Subgroups — #17005186

Order 4352.

544 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14539370	C₂	1	1	yes	maximal, characteristic
2	#14537878	C₂	1	2	yes	maximal
3	#14527657	C₂	1	4	yes	maximal, complemented
4	#14527636	C₂	1	4	yes	maximal, complemented
5	#14527547	C₂	1	1	yes	maximal, characteristic
6	#14527258	C₂	1	2	yes	maximal
7	C₁₇×#1452	C₂	1	1	yes	maximal, characteristic
8	#11763366	C₂²	1	2	yes	—
9	#11763214	C₄	1	1	yes	characteristic
10	#11763189	C₄	1	1	yes	characteristic
11	#11763160	C₄	1	2	yes	—
12	#11763050	C₄	1	2	yes	—
13	#11763050	C₄	1	2	yes	—
14	#11762939	C₂²	1	4	yes	—
15	#11762938	—	2	4	no	—
16	C₈×Dic₃₄	C₂²	1	4	yes	—
17	C₄×Dic₆₈	C₂²	1	2	yes	—
18	C₄×Dic₆₈	C₂²	1	2	yes	—
19	#11762867	—	2	4	no	—
20	#11762866	C₂²	1	2	yes	—
21	#11762562	C₂²	1	2	yes	—
22	#11762560	C₂²	1	2	yes	—
23	#11762541	C₂²	1	8	yes	—
24	C₁₇×#431	C₂²	1	1	yes	characteristic
25	C₁₇×#431	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
27	C₁₇×#322	C₂²	1	2	yes	—
28	C₁₇×#322	C₂²	1	2	yes	—
29	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
32	#10587612	—	2	2	no	—
33	#10587610	—	2	2	no	—
34	#10587610	C₂×C₄	1	4	yes	—
35	C₂×Dic₆₈	C₂×C₄	1	2	yes	—
36	C₂×Dic₆₈	C₂×C₄	1	2	yes	—
37	#10587556	—	2	2	no	—
38	#10587555	C₂×C₄	1	2	yes	—
39	#10587555	C₂×C₄	1	2	yes	—
40	C₄×Dic₃₄	—	2	2	no	—
41	C₄×Dic₃₄	C₂³	1	4	yes	—
42	C₁₇×C₄⋊C₈	—	2	1	no	—
43	C₁₇×C₄⋊C₈	D₄	1	2	yes	—
44	C₁₇×C₂²⋊C₈	—	2	1	no	—
45	C₁₇×C₂²⋊C₈	D₄	1	2	yes	—
46	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₁₇	C₂³	1	2	yes	—
49	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
50	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
51	C₄×C₈×C₁₇	C₂³	1	2	yes	—
52	C₂²⋊#3622	—	2	2	no	—
53	#10587489	C₂×C₄	1	2	yes	—
54	#10587489	C₂×C₄	1	2	yes	—
55	#10587487	C₂×C₄	1	4	yes	—
56	#10587487	C₂×C₄	1	4	yes	—
57	#10587486	—	2	4	no	—
58	#10587485	—	2	4	no	—
59	#10587485	—	2	4	no	—
60	C₈×Dic₁₇	—	2	4	no	—
61	C₄⋊#3622	C₂³	1	4	yes	—
62	#10587474	—	2	2	no	—
63	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
66	C₂×Dic₃₄	—	2	2	no	—
67	C₂×Dic₃₄	C₂²×C₄	1	4	yes	—
68	C₂×C₈×C₁₇	—	2	2	no	—
69	C₂×C₈×C₁₇	—	4	1	no	—
70	C₂×C₈×C₁₇	—	4	1	no	—
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	1	yes	characteristic
75	C₂×C₈×C₁₇	C₂×D₄	1	4	yes	—
76	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₁₇	#41	1	1	yes	characteristic
79	C₄²×C₁₇	C₂⁴	1	1	yes	characteristic
80	C₂²⋊Dic₁₇	—	2	4	no	—
81	C₄⋊Dic₁₇	—	2	2	no	—
82	C₄⋊Dic₁₇	—	2	2	no	—
83	C₄⋊Dic₁₇	—	2	2	no	—
84	C₄⋊Dic₁₇	C₂²×C₄	1	4	yes	—
85	#63225	—	2	4	no	—
86	#63225	—	2	4	no	—
87	C₄×Dic₁₇	—	2	4	no	—
88	C₂×#3622	—	2	4	no	—
89	Dic₆₈	—	4	2	no	—
90	Dic₆₈	—	4	2	no	—
91	#17991	—	17	1	no	maximal, Sylow
92	C₂³×C₁₇	C₄×D₄	1	1	yes	characteristic
