Subgroups — #17186964

Order 4352.

826 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	C₂²×#10587483	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762764	C₂	1	1	yes	maximal, characteristic
3	#14530383	C₂	1	4	yes	maximal, complemented
4	#14530380	C₂	1	4	yes	maximal, complemented
5	M₃₂×Dic₁₇	C₂	1	4	yes	maximal, complemented
6	C₂²×C₁₇×M₃₂	C₂	1	1	yes	maximal, characteristic
7	C₂²×#63223	C₄	1	1	yes	characteristic
8	C₂×#10587612	C₄	1	1	yes	characteristic
9	C₂×#10587483	C₄	1	8	yes	complemented
10	C₂×#10587483	C₂²	1	2	yes	—
11	C₂×#10587483	C₂²	1	2	yes	—
12	C₂²×#63214	C₂²	1	2	yes	—
13	#11762764	C₄	1	2	yes	—
14	#11762764	C₄	1	2	yes	—
15	#11762764	C₂²	1	2	yes	—
16	#11762764	C₂²	1	2	yes	—
17	C₂×#10587486	C₄	1	2	yes	—
18	C₂×C₈×Dic₁₇	C₂²	1	2	yes	—
19	#11762584	C₂²	1	8	yes	complemented
20	C₁₆×Dic₁₇	C₂²	1	8	yes	complemented
21	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
22	C₂×C₁₇×M₃₂	C₄	1	8	yes	complemented
23	C₂×C₁₇×M₃₂	C₂²	1	2	yes	—
24	C₂×C₁₇×M₃₂	C₂²	1	2	yes	—
25	C₂²×C₁₆×C₁₇	C₂²	1	2	yes	—
26	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₇	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₁₇	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
35	C₂×C₂²⋊Dic₁₇	C₈	1	2	yes	—
36	#10587612	C₈	1	2	yes	—
37	#10587612	C₈	1	2	yes	—
38	#10587612	C₂×C₄	1	2	yes	—
39	#10587612	C₂×C₄	1	2	yes	—
40	C₂×C₄⋊Dic₁₇	C₈	1	2	yes	—
41	C₂×C₄×Dic₁₇	C₂×C₄	1	2	yes	—
42	C₂×#63223	C₈	1	8	yes	—
43	C₂×#63223	C₂×C₄	1	2	yes	—
44	C₂×#63223	C₂×C₄	1	2	yes	—
45	C₂²×#3622	C₂×C₄	1	2	yes	—
46	C₁₇×M₃₂	C₂×C₄	1	8	yes	—
47	C₁₇×M₃₂	C₂×C₄	1	8	yes	complemented
48	C₂×C₁₆×C₁₇	C₂×C₄	1	8	yes	complemented
49	C₂×C₁₆×C₁₇	C₂³	1	4	yes	—
50	#10587486	C₂×C₄	1	4	yes	—
51	#10587486	C₂×C₄	1	4	yes	—
52	C₈×Dic₁₇	C₂×C₄	1	4	yes	—
53	C₈×Dic₁₇	C₂³	1	4	yes	—
54	#10587483	C₂×C₄	1	8	yes	—
55	#10587483	C₂×C₄	1	8	yes	complemented
56	C₂×#63214	C₂×C₄	1	8	yes	complemented
57	C₂×#63214	C₂³	1	4	yes	—
58	C₂⁴×C₁₇	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₇	C₄²	1	4	yes	—
60	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₇	C₂×C₈	1	4	yes	—
64	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×Dic₁₇	C₂×C₈	1	4	yes	—
68	C₂×C₈×C₁₇	C₄²	1	4	yes	—
69	C₂×C₈×C₁₇	C₄²	1	4	yes	—
70	C₂×C₈×C₁₇	C₄²	1	4	yes	—
71	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
81	C₂²⋊Dic₁₇	C₂×C₈	1	4	yes	—
