Subgroups — #14538895

Order 2176.

392 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763343	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#63252	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763151	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763134	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763117	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763110	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763032	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763025	C₂	1	1	yes	maximal, characteristic, complemented
9	#11762973	C₂	1	1	yes	maximal, characteristic, complemented
10	#11762956	C₂	1	1	yes	maximal, characteristic
11	#11762922	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762917	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762911	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762911	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#432	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#63252	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
18	#10587616	C₂²	1	1	yes	characteristic
19	#10587615	—	2	1	no	—
20	C₄×C₂²⋊D₁₇	—	2	1	no	—
21	C₄×C₂²⋊D₁₇	—	2	1	no	—
22	C₄×C₂²⋊D₁₇	—	2	1	no	—
23	#10587612	C₂²	1	1	yes	characteristic
24	#10587610	C₂²	1	1	yes	characteristic
25	C₂×Dic₆₈	C₂²	1	1	yes	characteristic
26	C₂×D₁₃₆	C₂²	1	1	yes	characteristic
27	C₂×#63219	C₂²	1	1	yes	characteristic
28	C₂×#63219	C₂²	1	1	yes	characteristic
29	#10587562	—	2	1	no	—
30	#10587561	C₂²	1	1	yes	characteristic
31	C₄×D₆₈	C₂²	1	1	yes	characteristic
32	C₄×D₆₈	C₂²	1	1	yes	characteristic
33	C₄×D₆₈	C₂²	1	1	yes	characteristic
34	#10587558	—	2	1	no	—
35	#10587558	—	2	1	no	—
36	C₄²×D₁₇	—	2	1	no	—
37	#10587556	C₂²	1	1	yes	characteristic
38	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
39	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
40	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
41	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
42	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₁₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
44	C₁₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
45	C₁₇×#97	C₂²	1	1	yes	characteristic
46	C₁₇×#97	C₂²	1	1	yes	characteristic
47	#10587492	C₂²	1	1	yes	characteristic
48	#10587492	C₂²	1	1	yes	characteristic
49	#10587492	C₂²	1	1	yes	characteristic
50	#10587492	C₂²	1	1	yes	characteristic
51	#10587489	C₂²	1	1	yes	characteristic
52	#10587489	C₂²	1	1	yes	characteristic
53	#10587488	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	#10587487	C₂²	1	1	yes	characteristic
58	#10587487	C₂²	1	1	yes	characteristic
59	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₇	—	2	1	no	—
63	C₂×C₂²⋊D₁₇	—	2	1	no	—
64	#63252	—	2	2	no	—
65	#63252	—	2	2	no	—
66	C₂×D₆₈	C₂³	1	1	yes	characteristic
67	C₂×D₆₈	C₂³	1	1	yes	characteristic
68	C₂×C₄×D₁₇	—	2	1	no	—
69	C₂×C₄×D₁₇	—	2	1	no	—
70	C₂×C₄×D₁₇	—	2	1	no	—
71	C₂×C₄×D₁₇	—	2	1	no	—
72	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