93	C₂×C₄×C₁₇	—	2	1	no	—
94	C₂×C₄×C₁₇	—	2	1	no	—
95	C₂×C₄×C₁₇	—	2	2	no	—
96	C₂×C₄×C₁₇	—	4	1	no	—
97	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
103	C₂×Dic₁₇	—	2	4	no	—
104	C₂×Dic₁₇	—	2	4	no	—
105	Dic₃₄	—	2	4	no	—
106	Dic₃₄	—	2	4	no	—
107	Dic₃₄	—	8	2	no	—
108	C₈×C₁₇	—	2	1	no	—
109	C₈×C₁₇	—	2	1	no	—
110	C₈×C₁₇	—	2	1	no	—
111	C₈×C₁₇	—	2	1	no	—
112	C₈×C₁₇	—	2	4	no	—
113	C₈×C₁₇	—	4	2	no	—
114	#3622	—	4	4	no	—
115	#3170	—	17	1	no	—
116	#2985	—	17	2	no	—
117	#1587	—	17	4	no	—
118	#1584	—	17	4	no	—
119	#1568	—	17	1	no	—
120	#1497	—	17	2	no	—
121	#1452	D₁₇	1	1	yes	characteristic
122	C₂²×C₁₇	—	2	1	no	—
123	C₂²×C₁₇	—	2	1	no	—
124	C₂²×C₁₇	—	2	1	no	—
125	C₂²×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
126	C₄×C₁₇	—	2	1	no	—
127	C₄×C₁₇	—	2	1	no	—
128	C₄×C₁₇	—	2	1	no	—
129	C₄×C₁₇	—	2	1	no	—
130	C₄×C₁₇	#568	1	1	yes	characteristic
131	C₄×C₁₇	#568	1	1	yes	characteristic
132	C₄×C₁₇	#578	1	2	yes	—
133	Dic₁₇	—	4	4	no	—
134	Dic₁₇	—	4	4	no	—
135	#533	—	17	2	no	—
136	#500	—	17	1	no	—
137	#494	—	17	1	no	—
138	#487	—	17	2	no	—
139	#467	—	17	2	no	—
140	#467	—	17	2	no	—
141	#446	—	17	4	no	—
142	#446	—	34	4	no	—
143	C₈×Q₈	—	17	4	no	—
144	C₄×Q₁₆	—	17	2	no	—
145	C₄×Q₁₆	—	17	2	no	—
146	#433	—	34	4	no	—
147	#432	—	17	2	no	—
148	#431	C₂×D₁₇	1	1	yes	characteristic
149	#431	C₂×D₁₇	1	1	yes	characteristic
150	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
151	#335	—	17	2	no	—
152	#334	—	17	2	no	—
153	Q₈⋊C₈	—	17	8	no	—
154	#322	C₂×D₁₇	1	2	yes	—
155	#322	C₂×D₁₇	1	2	yes	—
156	C₂×C₁₇	—	4	1	no	—
157	C₂×C₁₇	#2932	1	1	yes	characteristic
158	C₂×C₁₇	#2950	1	1	yes	characteristic
159	C₂×C₁₇	#2957	1	1	yes	characteristic
160	C₂×Q₁₆	—	17	2	no	—
161	C₂×Q₁₆	—	17	2	no	—
162	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
163	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
164	C₄⋊Q₈	—	17	2	no	—
165	C₄⋊Q₈	—	17	2	no	—
166	#125	—	34	2	no	—
167	C₂²⋊Q₈	—	17	4	no	—
168	C₂²⋊Q₈	—	34	2	no	—
169	C₄×Q₈	—	17	4	no	—
170	C₄×Q₈	—	34	2	no	—
171	#117	—	34	2	no	—
172	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
173	#107	—	17	2	no	—
174	#107	—	17	2	no	—
175	C₄⋊C₈	—	2	1	no	—
176	C₄⋊C₈	—	17	4	no	—
177	C₄⋊C₈	—	34	4	no	—
178	C₄⋊C₈	—	34	4	no	—
179	C₄⋊C₈	D₆₈	1	2	yes	—
180	#103	—	17	4	no	—
181	#103	—	17	4	no	—
182	C₂²⋊C₈	—	2	1	no	—
183	C₂²⋊C₈	—	34	2	no	—
184	C₂²⋊C₈	D₆₈	1	2	yes	—
185	#97	—	34	4	no	—
186	#97	—	34	2	no	—
187	C₄×C₈	—	34	4	no	—
188	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
189	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
190	C₄×C₈	C₂²×D₁₇	1	2	yes	—
191	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
192	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
193	C₄×C₈	C₂²×D₁₇	1	2	yes	—
