Subgroups — #14541374

Order 2176.

386 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763523	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763459	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763206	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763131	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763130	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763126	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763125	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763097	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763087	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763059	C₂	1	1	yes	maximal, characteristic, complemented
11	D₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762885	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762851	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762848	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#469	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#63254	—	2	1	no	—
17	C₂×D₈×C₁₇	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
19	C₁₇×#121	C₂²	1	1	yes	characteristic
20	C₁₇×#121	C₂²	1	1	yes	characteristic
21	#10587625	—	2	1	no	—
22	#10587624	C₂²	1	1	yes	characteristic
23	#10587624	C₂²	1	1	yes	characteristic
24	#10587622	—	2	1	no	—
25	#10587621	—	2	1	no	—
26	D₄×Dic₁₇	C₂²	1	1	yes	characteristic
27	D₄×Dic₁₇	C₂²	1	1	yes	characteristic
28	C₄×C₂²⋊D₁₇	C₂²	1	1	yes	characteristic
29	#10587612	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
31	C₂×#63218	C₂²	1	1	yes	characteristic
32	#10587578	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
34	#10587574	C₂²	1	1	yes	characteristic
35	#10587574	C₂²	1	1	yes	characteristic
36	#10587571	—	2	1	no	—
37	#10587569	—	2	1	no	—
38	#10587564	—	2	1	no	—
39	#10587563	—	2	1	no	—
40	C₁₇×#106	C₂²	1	1	yes	characteristic
41	C₁₇×#102	C₂²	1	1	yes	characteristic
42	C₁₇×#102	C₂²	1	1	yes	characteristic
43	#10587503	C₂²	1	1	yes	characteristic
44	#10587503	C₂²	1	1	yes	characteristic
45	#10587503	C₂²	1	1	yes	characteristic
46	#10587503	C₂²	1	1	yes	characteristic
47	C₂²⋊#3622	C₂²	1	1	yes	characteristic
48	#10587491	C₂²	1	1	yes	characteristic
49	#10587489	C₂²	1	1	yes	characteristic
50	#10587489	C₂²	1	1	yes	characteristic
51	#10587488	C₂²	1	1	yes	characteristic
52	#10587488	C₂²	1	1	yes	characteristic
53	#10587485	C₂²	1	1	yes	characteristic
54	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
55	#10587479	C₂²	1	1	yes	characteristic
56	#10587479	C₂²	1	1	yes	characteristic
57	#10587479	C₂²	1	1	yes	characteristic
58	#10587478	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₁₇	—	2	1	no	—
60	C₂×D₄×C₁₇	—	2	1	no	—
61	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₁₇	—	2	1	no	—
65	C₂×C₂²⋊D₁₇	—	2	1	no	—
