Subgroups — #14539677

Order 2176.

378 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763383	C₂	1	1	yes	maximal, characteristic
2	#11763382	C₂	1	1	yes	maximal, characteristic
3	#11763151	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763149	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763148	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763147	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763066	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763064	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763049	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763047	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762965	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762963	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762963	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762774	C₂	1	1	yes	maximal, characteristic
15	C₁₇×#451	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Q₈×C₁₇	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₁₇	C₂²	1	1	yes	characteristic
18	C₂×#63252	C₂²	1	1	yes	characteristic
19	C₂×Q₁₆×C₁₇	C₂²	1	1	yes	characteristic
20	C₂×Q₁₆×C₁₇	C₂²	1	1	yes	characteristic
21	C₁₇×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
22	#10587633	—	2	1	no	—
23	#10587632	C₂²	1	1	yes	characteristic
24	#10587632	C₂²	1	1	yes	characteristic
25	#10587632	C₂²	1	1	yes	characteristic
26	#10587632	C₂²	1	1	yes	characteristic
27	#10587632	C₂²	1	1	yes	characteristic
28	#10587630	—	2	1	no	—
29	C₂×#63231	C₂²	1	1	yes	characteristic
30	C₂×#63231	C₂²	1	1	yes	characteristic
31	#10587615	—	2	1	no	—
32	#10587615	—	2	1	no	—
33	#10587612	C₂²	1	1	yes	characteristic
34	#10587610	C₂²	1	1	yes	characteristic
35	C₂×#63218	C₂²	1	1	yes	characteristic
36	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
37	#10587583	C₂²	1	1	yes	characteristic
38	#10587580	—	2	1	no	—
39	#10587578	C₂²	1	1	yes	characteristic
40	#10587574	—	2	1	no	—
41	#10587571	—	2	1	no	—
42	#10587569	—	2	1	no	—
43	C₁₇×#103	C₂²	1	1	yes	characteristic
44	C₁₇×#103	C₂²	1	1	yes	characteristic
45	C₁₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
46	#10587506	C₂²	1	1	yes	characteristic
47	#10587506	C₂²	1	1	yes	characteristic
48	#10587506	C₂²	1	1	yes	characteristic
49	#10587506	C₂²	1	1	yes	characteristic
50	C₂²⋊#3622	C₂²	1	1	yes	characteristic
51	#10587491	C₂²	1	1	yes	characteristic
52	#10587491	C₂²	1	1	yes	characteristic
53	#10587489	C₂²	1	1	yes	characteristic
54	#10587488	C₂²	1	1	yes	characteristic
55	#10587487	C₂²	1	1	yes	characteristic
56	#10587487	C₂²	1	1	yes	characteristic
57	#10587479	C₂²	1	1	yes	characteristic
58	#10587478	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	—	2	1	no	—
60	C₂×Q₈×C₁₇	—	4	1	no	—
61	C₂×Q₈×C₁₇	—	4	1	no	—
62	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
63	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
64	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₁₇	—	2	1	no	—
