Subgroups — #17043196

Order 4352.

670 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14539308	C₂	1	1	yes	maximal, characteristic
2	#14537721	C₂	1	2	yes	maximal
3	#14529760	C₂	1	2	yes	maximal
4	#14529535	C₂	1	2	yes	maximal
5	#14529530	C₂	1	2	yes	maximal
6	#14528671	C₂	1	2	yes	maximal
7	#14528666	C₂	1	2	yes	maximal
8	#97×Dic₁₇	C₂	1	1	yes	maximal, characteristic
9	C₁₇×#1849	C₂	1	1	yes	maximal, characteristic
10	#11763366	C₂²	1	2	yes	—
11	#11763218	C₄	1	8	yes	complemented
12	#11762938	—	2	8	no	—
13	#11762847	C₂²	1	2	yes	—
14	#11762847	C₂²	1	2	yes	—
15	#11762837	C₂²	1	4	yes	—
16	#11762767	C₂²	1	4	yes	—
17	C₂×#10587474	C₂²	1	1	yes	characteristic
18	C₂×#10587486	C₂²	1	1	yes	characteristic
19	C₂×#10587486	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
21	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
22	C₄²×Dic₁₇	C₂²	1	1	yes	characteristic
23	#11762570	C₂²	1	4	yes	—
24	#11762570	C₂²	1	4	yes	—
25	C₂×C₁₇×#108	C₂²	1	2	yes	—
26	C₂×C₁₇×#108	C₂²	1	2	yes	—
27	C₁₇×#423	C₂²	1	2	yes	—
28	C₂×C₁₇×#97	C₂²	1	1	yes	characteristic
29	C₂×M₁₆×C₁₇	C₂³	1	4	yes	—
30	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
33	#10587612	C₂³	1	4	yes	—
34	#10587610	—	2	4	no	—
35	C₂×C₄×Dic₁₇	—	2	2	no	—
36	C₂×C₄×Dic₁₇	C₂³	1	1	yes	characteristic
37	C₂×C₄×Dic₁₇	C₂³	1	1	yes	characteristic
38	C₂²×#3622	C₂³	1	1	yes	characteristic
39	C₂²×#3622	C₂³	1	1	yes	characteristic
40	#10587556	C₂×C₄	1	4	yes	—
41	#10587555	C₂×C₄	1	4	yes	—
42	C₄×Dic₃₄	—	2	4	no	—
43	C₁₇×#108	—	2	4	no	—
44	C₁₇×#108	—	2	4	no	—
45	C₁₇×C₄⋊C₈	—	2	4	no	—
46	C₁₇×#97	C₂×C₄	1	4	yes	—
47	#10587489	C₂×C₄	1	4	yes	—
48	#10587489	C₂×C₄	1	4	yes	—
49	#10587488	C₂×C₄	1	4	yes	—
50	#10587488	C₂×C₄	1	4	yes	—
51	#10587486	—	2	4	no	—
52	#10587486	—	2	2	no	—
53	#10587486	Q₈	1	4	yes	complemented
54	#10587485	—	2	8	no	—
55	C₈×Dic₁₇	—	2	4	no	—
56	C₈×Dic₁₇	—	2	2	no	—
57	C₈×Dic₁₇	Q₈	1	4	yes	complemented
58	#10587474	D₄	1	4	yes	—
59	C₂²×C₄×C₁₇	#41	1	2	yes	—
60	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
61	C₂²×Dic₁₇	—	2	1	no	—
62	C₂²×Dic₁₇	—	2	1	no	—
63	C₂×Dic₃₄	—	2	4	no	—
64	M₁₆×C₁₇	—	2	4	no	—
65	M₁₆×C₁₇	—	2	4	no	—
66	C₂×C₈×C₁₇	—	2	4	no	—
67	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
70	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
71	C₂×C₈×C₁₇	#41	1	2	yes	—
72	C₂×C₈×C₁₇	#41	1	2	yes	—
73	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
74	C₄²×C₁₇	C₂×D₄	1	2	yes	—
75	C₂²⋊Dic₁₇	—	2	4	no	—
76	C₄⋊Dic₁₇	—	2	4	no	—
77	C₄⋊Dic₁₇	C₂²×C₄	1	8	yes	—
78	#63225	—	2	8	no	—
79	C₄×Dic₁₇	—	2	4	no	—
80	C₄×Dic₁₇	—	2	1	no	—
81	C₄×Dic₁₇	—	2	1	no	—
82	C₄×Dic₁₇	—	4	4	no	—
83	C₄×Dic₁₇	C₂×Q₈	1	2	yes	—
84	C₄×Dic₁₇	C₂×Q₈	1	2	yes	—
85	C₂×#3622	—	2	1	no	—
86	C₂×#3622	—	2	1	no	—
87	C₂×#3622	C₂×D₄	1	2	yes	—
88	C₂×#3622	C₂×D₄	1	2	yes	—
