Subgroups — #14539865

Order 2176.

364 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763419	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763380	C₂	1	1	yes	maximal, characteristic
3	#11763200	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763187	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763176	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763162	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763030	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763029	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763018	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763017	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762971	C₂	1	1	yes	maximal, characteristic
12	#11762967	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762964	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762867	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#452	C₂	1	1	yes	maximal, characteristic
16	C₂×#41×C₁₇	C₂²	1	1	yes	characteristic
17	C₂²×Dic₃₄	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₁₇	C₂²	1	1	yes	characteristic
19	C₂×SD₁₆×C₁₇	C₂²	1	1	yes	characteristic
20	C₁₇×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
21	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
22	#10587630	C₂²	1	1	yes	characteristic
23	#10587630	C₂²	1	1	yes	characteristic
24	#10587630	C₂²	1	1	yes	characteristic
25	C₂×#63231	C₂²	1	1	yes	characteristic
26	C₂×#63231	C₂²	1	1	yes	characteristic
27	C₂×#63231	C₂²	1	1	yes	characteristic
28	#10587622	C₂²	1	1	yes	characteristic
29	#10587622	C₂²	1	1	yes	characteristic
30	#10587621	—	2	1	no	—
31	#10587621	—	2	1	no	—
32	C₂×#63229	C₂²	1	1	yes	characteristic
33	#10587612	C₂²	1	1	yes	characteristic
34	#10587610	—	2	1	no	—
35	#10587610	—	2	1	no	—
36	#10587610	C₂²	1	1	yes	characteristic
37	C₂×Dic₆₈	C₂²	1	1	yes	characteristic
38	C₂×Dic₆₈	C₂²	1	1	yes	characteristic
39	#10587576	—	2	1	no	—
40	#10587576	C₂²	1	1	yes	characteristic
41	#10587575	—	2	1	no	—
42	#10587573	C₂²	1	1	yes	characteristic
43	#10587565	—	2	1	no	—
44	#10587565	—	2	1	no	—
45	C₁₇×#103	C₂²	1	1	yes	characteristic
46	C₁₇×#102	C₂²	1	1	yes	characteristic
47	C₁₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
48	#10587506	C₂²	1	1	yes	characteristic
49	#10587503	C₂²	1	1	yes	characteristic
50	C₂²⋊#3622	C₂²	1	1	yes	characteristic
51	#10587487	C₂²	1	1	yes	characteristic
52	#10587487	C₂²	1	1	yes	characteristic
53	#10587486	C₂²	1	1	yes	characteristic
54	#10587485	C₂²	1	1	yes	characteristic
55	#10587485	C₂²	1	1	yes	characteristic
56	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
57	#10587481	C₂²	1	1	yes	characteristic
58	#10587481	C₂²	1	1	yes	characteristic
59	#41×C₁₇	—	4	1	no	—
60	#41×C₁₇	—	4	1	no	—
61	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
62	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
63	C₂×D₄×C₁₇	—	2	1	no	—