66	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊D₁₇	D₄	1	1	yes	characteristic
68	C₂×C₂²⋊D₁₇	D₄	1	1	yes	characteristic
69	Q₈×D₁₇	—	4	1	no	—
70	Q₈×D₁₇	—	4	1	no	—
71	#63252	—	4	1	no	—
72	#63252	—	4	1	no	—
73	C₂×D₆₈	D₄	1	1	yes	characteristic
74	C₂×C₄×D₁₇	—	2	1	no	—
75	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
76	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
77	C₂×Dic₃₄	—	2	1	no	—
78	C₂×Dic₃₄	D₄	1	1	yes	characteristic
79	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
80	Q₁₆×C₁₇	—	4	1	no	—
81	Q₁₆×C₁₇	—	4	1	no	—
82	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
83	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
84	C₄⋊C₄×C₁₇	—	2	1	no	—
85	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
86	C₂²⋊C₄×C₁₇	—	2	1	no	—
87	C₂²⋊Dic₁₇	—	2	1	no	—
88	C₂²⋊Dic₁₇	—	2	1	no	—
89	#63231	—	4	1	no	—
90	#63231	—	4	1	no	—
91	#63227	—	2	1	no	—
92	#63227	—	2	1	no	—
93	#63227	—	2	1	no	—
94	#63227	—	2	1	no	—
95	#63227	—	2	1	no	—
96	#63227	—	2	1	no	—
97	#63227	—	2	1	no	—
98	C₄⋊Dic₁₇	—	2	1	no	—
99	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
100	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
101	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
102	#63225	—	2	1	no	—
103	#63225	—	2	1	no	—
104	#63225	—	2	1	no	—
105	#63225	—	2	1	no	—
106	#63225	—	2	1	no	—
107	#63225	—	2	1	no	—
108	C₄×Dic₁₇	—	2	1	no	—
109	C₄×Dic₁₇	—	2	1	no	—
110	C₂×#3622	C₂³	1	1	yes	characteristic
111	C₂×#3622	C₂³	1	1	yes	characteristic
112	#63218	—	4	1	no	—
113	C₈×D₁₇	—	4	1	no	—
114	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
115	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
116	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
117	Q₈×C₁₇	—	2	1	no	—
118	Q₈×C₁₇	—	2	1	no	—
119	Q₈×C₁₇	—	2	1	no	—
120	Q₈×C₁₇	—	2	1	no	—
121	Q₈×C₁₇	—	2	1	no	—
122	Q₈×C₁₇	—	2	1	no	—
123	Q₈×C₁₇	—	4	2	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	—	2	1	no	—
129	C₂×C₄×C₁₇	—	2	1	no	—
130	C₂×C₄×C₁₇	—	4	2	no	—
131	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
133	C₂²⋊D₁₇	—	4	1	no	—
134	C₂²⋊D₁₇	—	4	1	no	—
135	C₂×Dic₁₇	—	2	1	no	—
136	C₂×Dic₁₇	—	2	1	no	—
137	C₂×Dic₁₇	—	2	1	no	—
138	C₂×Dic₁₇	—	2	1	no	—
139	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
140	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
141	D₆₈	—	4	1	no	—
142	C₄×D₁₇	—	2	1	no	—
143	C₄×D₁₇	—	2	1	no	—
144	C₄×D₁₇	—	2	1	no	—
145	C₄×D₁₇	—	2	1	no	—
146	C₄×D₁₇	—	4	2	no	—
147	Dic₃₄	—	2	1	no	—
148	Dic₃₄	—	2	1	no	—
149	Dic₃₄	—	4	1	no	—
150	Dic₃₄	—	4	2	no	—
151	C₈×C₁₇	—	2	1	no	—
152	C₈×C₁₇	—	2	1	no	—
153	#3622	—	2	1	no	—
154	#3622	—	2	1	no	—
155	#3202	—	17	1	no	maximal, Sylow
156	C₂²×C₁₇	—	2	1	no	—
157	C₂²×C₁₇	—	2	1	no	—
158	C₂²×C₁₇	—	2	1	no	—
159	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
160	C₂×D₁₇	—	2	1	no	—
161	C₂×D₁₇	—	2	1	no	—
162	C₂×D₁₇	—	2	1	no	—
163	C₂×D₁₇	—	2	1	no	—
164	C₂×D₁₇	—	2	1	no	—
165	C₂×D₁₇	—	2	1	no	—
166	C₄×C₁₇	—	2	1	no	—
167	C₄×C₁₇	—	4	1	no	—
168	C₄×C₁₇	—	4	1	no	—