82	C₂²⋊Dic₁₇	C₂×C₈	1	4	yes	—
83	C₄⋊Dic₁₇	C₂×C₈	1	4	yes	—
84	C₄⋊Dic₁₇	C₂×C₈	1	4	yes	—
85	C₄×Dic₁₇	C₂×C₈	1	4	yes	—
86	C₄×Dic₁₇	C₂²×C₄	1	4	yes	—
87	#63223	C₂×C₈	1	8	yes	—
88	#63223	C₂×C₈	1	8	yes	—
89	C₂×#3622	C₂×C₈	1	8	yes	—
90	C₂×#3622	C₂²×C₄	1	4	yes	—
91	C₁₆×C₁₇	C₂²×C₄	1	8	yes	—
92	#63214	C₂²×C₄	1	8	yes	—
93	#33324	—	17	1	no	maximal, Sylow
94	C₂³×C₁₇	—	4	2	no	—
95	C₂³×C₁₇	C₄×C₈	1	4	yes	—
96	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
97	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
98	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₁₇	C₄×C₈	1	4	yes	—
100	C₂×C₄×C₁₇	C₄×C₈	1	4	yes	—
101	C₂×C₄×C₁₇	C₄×C₈	1	4	yes	—
102	C₂×C₄×C₁₇	C₂×C₄²	1	2	yes	—
103	C₂×C₄×C₁₇	C₂×C₄²	1	2	yes	—
104	C₂×C₄×C₁₇	C₂×C₄²	1	2	yes	—
105	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
107	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
108	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
112	C₂×Dic₁₇	—	4	4	no	—
113	C₂×Dic₁₇	C₂²×C₈	1	8	yes	—
114	C₈×C₁₇	C₂×C₄²	1	2	yes	—
115	C₈×C₁₇	C₂×C₄²	1	2	yes	—
116	C₈×C₁₇	C₂×C₄²	1	2	yes	—
117	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
118	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
119	#3622	C₂²×C₈	1	8	yes	—
120	C₂²×M₃₂	—	17	1	no	—
121	C₂²×M₃₂	D₁₇	1	1	yes	characteristic
122	C₂×#405	—	17	1	no	—
123	#2108	—	17	4	no	—
124	#2108	—	17	4	no	—
125	C₄×M₃₂	—	17	4	no	—
126	C₂²×C₁₇	—	2	2	no	—
127	C₂²×C₁₇	—	2	2	no	—
128	C₂²×C₁₇	—	2	2	no	—
129	C₂²×C₁₇	—	4	4	no	—
130	C₂²×C₁₇	C₂×C₄×C₈	1	2	yes	—
131	C₂²×C₁₇	C₂×C₄×C₈	1	2	yes	—
132	C₂²×C₁₇	C₂×C₄×C₈	1	2	yes	—
133	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
134	C₄×C₁₇	C₂×C₄×C₈	1	2	yes	—
135	C₄×C₁₇	C₂×C₄×C₈	1	2	yes	—
136	C₄×C₁₇	C₂×C₄×C₈	1	2	yes	—
137	C₄×C₁₇	C₂²×C₄²	1	1	yes	characteristic
138	C₄×C₁₇	C₂³×C₈	1	1	yes	characteristic
139	Dic₁₇	—	2	8	no	—
140	C₂²×M₁₆	—	17	1	no	—
141	C₂³×C₈	C₂×D₁₇	1	1	yes	characteristic
142	C₂×#117	—	17	1	no	—
143	C₂×M₃₂	—	17	8	no	—
144	C₂×M₃₂	—	17	2	no	—
145	C₂×M₃₂	—	17	2	no	—
146	C₂×M₃₂	Dic₁₇	1	8	yes	complemented
147	C₂×M₃₂	C₂×D₁₇	1	2	yes	—
148	C₂×M₃₂	C₂×D₁₇	1	2	yes	—
149	C₂²×C₁₆	—	17	2	no	—
150	C₂²×C₁₆	C₂×D₁₇	1	2	yes	—
151	#405	—	17	2	no	—
152	#405	—	17	2	no	—
153	#405	—	17	2	no	—
154	#405	—	17	2	no	—
155	C₂×#97	—	17	2	no	—
156	C₂×C₄×C₈	—	17	2	no	—
157	#346	—	17	8	no	—
158	C₄×C₁₆	—	17	8	no	—
