Subgroups — #14540767

Order 2176.

394 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	#11763509	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763462	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763177	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763149	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763138	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763110	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763052	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763035	C₂	1	1	yes	maximal, characteristic, complemented
9	#11762953	C₂	1	1	yes	maximal, characteristic, complemented
10	#11762953	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762919	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762879	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762809	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762804	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#461	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#63256	C₂²	1	1	yes	characteristic
17	C₂×#63256	C₂²	1	1	yes	characteristic
18	C₂×SD₁₆×C₁₇	C₂²	1	1	yes	characteristic
19	C₂×SD₁₆×C₁₇	C₂²	1	1	yes	characteristic
20	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
21	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
22	#10587632	C₂²	1	1	yes	characteristic
23	#10587632	C₂²	1	1	yes	characteristic
24	#10587624	C₂²	1	1	yes	characteristic
25	#10587624	C₂²	1	1	yes	characteristic
26	C₂×#63228	C₂²	1	1	yes	characteristic
27	#10587615	—	2	1	no	—
28	C₄×C₂²⋊D₁₇	—	2	1	no	—
29	C₂×#63218	C₂²	1	1	yes	characteristic
30	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
31	#10587581	—	2	1	no	—
32	#10587580	—	2	1	no	—
33	#10587580	—	2	1	no	—
34	#10587578	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₁₇	—	2	1	no	—
36	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
37	#10587570	—	2	1	no	—
38	#10587565	—	2	1	no	—
39	C₄×D₆₈	C₂²	1	1	yes	characteristic
40	C₄×D₆₈	C₂²	1	1	yes	characteristic
41	#10587556	C₂²	1	1	yes	characteristic
42	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₁₇×#103	C₂²	1	1	yes	characteristic
44	C₁₇×#103	C₂²	1	1	yes	characteristic
45	#10587506	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	#10587503	C₂²	1	1	yes	characteristic
50	#10587492	C₂²	1	1	yes	characteristic
51	#10587492	C₂²	1	1	yes	characteristic
52	#10587491	C₂²	1	1	yes	characteristic
53	#10587491	C₂²	1	1	yes	characteristic
54	#10587489	C₂²	1	1	yes	characteristic
55	#10587488	C₂²	1	1	yes	characteristic
56	#10587480	C₂²	1	1	yes	characteristic
57	#10587479	C₂²	1	1	yes	characteristic
58	#10587474	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
60	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₇	—	2	1	no	—
63	C₂×C₂²⋊D₁₇	—	2	1	no	—
64	C₂×C₂²⋊D₁₇	—	2	1	no	—
65	#63256	—	2	2	no	—
66	#63256	—	2	2	no	—
67	#63256	—	4	1	no	—
68	#63256	—	4	1	no	—
69	C₂×D₆₈	—	2	1	no	—
70	C₂×D₆₈	—	2	1	no	—
71	C₂×D₆₈	D₄	1	1	yes	characteristic
72	C₂×D₆₈	C₂³	1	1	yes	characteristic
73	C₂×C₄×D₁₇	—	2	1	no	—
74	C₂×C₄×D₁₇	—	2	1	no	—
75	C₂×C₄×D₁₇	—	2	1	no	—
76	C₂×C₄×D₁₇	—	2	1	no	—
77	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
78	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
79	SD₁₆×C₁₇	—	4	1	no	—
80	SD₁₆×C₁₇	—	4	1	no	—
81	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
82	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
83	C₄⋊C₄×C₁₇	—	2	1	no	—
84	C₄⋊C₄×C₁₇	—	2	1	no	—
85	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
86	C₂²⋊C₄×C₁₇	—	2	1	no	—
87	C₄²×C₁₇	C₂³	1	1	yes	characteristic
88	C₂²⋊Dic₁₇	—	2	1	no	—
89	C₂²⋊Dic₁₇	—	2	1	no	—
90	#63228	—	2	2	no	—
91	#63227	—	2	1	no	—
92	#63227	—	2	1	no	—
