Subgroups — #14537777

Order 2176.

372 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763346	C₂	1	1	yes	maximal, characteristic
2	#11762940	C₂	1	1	yes	maximal, characteristic, complemented
3	#11762882	C₂	1	1	yes	maximal, characteristic, complemented
4	#11762882	C₂	1	1	yes	maximal, characteristic, complemented
5	#11762879	C₂	1	1	yes	maximal, characteristic, complemented
6	#11762879	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×D₆₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#11762874	C₂	1	1	yes	maximal, characteristic, complemented
9	#11762869	C₂	1	1	yes	maximal, characteristic, complemented
10	#11762869	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762861	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762766	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762763	C₂	1	1	yes	maximal, characteristic, complemented
14	#97×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#446	C₂	1	1	yes	maximal, characteristic, complemented
16	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
17	#10587633	C₄	1	1	yes	characteristic
18	#10587632	C₄	1	1	yes	characteristic
19	C₂×#63218	C₂²	1	1	yes	characteristic
20	C₂×#63218	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
23	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
24	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
25	#10587584	C₄	1	1	yes	characteristic
26	#10587584	C₄	1	1	yes	characteristic
27	#10587581	C₄	1	1	yes	characteristic
28	#10587580	C₄	1	1	yes	characteristic
29	#10587580	C₄	1	1	yes	characteristic
30	#10587576	C₄	1	1	yes	characteristic
31	C₄×D₆₈	C₂²	1	1	yes	characteristic
32	C₄×D₆₈	C₂²	1	1	yes	characteristic
33	C₄×D₆₈	C₂²	1	1	yes	characteristic
34	#10587558	C₂²	1	1	yes	characteristic
35	#10587558	C₂²	1	1	yes	characteristic
36	C₄²×D₁₇	C₂²	1	1	yes	characteristic
37	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
38	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
39	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
40	C₁₇×#97	C₂²	1	1	yes	characteristic
41	C₁₇×#97	C₂²	1	1	yes	characteristic
42	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
43	#10587491	C₂²	1	1	yes	characteristic
44	#10587491	C₂²	1	1	yes	characteristic
45	#10587491	C₂²	1	1	yes	characteristic
46	#10587491	C₂²	1	1	yes	characteristic
47	#10587491	C₂²	1	1	yes	characteristic
48	#10587491	C₂²	1	1	yes	characteristic
49	#10587486	C₂²	1	1	yes	characteristic
50	#10587486	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
52	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
53	C₄⋊#3622	C₂²	1	1	yes	characteristic
54	C₄⋊#3622	C₂²	1	1	yes	characteristic
55	C₄⋊#3622	C₂²	1	1	yes	characteristic
56	#10587474	C₂²	1	1	yes	characteristic
57	#10587474	C₂²	1	1	yes	characteristic
58	C₄×#3622	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
60	C₂×D₆₈	C₂×C₄	1	1	yes	characteristic
61	C₂×D₆₈	C₂×C₄	1	1	yes	characteristic