64	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₁₇	—	2	1	no	—
66	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
67	C₂²×Dic₁₇	—	2	1	no	—
68	C₂×Dic₃₄	—	2	1	no	—
69	C₂×Dic₃₄	—	4	1	no	—
70	C₂×Dic₃₄	—	4	1	no	—
71	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
72	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
73	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
74	Q₁₆×C₁₇	—	4	1	no	—
75	SD₁₆×C₁₇	—	4	1	no	—
76	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₁₇	—	2	1	no	—
79	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
80	C₂²⋊C₄×C₁₇	—	2	1	no	—
81	C₂²⋊Dic₁₇	—	2	1	no	—
82	C₂²⋊Dic₁₇	—	2	1	no	—
83	C₂²⋊Dic₁₇	—	2	1	no	—
84	C₂²⋊Dic₁₇	—	2	1	no	—
85	C₂²⋊Dic₁₇	—	2	1	no	—
86	C₂²⋊Dic₁₇	D₄	1	1	yes	characteristic
87	C₂²⋊Dic₁₇	D₄	1	1	yes	characteristic
88	#63231	—	4	1	no	—
89	#63231	—	4	1	no	—
90	#63231	—	4	1	no	—
91	#63229	—	4	1	no	—
92	C₄⋊Dic₁₇	—	2	1	no	—
93	C₄⋊Dic₁₇	—	2	1	no	—
94	C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
95	C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
96	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
97	#63225	—	2	1	no	—
98	#63225	—	2	1	no	—
99	#63225	—	2	1	no	—
100	#63225	—	2	1	no	—
101	#63225	—	2	1	no	—
102	#63225	—	2	1	no	—
103	#63225	—	2	1	no	—
104	#63225	—	2	1	no	—
105	#63225	—	2	1	no	—
106	C₄×Dic₁₇	—	2	1	no	—
107	C₄×Dic₁₇	—	2	1	no	—
108	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
109	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
110	C₂×#3622	C₂³	1	1	yes	characteristic
111	C₂×#3622	C₂³	1	1	yes	characteristic
112	Dic₆₈	—	4	1	no	—
113	Dic₆₈	—	4	1	no	—
114	C₂³×C₁₇	—	2	1	no	—
115	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
116	Q₈×C₁₇	—	2	1	no	—
117	Q₈×C₁₇	—	2	1	no	—
118	Q₈×C₁₇	—	2	1	no	—
119	Q₈×C₁₇	—	2	1	no	—
120	D₄×C₁₇	—	2	1	no	—
121	D₄×C₁₇	—	2	1	no	—
122	D₄×C₁₇	—	4	2	no	—
123	C₂×C₄×C₁₇	—	2	1	no	—
124	C₂×C₄×C₁₇	—	2	1	no	—
125	C₂×C₄×C₁₇	—	2	1	no	—
126	C₂×C₄×C₁₇	—	2	1	no	—
127	C₂×C₄×C₁₇	—	2	1	no	—
128	C₂×C₄×C₁₇	—	4	2	no	—
129	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
131	C₂×Dic₁₇	—	2	1	no	—
132	C₂×Dic₁₇	—	2	1	no	—
133	C₂×Dic₁₇	—	2	1	no	—
134	C₂×Dic₁₇	—	2	1	no	—
135	C₂×Dic₁₇	—	4	2	no	—
136	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
137	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
138	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
139	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
140	Dic₃₄	—	2	1	no	—
141	Dic₃₄	—	2	1	no	—
142	Dic₃₄	—	2	1	no	—
143	Dic₃₄	—	2	1	no	—
144	Dic₃₄	—	2	1	no	—
145	Dic₃₄	—	2	1	no	—
146	Dic₃₄	—	4	2	no	—
147	C₈×C₁₇	—	2	1	no	—
148	C₈×C₁₇	—	2	1	no	—
149	#3622	—	2	1	no	—
150	#3622	—	2	1	no	—
151	#3223	—	17	1	no	maximal, Sylow
152	C₂²×C₁₇	—	2	1	no	—
153	C₂²×C₁₇	—	2	1	no	—
154	C₂²×C₁₇	—	2	1	no	—
155	C₂²×C₁₇	—	4	1	no	—
156	C₂²×C₁₇	—	4	1	no	—
157	C₂²×C₁₇	—	4	1	no	—
158	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