159	C₂×C₁₇	—	2	2	no	—
160	C₂×C₁₇	—	2	2	no	—
161	C₂×C₁₇	—	2	2	no	—
162	C₂×C₁₇	C₂²×C₄×C₈	1	1	yes	characteristic
163	C₂×C₁₇	#3407	1	2	yes	—
164	C₂³×C₄	C₄×D₁₇	1	1	yes	characteristic
165	C₂×M₁₆	—	17	2	no	—
166	C₂×M₁₆	—	17	2	no	—
167	C₂×M₁₆	—	17	8	no	—
168	C₂²×C₈	—	17	2	no	—
169	C₂²×C₈	C₄×D₁₇	1	4	yes	—
170	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
171	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
172	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
173	C₂²×C₈	C₂×Dic₁₇	1	4	yes	—
174	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
175	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
176	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
177	#117	—	17	2	no	—
178	#117	—	17	2	no	—
179	#117	—	17	2	no	—
180	#117	—	17	2	no	—
181	C₂×C₄⋊C₄	—	17	2	no	—
182	C₂×C₂²⋊C₄	—	17	2	no	—
183	C₂×C₄²	—	17	2	no	—
184	M₃₂	—	17	8	no	—
185	M₃₂	—	17	8	no	—
186	M₃₂	C₂×Dic₁₇	1	8	yes	—
187	M₃₂	C₂×Dic₁₇	1	8	yes	complemented
188	C₂×C₁₆	—	17	8	no	—
189	C₂×C₁₆	—	17	4	no	—
190	C₂×C₁₆	C₂×Dic₁₇	1	8	yes	complemented
191	C₂×C₁₆	C₂²×D₁₇	1	4	yes	—
192	#97	—	17	4	no	—
193	#97	—	17	4	no	—
194	C₄×C₈	—	17	4	no	—
195	C₄×C₈	—	17	4	no	—
196	C₁₇	#33324	1	1	yes	characteristic, complemented, Sylow
197	C₂⁴	C₈×D₁₇	1	1	yes	characteristic
198	C₂²×C₄	—	17	4	no	—
199	C₂²×C₄	C₈×D₁₇	1	1	yes	characteristic
200	C₂²×C₄	C₈×D₁₇	1	1	yes	characteristic
201	C₂²×C₄	C₈×D₁₇	1	1	yes	characteristic
202	C₂²×C₄	C₈×D₁₇	1	4	yes	—
203	C₂²×C₄	C₄×Dic₁₇	1	4	yes	—
204	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
207	M₁₆	—	17	8	no	—
208	M₁₆	—	17	8	no	—
209	C₂×C₈	—	17	4	no	—
210	C₂×C₈	—	17	8	no	—
211	C₂×C₈	C₄×Dic₁₇	1	4	yes	—
212	C₂×C₈	C₄×Dic₁₇	1	4	yes	—
213	C₂×C₈	C₄×Dic₁₇	1	4	yes	—
214	C₂×C₈	C₂×C₄×D₁₇	1	2	yes	—
215	C₂×C₈	C₂×C₄×D₁₇	1	2	yes	—
216	C₂×C₈	C₂×C₄×D₁₇	1	2	yes	—
217	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
218	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
219	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
220	C₂×C₈	C₂²×Dic₁₇	1	2	yes	—
221	C₂×C₈	C₂²×Dic₁₇	1	2	yes	—
222	C₂×C₈	C₂²×Dic₁₇	1	2	yes	—
223	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
224	C₄⋊C₄	—	17	4	no	—
225	C₄⋊C₄	—	17	4	no	—
226	C₂²⋊C₄	—	17	4	no	—
227	C₂²⋊C₄	—	17	4	no	—
228	C₄²	—	17	4	no	—
229	C₄²	—	17	4	no	—
230	C₁₆	—	17	8	no	—
231	C₁₆	C₂²×Dic₁₇	1	8	yes	—