62	C₂×D₆₈	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
66	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
67	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
73	C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
74	C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
75	C₄²×C₁₇	C₂³	1	1	yes	characteristic
76	C₄²×C₁₇	C₂³	1	1	yes	characteristic
77	C₄²×C₁₇	C₂³	1	1	yes	characteristic
78	#63227	C₂×C₄	1	1	yes	characteristic
79	#63227	C₂×C₄	1	1	yes	characteristic
80	#63227	C₂×C₄	1	1	yes	characteristic
81	#63227	C₂×C₄	1	1	yes	characteristic
82	#63227	C₂×C₄	1	1	yes	characteristic
83	#63227	C₂×C₄	1	1	yes	characteristic
84	#63227	C₂×C₄	1	1	yes	characteristic
85	#63227	C₂×C₄	1	1	yes	characteristic
86	#63227	C₂×C₄	1	1	yes	characteristic
87	#63227	C₂×C₄	1	1	yes	characteristic
88	C₄⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
89	C₄⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
90	C₄⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
91	#63225	C₂×C₄	1	1	yes	characteristic
92	#63225	C₂×C₄	1	1	yes	characteristic
93	#63225	C₂×C₄	1	1	yes	characteristic
94	#63225	C₂×C₄	1	1	yes	characteristic
95	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
96	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
97	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
98	C₂×#3622	C₂³	1	1	yes	characteristic
99	C₂×#3622	C₂³	1	1	yes	characteristic
100	C₂×#3622	C₂³	1	1	yes	characteristic
101	C₂×#3622	C₂³	1	1	yes	characteristic
102	#63218	—	2	2	no	—
103	#63218	—	2	2	no	—
104	C₈×D₁₇	—	2	2	no	—
105	C₈×D₁₇	—	2	2	no	—
106	C₈×D₁₇	—	2	2	no	—
107	C₈×D₁₇	—	2	2	no	—
108	C₂²×D₁₇	C₂²×C₄	1	1	yes	characteristic
109	C₂²×D₁₇	C₂²×C₄	1	1	yes	characteristic
110	C₂²×D₁₇	C₂²×C₄	1	1	yes	characteristic
111	Q₈×C₁₇	—	4	1	no	—
112	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
119	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
120	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
121	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
122	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
123	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
124	D₆₈	—	2	2	no	—
125	D₆₈	—	2	2	no	—
126	D₆₈	—	4	1	no	—
127	C₄×D₁₇	—	2	2	no	—
128	C₄×D₁₇	—	2	1	no	—
129	C₄×D₁₇	—	2	1	no	—
130	C₄×D₁₇	—	2	1	no	—
131	C₄×D₁₇	—	2	1	no	—
132	C₄×D₁₇	—	2	1	no	—
133	C₄×D₁₇	—	2	1	no	—
134	C₄×D₁₇	—	4	1	no	—
135	C₈×C₁₇	—	2	1	no	—
136	C₈×C₁₇	—	2	1	no	—
137	C₈×C₁₇	—	2	1	no	—
138	C₈×C₁₇	#41	1	2	yes	—
139	#3622	—	2	1	no	—
140	#3622	—	2	1	no	—
141	#3622	—	2	1	no	—
142	#3622	#41	1	2	yes	—
143	#2968	—	17	1	no	maximal, Sylow
144	C₂²×C₁₇	C₂³×C₄	1	1	yes	characteristic
145	C₂×D₁₇	—	2	1	no	—
146	C₂×D₁₇	—	2	1	no	—
147	C₂×D₁₇	—	2	1	no	—
148	C₂×D₁₇	—	2	1	no	—
149	C₂×D₁₇	—	2	1	no	—
150	C₂×D₁₇	—	2	1	no	—
151	C₂×D₁₇	—	2	1	no	—
152	C₂×D₁₇	—	2	1	no	—
153	C₂×D₁₇	#131	1	2	yes	—
154	C₄×C₁₇	—	2	1	no	—
155	C₄×C₁₇	—	2	1	no	—
156	C₄×C₁₇	—	2	1	no	—
157	C₄×C₁₇	—	2	1	no	—