159	C₄×C₁₇	—	2	1	no	—
160	C₄×C₁₇	—	4	1	no	—
161	C₄×C₁₇	—	4	1	no	—
162	C₄×C₁₇	—	4	1	no	—
163	C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
164	C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
165	Dic₁₇	—	2	1	no	—
166	Dic₁₇	—	2	1	no	—
167	Dic₁₇	—	2	1	no	—
168	Dic₁₇	—	2	1	no	—
169	Dic₁₇	—	4	1	no	—
170	Dic₁₇	—	4	1	no	—
171	Dic₁₇	—	4	1	no	—
172	Dic₁₇	—	4	1	no	—
173	#541	—	17	1	no	—
174	#536	—	17	1	no	—
175	#497	—	17	1	no	—
176	#494	—	17	1	no	—
177	#491	—	17	1	no	—
178	#488	—	17	1	no	—
179	#464	—	17	1	no	—
180	#464	—	17	1	no	—
181	#462	—	17	1	no	—
182	#462	—	17	1	no	—
183	#452	—	17	1	no	—
184	#452	D₁₇	1	1	yes	characteristic
185	#451	—	17	1	no	—
186	#451	—	17	1	no	—
187	#433	—	17	1	no	—
188	C₂×C₁₇	—	4	1	no	—
189	C₂×C₁₇	—	8	1	no	—
190	C₂×C₁₇	#535	1	1	yes	characteristic
191	C₂×C₁₇	#578	1	1	yes	characteristic
192	C₂×C₁₇	#578	1	1	yes	characteristic
193	C₂×#41	C₂×D₁₇	1	1	yes	characteristic
194	C₂²×Q₈	—	17	1	no	—
195	C₂×Q₁₆	—	17	1	no	—
196	C₂×Q₁₆	—	17	1	no	—
197	C₂×Q₁₆	—	17	1	no	—
198	C₂×Q₁₆	—	17	1	no	—
199	C₂×Q₁₆	—	17	1	no	—
200	C₂×Q₁₆	C₂×D₁₇	1	1	yes	characteristic
201	C₂×SD₁₆	—	17	1	no	—
202	C₂×SD₁₆	C₂×D₁₇	1	1	yes	characteristic
203	C₄⋊Q₈	—	17	1	no	—
204	C₄⋊Q₈	—	17	1	no	—
205	C₄⋊Q₈	—	17	1	no	—
206	#126	—	34	1	no	—
207	#126	—	34	1	no	—
208	#125	—	17	1	no	—
209	#125	—	34	1	no	—
210	#125	—	34	1	no	—
211	#124	—	17	1	no	—
212	#124	—	17	1	no	—
213	#123	—	34	1	no	—
214	#123	—	34	1	no	—
215	C₂²⋊Q₈	—	17	1	no	—
216	C₂²⋊Q₈	—	34	1	no	—
217	C₂²⋊Q₈	—	34	1	no	—
218	C₂²⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
219	C₄×Q₈	—	17	1	no	—
220	C₄×Q₈	—	17	1	no	—
221	#117	—	17	1	no	—
222	C₄⋊C₈	—	17	1	no	—
223	C₄⋊C₈	—	17	1	no	—
224	#103	—	17	1	no	—
225	#103	—	17	1	no	—
226	#103	—	17	1	no	—
227	#103	—	17	1	no	—
228	#103	—	17	1	no	—
229	#103	C₂×D₁₇	1	1	yes	characteristic
230	#102	—	17	1	no	—
231	#102	C₂×D₁₇	1	1	yes	characteristic
232	C₂²⋊C₈	—	17	1	no	—
233	C₂²⋊C₈	C₂×D₁₇	1	1	yes	characteristic
234	#97	—	17	1	no	—
235	C₄×C₈	—	17	1	no	—
236	C₁₇	#3223	1	1	yes	characteristic, complemented, Sylow
237	#41	—	4	1	no	—
238	#41	—	4	1	no	—
239	C₂×Q₈	—	17	1	no	—
240	C₂×Q₈	—	17	1	no	—
241	C₂×Q₈	—	17	1	no	—
242	C₂×Q₈	—	34	1	no	—
243	C₂×Q₈	—	68	1	no	—
244	C₂×Q₈	—	68	1	no	—
245	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
246	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	34	1	no	—
251	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
252	Q₁₆	—	4	1	no	—
253	Q₁₆	—	68	1	no	—
254	Q₁₆	—	68	1	no	—
255	Q₁₆	—	68	1	no	—
256	Q₁₆	—	68	1	no	—
257	Q₁₆	—	68	1	no	—
258	SD₁₆	—	4	1	no	—
259	SD₁₆	—	68	1	no	—
260	C₂×C₈	—	17	1	no	—
261	C₂×C₈	—	17	1	no	—
262	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
263	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	17	1	no	—
266	C₄⋊C₄	—	17	1	no	—
267	C₄⋊C₄	—	17	1	no	—
268	C₄⋊C₄	—	34	1	no	—
269	C₄⋊C₄	—	34	1	no	—
