Subgroups — #17978712

Order 4352.

552 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14545200	C₂	1	2	yes	maximal, complemented
2	#14545165	C₂	1	4	yes	maximal, complemented
3	#14545163	C₂	1	2	yes	maximal, complemented
4	#14545147	C₂	1	2	yes	maximal, complemented
5	#14545145	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#10587538	C₂	1	1	yes	maximal, characteristic, complemented
7	#14544985	C₂	1	2	yes	maximal, complemented
8	#14537740	C₂	1	1	yes	maximal, characteristic
9	C₂×#10587538	C₂²	1	2	yes	—
10	C₂×#10587529	C₂²	1	4	yes	—
11	#11763730	C₂²	1	4	yes	—
12	C₄⋊#63216	C₂²	1	2	yes	—
13	#11763704	C₂²	1	4	yes	—
14	C₂²⋊#63216	C₂²	1	2	yes	—
15	#11763698	C₂²	1	4	yes	—
16	#11763697	C₂²	1	1	yes	characteristic
17	#11763697	C₂²	1	2	yes	—
18	#11763697	C₂²	1	1	yes	characteristic
19	C₄×#63216	C₂²	1	2	yes	—
20	C₄×#63252	C₄	1	1	yes	characteristic
21	#11762933	C₄	1	1	yes	characteristic
22	#11762933	C₂²	1	1	yes	characteristic
23	#11762874	C₄	1	1	yes	characteristic
24	#11762874	C₂²	1	1	yes	characteristic
25	#11762872	C₄	1	2	yes	—
26	#11762872	C₂²	1	2	yes	—
27	#11762860	C₄	1	2	yes	—
28	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
29	C₄×C₈×D₁₇	C₂²	1	2	yes	—
30	C₁₇×#431	C₄	1	1	yes	characteristic
31	C₂×#63252	C₈	1	1	yes	characteristic
32	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
33	C₄×C₂²⋊D₁₇	C₈	1	2	yes	—
34	C₄×C₂²⋊D₁₇	C₂×C₄	1	2	yes	—
35	#10587612	C₈	1	1	yes	characteristic
36	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
37	C₂²×#3622	C₂³	1	1	yes	characteristic
38	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
39	C₂×C₈×D₁₇	C₂³	1	2	yes	—
40	C₄×D₆₈	C₈	1	1	yes	characteristic
41	C₄×D₆₈	C₂×C₄	1	1	yes	characteristic
42	#10587558	C₈	1	2	yes	—
43	C₄²×D₁₇	C₂×C₄	1	2	yes	—
44	C₄×Dic₃₄	C₈	1	1	yes	characteristic
45	C₄×Dic₃₄	C₂×C₄	1	1	yes	characteristic
46	#10587538	—	2	4	no	—
47	C₂×#63216	C₂³	1	4	yes	—
48	C₂×#63216	C₂³	1	2	yes	—
49	C₂×#63216	C₂³	1	2	yes	—
50	#10587529	—	2	8	no	—
51	C₁₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
52	C₁₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₁₇×C₂²⋊C₈	C₂×C₄	1	2	yes	—
54	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
55	#10587491	C₂×C₄	1	2	yes	—
56	#10587491	C₂×C₄	1	2	yes	—
57	#10587485	C₂×C₄	1	2	yes	—
58	#10587485	C₂×C₄	1	2	yes	—
59	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
60	C₈×Dic₁₇	C₂³	1	2	yes	—
61	C₄×#3622	C₂×C₄	1	1	yes	characteristic