158	C₄×C₁₇	—	2	1	no	—
159	C₄×C₁₇	—	2	1	no	—
160	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
161	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
162	Dic₁₇	—	2	1	no	—
163	Dic₁₇	—	2	1	no	—
164	Dic₁₇	—	2	1	no	—
165	Dic₁₇	—	2	1	no	—
166	Dic₁₇	—	2	1	no	—
167	#529	—	17	1	no	—
168	#446	—	17	1	no	—
169	#446	D₁₇	1	1	yes	characteristic, complemented
170	#435	—	17	1	no	—
171	#435	—	17	1	no	—
172	#435	—	17	1	no	—
173	#435	—	17	1	no	—
174	C₈×D₄	—	17	1	no	—
175	C₈×D₄	—	17	1	no	—
176	#433	—	17	1	no	—
177	#433	—	17	1	no	—
178	#431	—	17	1	no	—
179	#405	—	17	1	no	—
180	#405	—	17	1	no	—
181	C₂×#97	—	17	1	no	—
182	C₂×C₁₇	C₄×#41	1	1	yes	characteristic
183	C₂×C₁₇	C₂×#131	1	1	yes	characteristic
184	C₂×C₁₇	#568	1	1	yes	characteristic
185	D₁₇	—	2	2	no	—
186	D₁₇	—	4	1	no	—
187	D₁₇	—	4	1	no	—
188	C₂×M₁₆	—	17	1	no	—
189	C₂×M₁₆	—	17	1	no	—
190	C₂²×C₈	—	17	1	no	—
191	C₂²×C₈	—	17	1	no	—
192	C₂²×C₈	—	17	1	no	—
193	C₂²×C₈	—	17	1	no	—
194	#126	—	17	1	no	—
195	#126	—	17	1	no	—
196	#125	—	17	1	no	—
197	#124	—	17	1	no	—
198	#123	—	17	1	no	—
199	#123	—	17	1	no	—
200	C₂²⋊Q₈	—	17	1	no	—
201	#121	—	17	1	no	—
202	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
203	C₄×D₄	—	17	1	no	—
204	C₄×D₄	—	17	1	no	—
205	C₄×D₄	—	17	1	no	—
206	#117	—	17	1	no	—
207	#117	—	17	1	no	—
208	C₂×C₄²	—	17	1	no	—
209	C₄⋊C₈	—	17	1	no	—
210	C₄⋊C₈	—	17	1	no	—
211	C₄⋊C₈	—	17	1	no	—
212	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
213	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
214	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
215	C₂²⋊C₈	—	17	1	no	—
216	C₂²⋊C₈	—	17	1	no	—
217	C₂²⋊C₈	—	17	1	no	—
218	C₂²⋊C₈	—	17	1	no	—
219	C₂²⋊C₈	—	17	1	no	—
220	C₂²⋊C₈	—	17	1	no	—
221	#97	—	17	1	no	—
222	#97	—	17	1	no	—
223	#97	—	17	1	no	—
224	#97	—	17	1	no	—
225	#97	C₂×D₁₇	1	1	yes	characteristic
226	#97	C₂×D₁₇	1	1	yes	characteristic
227	C₄×C₈	—	17	1	no	—
228	C₄×C₈	—	17	1	no	—
229	C₄×C₈	—	17	1	no	—
230	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
231	C₁₇	#2968	1	1	yes	characteristic, complemented, Sylow
232	C₂×Q₈	C₄×D₁₇	1	1	yes	characteristic
233	C₂×D₄	—	17	1	no	—
234	C₂×D₄	—	17	1	no	—
235	C₂×D₄	—	17	1	no	—
236	C₂²×C₄	—	17	1	no	—
237	C₂²×C₄	—	17	1	no	—
238	C₂²×C₄	—	17	1	no	—
239	C₂²×C₄	—	17	1	no	—
240	C₂²×C₄	—	17	1	no	—
241	M₁₆	—	34	2	no	—
242	M₁₆	—	34	2	no	—
243	C₂×C₈	—	17	1	no	—
244	C₂×C₈	—	17	1	no	—
245	C₂×C₈	—	17	1	no	—
246	C₂×C₈	—	17	1	no	—
247	C₂×C₈	—	34	2	no	—
248	C₂×C₈	—	34	2	no	—
249	C₂×C₈	—	34	2	no	—
250	C₂×C₈	—	34	2	no	—
251	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
252	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
253	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
254	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
255	C₄⋊C₄	—	17	1	no	—
256	C₄⋊C₄	—	17	1	no	—
257	C₄⋊C₄	—	17	1	no	—
258	C₄⋊C₄	—	17	1	no	—
259	C₄⋊C₄	—	17	1	no	—
260	C₄⋊C₄	—	17	1	no	—
261	C₄⋊C₄	—	17	1	no	—
