Subgroups — #14540776

Order 2176.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763466	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763460	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763167	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763144	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763121	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763102	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763055	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763044	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763023	C₂	1	1	yes	maximal, characteristic, complemented
10	#11762954	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762949	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762912	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762878	C₂	1	1	yes	maximal, characteristic, complemented
14	#103×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#417	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#41×C₁₇	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₁₇	C₂²	1	1	yes	characteristic
18	C₂×#63252	—	2	1	no	—
19	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
21	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
22	C₂×#63230	C₂²	1	1	yes	characteristic
23	#10587624	—	2	1	no	—
24	#10587624	C₂²	1	1	yes	characteristic
25	#10587622	—	2	1	no	—
26	#10587622	C₂²	1	1	yes	characteristic
27	C₂×#63228	C₂²	1	1	yes	characteristic
28	#10587616	C₂²	1	1	yes	characteristic
29	#10587615	—	2	1	no	—
30	C₄×C₂²⋊D₁₇	—	2	1	no	—
31	C₄×C₂²⋊D₁₇	C₂²	1	1	yes	characteristic
32	#10587610	—	2	1	no	—
33	#10587610	C₂²	1	1	yes	characteristic
34	C₂×#63219	C₂²	1	1	yes	characteristic
35	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
36	#10587582	—	2	1	no	—
37	#10587581	C₂²	1	1	yes	characteristic
38	#10587580	—	2	1	no	—
39	#10587579	C₂²	1	1	yes	characteristic
40	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
41	#10587574	C₂²	1	1	yes	characteristic
42	C₁₇×#103	C₂²	1	1	yes	characteristic
43	C₁₇×#103	C₂²	1	1	yes	characteristic
44	C₁₇×#102	C₂²	1	1	yes	characteristic
45	C₁₇×#102	C₂²	1	1	yes	characteristic
46	#10587506	C₂²	1	1	yes	characteristic
47	#10587503	C₂²	1	1	yes	characteristic
48	C₂²⋊#3622	C₂²	1	1	yes	characteristic
49	#10587492	C₂²	1	1	yes	characteristic
50	#10587491	C₂²	1	1	yes	characteristic
51	#10587489	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	#10587481	C₂²	1	1	yes	characteristic
56	#10587481	C₂²	1	1	yes	characteristic
57	#10587478	C₂²	1	1	yes	characteristic
58	#10587478	C₂²	1	1	yes	characteristic
59	#41×C₁₇	—	2	2	no	—
60	#41×C₁₇	—	4	1	no	—
61	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₁₇	—	2	1	no	—
63	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₁₇	—	2	1	no	—
65	C₂²×C₄×C₁₇	—	2	1	no	—
66	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊D₁₇	—	2	1	no	—