62	C₄×#3622	C₂×C₄	1	1	yes	characteristic
63	C₄×#3622	C₂³	1	1	yes	characteristic
64	C₄×#3622	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
66	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₂²⋊D₁₇	C₂×C₈	1	2	yes	—
69	#63252	—	2	4	no	—
70	C₂×D₆₈	C₂×C₈	1	1	yes	characteristic
71	C₂×C₄×D₁₇	C₂×C₈	1	2	yes	—
72	C₂×C₄×D₁₇	C₂×C₈	1	2	yes	—
73	C₂×C₄×D₁₇	C₂²×C₄	1	2	yes	—
74	C₂×Dic₃₄	C₂×C₈	1	1	yes	characteristic
75	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
77	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
78	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
79	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
81	C₂²⋊Dic₁₇	C₂×C₈	1	2	yes	—
82	#63227	C₂×C₈	1	2	yes	—
83	#63227	C₂×C₈	1	2	yes	—
84	C₄⋊Dic₁₇	C₂×C₈	1	1	yes	characteristic
85	C₄⋊Dic₁₇	C₂×C₈	1	1	yes	characteristic
86	#63225	C₂×C₈	1	2	yes	—
87	#63225	C₂×C₈	1	2	yes	—
88	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
89	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
90	C₄×Dic₁₇	C₂²×C₄	1	2	yes	—
91	C₂×#3622	—	2	2	no	—
92	C₂×#3622	—	2	2	no	—
93	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
94	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
95	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
96	C₂×#3622	C₂⁴	1	1	yes	characteristic
97	C₈×D₁₇	—	2	4	no	—
98	C₈×D₁₇	—	2	4	no	—
99	#63216	—	2	4	no	—
100	#63216	—	2	2	no	—
101	#63216	#41	1	4	yes	complemented
102	#33484	—	17	1	no	maximal, Sylow
103	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
104	C₂²×D₁₇	C₂²×C₈	1	2	yes	—
105	C₂×C₄×C₁₇	—	2	1	no	—
106	C₂×C₄×C₁₇	—	2	1	no	—
107	C₂×C₄×C₁₇	—	2	2	no	—
108	C₂×C₄×C₁₇	—	4	1	no	—
109	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
115	C₂²⋊D₁₇	—	2	4	no	—
116	C₂×Dic₁₇	C₂²×C₈	1	2	yes	—
117	C₂×Dic₁₇	C₂²×C₈	1	2	yes	—
118	C₂×Dic₁₇	C₂²×C₈	1	2	yes	—
119	D₆₈	M₃₂	1	4	yes	complemented
120	C₄×D₁₇	—	2	4	no	—
121	C₄×D₁₇	—	2	2	no	—
122	C₄×D₁₇	—	2	2	no	—
123	C₄×D₁₇	—	4	2	no	—
124	Dic₃₄	—	2	2	no	—
125	C₈×C₁₇	—	2	2	no	—
126	C₈×C₁₇	—	2	2	no	—
127	#3622	—	2	1	no	—
128	#3622	—	2	1	no	—
129	#3622	#131	1	2	yes	—
130	#3622	C₂×#41	1	2	yes	—
131	C₈×#41	—	17	1	no	—
132	#2183	—	17	2	no	—
133	#2168	—	17	4	no	—
134	D₄×C₁₆	—	17	2	no	—
135	#2163	—	17	1	no	—
136	#2163	—	17	2	no	—
137	C₄×M₃₂	—	17	1	no	—
138	C₂×#346	—	17	2	no	—
139	C₂²×C₁₇	—	2	1	no	—
140	C₂²×C₁₇	—	2	1	no	—
141	C₂²×C₁₇	—	2	1	no	—
142	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
