Subgroups — #14541844

Order 2176.

368 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	#11763532	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763204	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763196	C₂	1	1	yes	maximal, characteristic
4	#11763191	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763031	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763028	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763024	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763021	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763014	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763006	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762940	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762919	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762914	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#63229	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#559	C₂	1	1	yes	maximal, characteristic, complemented
16	C₁₇×#127	C₂²	1	1	yes	characteristic
17	C₁₇×#124	—	2	1	no	—
18	C₁₇×#124	C₂²	1	1	yes	characteristic
19	C₁₇×#124	C₂²	1	1	yes	characteristic
20	C₁₇×#123	—	2	1	no	—
21	C₁₇×#123	—	2	1	no	—
22	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
23	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
24	C₄×D₄×C₁₇	—	2	1	no	—
25	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
27	#10587632	—	2	1	no	—
28	C₂×#63231	C₂²	1	1	yes	characteristic
29	C₂×#63231	C₂²	1	1	yes	characteristic
30	C₂×#63230	C₂²	1	1	yes	characteristic
31	C₂×#63229	C₂²	1	1	yes	characteristic
32	C₂×#63229	C₂²	1	1	yes	characteristic
33	C₂×#63229	C₂²	1	1	yes	characteristic
34	C₂×#63229	C₂²	1	1	yes	characteristic
35	C₂×#63228	C₂²	1	1	yes	characteristic
36	C₂×#63228	C₂²	1	1	yes	characteristic
37	#10587584	—	2	1	no	—
38	#10587584	—	2	1	no	—
39	#10587578	—	2	1	no	—
40	#10587561	C₂²	1	1	yes	characteristic
41	#10587561	C₂²	1	1	yes	characteristic
42	C₄×D₆₈	C₂²	1	1	yes	characteristic
43	#10587555	C₂²	1	1	yes	characteristic
44	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
45	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
46	#10587506	C₂²	1	1	yes	characteristic
47	#10587503	C₂²	1	1	yes	characteristic
48	#10587503	C₂²	1	1	yes	characteristic
49	#10587481	C₂²	1	1	yes	characteristic
50	#10587481	C₂²	1	1	yes	characteristic
51	#10587480	C₂²	1	1	yes	characteristic
52	#10587479	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	D₄	1	1	yes	characteristic
60	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₁₇	—	2	1	no	—
62	C₂×D₄×C₁₇	—	2	1	no	—
63	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
64	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₁₇	—	2	1	no	—
66	C₂²×C₄×C₁₇	—	2	1	no	—
67	C₂×D₆₈	C₂³	1	1	yes	characteristic
68	C₂×C₄×D₁₇	—	2	1	no	—