93	#63227	—	2	1	no	—
94	#63227	—	2	1	no	—
95	#63227	D₄	1	1	yes	characteristic
96	#63227	D₄	1	1	yes	characteristic
97	C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
98	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
99	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
100	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
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	C₂×#3622	C₂³	1	1	yes	characteristic
108	C₂×#3622	C₂³	1	1	yes	characteristic
109	#63218	—	4	1	no	—
110	C₈×D₁₇	—	4	1	no	—
111	C₂³×C₁₇	—	2	1	no	—
112	C₂²×D₁₇	—	2	1	no	—
113	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
114	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
115	Q₈×C₁₇	—	2	1	no	—
116	Q₈×C₁₇	—	2	1	no	—
117	Q₈×C₁₇	—	2	1	no	—
118	Q₈×C₁₇	—	2	1	no	—
119	D₄×C₁₇	—	2	1	no	—
120	D₄×C₁₇	—	2	1	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	—	8	1	no	—
125	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
128	C₂²⋊D₁₇	—	4	2	no	—
129	C₂²⋊D₁₇	—	8	1	no	—
130	C₂×Dic₁₇	—	2	1	no	—
131	C₂×Dic₁₇	—	2	1	no	—
132	C₂×Dic₁₇	—	2	1	no	—
133	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
135	D₆₈	—	2	1	no	—
136	D₆₈	—	4	1	no	—
137	D₆₈	—	4	2	no	—
138	D₆₈	—	4	2	no	—
139	D₆₈	#41	1	2	yes	—
140	C₄×D₁₇	—	2	1	no	—
141	C₄×D₁₇	—	2	1	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	C₄×D₁₇	—	8	1	no	—
148	C₈×C₁₇	—	2	1	no	—
149	C₈×C₁₇	—	2	1	no	—
150	#3622	—	2	1	no	—
151	#3622	—	2	1	no	—
152	#3307	—	17	1	no	maximal, Sylow
153	C₂²×C₁₇	—	4	1	no	—
154	C₂²×C₁₇	—	4	1	no	—
155	C₂²×C₁₇	—	4	1	no	—
156	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
157	C₂×D₁₇	—	2	1	no	—
158	C₂×D₁₇	—	2	1	no	—
159	C₂×D₁₇	—	2	2	no	—
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	—	4	1	no	—
165	C₂×D₁₇	—	4	1	no	—
166	C₄×C₁₇	—	2	1	no	—
167	C₄×C₁₇	—	2	1	no	—
168	C₄×C₁₇	—	4	1	no	—
169	C₄×C₁₇	—	4	1	no	—
170	C₄×C₁₇	—	4	1	no	—
171	C₄×C₁₇	C₂×#41	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	#553	—	17	1	no	—
179	#547	—	17	1	no	—
180	#491	—	17	1	no	—
181	#485	—	17	1	no	—
182	#483	—	17	1	no	—
183	#478	—	17	1	no	—
184	#468	—	17	1	no	—
185	#465	—	17	1	no	—
186	#461	D₁₇	1	1	yes	characteristic, complemented
187	#449	—	17	1	no	—
188	#449	—	17	1	no	—
189	#442	—	17	1	no	—
190	#435	—	17	1	no	—
191	#417	—	17	1	no	—
192	#416	—	17	1	no	—
193	C₂×C₁₇	—	8	1	no	—
194	C₂×C₁₇	#546	1	1	yes	characteristic
195	C₂×C₁₇	#575	1	1	yes	characteristic
196	C₂×C₁₇	#577	1	1	yes	characteristic
197	D₁₇	—	4	1	no	—
198	D₁₇	—	4	1	no	—
199	D₁₇	—	4	2	no	—
200	C₂×#41	—	17	1	no	—
201	C₂×#41	—	17	1	no	—
202	C₂×SD₁₆	C₂×D₁₇	1	1	yes	characteristic
203	C₂×SD₁₆	C₂×D₁₇	1	1	yes	characteristic
204	C₂×D₈	—	17	1	no	—
205	C₂×M₁₆	—	17	1	no	—
206	C₂²×C₈	—	17	1	no	—
207	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
208	#126	—	34	1	no	—
209	#125	—	17	1	no	—
210	#123	—	34	1	no	—
211	#123	—	34	1	no	—
212	#123	—	34	1	no	—
213	C₂²⋊Q₈	—	17	1	no	—
214	C₂²⋊Q₈	—	17	1	no	—
215	#121	—	17	1	no	—
216	#121	—	17	1	no	—
217	#121	—	34	1	no	—
218	#121	—	34	1	no	—
219	C₄×D₄	—	17	1	no	—
220	C₄×D₄	—	17	1	no	—
221	C₄×D₄	—	34	1	no	—
222	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
223	#117	—	17	1	no	—
224	C₂×C₄⋊C₄	—	17	1	no	—
225	C₂×C₄⋊C₄	—	34	1	no	—
226	#107	—	17	1	no	—
227	#106	—	17	1	no	—
228	#106	—	17	1	no	—
229	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
230	#103	—	17	1	no	—
231	#103	—	17	1	no	—
232	#103	—	17	1	no	—
233	#103	—	17	1	no	—
234	#103	C₂×D₁₇	1	1	yes	characteristic
235	#103	C₂×D₁₇	1	1	yes	characteristic
236	#102	—	17	1	no	—
237	#102	—	17	1	no	—
238	#102	—	17	1	no	—
239	#102	—	17	1	no	—
240	C₂²⋊C₈	—	17	1	no	—
241	C₂²⋊C₈	—	17	1	no	—
242	#97	—	17	1	no	—
