Subgroups — #17052799

Order 4352.

862 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14537668	C₂	1	2	yes	maximal
2	#14537379	C₂	1	2	yes	maximal, complemented
3	#14529565	C₂	1	2	yes	maximal, complemented
4	#14529012	C₂	1	4	yes	maximal, complemented
5	#14528941	C₂	1	1	yes	maximal, characteristic, complemented
6	#14528843	C₂	1	2	yes	maximal, complemented
7	#14528663	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₇×#443	C₂	1	1	yes	maximal, characteristic
9	#11763292	C₂²	1	4	yes	—
10	#11762921	—	2	2	no	—
11	#11762916	—	2	2	no	—
12	SD₁₆×Dic₁₇	—	2	4	no	—
13	C₂×#10587490	C₂²	1	1	yes	characteristic
14	#11762851	—	2	4	no	—
15	#11762851	C₂²	1	4	yes	—
16	#11762847	—	2	2	no	—
17	#11762847	C₂²	1	1	yes	characteristic
18	C₂×#10587488	—	2	2	no	—
19	C₂×#10587488	C₂²	1	2	yes	—
20	C₂×#10587488	C₂²	1	2	yes	—
21	#11762810	C₂²	1	4	yes	—
22	#11762764	C₂²	1	2	yes	—
23	#11762679	C₂²	1	2	yes	—
24	#11762577	C₂²	1	4	yes	—
25	#11762569	C₂²	1	2	yes	—
26	C₂×C₁₇×#135	C₂²	1	1	yes	characteristic
27	C₂×C₁₇×#131	C₂²	1	2	yes	—
28	C₁₇×#443	C₄	1	8	yes	complemented
29	C₂×C₁₇×#108	C₂²	1	1	yes	characteristic
30	C₂×C₁₇×#104	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
32	C₂×#41×C₁₇	C₂³	1	2	yes	—
33	C₁₇×#135	C₂×C₄	1	4	yes	—
34	C₁₇×#135	Q₈	1	4	yes	—
35	C₂×Q₁₆×C₁₇	D₄	1	1	yes	characteristic
36	C₂×SD₁₆×C₁₇	D₄	1	2	yes	—
37	C₂×D₈×C₁₇	D₄	1	1	yes	characteristic
38	C₁₇×#131	—	2	4	no	—
39	C₁₇×#131	C₂×C₄	1	8	yes	complemented
40	C₂×M₁₆×C₁₇	—	2	2	no	—
41	C₂×M₁₆×C₁₇	C₂³	1	2	yes	—
42	C₂²×C₈×C₁₇	—	2	2	no	—
43	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
46	Q₈×Dic₁₇	—	2	4	no	—
47	D₄×Dic₁₇	—	2	4	no	—
48	D₄×Dic₁₇	—	4	4	no	—
49	#10587612	—	2	4	no	—
50	#10587612	C₂³	1	2	yes	—
51	C₂×C₄⋊Dic₁₇	—	2	4	no	—
52	C₂×C₄⋊Dic₁₇	—	4	2	no	—
53	C₂×C₄⋊Dic₁₇	C₂³	1	4	yes	—
54	C₂×#63223	C₂³	1	2	yes	—
55	C₁₇×#108	C₂×C₄	1	4	yes	complemented
56	C₁₇×#104	C₂×C₄	1	8	yes	complemented
57	C₄×C₈×C₁₇	C₂×C₄	1	4	yes	complemented
58	#10587506	—	2	4	no	—
59	#10587503	—	2	4	no	—
60	#10587490	—	2	2	no	—
61	#10587489	—	2	4	no	—
62	#10587489	D₄	1	2	yes	—
63	#10587488	—	2	4	no	—
64	#10587488	—	2	2	no	—
65	#10587488	—	2	2	no	—
66	#10587488	—	2	2	no	—
67	#10587488	—	2	2	no	—
68	#10587488	D₄	1	2	yes	—
69	#10587486	—	2	2	no	—
70	C₈×Dic₁₇	—	2	2	no	—
71	#41×C₁₇	—	2	2	no	—
72	#41×C₁₇	—	2	2	no	—
73	#41×C₁₇	C₂²×C₄	1	4	yes	—
74	#41×C₁₇	C₂×Q₈	1	4	yes	—
75	C₂×Q₈×C₁₇	C₂×D₄	1	2	yes	—
76	C₂×D₄×C₁₇	—	2	2	no	—
77	C₂×D₄×C₁₇	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₁₇	—	2	1	no	—
79	C₂²×C₄×C₁₇	—	2	2	no	—
80	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
81	C₂²×Dic₁₇	—	2	4	no	—
82	C₂²×Dic₁₇	—	4	4	no	—
83	Q₁₆×C₁₇	C₄⋊C₄	1	4	yes	complemented