69	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
70	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₁₇	—	2	1	no	—
72	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
73	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
74	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
75	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
76	C₂²⋊C₄×C₁₇	—	2	1	no	—
77	C₂²⋊C₄×C₁₇	—	2	1	no	—
78	C₂²⋊C₄×C₁₇	—	2	1	no	—
79	C₂²⋊C₄×C₁₇	—	2	1	no	—
80	C₂²⋊C₄×C₁₇	—	2	1	no	—
81	C₂²⋊C₄×C₁₇	—	2	1	no	—
82	C₄²×C₁₇	—	2	1	no	—
83	C₄²×C₁₇	C₂³	1	1	yes	characteristic
84	C₄²×C₁₇	C₂³	1	1	yes	characteristic
85	C₄²×C₁₇	C₂³	1	1	yes	characteristic
86	#63231	—	2	2	no	—
87	#63231	—	4	1	no	—
88	#63230	—	4	1	no	—
89	#63229	—	2	2	no	—
90	#63229	—	4	1	no	—
91	#63229	—	4	1	no	—
92	#63229	—	4	1	no	—
93	#63228	—	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	—	2	1	no	—
99	#63227	—	2	1	no	—
100	C₄⋊Dic₁₇	—	2	1	no	—
101	C₄⋊Dic₁₇	—	2	1	no	—
102	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
103	#63225	—	2	1	no	—
104	#63225	—	2	1	no	—
105	C₄×Dic₁₇	—	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	C₂×#3622	C₂³	1	1	yes	characteristic
110	C₂×#3622	C₂³	1	1	yes	characteristic
111	C₂³×C₁₇	—	2	1	no	—
112	C₂³×C₁₇	—	2	1	no	—
113	C₂²×D₁₇	—	2	1	no	—
114	Q₈×C₁₇	—	2	1	no	—
115	Q₈×C₁₇	—	2	1	no	—
116	Q₈×C₁₇	—	4	1	no	—
117	D₄×C₁₇	—	2	1	no	—
118	D₄×C₁₇	—	2	1	no	—
119	D₄×C₁₇	—	2	1	no	—
120	D₄×C₁₇	—	2	1	no	—
121	D₄×C₁₇	—	4	2	no	—
122	D₄×C₁₇	—	8	1	no	—
123	C₂×C₄×C₁₇	—	2	1	no	—
124	C₂×C₄×C₁₇	—	2	1	no	—
125	C₂×C₄×C₁₇	—	8	1	no	—
126	C₂×C₄×C₁₇	—	8	1	no	—
127	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
134	C₂×Dic₁₇	—	2	1	no	—
135	C₂×Dic₁₇	—	2	1	no	—
136	C₂×Dic₁₇	—	2	1	no	—
137	D₆₈	—	2	1	no	—
138	D₆₈	—	2	1	no	—
139	C₄×D₁₇	—	4	2	no	—
140	Dic₃₄	—	2	1	no	—
141	Dic₃₄	—	2	1	no	—
142	Dic₃₄	—	2	1	no	—
143	Dic₃₄	—	2	1	no	—
144	#3622	—	2	1	no	—
145	#3622	—	2	1	no	—
146	#3622	—	2	1	no	—
147	#3622	#41	1	2	yes	—
148	#3401	—	17	1	no	maximal, Sylow
149	C₂²×C₁₇	—	4	1	no	—
150	C₂²×C₁₇	—	4	1	no	—
151	C₂²×C₁₇	—	4	1	no	—
152	C₂²×C₁₇	—	4	1	no	—
153	C₂²×C₁₇	—	4	1	no	—
154	C₂²×C₁₇	—	4	1	no	—
155	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
156	C₂×D₁₇	—	4	1	no	—
157	C₂×D₁₇	—	4	1	no	—
158	C₂×D₁₇	—	4	1	no	—
159	C₄×C₁₇	—	2	1	no	—
160	C₄×C₁₇	—	2	1	no	—
161	C₄×C₁₇	—	2	1	no	—
162	C₄×C₁₇	—	2	1	no	—
163	C₄×C₁₇	—	2	1	no	—
164	C₄×C₁₇	—	2	1	no	—
165	C₄×C₁₇	—	4	1	no	—
166	C₄×C₁₇	—	4	1	no	—
167	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
168	C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
169	Dic₁₇	—	4	1	no	—
170	Dic₁₇	—	4	1	no	—
171	Dic₁₇	—	4	1	no	—
172	#559	D₁₇	1	1	yes	characteristic, complemented
173	#556	—	17	1	no	—
174	#497	—	17	1	no	—
175	#496	—	17	1	no	—
176	#495	—	17	1	no	—
177	#464	—	17	1	no	—
178	#464	—	17	1	no	—
179	#463	—	17	1	no	—