89	C₂×#3622	C₂×Q₈	1	2	yes	—
90	C₂×#3622	C₂×Q₈	1	2	yes	—
91	#21736	—	17	1	no	maximal, Sylow
92	C₂³×C₁₇	C₂×#41	1	1	yes	characteristic
93	C₂×C₄×C₁₇	—	4	2	no	—
94	C₂×C₄×C₁₇	C₄×D₄	1	4	yes	—
95	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
98	C₂×C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
99	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
100	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
101	C₂×Dic₁₇	—	2	4	no	—
102	C₂×Dic₁₇	—	2	2	no	—
103	C₂×Dic₁₇	—	2	8	no	—
104	C₂×Dic₁₇	—	2	2	no	—
105	C₂×Dic₁₇	—	4	2	no	—
106	Dic₃₄	—	8	4	no	—
107	C₈×C₁₇	—	4	4	no	—
108	C₈×C₁₇	C₄×Q₈	1	4	yes	—
109	C₈×C₁₇	C₄×Q₈	1	4	yes	—
110	#3622	—	2	2	no	—
111	#3622	C₄⋊Q₈	1	4	yes	—
112	#3162	—	17	1	no	—
113	#2961	—	17	2	no	—
114	#1994	—	17	2	no	—
115	#1946	—	17	2	no	—
116	#1945	—	17	2	no	—
117	#1849	D₁₇	1	1	yes	characteristic
118	#1773	—	17	2	no	—
119	#1772	—	17	2	no	—
120	C₄×#97	—	17	1	no	—
121	C₂²×C₁₇	—	2	2	no	—
122	C₂²×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
123	C₂²×C₁₇	#533	1	1	yes	characteristic
124	C₂²×C₁₇	#533	1	1	yes	characteristic
125	C₄×C₁₇	—	2	4	no	—
126	C₄×C₁₇	C₂×C₄×Q₈	1	1	yes	characteristic
127	C₄×C₁₇	C₂×C₄×Q₈	1	1	yes	characteristic
128	C₄×C₁₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
129	C₄×C₁₇	#532	1	1	yes	characteristic
130	Dic₁₇	—	2	4	no	—
131	Dic₁₇	—	4	2	no	—
132	Dic₁₇	—	4	8	no	—
133	#533	—	17	2	no	—
134	#501	—	17	8	no	—
135	#446	—	34	8	no	—
136	C₂×#108	C₂×D₁₇	1	2	yes	—
137	C₂×#108	C₂×D₁₇	1	2	yes	—
138	#427	—	17	2	no	—
139	#427	—	17	2	no	—
140	#424	—	17	4	no	—
141	#423	C₂×D₁₇	1	2	yes	—
142	#405	—	17	4	no	—
143	C₂×#97	—	17	1	no	—
144	C₂×#97	—	17	1	no	—
145	C₂×#97	—	17	1	no	—
146	C₂×#97	C₂×D₁₇	1	1	yes	characteristic
147	C₂×C₄×C₈	—	17	1	no	—
148	C₂×C₄×C₈	—	17	1	no	—
149	C₄³	—	17	1	no	—
150	#339	—	17	4	no	—
151	#339	—	17	4	no	—
152	C₂×C₁₇	—	2	1	no	—
153	C₂×C₁₇	—	2	1	no	—
154	C₂×C₁₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
155	C₂×C₁₇	#2956	1	2	yes	—
156	C₂×M₁₆	C₂²×D₁₇	1	4	yes	—
157	C₂²×C₈	—	17	1	no	—
158	C₂²×C₈	—	17	1	no	—
159	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
160	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
161	C₄⋊Q₈	—	17	4	no	—
162	#125	—	17	4	no	—
163	C₂²⋊Q₈	—	34	4	no	—
164	C₄×Q₈	—	34	4	no	—
165	#117	—	17	4	no	—
166	C₂×C₄²	—	17	1	no	—
167	C₂×C₄²	—	17	1	no	—
168	C₂×C₄²	—	34	2	no	—
169	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
170	#108	—	2	4	no	—
171	#108	—	2	4	no	—
172	#107	—	17	4	no	—
173	#107	—	17	4	no	—
174	#106	—	17	4	no	—
175	#106	—	17	4	no	—
176	C₄⋊C₈	—	2	4	no	—
177	C₄⋊C₈	—	34	8	no	—
178	#97	—	17	4	no	—
179	#97	—	17	4	no	—
180	#97	—	34	4	no	—
181	#97	—	34	2	no	—
182	#97	C₄×D₁₇	1	4	yes	—
183	C₄×C₈	—	17	4	no	—
184	C₄×C₈	—	34	4	no	—
185	C₄×C₈	—	34	2	no	—