84	SD₁₆×C₁₇	C₄⋊C₄	1	8	yes	complemented
85	D₈×C₁₇	C₄⋊C₄	1	4	yes	complemented
86	M₁₆×C₁₇	—	2	2	no	—
87	M₁₆×C₁₇	—	2	8	no	—
88	M₁₆×C₁₇	C₂²×C₄	1	4	yes	—
89	C₂×C₈×C₁₇	—	2	2	no	—
90	C₂×C₈×C₁₇	—	2	8	no	—
91	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
98	C₂×C₈×C₁₇	#41	1	2	yes	—
99	C₄²×C₁₇	—	2	1	no	—
100	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
101	C₂²⋊Dic₁₇	—	2	2	no	—
102	C₂²⋊Dic₁₇	—	4	4	no	—
103	C₄⋊Dic₁₇	—	2	4	no	—
104	C₄⋊Dic₁₇	—	2	4	no	—
105	C₄⋊Dic₁₇	—	4	4	no	—
106	C₄⋊Dic₁₇	—	8	4	no	—
107	C₄⋊Dic₁₇	C₂×D₄	1	4	yes	—
108	C₄×Dic₁₇	—	2	2	no	—
109	C₄×Dic₁₇	—	4	4	no	—
110	#63223	—	2	2	no	—
111	#63223	—	2	2	no	—
112	C₂×#3622	—	2	2	no	—
113	#22689	—	17	1	no	maximal, Sylow
114	C₂³×C₁₇	—	2	2	no	—
115	C₂³×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
116	Q₈×C₁₇	—	2	2	no	—
117	Q₈×C₁₇	C₂×C₄⋊C₄	1	4	yes	—
118	D₄×C₁₇	—	2	4	no	—
119	D₄×C₁₇	—	2	4	no	—
120	D₄×C₁₇	—	2	2	no	—
121	D₄×C₁₇	C₂×C₄⋊C₄	1	4	yes	—
122	C₂×C₄×C₁₇	—	2	4	no	—
123	C₂×C₄×C₁₇	—	2	2	no	—
124	C₂×C₄×C₁₇	—	2	2	no	—
125	C₂×C₄×C₁₇	—	2	4	no	—
126	C₂×C₄×C₁₇	—	2	4	no	—
127	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
131	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
132	C₂×C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
133	C₂×Dic₁₇	—	2	4	no	—
134	C₂×Dic₁₇	—	4	4	no	—
135	C₂×Dic₁₇	—	8	4	no	—
136	C₂×Dic₁₇	—	16	4	no	—
137	C₈×C₁₇	—	2	4	no	—
138	C₈×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
139	C₈×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
140	C₈×C₁₇	C₄×D₄	1	2	yes	—
141	C₈×C₁₇	C₄×D₄	1	2	yes	—
142	#3622	—	4	2	no	—
143	C₂×#443	D₁₇	1	1	yes	characteristic
144	#2952	—	17	2	no	—
145	#2912	—	17	2	no	—
146	#1951	—	17	2	no	—
147	#1859	—	17	4	no	—
148	#1844	—	17	1	no	—
149	#1815	—	17	2	no	—
150	#1772	—	17	1	no	—
151	C₂²×C₁₇	—	2	2	no	—
152	C₂²×C₁₇	—	2	4	no	—
153	C₂²×C₁₇	—	4	4	no	—
154	C₂²×C₁₇	#520	1	1	yes	characteristic
155	C₂²×C₁₇	D₄²	1	1	yes	characteristic
156	C₂²×C₁₇	#554	1	1	yes	characteristic
157	C₄×C₁₇	—	2	4	no	—
158	C₄×C₁₇	—	2	4	no	—
159	C₄×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
160	C₄×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
161	C₄×C₁₇	#547	1	1	yes	characteristic
162	C₄×C₁₇	D₄×Q₈	1	1	yes	characteristic
163	Dic₁₇	—	4	4	no	—
164	Dic₁₇	—	8	4	no	—
165	C₂×#135	C₂×D₁₇	1	1	yes	characteristic
166	C₂×#131	C₂×D₁₇	1	2	yes	—
167	#520	—	17	4	no	—
168	#443	Dic₁₇	1	8	yes	complemented
169	#442	—	34	2	no	—
170	#441	—	34	2	no	—
171	C₄×SD₁₆	—	34	4	no	—
172	C₂×#108	—	17	1	no	—
173	C₂×#108	C₂×D₁₇	1	1	yes	characteristic
174	#428	—	17	4	no	—
175	#428	—	34	4	no	—
176	#427	—	17	1	no	—
177	#427	—	34	2	no	—
178	C₂×#106	—	17	2	no	—