180	#463	—	17	1	no	—
181	#461	—	17	1	no	—
182	#460	—	17	1	no	—
183	#446	—	17	1	no	—
184	#442	—	17	1	no	—
185	#441	—	17	1	no	—
186	C₄×SD₁₆	—	17	1	no	—
187	C₂×C₁₇	—	8	1	no	—
188	C₂×C₁₇	—	8	1	no	—
189	C₂×C₁₇	#550	1	1	yes	characteristic
190	C₂×C₁₇	#575	1	1	yes	characteristic
191	C₂×C₁₇	#577	1	1	yes	characteristic
192	D₁₇	—	8	1	no	—
193	C₂×Q₁₆	—	17	1	no	—
194	C₂×Q₁₆	—	17	1	no	—
195	C₂×SD₁₆	—	17	1	no	—
196	C₂×SD₁₆	—	17	1	no	—
197	C₂×SD₁₆	—	17	1	no	—
198	C₂×SD₁₆	—	17	1	no	—
199	C₂×SD₁₆	—	17	1	no	—
200	C₂×D₈	—	17	1	no	—
201	C₂×D₈	—	17	1	no	—
202	C₄⋊Q₈	—	17	1	no	—
203	#127	C₂×D₁₇	1	1	yes	characteristic
204	#126	—	34	1	no	—
205	#126	—	34	1	no	—
206	#124	—	2	1	no	—
207	#124	—	17	1	no	—
208	#124	—	17	1	no	—
209	#124	C₂×D₁₇	1	1	yes	characteristic
210	#124	C₂×D₁₇	1	1	yes	characteristic
211	#123	—	2	1	no	—
212	#123	—	2	1	no	—
213	C₂²⋊Q₈	—	34	1	no	—
214	C₄×Q₈	—	17	1	no	—
215	C₄×Q₈	—	17	1	no	—
216	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
217	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
218	C₄×D₄	—	2	1	no	—
219	C₄×D₄	—	17	1	no	—
220	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
221	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
222	#117	—	34	1	no	—
223	#106	—	17	1	no	—
224	C₄⋊C₈	—	17	1	no	—
225	C₄⋊C₈	—	17	1	no	—
226	C₄⋊C₈	—	17	1	no	—
227	#103	—	17	1	no	—
228	#103	—	17	1	no	—
229	#103	—	17	1	no	—
230	#102	—	17	1	no	—
231	#102	—	17	1	no	—
232	#102	—	17	1	no	—
233	#97	—	17	1	no	—
234	#97	—	17	1	no	—
235	C₄×C₈	—	17	1	no	—
236	C₁₇	#3401	1	1	yes	characteristic, complemented, Sylow
237	C₂×Q₈	—	17	1	no	—
238	C₂×Q₈	—	17	1	no	—
239	C₂×Q₈	C₂²⋊D₁₇	1	1	yes	characteristic
240	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
241	C₂×D₄	—	2	1	no	—
242	C₂×D₄	—	2	1	no	—
243	C₂×D₄	—	17	1	no	—
244	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
245	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	2	1	no	—
248	C₂²×C₄	—	34	1	no	—
249	Q₁₆	—	34	2	no	—
250	Q₁₆	—	68	1	no	—
251	SD₁₆	—	34	2	no	—
252	SD₁₆	—	68	1	no	—
253	SD₁₆	—	68	1	no	—
254	SD₁₆	—	68	1	no	—
255	SD₁₆	—	68	1	no	—
256	D₈	—	34	2	no	—
257	D₈	—	68	1	no	—
258	C₂×C₈	—	17	1	no	—
259	C₂×C₈	—	17	1	no	—
260	C₂×C₈	—	17	1	no	—
261	C₂×C₈	—	17	1	no	—
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	17	1	no	—
264	C₄⋊C₄	—	34	1	no	—
265	C₄⋊C₄	—	34	1	no	—
266	C₄⋊C₄	—	34	1	no	—
267	C₄⋊C₄	—	34	1	no	—
268	C₄⋊C₄	C₂²⋊D₁₇	1	1	yes	characteristic
269	C₄⋊C₄	C₂²⋊D₁₇	1	1	yes	characteristic
270	C₄⋊C₄	C₂²⋊D₁₇	1	1	yes	characteristic
271	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
272	C₂²⋊C₄	—	2	1	no	—
273	C₂²⋊C₄	—	2	1	no	—
274	C₂²⋊C₄	—	2	1	no	—
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	2	1	no	—
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	34	1	no	—
279	C₂²⋊C₄	—	34	1	no	—
280	C₂²⋊C₄	—	34	1	no	—
281	C₂²⋊C₄	—	34	1	no	—