66	C₂×C₂²⋊D₁₇	D₄	1	1	yes	characteristic
67	C₂²×Dic₁₇	—	2	1	no	—
68	C₂²×Dic₁₇	—	2	1	no	—
69	C₂²×Dic₁₇	—	2	1	no	—
70	#63254	—	4	4	no	—
71	C₂×C₄×D₁₇	—	2	1	no	—
72	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
73	C₂×Dic₃₄	—	2	1	no	—
74	C₂×Dic₃₄	—	2	1	no	—
75	D₈×C₁₇	—	2	2	no	—
76	M₁₆×C₁₇	—	2	2	no	—
77	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
78	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
80	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
81	C₂²⋊C₄×C₁₇	—	2	1	no	—
82	C₂²⋊C₄×C₁₇	—	2	1	no	—
83	C₂²⋊Dic₁₇	—	2	1	no	—
84	C₂²⋊Dic₁₇	—	2	1	no	—
85	C₂²⋊Dic₁₇	—	2	1	no	—
86	C₂²⋊Dic₁₇	—	2	1	no	—
87	C₂²⋊Dic₁₇	—	2	1	no	—
88	C₂²⋊Dic₁₇	D₄	1	1	yes	characteristic
89	#63227	—	2	1	no	—
90	#63227	D₄	1	1	yes	characteristic
91	C₄⋊Dic₁₇	—	2	1	no	—
92	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
93	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
94	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
95	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
96	#63225	—	2	1	no	—
97	#63225	—	2	1	no	—
98	#63225	—	2	1	no	—
99	#63225	D₄	1	1	yes	characteristic
100	C₄×Dic₁₇	—	2	1	no	—
101	C₄×Dic₁₇	—	2	1	no	—
102	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
103	C₂×#3622	C₂³	1	1	yes	characteristic
104	C₂×#3622	C₂³	1	1	yes	characteristic
105	#63218	—	2	2	no	—
106	C₂³×C₁₇	—	2	1	no	—
107	C₂³×C₁₇	—	2	1	no	—
108	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
110	D₄×C₁₇	—	2	1	no	—
111	D₄×C₁₇	—	2	1	no	—
112	D₄×C₁₇	—	2	1	no	—
113	D₄×C₁₇	—	2	1	no	—
114	D₄×C₁₇	—	4	2	no	—
115	D₄×C₁₇	—	8	1	no	—
116	C₂×C₄×C₁₇	—	2	1	no	—
117	C₂×C₄×C₁₇	—	2	1	no	—
118	C₂×C₄×C₁₇	—	2	1	no	—
119	C₂×C₄×C₁₇	—	2	1	no	—
120	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
122	C₂²⋊D₁₇	—	2	2	no	—
123	C₂²⋊D₁₇	—	4	2	no	—
124	C₂²⋊D₁₇	—	8	1	no	—
125	C₂×Dic₁₇	—	2	1	no	—
126	C₂×Dic₁₇	—	2	1	no	—
127	C₂×Dic₁₇	—	2	1	no	—
128	C₂×Dic₁₇	—	2	1	no	—
129	C₂×Dic₁₇	—	4	2	no	—
130	C₂×Dic₁₇	—	4	2	no	—
131	C₂×Dic₁₇	—	4	2	no	—
132	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
133	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
135	C₄×D₁₇	—	2	1	no	—
136	C₄×D₁₇	—	2	1	no	—
137	C₄×D₁₇	—	4	2	no	—
138	Dic₃₄	—	4	2	no	—
139	Dic₃₄	—	8	1	no	—
140	C₈×C₁₇	—	2	1	no	—
141	C₈×C₁₇	#41	1	2	yes	complemented
142	#3622	—	2	1	no	—
143	#3622	—	2	1	no	—
144	#3354	—	17	1	no	maximal, Sylow
145	C₂²×C₁₇	—	2	1	no	—
146	C₂²×C₁₇	—	2	1	no	—
147	C₂²×C₁₇	—	2	1	no	—
148	C₂²×C₁₇	—	4	1	no	—
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	C₂²×D₄	1	1	yes	characteristic
155	C₂×D₁₇	—	2	1	no	—
156	C₂×D₁₇	—	2	1	no	—
157	C₂×D₁₇	—	2	1	no	—
158	C₄×C₁₇	—	2	1	no	—
159	C₄×C₁₇	—	4	1	no	—
160	C₄×C₁₇	—	4	1	no	—
161	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
162	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
163	Dic₁₇	—	2	1	no	—