169	C₄×C₁₇	—	4	1	no	—
170	C₄×C₁₇	—	4	1	no	—
171	C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
172	C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
173	Dic₁₇	—	2	1	no	—
174	Dic₁₇	—	2	1	no	—
175	Dic₁₇	—	4	1	no	—
176	Dic₁₇	—	4	1	no	—
177	Dic₁₇	—	4	1	no	—
178	Dic₁₇	—	4	1	no	—
179	#536	—	17	1	no	—
180	#536	—	17	1	no	—
181	#485	—	17	1	no	—
182	#485	—	17	1	no	—
183	#485	—	17	1	no	—
184	#485	—	17	1	no	—
185	#470	—	17	1	no	—
186	#470	—	17	1	no	—
187	#467	—	17	1	no	—
188	#467	—	17	1	no	—
189	#451	—	17	1	no	—
190	#451	—	17	1	no	—
191	#451	—	17	1	no	—
192	#451	D₁₇	1	1	yes	characteristic, complemented
193	#408	—	17	1	no	—
194	C₂×C₁₇	—	4	1	no	—
195	C₂×C₁₇	#534	1	1	yes	characteristic
196	C₂×C₁₇	#578	1	1	yes	characteristic
197	C₂×C₁₇	#578	1	1	yes	characteristic
198	D₁₇	—	4	1	no	—
199	D₁₇	—	4	1	no	—
200	C₂×#41	—	17	1	no	—
201	C₂²×Q₈	—	17	1	no	—
202	C₂²×Q₈	C₂×D₁₇	1	1	yes	characteristic
203	C₂×Q₁₆	—	17	1	no	—
204	C₂×Q₁₆	—	17	1	no	—
205	C₂×Q₁₆	C₂×D₁₇	1	1	yes	characteristic
206	C₂×Q₁₆	C₂×D₁₇	1	1	yes	characteristic
207	C₂×M₁₆	—	17	1	no	—
208	C₂²×C₈	—	17	1	no	—
209	C₄⋊Q₈	—	34	1	no	—
210	C₄⋊Q₈	—	34	1	no	—
211	#124	—	34	1	no	—
212	#124	—	34	1	no	—
213	#123	—	34	1	no	—
214	#123	—	34	1	no	—
215	#123	—	34	1	no	—
216	#123	—	34	1	no	—
217	C₂²⋊Q₈	—	17	1	no	—
218	C₂²⋊Q₈	—	17	1	no	—
219	C₂²⋊Q₈	—	17	1	no	—
220	C₂²⋊Q₈	—	17	1	no	—
221	C₂²⋊Q₈	—	17	1	no	—
222	C₂²⋊Q₈	—	17	1	no	—
223	C₂²⋊Q₈	—	17	1	no	—
224	C₂²⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
225	#117	—	17	1	no	—
226	#117	—	17	1	no	—
227	#107	—	17	1	no	—
228	#107	—	17	1	no	—
229	#106	—	17	1	no	—
230	#106	—	17	1	no	—
231	#103	—	17	1	no	—
232	#103	—	17	1	no	—
233	#103	—	17	1	no	—
234	#103	—	17	1	no	—
235	#103	—	17	1	no	—
236	#103	—	17	1	no	—
237	#103	C₂×D₁₇	1	1	yes	characteristic
238	#103	C₂×D₁₇	1	1	yes	characteristic
239	C₂²⋊C₈	—	17	1	no	—
240	C₂²⋊C₈	—	17	1	no	—
241	C₂²⋊C₈	—	17	1	no	—
242	C₂²⋊C₈	C₂×D₁₇	1	1	yes	characteristic
243	C₁₇	#3202	1	1	yes	characteristic, complemented, Sylow
244	#41	—	68	1	no	—
245	#41	—	68	1	no	—
246	C₂×Q₈	—	2	1	no	—
247	C₂×Q₈	—	4	1	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×Q₈	—	17	1	no	—
250	C₂×Q₈	—	17	1	no	—
251	C₂×Q₈	—	34	1	no	—
252	C₂×Q₈	—	68	1	no	—
253	C₂×Q₈	—	68	1	no	—
254	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
255	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
256	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
257	C₂×D₄	—	17	1	no	—
258	C₂×D₄	—	17	1	no	—
259	C₂×D₄	—	17	1	no	—
260	C₂²×C₄	—	2	1	no	—
261	C₂²×C₄	—	17	1	no	—
262	C₂²×C₄	—	17	1	no	—
263	C₂²×C₄	—	34	1	no	—
264	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
265	Q₁₆	—	4	1	no	—
266	Q₁₆	—	4	1	no	—
267	Q₁₆	—	68	1	no	—
268	Q₁₆	—	68	1	no	—
269	M₁₆	—	68	1	no	—
270	C₂×C₈	—	17	1	no	—
271	C₂×C₈	—	17	1	no	—
272	C₂×C₈	—	68	1	no	—
273	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
274	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	17	1	no	—
277	C₄⋊C₄	—	17	1	no	—