243	C₁₇	#3307	1	1	yes	characteristic, complemented, Sylow
244	#41	—	34	2	no	—
245	#41	—	34	2	no	—
246	#41	—	68	1	no	—
247	#41	—	68	1	no	—
248	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
249	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
250	C₂×D₄	—	17	1	no	—
251	C₂×D₄	—	17	1	no	—
252	C₂×D₄	—	34	1	no	—
253	C₂×D₄	—	34	1	no	—
254	C₂×D₄	—	34	1	no	—
255	C₂×D₄	—	34	1	no	—
256	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
257	C₂²×C₄	—	2	1	no	—
258	C₂²×C₄	—	17	1	no	—
259	C₂²×C₄	—	17	1	no	—
260	C₂²×C₄	—	34	1	no	—
261	C₂²×C₄	—	34	1	no	—
262	C₂²×C₄	—	34	1	no	—
263	C₂²×C₄	—	34	1	no	—
264	SD₁₆	—	4	1	no	—
265	SD₁₆	—	4	1	no	—
266	D₈	—	34	2	no	—
267	M₁₆	—	68	1	no	—
268	C₂×C₈	—	17	1	no	—
269	C₂×C₈	—	17	1	no	—
270	C₂×C₈	—	68	1	no	—
271	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
272	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	17	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	C₂²×D₁₇	1	1	yes	characteristic
285	C₂²⋊C₄	—	2	1	no	—
286	C₂²⋊C₄	—	17	1	no	—
287	C₂²⋊C₄	—	17	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	C₄²	—	34	1	no	—
295	C₄²	C₂²×D₁₇	1	1	yes	characteristic
296	C₂³	—	2	1	no	—
297	C₂³	—	17	1	no	—
298	C₂³	—	17	1	no	—
299	C₂³	—	34	1	no	—
300	Q₈	—	2	1	no	—
301	Q₈	—	2	1	no	—
302	Q₈	—	2	1	no	—
303	Q₈	—	2	1	no	—
304	D₄	—	2	1	no	—
305	D₄	—	2	1	no	—
306	D₄	—	17	2	no	—
307	D₄	—	34	1	no	—
308	D₄	—	68	1	no	—
309	D₄	—	68	2	no	—
310	D₄	—	68	2	no	—
311	D₄	—	68	2	no	—
312	D₄	—	136	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	8	1	no	—
317	C₂×C₄	—	17	1	no	—
318	C₂×C₄	—	17	1	no	—
319	C₂×C₄	—	34	1	no	—
320	C₂×C₄	—	34	1	no	—
321	C₂×C₄	—	34	1	no	—
322	C₂×C₄	—	34	1	no	—
323	C₂×C₄	—	34	1	no	—
324	C₂×C₄	—	34	1	no	—
325	C₂×C₄	—	34	1	no	—
326	C₂×C₄	—	68	2	no	—
327	C₂×C₄	—	68	2	no	—
328	C₂×C₄	—	68	2	no	—
329	C₂×C₄	—	136	1	no	—
330	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
331	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
332	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	34	1	no	—
336	C₈	—	34	1	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	4	1	no	—
339	C₂²	—	4	1	no	—
340	C₂²	—	34	1	no	—
341	C₂²	—	34	1	no	—
342	C₂²	—	34	2	no	—
343	C₂²	—	34	1	no	—
344	C₂²	—	34	1	no	—
345	C₂²	—	34	1	no	—
346	C₂²	—	34	1	no	—
347	C₂²	—	68	1	no	—
348	C₂²	—	68	1	no	—
349	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	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	#10587691	1	1	yes	characteristic
361	C₄	#41×D₁₇	1	1	yes	characteristic
362	C₂	—	8	1	no	—
363	C₂	—	68	1	no	—
364	C₂	—	68	1	no	—
365	C₂	—	68	2	no	—
366	C₂	#11763456	1	1	yes	characteristic
367	C₂	#11763608	1	1	yes	characteristic
368	C₂	#11763615	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×#461	C₂
socle	C₂²×C₁₇	C₂²×D₄
hypercenter	#461	D₁₇
hypocenter	C₁₇	#3307
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #461

Complements

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

N	H ≅ G/N	type
1. #11763509	C₂	semidirect
2. #11763462	C₂	semidirect
3. #11763177	C₂	semidirect
4. #11763149	C₂	semidirect
5. #11763138	C₂	semidirect
6. #11763110	C₂	semidirect
7. #11763052	C₂	semidirect
8. #11763035	C₂	semidirect
9. #11762953	C₂	semidirect
10. #11762953	C₂	semidirect
11. #11762919	C₂	semidirect
12. #11762879	C₂	semidirect
13. #11762809	C₂	semidirect
14. #11762804	C₂	semidirect
15. C₁₇×#461	C₂	semidirect
186. #461	D₁₇	semidirect
243. C₁₇	#3307	semidirect

Maximal subgroups

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

subgroup	index
#11763509	2
#11763462	2
#11763177	2
#11763149	2
#11763138	2
#11763110	2
#11763052	2
#11763035	2
#11762953	2
#11762919	2
#11762879	2
#11762809	2
#11762804	2
C₁₇×#461	2
#3307	17