164	Dic₁₇	—	2	1	no	—
165	Dic₁₇	—	4	1	no	—
166	Dic₁₇	—	4	1	no	—
167	Dic₁₇	—	4	1	no	—
168	Dic₁₇	—	4	1	no	—
169	Dic₁₇	AGL(1,ℤ/8)	1	2	yes	—
170	#555	—	17	1	no	—
171	#547	—	17	1	no	—
172	#498	—	17	1	no	—
173	#482	—	17	1	no	—
174	#482	—	17	1	no	—
175	#481	—	17	1	no	—
176	#481	—	17	1	no	—
177	#476	—	17	1	no	—
178	#474	—	17	1	no	—
179	#469	—	17	1	no	—
180	#469	D₁₇	1	1	yes	characteristic, complemented
181	C₄×D₈	—	17	1	no	—
182	#436	—	17	1	no	—
183	#428	—	17	1	no	—
184	#428	—	17	1	no	—
185	C₂×C₁₇	—	4	1	no	—
186	C₂×C₁₇	—	8	1	no	—
187	C₂×C₁₇	—	8	1	no	—
188	C₂×C₁₇	#547	1	1	yes	characteristic
189	C₂×C₁₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
190	C₂×C₁₇	#577	1	1	yes	characteristic
191	D₁₇	—	4	1	no	—
192	C₂×#41	—	34	1	no	—
193	C₂×D₈	C₂×D₁₇	1	1	yes	characteristic
194	C₂×M₁₆	—	17	1	no	—
195	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic
196	C₄⋊Q₈	—	17	1	no	—
197	C₄⋊Q₈	—	17	1	no	—
198	#124	—	34	1	no	—
199	#124	—	34	1	no	—
200	#123	—	34	1	no	—
201	#123	—	34	1	no	—
202	C₂²⋊Q₈	—	34	1	no	—
203	#121	—	17	1	no	—
204	#121	—	17	1	no	—
205	#121	—	34	1	no	—
206	#121	C₂×D₁₇	1	1	yes	characteristic
207	#121	C₂×D₁₇	1	1	yes	characteristic
208	C₄×D₄	—	17	1	no	—
209	C₄×D₄	—	17	1	no	—
210	C₄×D₄	—	17	1	no	—
211	#117	—	17	1	no	—
212	#117	—	17	1	no	—
213	#117	—	34	1	no	—
214	C₂×C₄⋊C₄	—	17	1	no	—
215	C₂×C₄⋊C₄	—	17	1	no	—
216	#107	—	17	1	no	—
217	#107	—	17	1	no	—
218	#107	—	17	1	no	—
219	#106	—	17	1	no	—
220	#106	—	17	1	no	—
221	#106	—	17	1	no	—
222	#106	—	17	1	no	—
223	#106	—	17	1	no	—
224	#106	C₂×D₁₇	1	1	yes	characteristic
225	C₄⋊C₈	—	17	1	no	—
226	#102	—	17	1	no	—
227	#102	—	17	1	no	—
228	#102	—	17	1	no	—
229	#102	—	17	1	no	—
230	#102	C₂×D₁₇	1	1	yes	characteristic
231	#102	C₂×D₁₇	1	1	yes	characteristic
232	C₂²⋊C₈	—	17	1	no	—
233	C₂²⋊C₈	—	17	1	no	—
234	C₄×C₈	—	17	1	no	—
235	C₁₇	#3354	1	1	yes	characteristic, complemented, Sylow
236	#41	—	68	4	no	—
237	C₂×Q₈	—	34	1	no	—
238	C₂×Q₈	—	34	1	no	—
239	C₂×D₄	—	2	1	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	—	17	1	no	—
242	C₂×D₄	—	34	1	no	—
243	C₂×D₄	—	34	1	no	—
244	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
245	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
246	C₂²×C₄	—	17	1	no	—
247	C₂²×C₄	—	34	1	no	—
248	C₂²×C₄	—	34	1	no	—
249	C₂²×C₄	—	34	1	no	—
250	C₂²×C₄	—	34	1	no	—
251	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
252	D₈	—	2	2	no	—
253	M₁₆	—	2	2	no	—
254	M₁₆	—	34	2	no	—
255	C₂×C₈	—	17	1	no	—
256	C₂×C₈	—	17	1	no	—
257	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
258	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
259	C₄⋊C₄	—	17	1	no	—
260	C₄⋊C₄	—	17	1	no	—
261	C₄⋊C₄	—	17	1	no	—
262	C₄⋊C₄	—	17	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	C₂²⋊C₄	—	2	1	no	—
271	C₂²⋊C₄	—	2	1	no	—
272	C₂²⋊C₄	—	17	1	no	—