143	C₂×D₁₇	—	2	2	no	—
144	C₂×D₁₇	—	2	2	no	—
145	C₂×D₁₇	C₂×M₃₂	1	4	yes	—
146	C₄×C₁₇	—	2	1	no	—
147	C₄×C₁₇	—	2	1	no	—
148	C₄×C₁₇	—	2	1	no	—
149	C₄×C₁₇	—	2	1	no	—
150	C₄×C₁₇	C₂×M₃₂	1	2	yes	—
151	C₄×C₁₇	C₄×#41	1	1	yes	characteristic
152	C₄×C₁₇	C₂×#131	1	1	yes	characteristic
153	Dic₁₇	—	2	2	no	—
154	Dic₁₇	—	2	2	no	—
155	Dic₁₇	—	2	2	no	—
156	C₄×#41	—	17	1	no	—
157	C₂×M₃₂	—	17	2	no	—
158	C₂²×C₁₆	—	17	4	no	—
159	C₈×Q₈	—	17	1	no	—
160	C₈×Q₈	—	17	1	no	—
161	C₈×D₄	—	17	1	no	—
162	C₈×D₄	—	17	1	no	—
163	C₈×D₄	—	17	2	no	—
164	C₈×D₄	—	17	2	no	—
165	#431	—	17	2	no	—
166	#431	#595	1	1	yes	characteristic
167	C₂×C₄×C₈	—	17	1	no	—
168	C₂×C₄×C₈	—	17	2	no	—
169	C₄⋊C₁₆	—	17	2	no	—
170	C₄⋊C₁₆	—	17	4	no	—
171	C₂²⋊C₁₆	—	17	4	no	—
172	C₂²⋊C₁₆	—	17	2	no	—
173	#346	—	17	1	no	—
174	#346	—	17	2	no	—
175	#346	—	17	4	no	—
176	#346	—	17	1	no	—
177	C₄×C₁₆	—	17	2	no	—
178	C₂×C₁₇	—	4	1	no	—
179	C₂×C₁₇	C₈×#41	1	1	yes	characteristic
180	C₂×C₁₇	C₂²×M₃₂	1	1	yes	characteristic
181	C₂×C₁₇	#3407	1	1	yes	characteristic
182	D₁₇	—	2	4	no	—
183	C₂×#41	—	17	1	no	—
184	C₂²×C₈	—	17	1	no	—
185	C₂²×C₈	—	17	1	no	—
186	C₂²×C₈	—	17	2	no	—
187	C₂²×C₈	—	17	2	no	—
188	C₄×Q₈	—	17	1	no	—
189	C₄×Q₈	—	17	1	no	—
190	C₄×D₄	—	17	2	no	—
191	C₄×D₄	—	17	2	no	—
192	C₄×D₄	—	17	1	no	—
193	C₄×D₄	—	17	1	no	—
194	#117	—	17	1	no	—
195	#117	—	17	2	no	—
196	C₂×C₄²	—	17	2	no	—
197	C₂×C₄²	C₂×#595	1	1	yes	characteristic
198	M₃₂	—	34	4	no	—
199	C₂×C₁₆	—	17	2	no	—
200	C₂×C₁₆	—	17	2	no	—
201	C₂×C₁₆	—	17	4	no	—
202	C₂×C₁₆	—	34	8	no	—
203	C₄⋊C₈	—	17	2	no	—
204	C₄⋊C₈	—	17	2	no	—
205	C₄⋊C₈	C₂×#595	1	1	yes	characteristic
206	C₄⋊C₈	C₂×#595	1	1	yes	characteristic
207	C₂²⋊C₈	—	17	2	no	—
208	C₂²⋊C₈	—	17	2	no	—
209	C₂²⋊C₈	C₂×#595	1	2	yes	—
210	C₄×C₈	—	17	2	no	—
211	C₄×C₈	—	17	2	no	—
212	C₄×C₈	—	17	1	no	—
213	C₄×C₈	—	17	1	no	—
214	C₄×C₈	—	17	1	no	—
215	C₄×C₈	—	17	1	no	—
216	C₄×C₈	C₂×#595	1	2	yes	—
217	C₁₇	#33484	1	1	yes	characteristic, complemented, Sylow
218	#41	—	34	4	no	—
219	C₂×Q₈	—	17	1	no	—
220	C₂×D₄	—	17	2	no	—
221	C₂×D₄	—	17	1	no	—
222	C₂²×C₄	—	17	2	no	—
223	C₂²×C₄	—	17	2	no	—
224	C₂²×C₄	—	17	2	no	—
225	C₂²×C₄	#63242	1	1	yes	characteristic
226	C₂²×C₄	#63242	1	1	yes	characteristic
227	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
228	C₂×C₈	—	17	1	no	—
229	C₂×C₈	—	17	1	no	—
230	C₂×C₈	—	17	1	no	—
231	C₂×C₈	—	17	1	no	—
232	C₂×C₈	—	34	4	no	—
233	C₂×C₈	—	34	4	no	—
234	C₂×C₈	—	34	2	no	—
235	C₂×C₈	—	34	2	no	—