282	C₂²⋊C₄	—	34	1	no	—
283	C₄²	—	2	1	no	—
284	C₄²	—	34	1	no	—
285	C₄²	—	34	1	no	—
286	C₄²	C₂²×D₁₇	1	1	yes	characteristic
287	C₄²	C₂²×D₁₇	1	1	yes	characteristic
288	C₄²	C₂²×D₁₇	1	1	yes	characteristic
289	C₂³	—	2	1	no	—
290	C₂³	—	2	1	no	—
291	C₂³	—	34	1	no	—
292	Q₈	—	2	1	no	—
293	Q₈	—	2	1	no	—
294	Q₈	—	4	1	no	—
295	Q₈	—	34	1	no	—
296	Q₈	—	34	1	no	—
297	Q₈	—	34	1	no	—
298	Q₈	—	34	1	no	—
299	D₄	—	2	1	no	—
300	D₄	—	2	1	no	—
301	D₄	—	2	1	no	—
302	D₄	—	2	1	no	—
303	D₄	—	4	2	no	—
304	D₄	—	8	1	no	—
305	D₄	—	34	1	no	—
306	D₄	—	34	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	8	1	no	—
310	C₂×C₄	—	8	1	no	—
311	C₂×C₄	—	34	1	no	—
312	C₂×C₄	—	34	1	no	—
313	C₂×C₄	—	34	1	no	—
314	C₂×C₄	—	68	2	no	—
315	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
316	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
317	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
318	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
319	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
320	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
321	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
322	C₈	—	17	2	no	—
323	C₈	—	34	1	no	—
324	C₈	—	34	1	no	—
325	C₈	—	34	1	no	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	68	1	no	—
333	C₂²	—	68	1	no	—
334	C₂²	—	68	1	no	—
335	C₂²	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	4	1	no	—
343	C₄	—	4	1	no	—
344	C₄	—	68	1	no	—
345	C₄	—	68	1	no	—
346	C₄	—	68	1	no	—
347	C₄	#41×D₁₇	1	1	yes	characteristic
348	C₄	#10587696	1	1	yes	characteristic
349	C₂	—	8	1	no	—
350	C₂	—	8	1	no	—
351	C₂	—	136	1	no	—
352	C₂	#11763478	1	1	yes	characteristic
353	C₂	#11763607	1	1	yes	characteristic
354	C₂	#11763620	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₇
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#559	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#11763196	C₂
hypercenter	#559	D₁₇
hypocenter	C₁₇	#3401
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #559

Complements

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

N	H ≅ G/N	type
1. #11763532	C₂	semidirect
2. #11763204	C₂	semidirect
4. #11763191	C₂	semidirect
5. #11763031	C₂	semidirect
6. #11763028	C₂	semidirect
7. #11763024	C₂	semidirect
8. #11763021	C₂	semidirect
9. #11763014	C₂	semidirect
10. #11763006	C₂	semidirect
11. #11762940	C₂	semidirect
12. #11762919	C₂	semidirect
13. #11762914	C₂	semidirect
14. C₄×#63229	C₂	semidirect
15. C₁₇×#559	C₂	semidirect
172. #559	D₁₇	semidirect
236. C₁₇	#3401	semidirect

Maximal subgroups

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

subgroup	index
#11763532	2
#11763204	2
#11763196	2
#11763191	2
#11763031	2
#11763028	2
#11763024	2
#11763021	2
#11763014	2
#11763006	2
#11762940	2
#11762919	2
#11762914	2
C₄×#63229	2
C₁₇×#559	2
#3401	17