273	C₂²⋊C₄	—	17	1	no	—
274	C₂²⋊C₄	—	34	1	no	—
275	C₂²⋊C₄	—	34	1	no	—
276	C₂²⋊C₄	—	34	1	no	—
277	C₂²⋊C₄	—	34	1	no	—
278	C₂²⋊C₄	—	34	1	no	—
279	C₂²⋊C₄	—	34	1	no	—
280	C₄²	—	17	1	no	—
281	C₄²	—	34	1	no	—
282	C₄²	—	34	1	no	—
283	C₂³	—	2	1	no	—
284	C₂³	—	2	1	no	—
285	C₂³	—	17	1	no	—
286	C₂³	D₄×D₁₇	1	1	yes	characteristic
287	Q₈	—	68	2	no	—
288	Q₈	—	136	1	no	—
289	D₄	—	2	1	no	—
290	D₄	—	2	1	no	—
291	D₄	—	2	1	no	—
292	D₄	—	2	1	no	—
293	D₄	—	4	2	no	—
294	D₄	—	8	1	no	—
295	D₄	—	34	2	no	—
296	D₄	—	68	2	no	—
297	D₄	—	136	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	17	1	no	—
303	C₂×C₄	—	17	1	no	—
304	C₂×C₄	—	17	1	no	—
305	C₂×C₄	—	34	1	no	—
306	C₂×C₄	—	34	1	no	—
307	C₂×C₄	—	34	1	no	—
308	C₂×C₄	—	34	1	no	—
309	C₂×C₄	—	34	1	no	—
310	C₂×C₄	—	34	1	no	—
311	C₂×C₄	—	68	2	no	—
312	C₂×C₄	—	68	2	no	—
313	C₂×C₄	—	68	2	no	—
314	C₂×C₄	—	68	2	no	—
315	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
316	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
317	C₈	—	2	1	no	—
318	C₈	—	34	1	no	—
319	C₈	—	34	1	no	—
320	C₈	#63254	1	2	yes	complemented
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	34	1	no	—
331	C₂²	—	34	1	no	—
332	C₂²	—	34	1	no	—
333	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
334	C₄	—	2	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	17	2	no	—
338	C₄	—	34	1	no	—
339	C₄	—	34	1	no	—
340	C₄	—	68	1	no	—
341	C₄	—	68	1	no	—
342	C₄	—	68	1	no	—
343	C₄	—	68	1	no	—
344	C₄	C₂×#63254	1	1	yes	characteristic
345	C₄	#10587697	1	1	yes	characteristic
346	C₂	—	4	1	no	—
347	C₂	—	8	1	no	—
348	C₂	—	8	1	no	—
349	C₂	—	68	1	no	—
350	C₂	#11763459	1	1	yes	characteristic
351	C₂	AGL(1,ℤ/8)×D₁₇	1	1	yes	characteristic
352	C₂	#11763619	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×#469	C₂
socle	C₂²×C₁₇	C₂²×D₄
hypercenter	#469	D₁₇
hypocenter	C₁₇	#3354
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #469

Complements

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

N	H ≅ G/N	type
1. #11763523	C₂	semidirect
2. #11763459	C₂	semidirect
3. #11763206	C₂	semidirect
4. #11763131	C₂	semidirect
5. #11763130	C₂	semidirect
6. #11763126	C₂	semidirect
7. #11763125	C₂	semidirect
8. #11763097	C₂	semidirect
9. #11763087	C₂	semidirect
10. #11763059	C₂	semidirect
11. D₈×Dic₁₇	C₂	semidirect
12. #11762885	C₂	semidirect
13. #11762851	C₂	semidirect
14. #11762848	C₂	semidirect
15. C₁₇×#469	C₂	semidirect
141. C₈×C₁₇	#41	semidirect
180. #469	D₁₇	semidirect
235. C₁₇	#3354	semidirect
320. C₈	#63254	semidirect

Maximal subgroups

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

subgroup	index
#11763523	2
#11763459	2
#11763206	2
#11763131	2
#11763130	2
#11763126	2
#11763125	2
#11763097	2
#11763087	2
#11763059	2
D₈×Dic₁₇	2
#11762885	2
#11762851	2
#11762848	2
C₁₇×#469	2
#3354	17