68	C₂×C₂²⋊D₁₇	—	2	1	no	—
69	C₂×C₂²⋊D₁₇	D₄	1	1	yes	characteristic
70	Q₈×D₁₇	—	2	2	no	—
71	Q₈×D₁₇	—	4	1	no	—
72	#63252	—	4	4	no	—
73	C₂×D₆₈	—	2	1	no	—
74	C₂×D₆₈	C₂³	1	1	yes	characteristic
75	C₂×C₄×D₁₇	—	2	1	no	—
76	C₂×C₄×D₁₇	—	2	1	no	—
77	C₂×C₄×D₁₇	—	2	1	no	—
78	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
79	C₂×Dic₃₄	—	2	1	no	—
80	C₂×Dic₃₄	—	2	1	no	—
81	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
82	M₁₆×C₁₇	—	4	1	no	—
83	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
84	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
85	C₄⋊C₄×C₁₇	—	2	1	no	—
86	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
87	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
88	C₂²⋊Dic₁₇	—	2	1	no	—
89	C₂²⋊Dic₁₇	—	2	1	no	—
90	C₂²⋊Dic₁₇	—	2	1	no	—
91	C₂²⋊Dic₁₇	—	2	1	no	—
92	C₂²⋊Dic₁₇	D₄	1	1	yes	characteristic
93	#63230	—	2	2	no	—
94	#63228	—	4	1	no	—
95	#63227	—	2	1	no	—
96	#63227	—	2	1	no	—
97	#63227	—	2	1	no	—
98	#63227	D₄	1	1	yes	characteristic
99	C₄⋊Dic₁₇	—	2	1	no	—
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	D₄	1	1	yes	characteristic
106	C₄×Dic₁₇	—	2	1	no	—
107	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
108	C₂×#3622	C₂³	1	1	yes	characteristic
109	C₂×#3622	C₂³	1	1	yes	characteristic
110	#63219	—	4	1	no	—
111	C₈×D₁₇	—	4	1	no	—
112	C₂³×C₁₇	—	2	1	no	—
113	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
114	C₂²×D₁₇	—	2	1	no	—
115	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
116	Q₈×C₁₇	—	2	1	no	—
117	Q₈×C₁₇	#41	1	2	yes	complemented
118	D₄×C₁₇	—	2	1	no	—
119	D₄×C₁₇	—	2	1	no	—
120	D₄×C₁₇	—	4	2	no	—
121	C₂×C₄×C₁₇	—	2	1	no	—
122	C₂×C₄×C₁₇	—	2	1	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	—	4	2	no	—
127	C₂×C₄×C₁₇	—	4	2	no	—
128	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
130	C₂²⋊D₁₇	—	2	2	no	—
131	C₂²⋊D₁₇	—	4	2	no	—
132	C₂²⋊D₁₇	—	4	2	no	—
133	C₂×Dic₁₇	—	2	1	no	—
134	C₂×Dic₁₇	—	2	1	no	—
135	C₂×Dic₁₇	—	2	1	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	D₆₈	—	2	1	no	—
140	D₆₈	—	2	1	no	—
141	D₆₈	—	4	2	no	—
142	C₄×D₁₇	—	2	1	no	—
143	C₄×D₁₇	—	2	1	no	—
144	C₄×D₁₇	—	4	2	no	—
145	C₄×D₁₇	—	4	2	no	—
146	C₄×D₁₇	—	4	2	no	—
147	Dic₃₄	—	2	1	no	—
148	Dic₃₄	—	2	1	no	—
149	Dic₃₄	—	4	2	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	#3308	—	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	—	4	1	no	—
160	C₂²×C₁₇	—	4	1	no	—
161	C₂²×C₁₇	—	4	1	no	—
162	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
163	C₂×D₁₇	—	2	1	no	—
164	C₂×D₁₇	—	2	1	no	—
165	C₂×D₁₇	—	4	1	no	—
166	C₂×D₁₇	—	4	1	no	—
167	C₂×D₁₇	—	4	1	no	—
168	C₂×D₁₇	AGL(1,ℤ/8)	1	2	yes	—
169	C₄×C₁₇	—	2	2	no	—
170	C₄×C₁₇	—	2	1	no	—
171	C₄×C₁₇	—	4	1	no	—
172	C₄×C₁₇	—	4	1	no	—
173	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
174	C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
