Subgroups — #16933701

Order 4352.

502 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14539212	C₂	1	1	yes	maximal, characteristic
2	#14527343	C₂	1	4	yes	maximal, complemented
3	#14527067	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₇×#1846	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#10587590	—	2	1	no	—
6	C₂×#10587590	—	2	1	no	—
7	#11763359	C₂²	1	2	yes	—
8	#11763193	—	2	1	no	—
9	#11763193	—	2	1	no	—
10	#11763189	C₄	1	1	yes	characteristic, complemented
11	#11763186	C₄	1	1	yes	characteristic, complemented
12	#11763182	C₄	1	2	yes	complemented
13	#11762862	C₂²	1	2	yes	—
14	#11762542	—	2	8	no	—
15	#11762530	—	2	1	no	—
16	C₈×#3622	—	2	1	no	—
17	C₂×C₁₇×#107	—	2	1	no	—
18	C₂×C₁₇×#106	—	2	1	no	—
19	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
20	C₁₇×#382	C₂²	1	2	yes	—
21	C₂×#63252	—	2	2	no	—
22	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
25	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
26	#10587616	—	2	4	no	—
27	C₂×Dic₆₈	—	2	1	no	—
28	C₂×Dic₆₈	—	2	1	no	—
29	#10587590	—	4	2	no	—
30	#10587590	—	4	2	no	—
31	#10587590	—	4	2	no	—
32	#10587590	—	4	2	no	—
33	C₂×D₁₃₆	—	2	1	no	—
34	C₂×D₁₃₆	—	2	1	no	—
35	C₂×#63219	—	2	2	no	—
36	C₂×#63219	—	2	2	no	—
37	#10587561	—	2	2	no	—
38	#10587560	C₂×C₄	1	2	yes	—
39	C₄×D₆₈	—	2	4	no	—
40	#10587555	C₂×C₄	1	2	yes	—
41	C₁₇×#107	—	4	2	no	—
42	C₁₇×#106	—	4	2	no	—
43	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
46	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
47	C₁₇×#95	—	2	4	no	—
48	C₂²⋊#3622	D₄	1	4	yes	—
49	C₄⋊#3622	—	2	2	no	—
50	C₄×#3622	—	2	2	no	—
51	C₂²×C₄×C₁₇	—	4	2	no	—
52	C₂²×C₄×C₁₇	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₂²⋊D₁₇	—	4	2	no	—
55	#63252	—	4	2	no	—
56	#63252	—	4	2	no	—
57	#63252	—	4	2	no	—
58	#63252	—	4	2	no	—
59	C₂×D₆₈	—	2	2	no	—
60	C₂×D₆₈	—	2	4	no	—
61	C₂×C₄×D₁₇	—	4	2	no	—
62	C₂×Dic₃₄	—	2	2	no	—
63	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₇	D₈	1	2	yes	—
68	C₂×C₈×C₁₇	D₈	1	2	yes	—
69	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
70	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
71	C₄⋊C₄×C₁₇	—	4	2	no	—
72	C₄⋊C₄×C₁₇	—	4	2	no	—
73	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
74	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₁₇	#41	1	2	yes	—
76	#63227	—	4	4	no	—
77	C₄⋊Dic₁₇	—	2	4	no	—
78	C₂×#3622	—	2	4	no	—
79	Dic₆₈	—	4	2	no	—
80	Dic₆₈	—	4	2	no	—
81	D₁₃₆	—	4	2	no	—
82	D₁₃₆	—	4	2	no	—
83	#63219	—	4	4	no	—
84	#63219	—	4	4	no	—
85	#9756	—	17	1	no	maximal, Sylow
86	C₂³×C₁₇	#127	1	1	yes	characteristic
87	C₂²×D₁₇	—	4	2	no	—
88	C₂×C₄×C₁₇	—	4	2	no	—
89	C₂×C₄×C₁₇	—	8	4	no	—
90	C₂×C₄×C₁₇	—	8	2	no	—
91	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
93	C₂×C₄×C₁₇	#121	1	2	yes	—
94	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
