Subgroups — #16924893

Order 4352.

518 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₆₈	C₂	1	1	yes	maximal, characteristic
2	C₄×#10587494	C₂	1	1	yes	maximal, characteristic
3	#14527104	C₂	1	1	yes	maximal, characteristic
4	#14527073	C₂	1	1	yes	maximal, characteristic, complemented
5	#14526635	C₂	1	1	yes	maximal, characteristic, complemented
6	#14526633	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#1451	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₁₇	C₄	1	2	yes	—
9	C₂×C₄×D₆₈	C₂²	1	1	yes	characteristic
10	C₈×D₆₈	—	2	1	no	—
11	C₈×D₆₈	—	2	1	no	—
12	C₈×D₆₈	—	2	1	no	—
13	C₈×D₆₈	—	2	1	no	—
14	#11762862	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#3622	C₂²	1	1	yes	characteristic
16	C₂×#10587494	C₄	1	2	yes	—
17	C₂×#10587491	C₄	1	2	yes	—
18	C₄×#63223	C₂²	1	1	yes	characteristic
19	#11762577	C₄	1	2	yes	—
20	#11762532	C₄	1	4	yes	—
21	#11762531	—	2	1	no	—
22	#11762530	—	2	1	no	—
23	C₁₇×#431	C₂²	1	1	yes	characteristic
24	C₄×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
26	C₁₇×#322	—	2	1	no	—
27	C₁₇×#322	—	2	1	no	—
28	C₂²×D₆₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×D₁₇	C₂×C₄	1	2	yes	—
30	C₂×M₁₆×C₁₇	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
34	C₂×#63227	C₂×C₄	1	2	yes	—
35	C₂×C₄⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
36	C₂×#63223	C₂×C₄	1	2	yes	—
37	C₂²×#3622	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₁₇	—	2	2	no	—
39	C₂×C₈×D₁₇	—	2	2	no	—
40	C₂×C₈×D₁₇	—	4	4	no	—
41	C₄×D₆₈	—	2	1	no	—
42	C₄×D₆₈	—	2	1	no	—
43	C₄×D₆₈	—	2	1	no	—
44	C₄×D₆₈	—	2	1	no	—
45	C₁₇×C₄⋊C₈	—	2	1	no	—
46	C₁₇×C₂²⋊C₈	C₂×C₄	1	2	yes	—
47	C₁₇×#97	—	2	1	no	—
48	C₄×C₈×C₁₇	—	2	1	no	—
49	C₄×C₈×C₁₇	—	2	1	no	—
50	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
51	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
52	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
53	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
54	C₂²⋊#3622	C₂×C₄	1	2	yes	—
55	#10587494	—	2	4	no	—
56	#10587491	—	2	2	no	—
57	#10587491	—	2	2	no	—
58	C₄⋊#3622	—	2	1	no	—
59	C₄⋊#3622	—	2	1	no	—
60	C₄⋊#3622	—	2	1	no	—
61	#10587474	—	2	1	no	—
62	C₄×#3622	—	2	1	no	—
63	C₄×#3622	—	2	1	no	—
64	C₂³×D₁₇	C₄²	1	2	yes	—
65	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×Dic₁₇	C₄²	1	2	yes	—
69	C₂×D₆₈	—	2	1	no	—
70	C₂×D₆₈	—	2	1	no	—
71	C₂×D₆₈	—	2	2	no	—
72	C₂×D₆₈	M₁₆	1	4	yes	—
73	C₂×C₄×D₁₇	—	2	2	no	—
74	C₂×C₄×D₁₇	—	2	2	no	—