194	C₁₇	#17991	1	1	yes	characteristic, complemented, Sylow
195	C₂×Q₈	—	17	4	no	—
196	C₂×Q₈	—	34	2	no	—
197	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
198	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
199	C₂²×C₄	#63252	1	1	yes	characteristic
200	Q₁₆	—	68	2	no	—
201	Q₁₆	—	68	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	4	1	no	—
204	C₂×C₈	—	4	1	no	—
205	C₂×C₈	—	34	4	no	—
206	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
207	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
208	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
209	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
210	C₂×C₈	C₂×D₆₈	1	4	yes	—
211	C₄⋊C₄	—	17	4	no	—
212	C₄⋊C₄	—	34	4	no	—
213	C₄⋊C₄	—	34	2	no	—
214	C₄⋊C₄	—	34	2	no	—
215	C₄⋊C₄	—	34	4	no	—
216	C₄⋊C₄	—	34	2	no	—
217	C₂²⋊C₄	—	34	4	no	—
218	C₄²	—	34	4	no	—
219	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
220	C₄²	C₂×D₆₈	1	1	yes	characteristic
221	C₄²	#63252	1	1	yes	characteristic
222	C₄²	C₂³×D₁₇	1	1	yes	characteristic
223	C₂³	C₄×D₆₈	1	1	yes	characteristic
224	Q₈	—	34	4	no	—
225	Q₈	—	34	4	no	—
226	Q₈	—	136	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	34	4	no	—
232	C₂×C₄	—	34	4	no	—
233	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
234	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
235	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
236	C₂×C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
237	C₂×C₄	C₂²×D₆₈	1	1	yes	characteristic
238	C₂×C₄	C₂×#63252	1	1	yes	characteristic
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	2	1	no	—
243	C₈	—	2	4	no	—
244	C₈	—	4	2	no	—
245	C₈	—	68	4	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	C₂×C₄×D₆₈	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	68	4	no	—
255	C₄	—	68	4	no	—
256	C₄	#11763575	1	1	yes	characteristic
257	C₄	#11763575	1	1	yes	characteristic
258	C₄	#11763626	1	2	yes	—
259	C₂	—	4	1	no	—
260	C₂	#14537503	1	1	yes	characteristic
261	C₂	#14537653	1	1	yes	characteristic
262	C₂	#14537702	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₆₈
commutator	C₂×C₄×C₁₇	C₂³×C₄
Frattini	C₄²	C₂³×D₁₇
Fitting	C₁₇×#1452	C₂
socle	C₂²×C₁₇	C₂×C₄×D₄
prime radical	C₂³×C₁₇	C₄×D₄
cosocle	C₄²×C₁₇	C₂⁴
hypercenter	#1452	D₁₇
hypocenter	C₁₇	#17991
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1452

Complements

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

N	H ≅ G/N	type
3. #14527657	C₂	semidirect
4. #14527636	C₂	semidirect
194. C₁₇	#17991	semidirect

Maximal subgroups

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

subgroup	index
#14539370	2
#14537878	2
#14527657	2
#14527636	2
#14527547	2
#14527258	2
C₁₇×#1452	2
#17991	17