95	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
98	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
99	C₂²⋊D₁₇	—	8	2	no	—
100	C₂²⋊D₁₇	—	8	2	no	—
101	C₂×Dic₁₇	—	4	2	no	—
102	D₆₈	—	4	2	no	—
103	D₆₈	—	4	2	no	—
104	D₆₈	—	8	4	no	—
105	C₄×D₁₇	—	8	2	no	—
106	C₄×D₁₇	—	8	2	no	—
107	Dic₃₄	—	4	2	no	—
108	Dic₃₄	—	4	2	no	—
109	C₈×C₁₇	#102	1	2	yes	—
110	C₈×C₁₇	#102	1	2	yes	—
111	C₈×C₁₇	#102	1	2	yes	—
112	C₈×C₁₇	#102	1	2	yes	—
113	#3622	—	4	4	no	—
114	#3147	—	17	1	no	—
115	#1846	D₁₇	1	1	yes	characteristic, complemented
116	#1517	—	17	4	no	—
117	#1450	—	17	1	no	—
118	C₂²×C₁₇	—	2	2	no	—
119	C₂²×C₁₇	#390	1	1	yes	characteristic
120	C₂²×C₁₇	#492	1	1	yes	characteristic
121	C₂²×C₁₇	#493	1	1	yes	characteristic
122	C₂×D₁₇	—	8	2	no	—
123	C₂×D₁₇	—	8	4	no	—
124	C₄×C₁₇	—	2	2	no	—
125	C₄×C₁₇	—	2	2	no	—
126	C₄×C₁₇	—	16	2	no	—
127	C₄×C₁₇	C₂×#102	1	1	yes	characteristic
128	C₄×C₁₇	C₂×#102	1	1	yes	characteristic
129	C₄×C₁₇	#486	1	1	yes	characteristic
130	C₄×C₁₇	#486	1	1	yes	characteristic
131	Dic₁₇	—	8	2	no	—
132	C₂×#135	—	34	1	no	—
133	C₂×#135	—	34	1	no	—
134	#532	—	17	2	no	—
135	#495	—	34	1	no	—
136	#495	—	34	1	no	—
137	#494	—	17	1	no	—
138	#493	—	17	1	no	—
139	#492	—	17	2	no	—
140	#431	—	17	2	no	—
141	C₂×#107	—	2	1	no	—
142	C₂×#106	—	2	1	no	—
143	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
144	#382	C₂×D₁₇	1	2	yes	—
145	#327	—	34	8	no	—
146	#322	—	34	1	no	—
147	C₈²	—	34	1	no	—
148	C₂×C₁₇	—	2	1	no	—
149	C₂×C₁₇	—	2	1	no	—
150	C₂×C₁₇	#1957	1	1	yes	characteristic
151	C₂×C₁₇	#1976	1	2	yes	—
152	D₁₇	—	16	2	no	—
153	C₂×#41	—	34	2	no	—
154	#135	—	68	2	no	—
155	#135	—	68	2	no	—
156	#135	—	68	2	no	—
157	#135	—	68	2	no	—
158	C₂×Q₁₆	—	34	1	no	—
159	C₂×Q₁₆	—	34	1	no	—
160	C₂×SD₁₆	—	34	2	no	—
161	C₂×SD₁₆	—	34	2	no	—
162	C₂×D₈	—	34	1	no	—
163	C₂×D₈	—	34	1	no	—
164	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
165	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
166	C₄⋊Q₈	—	17	2	no	—
167	#127	—	17	2	no	—
168	#124	—	34	2	no	—
169	#121	—	34	4	no	—
170	C₄×D₄	—	34	4	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
173	#107	—	4	2	no	—
174	#106	—	4	2	no	—
175	C₄⋊C₈	—	34	2	no	—
176	C₂²⋊C₈	—	17	4	no	—
177	C₄×C₈	—	34	2	no	—
178	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
179	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
180	C₄×C₈	D₆₈	1	1	yes	characteristic
181	C₄×C₈	D₆₈	1	1	yes	characteristic
182	#95	—	2	4	no	—
183	C₁₇	#9756	1	1	yes	characteristic, complemented, Sylow
184	#41	—	68	2	no	—
185	#41	—	68	2	no	—
186	#41	—	68	2	no	—
187	#41	—	68	2	no	—
188	C₂×Q₈	—	34	2	no	—
189	C₂×D₄	—	34	4	no	—
190	C₂×D₄	—	34	2	no	—
191	C₂×D₄	—	68	2	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	—	68	2	no	—
194	C₂²×C₄	D₄×D₁₇	1	2	yes	—
195	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