278	C₄⋊C₄	—	17	1	no	—
279	C₄⋊C₄	—	34	1	no	—
280	C₄⋊C₄	—	34	1	no	—
281	C₄⋊C₄	—	34	1	no	—
282	C₄⋊C₄	—	34	1	no	—
283	C₄⋊C₄	—	34	1	no	—
284	C₄⋊C₄	—	34	1	no	—
285	C₄⋊C₄	—	34	1	no	—
286	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
287	C₂²⋊C₄	—	2	1	no	—
288	C₂²⋊C₄	—	34	1	no	—
289	C₂²⋊C₄	—	34	1	no	—
290	C₂²⋊C₄	—	34	1	no	—
291	C₂²⋊C₄	—	34	1	no	—
292	C₂²⋊C₄	—	34	1	no	—
293	C₂²⋊C₄	—	34	1	no	—
294	C₂²⋊C₄	—	34	1	no	—
295	C₂²⋊C₄	—	34	1	no	—
296	C₂²⋊C₄	—	34	1	no	—
297	C₄²	—	34	1	no	—
298	C₄²	—	34	1	no	—
299	C₂³	—	17	1	no	—
300	C₂³	—	17	1	no	—
301	C₂³	D₄×D₁₇	1	1	yes	characteristic
302	Q₈	—	2	1	no	—
303	Q₈	—	2	1	no	—
304	Q₈	—	2	1	no	—
305	Q₈	—	2	1	no	—
306	Q₈	—	2	1	no	—
307	Q₈	—	2	1	no	—
308	Q₈	—	4	2	no	—
309	Q₈	—	34	1	no	—
310	Q₈	—	34	1	no	—
311	Q₈	—	68	1	no	—
312	Q₈	—	68	2	no	—
313	D₄	—	68	1	no	—
314	D₄	—	68	1	no	—
315	D₄	—	68	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	17	1	no	—
324	C₂×C₄	—	17	1	no	—
325	C₂×C₄	—	34	1	no	—
326	C₂×C₄	—	34	1	no	—
327	C₂×C₄	—	34	1	no	—
328	C₂×C₄	—	34	1	no	—
329	C₂×C₄	—	34	1	no	—
330	C₂×C₄	—	34	1	no	—
331	C₂×C₄	—	34	1	no	—
332	C₂×C₄	—	34	1	no	—
333	C₂×C₄	—	68	2	no	—
334	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
335	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
336	C₈	—	2	1	no	—
337	C₈	—	2	1	no	—
338	C₈	—	34	1	no	—
339	C₈	—	34	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	34	1	no	—
344	C₂²	—	34	1	no	—
345	C₂²	—	34	1	no	—
346	C₂²	—	34	1	no	—
347	C₂²	—	34	1	no	—
348	C₂²	—	34	1	no	—
349	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	4	1	no	—
352	C₄	—	4	1	no	—
353	C₄	—	4	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	34	1	no	—
356	C₄	—	34	1	no	—
357	C₄	—	68	1	no	—
358	C₄	—	68	1	no	—
359	C₄	—	68	1	no	—
360	C₄	—	68	1	no	—
361	C₄	#10587691	1	1	yes	characteristic
362	C₄	#10587691	1	1	yes	characteristic
363	C₂	—	4	1	no	—
364	C₂	—	68	1	no	—
365	C₂	—	68	1	no	—
366	C₂	#11763369	1	1	yes	characteristic
367	C₂	#11763622	1	1	yes	characteristic
368	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×#451	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂×#63252	C₂²
hypercenter	#451	D₁₇
hypocenter	C₁₇	#3202
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #451

Complements

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

N	H ≅ G/N	type
3. #11763151	C₂	semidirect
4. #11763149	C₂	semidirect
5. #11763148	C₂	semidirect
6. #11763147	C₂	semidirect
7. #11763066	C₂	semidirect
8. #11763064	C₂	semidirect
9. #11763049	C₂	semidirect
10. #11763047	C₂	semidirect
11. #11762965	C₂	semidirect
12. #11762963	C₂	semidirect
13. #11762963	C₂	semidirect
15. C₁₇×#451	C₂	semidirect
192. #451	D₁₇	semidirect
243. C₁₇	#3202	semidirect

Maximal subgroups

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

subgroup	index
#11763383	2
#11763382	2
#11763151	2
#11763149	2
#11763148	2
#11763147	2
#11763066	2
#11763064	2
#11763049	2
#11763047	2
#11762965	2
#11762963	2
#11762774	2
C₁₇×#451	2
#3202	17