175	Dic₁₇	—	2	1	no	—
176	Dic₁₇	—	2	1	no	—
177	Dic₁₇	—	2	1	no	—
178	Dic₁₇	—	4	1	no	—
179	Dic₁₇	—	4	1	no	—
180	Dic₁₇	—	4	1	no	—
181	#548	—	17	1	no	—
182	#547	—	17	1	no	—
183	#489	—	17	1	no	—
184	#484	—	17	1	no	—
185	#480	—	17	1	no	—
186	#477	—	17	1	no	—
187	#468	—	17	1	no	—
188	#466	—	17	1	no	—
189	#463	—	17	1	no	—
190	#449	—	17	1	no	—
191	#448	—	17	1	no	—
192	#440	—	17	1	no	—
193	#435	—	17	1	no	—
194	#417	D₁₇	1	1	yes	characteristic, complemented
195	C₂×#103	—	17	1	no	—
196	C₂×C₁₇	—	4	1	no	—
197	C₂×C₁₇	—	8	1	no	—
198	C₂×C₁₇	#546	1	1	yes	characteristic
199	C₂×C₁₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
200	C₂×C₁₇	#578	1	1	yes	characteristic
201	D₁₇	—	2	2	no	—
202	D₁₇	—	8	1	no	—
203	C₂×#41	—	34	1	no	—
204	C₂×#41	C₂×D₁₇	1	1	yes	characteristic
205	C₂²×Q₈	—	17	1	no	—
206	C₂×SD₁₆	—	17	1	no	—
207	C₂×SD₁₆	—	17	1	no	—
208	C₂×D₈	—	17	1	no	—
209	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic
210	C₂²×C₈	—	17	1	no	—
211	C₄⋊Q₈	—	17	1	no	—
212	#124	—	17	1	no	—
213	#124	—	34	1	no	—
214	#123	—	34	1	no	—
215	#123	—	34	1	no	—
216	C₂²⋊Q₈	—	17	1	no	—
217	C₂²⋊Q₈	—	34	1	no	—
218	C₂²⋊Q₈	—	34	1	no	—
219	#121	—	17	1	no	—
220	#121	—	17	1	no	—
221	#121	—	17	1	no	—
222	#121	—	34	1	no	—
223	C₄×Q₈	—	17	1	no	—
224	C₄×D₄	—	17	1	no	—
225	C₄×D₄	—	17	1	no	—
226	C₄×D₄	—	34	1	no	—
227	C₂×C₄⋊C₄	—	17	1	no	—
228	C₂×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
229	#107	—	17	1	no	—
230	#107	—	17	1	no	—
231	#107	—	17	1	no	—
232	C₄⋊C₈	—	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	#102	—	17	1	no	—
240	#102	—	17	1	no	—
241	#102	C₂×D₁₇	1	1	yes	characteristic
242	#102	C₂×D₁₇	1	1	yes	characteristic
243	C₂²⋊C₈	—	17	1	no	—
244	C₂²⋊C₈	—	17	1	no	—
245	#97	—	17	1	no	—
246	C₁₇	#3308	1	1	yes	characteristic, complemented, Sylow
247	#41	—	2	2	no	—
248	#41	—	4	1	no	—
249	#41	—	68	4	no	—
250	C₂×Q₈	—	17	1	no	—
251	C₂×Q₈	—	34	1	no	—
252	C₂×Q₈	—	34	1	no	—
253	C₂×Q₈	—	34	2	no	—
254	C₂×Q₈	—	68	1	no	—
255	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	—	17	1	no	—
258	C₂×D₄	—	17	1	no	—
259	C₂×D₄	—	34	1	no	—
260	C₂×D₄	—	34	1	no	—
261	C₂×D₄	—	34	1	no	—
262	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
263	C₂²×C₄	—	2	1	no	—
264	C₂²×C₄	—	2	1	no	—
265	C₂²×C₄	—	17	1	no	—
266	C₂²×C₄	—	34	1	no	—
267	C₂²×C₄	—	34	1	no	—
268	C₂²×C₄	—	34	1	no	—
269	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
270	SD₁₆	—	34	2	no	—
271	SD₁₆	—	68	1	no	—
272	D₈	—	68	1	no	—
273	M₁₆	—	4	1	no	—
274	C₂×C₈	—	17	1	no	—
275	C₂×C₈	—	17	1	no	—
276	C₂×C₈	—	68	1	no	—
277	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
278	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	17	1	no	—
281	C₄⋊C₄	—	17	1	no	—
282	C₄⋊C₄	—	17	1	no	—
283	C₄⋊C₄	—	34	1	no	—
284	C₄⋊C₄	—	34	1	no	—
285	C₄⋊C₄	—	34	1	no	—
286	C₄⋊C₄	—	34	1	no	—
287	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
