×C₁₇	1	2	yes	—
227	C₁₆	—	2	4	no	—
228	C₁₆	—	4	2	no	—
229	C₂³	—	2	1	no	—
230	C₂³	—	2	1	no	—
231	C₂³	—	2	2	no	—
232	C₂³	—	8	1	no	—
233	C₂³	C₂×M₁₆×C₁₇	1	1	yes	characteristic
234	Q₈	—	2	2	no	—
235	D₄	—	2	2	no	—
236	D₄	—	4	2	no	—
237	D₄	—	8	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	C₂×C₂²⋊C₄×C₁₇	1	1	yes	characteristic
248	C₂×C₄	C₄×D₄×C₁₇	1	2	yes	—
249	C₂×C₄	C₂×M₁₆×C₁₇	1	1	yes	characteristic
250	C₂×C₄	C₁₇×#131	1	2	yes	—
251	C₈	—	2	2	no	—
252	C₈	—	4	1	no	—
253	C₈	—	4	1	no	—
254	C₈	C₁₇×#121	1	2	yes	—
255	C₈	C₁₇×#124	1	1	yes	characteristic
256	C₈	C₁₇×#127	1	1	yes	characteristic
257	C₂²	—	4	1	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	4	2	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	1	no	—
262	C₂²	—	4	1	no	—
263	C₂²	—	8	2	no	—
264	C₂²	C₁₇×#407	1	1	yes	characteristic
265	C₂²	C₁₇×#435	1	2	yes	—
266	C₄	—	4	2	no	—
267	C₄	—	4	1	no	—
268	C₄	—	4	1	no	—
269	C₄	C₁₇×#390	1	1	yes	characteristic
270	C₄	C₁₇×#408	1	1	yes	characteristic
271	C₄	C₁₇×#435	1	2	yes	—
272	C₂	—	2	1	no	—
273	C₂	—	2	2	no	—
274	C₂	—	8	1	no	—
275	C₂	—	8	1	no	—
276	C₂	C₁₇×#1955	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₇	#390
commutator	C₂²×C₄	C₂²×C₄×C₁₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₇
socle	C₂×C₁₇	#1955
prime radical	C₁₇×#525	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₇ ◁ C₂²×C₈×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₇×#2153	C₂	semidirect
3. C₁₇×#2117	C₂	semidirect
4. C₂×C₁₇×#349	C₂	semidirect
5. C₁₇×#2110	C₂	semidirect
44. C₁₇×M₃₂	D₄	semidirect
90. #13201	C₁₇	direct (N × H)
120. #2153	C₂×C₁₇	semidirect
121. #2117	C₂×C₁₇	semidirect
122. C₂×#349	C₂×C₁₇	semidirect
123. #2110	C₂×C₁₇	semidirect
182. M₃₂	D₄×C₁₇	semidirect
190. C₁₇	#13201	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×#2930	2
C₁₇×#2153	2
C₁₇×#2117	2
C₂×C₁₇×#349	2
C₁₇×#2110	2
#13201	17

Subgroups — C17×#13201 #16870945

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#13201 #16870945