75	C₂×C₄×D₁₇	—	4	2	no	—
76	C₂×C₄×D₁₇	—	4	2	no	—
77	M₁₆×C₁₇	—	2	2	no	—
78	M₁₆×C₁₇	—	2	2	no	—
79	C₂×C₈×C₁₇	—	2	2	no	—
80	C₂×C₈×C₁₇	—	2	2	no	—
81	C₂×C₈×C₁₇	—	2	2	no	—
82	C₂×C₈×C₁₇	—	4	1	no	—
83	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
87	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₁₇	#41	1	1	yes	characteristic
90	C₄²×C₁₇	#41	1	1	yes	characteristic
91	#63227	—	2	2	no	—
92	#63227	—	2	2	no	—
93	C₄⋊Dic₁₇	—	2	1	no	—
94	C₄⋊Dic₁₇	—	2	1	no	—
95	#63223	—	2	2	no	—
96	#63223	—	2	2	no	—
97	C₂×#3622	—	2	2	no	—
98	C₂×#3622	—	2	2	no	—
99	C₂×#3622	—	2	2	no	—
100	C₂×#3622	—	4	2	no	—
101	C₈×D₁₇	—	4	4	no	—
102	C₈×D₁₇	—	4	4	no	—
103	#8164	—	17	1	no	maximal, Sylow
104	C₂³×C₁₇	C₂×C₄²	1	1	yes	characteristic
105	C₂²×D₁₇	—	2	2	no	—
106	C₂²×D₁₇	—	2	2	no	—
107	C₂²×D₁₇	—	4	2	no	—
108	C₂²×D₁₇	—	4	2	no	—
109	C₂²×D₁₇	#97	1	4	yes	—
110	C₂×C₄×C₁₇	—	2	1	no	—
111	C₂×C₄×C₁₇	—	2	1	no	—
112	C₂×C₄×C₁₇	—	4	1	no	—
113	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
115	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
120	C₂×C₄×C₁₇	#131	1	2	yes	—
121	C₂×Dic₁₇	—	2	2	no	—
122	C₂×Dic₁₇	—	4	2	no	—
123	D₆₈	—	2	4	no	—
124	D₆₈	—	2	4	no	—
125	C₄×D₁₇	—	4	4	no	—
126	C₄×D₁₇	—	4	2	no	—
127	C₄×D₁₇	—	4	2	no	—
128	C₈×C₁₇	—	2	1	no	—
129	C₈×C₁₇	—	2	1	no	—
130	C₈×C₁₇	—	2	1	no	—
131	C₈×C₁₇	—	4	2	no	—
132	C₈×C₁₇	—	4	2	no	—
133	C₈×C₁₇	#100	1	2	yes	complemented
134	#3622	—	4	2	no	—
135	#3622	—	4	2	no	—
136	#3622	—	4	2	no	—
137	C₂×C₈×D₄	—	17	1	no	—
138	C₄×#100	—	17	1	no	—
139	#1460	—	17	1	no	—
140	#1451	—	17	1	no	—
141	#1451	D₁₇	1	1	yes	characteristic, complemented
142	#1283	—	17	1	no	—
143	#1283	—	17	1	no	—
144	C₂²×C₁₇	—	2	1	no	—
145	C₂²×C₁₇	—	2	1	no	—
146	C₂²×C₁₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
147	C₂²×C₁₇	C₂×#97	1	1	yes	characteristic
148	C₂²×C₁₇	#405	1	1	yes	characteristic
149	C₂×D₁₇	—	2	4	no	—
150	C₂×D₁₇	—	4	4	no	—
151	C₂×D₁₇	—	4	2	no	—
152	C₂×D₁₇	—	4	2	no	—
153	C₂×D₁₇	—	4	4	no	—
154	C₄×C₁₇	—	2	1	no	—
155	C₄×C₁₇	—	2	1	no	—
156	C₄×C₁₇	—	2	1	no	—
157	C₄×C₁₇	C₂×#100	1	1	yes	characteristic
158	C₄×C₁₇	#413	1	1	yes	characteristic
159	C₄×C₁₇	#435	1	2	yes	—
160	C₄×C₁₇	#435	1	2	yes	—
161	Dic₁₇	—	4	2	no	—
162	C₂³×C₈	—	17	2	no	—
163	C₂×C₄×D₄	—	17	1	no	—
164	C₈×D₄	—	34	1	no	—
165	C₈×D₄	—	34	1	no	—
166	C₈×D₄	—	34	1	no	—
167	C₈×D₄	—	34	1	no	—
168	#431	—	17	1	no	—
169	#431	C₂×D₁₇	1	1	yes	characteristic
170	C₂×C₄⋊C₈	—	17	1	no	—
171	C₂×#100	—	17	2	no	—