288	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
289	C₂²⋊C₄	—	17	1	no	—
290	C₂²⋊C₄	—	17	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	C₂²⋊C₄	—	34	1	no	—
298	C₄²	—	17	1	no	—
299	C₄²	—	34	1	no	—
300	C₂³	—	2	1	no	—
301	C₂³	—	17	1	no	—
302	C₂³	—	34	1	no	—
303	C₂³	D₄×D₁₇	1	1	yes	characteristic
304	Q₈	—	2	1	no	—
305	Q₈	—	34	1	no	—
306	Q₈	—	34	1	no	—
307	Q₈	—	68	2	no	—
308	Q₈	—	68	2	no	—
309	Q₈	#63252	1	2	yes	complemented
310	D₄	—	2	1	no	—
311	D₄	—	2	1	no	—
312	D₄	—	4	2	no	—
313	D₄	—	34	2	no	—
314	D₄	—	34	1	no	—
315	D₄	—	34	1	no	—
316	D₄	—	68	2	no	—
317	D₄	—	68	2	no	—
318	D₄	—	68	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	17	1	no	—
327	C₂×C₄	—	17	1	no	—
328	C₂×C₄	—	17	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	—	34	1	no	—
334	C₂×C₄	—	68	2	no	—
335	C₂×C₄	—	68	2	no	—
336	C₂×C₄	—	68	2	no	—
337	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
338	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
339	C₈	—	2	1	no	—
340	C₈	—	2	1	no	—
341	C₈	—	34	1	no	—
342	C₈	—	34	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	4	1	no	—
347	C₂²	—	4	1	no	—
348	C₂²	—	4	1	no	—
349	C₂²	—	17	2	no	—
350	C₂²	—	34	1	no	—
351	C₂²	—	34	1	no	—
352	C₂²	—	68	1	no	—
353	C₂²	—	68	1	no	—
354	C₂²	—	68	1	no	—
355	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
356	C₄	—	2	1	no	—
357	C₄	—	2	2	no	—
358	C₄	—	4	1	no	—
359	C₄	—	4	1	no	—
360	C₄	—	34	1	no	—
361	C₄	—	34	1	no	—
362	C₄	—	34	1	no	—
363	C₄	—	68	1	no	—
364	C₄	—	68	1	no	—
365	C₄	—	68	1	no	—
366	C₄	C₂×#63252	1	1	yes	characteristic
367	C₄	#10587691	1	1	yes	characteristic
368	C₂	—	4	1	no	—
369	C₂	—	8	1	no	—
370	C₂	—	34	2	no	—
371	C₂	—	136	1	no	—
372	C₂	#11763449	1	1	yes	characteristic
373	C₂	AGL(1,ℤ/8)×D₁₇	1	1	yes	characteristic
374	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×#417	C₂
socle	C₂²×C₁₇	C₂²×D₄
hypercenter	#417	D₁₇
hypocenter	C₁₇	#3308
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #417

Complements

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

N	H ≅ G/N	type
1. #11763466	C₂	semidirect
2. #11763460	C₂	semidirect
3. #11763167	C₂	semidirect
4. #11763144	C₂	semidirect
5. #11763121	C₂	semidirect
6. #11763102	C₂	semidirect
7. #11763055	C₂	semidirect
8. #11763044	C₂	semidirect
9. #11763023	C₂	semidirect
10. #11762954	C₂	semidirect
11. #11762949	C₂	semidirect
12. #11762912	C₂	semidirect
13. #11762878	C₂	semidirect
14. #103×D₁₇	C₂	semidirect
15. C₁₇×#417	C₂	semidirect
117. Q₈×C₁₇	#41	semidirect
194. #417	D₁₇	semidirect
246. C₁₇	#3308	semidirect
309. Q₈	#63252	semidirect

Maximal subgroups

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

subgroup	index
#11763466	2
#11763460	2
#11763167	2
#11763144	2
#11763121	2
#11763102	2
#11763055	2
#11763044	2
#11763023	2
#11762954	2
#11762949	2
#11762912	2
#11762878	2
#103×D₁₇	2
C₁₇×#417	2
#3308	17