179	C₂×#106	—	17	2	no	—
180	C₂×#106	—	34	2	no	—
181	C₂×#104	C₂×D₁₇	1	2	yes	—
182	#417	—	17	4	no	—
183	#405	—	17	2	no	—
184	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
185	#384	—	17	2	no	—
186	#343	—	17	4	no	—
187	#339	—	17	2	no	—
188	C₂×C₁₇	—	2	1	no	—
189	C₂×C₁₇	—	2	1	no	—
190	C₂×C₁₇	—	4	4	no	—
191	C₂×C₁₇	D₄×C₄⋊C₄	1	1	yes	characteristic
192	C₂×C₁₇	#3291	1	2	yes	—
193	C₂×#41	C₂²×D₁₇	1	2	yes	—
194	#135	Dic₃₄	1	4	yes	—
195	#135	C₂×Dic₁₇	1	4	yes	—
196	C₂×Q₁₆	D₆₈	1	1	yes	characteristic
197	C₂×SD₁₆	D₆₈	1	2	yes	—
198	C₂×D₈	D₆₈	1	1	yes	characteristic
199	#131	—	2	4	no	—
200	#131	C₂×Dic₁₇	1	8	yes	complemented
201	C₂×M₁₆	—	2	2	no	—
202	C₂×M₁₆	—	17	2	no	—
203	C₂×M₁₆	C₂²×D₁₇	1	2	yes	—
204	C₂²×C₈	—	2	2	no	—
205	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
206	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
207	C₄×Q₈	—	34	4	no	—
208	C₄×D₄	—	34	4	no	—
209	C₄×D₄	—	68	4	no	—
210	#117	—	17	2	no	—
211	#117	—	34	4	no	—
212	C₂×C₄⋊C₄	—	17	4	no	—
213	C₂×C₄⋊C₄	—	34	4	no	—
214	C₂×C₄⋊C₄	—	68	2	no	—
215	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
216	#108	—	34	2	no	—
217	#108	C₂×Dic₁₇	1	4	yes	complemented
218	#107	—	17	2	no	—
219	#107	—	34	4	no	—
220	#106	—	17	2	no	—
221	#106	—	34	4	no	—
222	#106	—	34	2	no	—
223	#106	—	34	2	no	—
224	#106	—	34	2	no	—
225	#106	—	34	2	no	—
226	#104	C₂×Dic₁₇	1	8	yes	complemented
227	#103	—	34	4	no	—
228	#102	—	34	4	no	—
229	#97	—	34	2	no	—
230	C₄×C₈	—	34	2	no	—
231	C₄×C₈	C₂×Dic₁₇	1	4	yes	complemented
232	C₁₇	#22689	1	1	yes	characteristic, complemented, Sylow
233	#41	—	2	2	no	—
234	#41	—	2	2	no	—
235	#41	C₂×Dic₃₄	1	4	yes	—
236	#41	C₂²×Dic₁₇	1	4	yes	—
237	C₂×Q₈	C₂×D₆₈	1	2	yes	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	C₂×D₆₈	1	2	yes	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	34	4	no	—
243	C₂²×C₄	—	68	4	no	—
244	C₂²×C₄	C₂³×D₁₇	1	1	yes	characteristic
245	Q₁₆	C₄⋊Dic₁₇	1	4	yes	complemented
246	SD₁₆	C₄⋊Dic₁₇	1	8	yes	complemented
247	D₈	C₄⋊Dic₁₇	1	4	yes	complemented
248	M₁₆	—	2	2	no	—
249	M₁₆	—	2	8	no	—
250	M₁₆	—	34	2	no	—
251	M₁₆	—	34	2	no	—
252	M₁₆	C₂²×Dic₁₇	1	4	yes	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	8	no	—
255	C₂×C₈	—	34	2	no	—
256	C₂×C₈	C₂×Dic₃₄	1	2	yes	—
257	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
258	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
259	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
260	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
261	C₂×C₈	#63254	1	2	yes	—
262	C₂×C₈	C₂²×Dic₁₇	1	2	yes	—
263	C₂×C₈	C₂²×Dic₁₇	1	2	yes	—
264	C₄⋊C₄	—	17	4	no	—
265	C₄⋊C₄	—	34	4	no	—
266	C₄⋊C₄	—	34	4	no	—
267	C₄⋊C₄	—	68	4	no	—
268	C₄⋊C₄	—	136	4	no	—