172	C₂×C₂²⋊C₈	—	17	2	no	—
173	C₄×M₁₆	—	17	1	no	—
174	C₄×M₁₆	C₂×D₁₇	1	1	yes	characteristic
175	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
176	#343	—	17	2	no	—
177	#323	—	17	4	no	—
178	#322	—	2	1	no	—
179	#322	—	2	1	no	—
180	#322	—	34	1	no	—
181	#322	—	34	1	no	—
182	C₂×C₁₇	—	2	1	no	—
183	C₂×C₁₇	—	2	1	no	—
184	C₂×C₁₇	#1752	1	1	yes	characteristic
185	C₂×C₁₇	C₄×#100	1	1	yes	characteristic
186	C₂×C₁₇	#1757	1	1	yes	characteristic
187	D₁₇	—	4	4	no	—
188	C₂²×D₄	—	17	1	no	—
189	C₂³×C₄	—	17	2	no	—
190	C₂×M₁₆	—	17	2	no	—
191	C₂×M₁₆	C₄×D₁₇	1	2	yes	—
192	C₂²×C₈	—	17	2	no	—
193	C₂²×C₈	—	34	2	no	—
194	C₂²×C₈	—	34	2	no	—
195	C₂²×C₈	—	68	4	no	—
196	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
197	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
198	C₄×D₄	—	34	1	no	—
199	C₄×D₄	—	34	1	no	—
200	C₄×D₄	—	34	1	no	—
201	C₄×D₄	—	34	1	no	—
202	C₂×C₄⋊C₄	—	17	1	no	—
203	C₂×C₂²⋊C₄	—	17	2	no	—
204	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
205	C₄⋊C₈	—	2	1	no	—
206	C₄⋊C₈	—	34	1	no	—
207	C₄⋊C₈	—	34	1	no	—
208	C₄⋊C₈	—	34	1	no	—
209	#100	—	34	4	no	—
210	C₂²⋊C₈	—	17	2	no	—
211	C₂²⋊C₈	—	34	2	no	—
212	C₂²⋊C₈	—	34	2	no	—
213	C₂²⋊C₈	C₄×D₁₇	1	2	yes	—
214	#97	—	2	1	no	—
215	#97	—	34	1	no	—
216	C₄×C₈	—	2	1	no	—
217	C₄×C₈	—	2	1	no	—
218	C₄×C₈	—	34	1	no	—
219	C₄×C₈	—	34	1	no	—
220	C₄×C₈	D₆₈	1	1	yes	characteristic
221	C₄×C₈	D₆₈	1	1	yes	characteristic
222	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
223	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
224	C₁₇	#8164	1	1	yes	characteristic, complemented, Sylow
225	C₂⁴	—	17	2	no	—
226	C₂×D₄	—	17	4	no	—
227	C₂×D₄	—	34	1	no	—
228	C₂×D₄	—	34	2	no	—
229	C₂×D₄	—	34	1	no	—
230	C₂²×C₄	—	17	2	no	—
231	C₂²×C₄	—	34	2	no	—
232	C₂²×C₄	—	34	2	no	—
233	C₂²×C₄	—	68	2	no	—
234	C₂²×C₄	—	68	2	no	—
235	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
238	M₁₆	—	2	2	no	—
239	M₁₆	—	2	2	no	—
240	M₁₆	—	34	2	no	—
241	M₁₆	—	34	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	—	34	2	no	—
247	C₂×C₈	—	34	2	no	—
248	C₂×C₈	—	34	2	no	—
249	C₂×C₈	—	68	4	no	—
250	C₂×C₈	—	68	4	no	—
251	C₂×C₈	—	68	2	no	—
252	C₂×C₈	#63227	1	1	yes	characteristic
253	C₂×C₈	#63227	1	1	yes	characteristic
254	C₂×C₈	#63227	1	1	yes	characteristic
255	C₂×C₈	#63227	1	1	yes	characteristic
256	C₄⋊C₄	—	34	1	no	—
257	C₄⋊C₄	—	34	1	no	—
258	C₂²⋊C₄	—	34	2	no	—
259	C₂²⋊C₄	—	34	2	no	—
260	C₄²	C₂×D₆₈	1	1	yes	characteristic
261	C₄²	#63252	1	1	yes	characteristic