232	C₂³	—	4	2	no	—
233	C₂³	C₈×Dic₁₇	1	4	yes	—
234	C₂³	C₂×C₈×D₁₇	1	1	yes	characteristic
235	C₂³	C₂×C₈×D₁₇	1	1	yes	characteristic
236	C₂³	C₂×C₈×D₁₇	1	1	yes	characteristic
237	C₂×C₄	—	17	8	no	—
238	C₂×C₄	—	68	4	no	—
239	C₂×C₄	C₈×Dic₁₇	1	4	yes	—
240	C₂×C₄	C₈×Dic₁₇	1	4	yes	—
241	C₂×C₄	C₈×Dic₁₇	1	4	yes	—
242	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
243	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
244	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
245	C₂×C₄	C₂×C₈×D₁₇	1	2	yes	—
246	C₂×C₄	C₂×C₈×D₁₇	1	2	yes	—
247	C₂×C₄	C₂×C₈×D₁₇	1	2	yes	—
248	C₂×C₄	C₂×C₄×Dic₁₇	1	2	yes	—
249	C₂×C₄	C₂×C₄×Dic₁₇	1	2	yes	—
250	C₂×C₄	C₂×C₄×Dic₁₇	1	2	yes	—
251	C₂×C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
252	C₈	—	17	8	no	—
253	C₈	C₂×C₄×Dic₁₇	1	2	yes	—
254	C₈	C₂×C₄×Dic₁₇	1	2	yes	—
255	C₈	C₂×C₄×Dic₁₇	1	2	yes	—
256	C₈	C₂²×C₄×D₁₇	1	1	yes	characteristic
257	C₈	C₂³×Dic₁₇	1	1	yes	characteristic
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	4	4	no	—
262	C₂²	C₂×C₈×Dic₁₇	1	2	yes	—
263	C₂²	C₂×C₈×Dic₁₇	1	2	yes	—
264	C₂²	C₂×C₈×Dic₁₇	1	2	yes	—
265	C₂²	C₂²×C₈×D₁₇	1	1	yes	characteristic
266	C₄	—	34	8	no	—
267	C₄	C₂×C₈×Dic₁₇	1	2	yes	—
268	C₄	C₂×C₈×Dic₁₇	1	2	yes	—
269	C₄	C₂×C₈×Dic₁₇	1	2	yes	—
270	C₄	C₂²×C₄×Dic₁₇	1	1	yes	characteristic
271	C₄	C₂²×C₈×D₁₇	1	1	yes	characteristic
272	C₂	—	2	2	no	—
273	C₂	—	2	2	no	—
274	C₂	—	2	2	no	—
275	C₂	C₂²×C₈×Dic₁₇	1	1	yes	characteristic
276	C₂	#14541886	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₇
commutator	C₂×C₁₇	C₂²×C₄×C₈
Fitting	C₂²×C₁₇×M₃₂	C₂
socle	C₂²×C₁₇	C₂³×C₈
prime radical	C₂⁴×C₁₇	C₂×C₈
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	C₂²×M₃₂	D₁₇
hypocenter	C₁₇	#33324
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. #14530383	C₂	semidirect
4. #14530380	C₂	semidirect
5. M₃₂×Dic₁₇	C₂	semidirect
9. C₂×#10587483	C₄	semidirect
19. #11762584	C₂²	semidirect
20. C₁₆×Dic₁₇	C₂²	semidirect
22. C₂×C₁₇×M₃₂	C₄	semidirect
47. C₁₇×M₃₂	C₂×C₄	semidirect
48. C₂×C₁₆×C₁₇	C₂×C₄	semidirect
55. #10587483	C₂×C₄	semidirect
56. C₂×#63214	C₂×C₄	semidirect
146. C₂×M₃₂	Dic₁₇	semidirect
187. M₃₂	C₂×Dic₁₇	semidirect
190. C₂×C₁₆	C₂×Dic₁₇	semidirect
196. C₁₇	#33324	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#10587483	2
C₂×#11762764	2
#14530383	2
#14530380	2
M₃₂×Dic₁₇	2
C₂²×C₁₇×M₃₂	2
#33324	17