262	C₄⋊C₄	C₄×D₁₇	1	1	yes	characteristic
263	C₄⋊C₄	C₄×D₁₇	1	1	yes	characteristic
264	C₄⋊C₄	C₄×D₁₇	1	1	yes	characteristic
265	C₂²⋊C₄	—	17	1	no	—
266	C₂²⋊C₄	—	17	1	no	—
267	C₂²⋊C₄	—	17	1	no	—
268	C₂²⋊C₄	—	17	1	no	—
269	C₂²⋊C₄	—	17	1	no	—
270	C₂²⋊C₄	—	17	1	no	—
271	C₂²⋊C₄	—	17	1	no	—
272	C₂²⋊C₄	—	17	1	no	—
273	C₂²⋊C₄	—	17	1	no	—
274	C₂²⋊C₄	—	17	1	no	—
275	C₄²	—	17	1	no	—
276	C₄²	—	17	1	no	—
277	C₄²	—	17	1	no	—
278	C₄²	C₂²×D₁₇	1	1	yes	characteristic
279	C₄²	C₂²×D₁₇	1	1	yes	characteristic
280	C₄²	C₂²×D₁₇	1	1	yes	characteristic
281	C₂³	—	17	1	no	—
282	C₂³	—	17	1	no	—
283	C₂³	—	17	1	no	—
284	Q₈	—	4	1	no	—
285	D₄	—	34	2	no	—
286	D₄	—	34	2	no	—
287	D₄	—	68	1	no	—
288	C₂×C₄	—	17	1	no	—
289	C₂×C₄	—	17	1	no	—
290	C₂×C₄	—	17	1	no	—
291	C₂×C₄	—	17	1	no	—
292	C₂×C₄	—	17	1	no	—
293	C₂×C₄	—	34	2	no	—
294	C₂×C₄	—	34	1	no	—
295	C₂×C₄	—	34	1	no	—
296	C₂×C₄	—	34	1	no	—
297	C₂×C₄	—	34	1	no	—
298	C₂×C₄	—	34	1	no	—
299	C₂×C₄	—	34	1	no	—
300	C₂×C₄	—	68	1	no	—
301	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
302	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
303	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
304	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
305	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
306	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	17	2	no	—
312	C₈	—	34	1	no	—
313	C₈	—	34	1	no	—
314	C₈	—	34	1	no	—
315	C₈	#63256	1	2	yes	—
316	C₂²	—	17	2	no	—
317	C₂²	—	34	1	no	—
318	C₂²	—	34	1	no	—
319	C₂²	—	34	1	no	—
320	C₂²	—	34	1	no	—
321	C₂²	—	34	1	no	—
322	C₂²	—	34	1	no	—
323	C₂²	—	34	1	no	—
324	C₂²	—	34	1	no	—
325	C₂²	C₂²×C₄×D₁₇	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	34	1	no	—
333	C₄	—	34	1	no	—
334	C₄	—	34	1	no	—
335	C₄	—	34	1	no	—
336	C₄	—	34	1	no	—
337	C₄	C₂×#63256	1	1	yes	characteristic
338	C₄	#41×D₁₇	1	1	yes	characteristic
339	C₂	—	34	2	no	—
340	C₂	—	68	1	no	—
341	C₂	—	68	1	no	—
342	C₂	C₄×#63256	1	1	yes	characteristic
343	C₂	#131×D₁₇	1	1	yes	characteristic
344	C₂	#11763575	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₇
commutator, socle	C₂²×C₁₇	C₂³×C₄
Fitting	C₁₇×#446	C₂
prime radical	#10587581	C₄
cosocle	C₂×C₄×C₁₇	C₂⁴
hypercenter	#446	D₁₇
hypocenter	C₁₇	#2968
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #11762940	C₂	semidirect
3. #11762882	C₂	semidirect
4. #11762882	C₂	semidirect
5. #11762879	C₂	semidirect
6. #11762879	C₂	semidirect
7. C₈×D₆₈	C₂	semidirect
8. #11762874	C₂	semidirect
9. #11762869	C₂	semidirect
10. #11762869	C₂	semidirect
11. #11762861	C₂	semidirect
12. #11762766	C₂	semidirect
13. #11762763	C₂	semidirect
14. #97×D₁₇	C₂	semidirect
15. C₁₇×#446	C₂	semidirect
169. #446	D₁₇	semidirect
231. C₁₇	#2968	semidirect

Maximal subgroups

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

subgroup	index
#11763346	2
#11762940	2
#11762882	2
#11762879	2
C₈×D₆₈	2
#11762874	2
#11762869	2
#11762861	2
#11762766	2
#11762763	2
#97×D₁₇	2
C₁₇×#446	2
#2968	17