262	C₄²	#63252	1	1	yes	characteristic
263	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
264	C₂³	—	17	4	no	—
265	C₂³	—	34	2	no	—
266	C₂³	—	34	2	no	—
267	C₂³	—	68	2	no	—
268	C₂³	—	68	2	no	—
269	C₂³	C₄²×D₁₇	1	1	yes	characteristic
270	D₄	—	34	4	no	—
271	D₄	—	34	4	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	34	2	no	—
276	C₂×C₄	—	68	4	no	—
277	C₂×C₄	—	68	2	no	—
278	C₂×C₄	—	68	2	no	—
279	C₂×C₄	—	68	2	no	—
280	C₂×C₄	#10587558	1	1	yes	characteristic
281	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
282	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
283	C₂×C₄	M₁₆×D₁₇	1	2	yes	—
284	C₂×C₄	#10587593	1	2	yes	—
285	C₂×C₄	C₂×#63227	1	1	yes	characteristic
286	C₂×C₄	C₄×C₂²⋊D₁₇	1	1	yes	characteristic
287	C₂×C₄	C₄×C₂²⋊D₁₇	1	1	yes	characteristic
288	C₈	—	2	1	no	—
289	C₈	—	2	1	no	—
290	C₈	—	2	1	no	—
291	C₈	—	4	2	no	—
292	C₈	—	4	2	no	—
293	C₈	—	68	2	no	—
294	C₈	—	68	2	no	—
295	C₈	—	68	2	no	—
296	C₈	#10587494	1	2	yes	complemented
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	34	4	no	—
300	C₂²	—	68	4	no	—
301	C₂²	—	68	4	no	—
302	C₂²	—	68	2	no	—
303	C₂²	—	68	2	no	—
304	C₂²	C₄×#63227	1	1	yes	characteristic
305	C₂²	#97×D₁₇	1	1	yes	characteristic
306	C₂²	#11762766	1	1	yes	characteristic
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	68	2	no	—
311	C₄	C₂×#10587494	1	1	yes	characteristic
312	C₄	#11762791	1	1	yes	characteristic
313	C₄	#11762878	1	2	yes	—
314	C₄	#11762882	1	2	yes	—
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	68	4	no	—
318	C₂	#14528561	1	1	yes	characteristic
319	C₂	C₄×#10587494	1	1	yes	characteristic
320	C₂	#14528585	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#63227
commutator	C₂³×C₁₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1451	C₂
socle	C₂²×C₁₇	C₄×C₂²⋊C₄
prime radical	C₂²×D₆₈	C₂×C₄
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#1451	D₁₇
hypocenter	C₁₇	#8164
LCS	G ▷ C₂³×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1451

Complements

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

N	H ≅ G/N	type
4. #14527073	C₂	semidirect
5. #14526635	C₂	semidirect
6. #14526633	C₂	semidirect
7. C₁₇×#1451	C₂	semidirect
133. C₈×C₁₇	#100	semidirect
141. #1451	D₁₇	semidirect
224. C₁₇	#8164	semidirect
296. C₈	#10587494	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₆₈	2
C₄×#10587494	2
#14527104	2
#14527073	2
#14526635	2
#14526633	2
C₁₇×#1451	2
#8164	17