186	C₁₇	#21736	1	1	yes	characteristic, complemented, Sylow
187	C₂×Q₈	—	34	4	no	—
188	C₂²×C₄	—	34	1	no	—
189	C₂²×C₄	—	34	1	no	—
190	C₂²×C₄	#63256	1	2	yes	—
191	C₂²×C₄	C₂³×D₁₇	1	1	yes	characteristic
192	M₁₆	—	2	4	no	—
193	M₁₆	—	2	4	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	17	2	no	—
196	C₂×C₈	—	17	2	no	—
197	C₂×C₈	—	17	2	no	—
198	C₂×C₈	—	17	2	no	—
199	C₂×C₈	—	34	1	no	—
200	C₂×C₈	—	34	1	no	—
201	C₂×C₈	C₂×C₄×D₁₇	1	2	yes	—
202	C₂×C₈	C₂×C₄×D₁₇	1	2	yes	—
203	C₂×C₈	#63254	1	2	yes	—
204	C₂×C₈	#63254	1	2	yes	—
205	C₂×C₈	Q₈×D₁₇	1	2	yes	—
206	C₂×C₈	Q₈×D₁₇	1	2	yes	—
207	C₄⋊C₄	—	17	8	no	—
208	C₄⋊C₄	—	34	8	no	—
209	C₄⋊C₄	—	34	4	no	—
210	C₂²⋊C₄	—	34	4	no	—
211	C₄²	—	17	2	no	—
212	C₄²	—	17	2	no	—
213	C₄²	—	34	4	no	—
214	C₄²	—	34	1	no	—
215	C₄²	—	34	1	no	—
216	C₄²	—	68	4	no	—
217	C₄²	C₂×C₄×D₁₇	1	2	yes	—
218	C₄²	D₄×D₁₇	1	2	yes	—
219	C₂³	C₂×#63256	1	1	yes	characteristic
220	Q₈	—	136	4	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	34	4	no	—
223	C₂×C₄	—	34	2	no	—
224	C₂×C₄	—	34	8	no	—
225	C₂×C₄	—	34	2	no	—
226	C₂×C₄	—	68	2	no	—
227	C₂×C₄	#10587579	1	4	yes	—
228	C₂×C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
229	C₂×C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
230	C₂×C₄	C₂×#63254	1	1	yes	characteristic
231	C₂×C₄	C₂×#63254	1	1	yes	characteristic
232	C₂×C₄	C₂×Q₈×D₁₇	1	1	yes	characteristic
233	C₂×C₄	C₂×Q₈×D₁₇	1	1	yes	characteristic
234	C₈	—	4	4	no	—
235	C₈	—	17	4	no	—
236	C₈	—	34	2	no	—
237	C₈	#10587573	1	4	yes	—
238	C₈	#10587573	1	4	yes	—
239	C₂²	—	2	2	no	—
240	C₂²	C₂×#10587579	1	1	yes	characteristic
241	C₂²	#11763364	1	1	yes	characteristic
242	C₂²	#11763364	1	1	yes	characteristic
243	C₄	—	2	4	no	—
244	C₄	—	34	4	no	—
245	C₄	—	68	2	no	—
246	C₄	—	68	8	no	—
247	C₄	C₂×#10587573	1	1	yes	characteristic
248	C₄	C₂×#10587573	1	1	yes	characteristic
249	C₄	C₄⋊Q₈×D₁₇	1	1	yes	characteristic
250	C₄	#11763357	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	2	1	no	—
253	C₂	#14531351	1	1	yes	characteristic
254	C₂	#14537693	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₇
commutator	C₂×C₄×C₁₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₇
Fitting	C₁₇×#1849	C₂
socle	C₂²×C₁₇	C₂×C₄×D₄
prime radical	C₂³×C₁₇	C₂×#41
cosocle	C₂²×C₄×C₁₇	C₂⁴
hypercenter	#1849	D₁₇
hypocenter	C₁₇	#21736
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1849

Complements

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

N	H ≅ G/N	type
11. #11763218	C₄	semidirect
53. #10587486	Q₈	semidirect
57. C₈×Dic₁₇	Q₈	semidirect
186. C₁₇	#21736	semidirect

Maximal subgroups

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

subgroup	index
#14539308	2
#14537721	2
#14529760	2
#14529535	2
#14529530	2
#14528671	2
#14528666	2
#97×Dic₁₇	2
C₁₇×#1849	2
#21736	17