270	C₄⋊C₄	—	34	1	no	—
271	C₄⋊C₄	—	34	1	no	—
272	C₄⋊C₄	—	34	1	no	—
273	C₄⋊C₄	—	34	1	no	—
274	C₄⋊C₄	—	34	1	no	—
275	C₄⋊C₄	—	34	1	no	—
276	C₄⋊C₄	—	34	1	no	—
277	C₄⋊C₄	—	34	1	no	—
278	C₄⋊C₄	—	34	1	no	—
279	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
280	C₂²⋊C₄	—	2	1	no	—
281	C₂²⋊C₄	—	17	1	no	—
282	C₂²⋊C₄	—	17	1	no	—
283	C₂²⋊C₄	—	34	1	no	—
284	C₂²⋊C₄	—	34	1	no	—
285	C₂²⋊C₄	—	34	1	no	—
286	C₂²⋊C₄	—	34	1	no	—
287	C₂²⋊C₄	—	34	1	no	—
288	C₄²	—	17	1	no	—
289	C₄²	—	17	1	no	—
290	C₄²	—	34	1	no	—
291	C₄²	—	34	1	no	—
292	C₂³	—	2	1	no	—
293	C₂³	D₄×D₁₇	1	1	yes	characteristic
294	Q₈	—	2	1	no	—
295	Q₈	—	2	1	no	—
296	Q₈	—	2	1	no	—
297	Q₈	—	2	1	no	—
298	Q₈	—	34	1	no	—
299	Q₈	—	34	1	no	—
300	Q₈	—	34	1	no	—
301	Q₈	—	34	1	no	—
302	Q₈	—	34	1	no	—
303	Q₈	—	34	1	no	—
304	Q₈	—	68	2	no	—
305	D₄	—	2	1	no	—
306	D₄	—	2	1	no	—
307	D₄	—	4	2	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	17	1	no	—
315	C₂×C₄	—	17	1	no	—
316	C₂×C₄	—	17	1	no	—
317	C₂×C₄	—	17	1	no	—
318	C₂×C₄	—	34	1	no	—
319	C₂×C₄	—	34	1	no	—
320	C₂×C₄	—	34	1	no	—
321	C₂×C₄	—	34	1	no	—
322	C₂×C₄	—	68	2	no	—
323	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
324	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
325	C₈	—	2	1	no	—
326	C₈	—	2	1	no	—
327	C₈	—	34	1	no	—
328	C₈	—	34	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	4	1	no	—
334	C₂²	—	4	1	no	—
335	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	4	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	34	1	no	—
341	C₄	—	34	1	no	—
342	C₄	—	34	1	no	—
343	C₄	—	34	1	no	—
344	C₄	—	68	1	no	—
345	C₄	—	68	1	no	—
346	C₄	—	68	1	no	—
347	C₄	—	68	1	no	—
348	C₄	#10587691	1	1	yes	characteristic
349	C₄	#10587691	1	1	yes	characteristic
350	C₂	—	4	1	no	—
351	C₂	—	8	1	no	—
352	C₂	#11763374	1	1	yes	characteristic
353	C₂	#11763623	1	1	yes	characteristic
354	C₂	#11763625	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₇
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#452	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂×#41×C₁₇	C₂²
hypercenter	#452	D₁₇
hypocenter	C₁₇	#3223
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #452

Complements

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

N	H ≅ G/N	type
1. #11763419	C₂	semidirect
3. #11763200	C₂	semidirect
4. #11763187	C₂	semidirect
5. #11763176	C₂	semidirect
6. #11763162	C₂	semidirect
7. #11763030	C₂	semidirect
8. #11763029	C₂	semidirect
9. #11763018	C₂	semidirect
10. #11763017	C₂	semidirect
12. #11762967	C₂	semidirect
13. #11762964	C₂	semidirect
14. #11762867	C₂	semidirect
236. C₁₇	#3223	semidirect

Maximal subgroups

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

subgroup	index
#11763419	2
#11763380	2
#11763200	2
#11763187	2
#11763176	2
#11763162	2
#11763030	2
#11763029	2
#11763018	2
#11763017	2
#11762971	2
#11762967	2
#11762964	2
#11762867	2
C₁₇×#452	2
#3223	17