196	Q₁₆	—	68	2	no	—
197	Q₁₆	—	68	2	no	—
198	SD₁₆	—	68	4	no	—
199	SD₁₆	—	68	4	no	—
200	D₈	—	68	2	no	—
201	D₈	—	68	2	no	—
202	C₂×C₈	—	34	4	no	—
203	C₂×C₈	#63227	1	1	yes	characteristic
204	C₂×C₈	#63227	1	1	yes	characteristic
205	C₂×C₈	#63227	1	1	yes	characteristic
206	C₂×C₈	#63227	1	1	yes	characteristic
207	C₂×C₈	#63228	1	2	yes	—
208	C₂×C₈	#63228	1	2	yes	—
209	C₂×C₈	#63230	1	2	yes	—
210	C₂×C₈	#63230	1	2	yes	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄⋊C₄	—	4	2	no	—
213	C₄⋊C₄	—	34	4	no	—
214	C₂²⋊C₄	—	68	4	no	—
215	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
216	C₄²	C₂×D₆₈	1	1	yes	characteristic
217	C₄²	#63256	1	2	yes	—
218	C₂³	—	68	2	no	—
219	C₂³	#10587626	1	1	yes	characteristic
220	Q₈	—	68	2	no	—
221	Q₈	—	68	2	no	—
222	D₄	—	68	2	no	—
223	D₄	—	68	2	no	—
224	D₄	—	136	4	no	—
225	D₄	—	136	2	no	—
226	D₄	—	136	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	8	4	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	—	68	2	no	—
231	C₂×C₄	—	136	2	no	—
232	C₂×C₄	—	136	2	no	—
233	C₂×C₄	#10587579	1	2	yes	—
234	C₂×C₄	#10587581	1	2	yes	—
235	C₂×C₄	C₂×#63227	1	1	yes	characteristic
236	C₂×C₄	C₂×#63228	1	1	yes	characteristic
237	C₂×C₄	C₂×#63228	1	1	yes	characteristic
238	C₂×C₄	C₂×#63230	1	1	yes	characteristic
239	C₂×C₄	C₂×#63230	1	1	yes	characteristic
240	C₂×C₄	#10587633	1	1	yes	characteristic
241	C₈	—	68	4	no	—
242	C₈	#10587480	1	2	yes	—
243	C₈	#10587480	1	2	yes	—
244	C₈	#10587480	1	2	yes	—
245	C₈	#10587480	1	2	yes	—
246	C₂²	—	2	2	no	—
247	C₂²	—	136	2	no	—
248	C₂²	—	136	4	no	—
249	C₂²	#11762710	1	1	yes	characteristic
250	C₂²	#11763183	1	1	yes	characteristic
251	C₂²	#11763185	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	16	2	no	—
255	C₄	—	136	2	no	—
256	C₄	C₂×#10587480	1	1	yes	characteristic
257	C₄	C₂×#10587480	1	1	yes	characteristic
258	C₄	#11763157	1	1	yes	characteristic
259	C₄	#11763157	1	1	yes	characteristic
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	272	2	no	—
263	C₂	#14529589	1	1	yes	characteristic
264	C₂	#14529680	1	2	yes	—

	subgroup	G/H
center	C₂²	#11762710
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1846	C₂
socle	C₂²×C₁₇	#390
prime radical	#14539212	C₂
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#1846	D₁₇
hypocenter	C₁₇	#9756
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1846

Complements

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

N	H ≅ G/N	type
2. #14527343	C₂	semidirect
3. #14527067	C₂	semidirect
4. C₁₇×#1846	C₂	semidirect
10. #11763189	C₄	semidirect
11. #11763186	C₄	semidirect
12. #11763182	C₄	semidirect
115. #1846	D₁₇	semidirect
183. C₁₇	#9756	semidirect

Maximal subgroups

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

subgroup	index
#14539212	2
#14527343	2
#14527067	2
C₁₇×#1846	2
#9756	17