236	C₂×C₈	C₂²×#595	1	2	yes	—
237	C₂×C₈	C₂²×#595	1	2	yes	—
238	C₄⋊C₄	—	17	2	no	—
239	C₄⋊C₄	—	17	2	no	—
240	C₄⋊C₄	—	17	1	no	—
241	C₄⋊C₄	—	17	1	no	—
242	C₂²⋊C₄	—	17	2	no	—
243	C₂²⋊C₄	—	17	2	no	—
244	C₂²⋊C₄	—	17	2	no	—
245	C₄²	—	17	2	no	—
246	C₄²	—	17	2	no	—
247	C₄²	—	17	2	no	—
248	C₄²	#63242	1	1	yes	characteristic
249	C₄²	#63242	1	1	yes	characteristic
250	C₄²	C₂²×#595	1	1	yes	characteristic
251	C₄²	C₂²×#595	1	1	yes	characteristic
252	C₁₆	—	17	4	no	—
253	C₁₆	—	34	4	no	—
254	C₁₆	—	34	2	no	—
255	C₂³	—	17	2	no	—
256	C₂³	C₂×#63242	1	1	yes	characteristic
257	Q₈	—	34	2	no	—
258	D₄	—	17	4	no	—
259	D₄	—	34	4	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	17	2	no	—
265	C₂×C₄	—	17	2	no	—
266	C₂×C₄	—	17	2	no	—
267	C₂×C₄	—	34	4	no	—
268	C₂×C₄	—	34	2	no	—
269	C₂×C₄	—	34	2	no	—
270	C₂×C₄	—	68	2	no	—
271	C₂×C₄	C₂×#63242	1	1	yes	characteristic
272	C₂×C₄	C₂×#63242	1	1	yes	characteristic
273	C₂×C₄	C₂×#63242	1	1	yes	characteristic
274	C₂×C₄	C₂×#63242	1	1	yes	characteristic
275	C₂×C₄	C₂×#63242	1	1	yes	characteristic
276	C₂×C₄	C₂³×#595	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	2	2	no	—
279	C₈	—	17	2	no	—
280	C₈	—	17	2	no	—
281	C₈	—	34	1	no	—
282	C₈	—	34	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	17	4	no	—
287	C₂²	—	34	2	no	—
288	C₂²	—	34	2	no	—
289	C₂²	C₂²×#63242	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	34	2	no	—
295	C₄	—	34	2	no	—
296	C₄	—	34	2	no	—
297	C₄	#11763865	1	2	yes	—
298	C₄	#41×#595	1	1	yes	characteristic
299	C₄	D₁₇⋊#131	1	1	yes	characteristic
300	C₂	—	4	1	no	—
301	C₂	—	34	4	no	—
302	C₂	#14545588	1	1	yes	characteristic
303	C₂	C₂×#11763865	1	1	yes	characteristic
304	C₂	#14545753	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#595
commutator, socle	C₂²×C₁₇	C₂³×C₈
Fitting	C₁₇×#431	C₄
prime radical	C₂×#63252	C₈
cosocle	C₂×#3622	C₂⁴
hypercenter	#431	#595
hypocenter	C₁₇	#33484
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
1. #14545200	C₂	semidirect
2. #14545165	C₂	semidirect
3. #14545163	C₂	semidirect
4. #14545147	C₂	semidirect
5. #14545145	C₂	semidirect
6. C₄×#10587538	C₂	semidirect
7. #14544985	C₂	semidirect
101. #63216	#41	semidirect
119. D₆₈	M₃₂	semidirect
217. C₁₇	#33484	semidirect

Maximal subgroups

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

subgroup	index
#14545200	2
#14545165	2
#14545163	2
#14545147	2
#14545145	2
C₄×#10587538	2
#14544985	2
#14537740	2
#33484	17