269	C₂²⋊C₄	—	34	2	no	—
270	C₂²⋊C₄	—	68	4	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	34	2	no	—
273	C₄²	—	68	4	no	—
274	C₄²	C₂²×Dic₁₇	1	2	yes	—
275	C₂³	—	2	2	no	—
276	C₂³	#10587696	1	1	yes	characteristic
277	Q₈	—	2	2	no	—
278	Q₈	C₂×C₄⋊Dic₁₇	1	4	yes	—
279	D₄	—	2	4	no	—
280	D₄	—	2	4	no	—
281	D₄	—	2	2	no	—
282	D₄	C₂×C₄⋊Dic₁₇	1	4	yes	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	272	4	no	—
289	C₂×C₄	—	34	4	no	—
290	C₂×C₄	—	68	4	no	—
291	C₂×C₄	—	136	4	no	—
292	C₂×C₄	C₂²×Dic₃₄	1	1	yes	characteristic
293	C₂×C₄	C₂²×D₆₈	1	1	yes	characteristic
294	C₂×C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
295	C₂×C₄	C₂×#63254	1	1	yes	characteristic
296	C₂×C₄	#10587697	1	1	yes	characteristic
297	C₂×C₄	C₂³×Dic₁₇	1	1	yes	characteristic
298	C₈	—	2	4	no	—
299	C₈	—	68	2	no	—
300	C₈	C₂×C₄⋊Dic₁₇	1	2	yes	—
301	C₈	C₂×C₄⋊Dic₁₇	1	2	yes	—
302	C₈	D₄×Dic₁₇	1	2	yes	—
303	C₈	D₄×Dic₁₇	1	2	yes	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	4	4	no	—
307	C₂²	#11763292	1	1	yes	characteristic
308	C₂²	D₄×D₆₈	1	1	yes	characteristic
309	C₂²	#11763516	1	1	yes	characteristic
310	C₄	—	2	4	no	—
311	C₄	—	2	4	no	—
312	C₄	—	68	4	no	—
313	C₄	—	136	4	no	—
314	C₄	C₂²×C₄⋊Dic₁₇	1	1	yes	characteristic
315	C₄	C₂×D₄×Dic₁₇	1	1	yes	characteristic
316	C₄	#11763463	1	1	yes	characteristic
317	C₄	D₄×Dic₃₄	1	1	yes	characteristic
318	C₂	—	2	1	no	—
319	C₂	—	2	1	no	—
320	C₂	—	4	4	no	—
321	C₂	D₄×C₄⋊Dic₁₇	1	1	yes	characteristic
322	C₂	#14540578	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₆₈
commutator	C₂×C₄×C₁₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₇
Fitting	C₂×C₁₇×#443	C₂
socle	C₂²×C₁₇	D₄²
prime radical	C₂×C₁₇×#135	C₂²
cosocle	C₂²×C₄×C₁₇	C₂⁴
hypercenter	C₂×#443	D₁₇
hypocenter	C₁₇	#22689
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
2. #14537379	C₂	semidirect
3. #14529565	C₂	semidirect
4. #14529012	C₂	semidirect
5. #14528941	C₂	semidirect
6. #14528843	C₂	semidirect
7. #14528663	C₂	semidirect
28. C₁₇×#443	C₄	semidirect
39. C₁₇×#131	C₂×C₄	semidirect
55. C₁₇×#108	C₂×C₄	semidirect
56. C₁₇×#104	C₂×C₄	semidirect
57. C₄×C₈×C₁₇	C₂×C₄	semidirect
83. Q₁₆×C₁₇	C₄⋊C₄	semidirect
84. SD₁₆×C₁₇	C₄⋊C₄	semidirect
85. D₈×C₁₇	C₄⋊C₄	semidirect
168. #443	Dic₁₇	semidirect
200. #131	C₂×Dic₁₇	semidirect
217. #108	C₂×Dic₁₇	semidirect
226. #104	C₂×Dic₁₇	semidirect
231. C₄×C₈	C₂×Dic₁₇	semidirect
232. C₁₇	#22689	semidirect
245. Q₁₆	C₄⋊Dic₁₇	semidirect
246. SD₁₆	C₄⋊Dic₁₇	semidirect
247. D₈	C₄⋊Dic₁₇	semidirect

Maximal subgroups

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

subgroup	index
#14537668	2
#14537379	2
#14529565	2
#14529012	2
#14528941	2
#14528843	2
#14528663	2
C₂×C₁₇×#443	2
#22689	17