73	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
76	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
78	C₄²×C₁₇	D₄	1	1	yes	characteristic
79	C₄²×C₁₇	D₄	1	1	yes	characteristic
80	C₄²×C₁₇	C₂³	1	1	yes	characteristic
81	C₄²×C₁₇	C₂³	1	1	yes	characteristic
82	C₂²⋊Dic₁₇	—	2	1	no	—
83	C₂²⋊Dic₁₇	—	2	1	no	—
84	#63227	—	2	1	no	—
85	#63227	—	2	1	no	—
86	#63227	—	2	1	no	—
87	#63227	—	2	1	no	—
88	#63227	—	2	1	no	—
89	#63227	—	2	1	no	—
90	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
91	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
92	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
93	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
94	#63225	—	2	1	no	—
95	#63225	—	2	1	no	—
96	#63225	—	2	1	no	—
97	#63225	—	2	1	no	—
98	#63225	—	2	1	no	—
99	#63225	—	2	1	no	—
100	C₄×Dic₁₇	—	2	1	no	—
101	C₄×Dic₁₇	—	2	1	no	—
102	C₄×Dic₁₇	—	2	1	no	—
103	C₄×Dic₁₇	—	2	1	no	—
104	Dic₆₈	—	2	2	no	—
105	D₁₃₆	—	2	2	no	—
106	#63219	—	2	2	no	—
107	#63219	—	2	2	no	—
108	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₁₇	—	2	1	no	—
110	C₂²×D₁₇	—	2	1	no	—
111	C₂×C₄×C₁₇	—	2	1	no	—
112	C₂×C₄×C₁₇	—	2	1	no	—
113	C₂×C₄×C₁₇	—	4	1	no	—
114	C₂×C₄×C₁₇	—	4	1	no	—
115	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
121	C₂²⋊D₁₇	—	2	4	no	—
122	C₂²⋊D₁₇	—	4	2	no	—
123	C₂×Dic₁₇	—	2	1	no	—
124	C₂×Dic₁₇	—	2	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	D₆₈	—	2	1	no	—
130	D₆₈	—	2	1	no	—
131	D₆₈	—	2	1	no	—
132	D₆₈	#41	1	2	yes	—
133	C₄×D₁₇	—	2	4	no	—
134	C₄×D₁₇	—	4	2	no	—
135	C₄×D₁₇	—	4	2	no	—
136	C₄×D₁₇	—	4	2	no	—
137	Dic₃₄	—	2	1	no	—
138	Dic₃₄	—	2	1	no	—
139	Dic₃₄	—	2	1	no	—
140	Dic₃₄	#41	1	2	yes	—
141	C₈×C₁₇	—	2	1	no	—
142	C₈×C₁₇	—	2	1	no	—
143	C₈×C₁₇	—	2	1	no	—
144	C₈×C₁₇	—	2	1	no	—
145	#3111	—	17	1	no	maximal, Sylow
146	C₂²×C₁₇	—	2	1	no	—
147	C₂²×C₁₇	—	2	1	no	—
148	C₂²×C₁₇	—	2	1	no	—
149	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
150	C₂×D₁₇	—	2	2	no	—
151	C₂×D₁₇	—	4	1	no	—
152	C₂×D₁₇	—	4	1	no	—
153	C₂×D₁₇	—	4	1	no	—
154	C₂×D₁₇	—	4	1	no	—
155	C₂×D₁₇	—	4	1	no	—
156	C₄×C₁₇	—	2	1	no	—
157	C₄×C₁₇	—	2	1	no	—
158	C₄×C₁₇	—	2	1	no	—
159	C₄×C₁₇	—	2	1	no	—
160	C₄×C₁₇	—	2	1	no	—
161	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
162	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
163	Dic₁₇	—	2	2	no	—
164	Dic₁₇	—	4	1	no	—
165	Dic₁₇	—	4	1	no	—
166	Dic₁₇	—	4	1	no	—
167	Dic₁₇	—	4	1	no	—
168	Dic₁₇	—	4	1	no	—
169	#529	—	17	1	no	—
170	C₄×#41	—	17	1	no	—
171	#485	—	17	1	no	—
172	#482	—	17	1	no	—
173	#479	—	17	1	no	—
174	#478	—	17	1	no	—
175	#464	—	17	1	no	—
176	#463	—	17	1	no	—
177	#452	—	17	1	no	—
178	#449	—	17	1	no	—
179	#442	—	17	1	no	—
180	#441	—	17	1	no	—
181	#440	—	17	1	no	—
182	#440	—	17	1	no	—
183	#432	D₁₇	1	1	yes	characteristic, complemented
184	C₂×C₁₇	—	4	1	no	—
185	C₂×C₁₇	#525	1	1	yes	characteristic
186	C₂×C₁₇	#575	1	1	yes	characteristic
187	C₂×C₁₇	#575	1	1	yes	characteristic
188	D₁₇	—	4	2	no	—
189	D₁₇	—	8	1	no	—
190	C₂×#41	—	17	1	no	—
191	C₂×Q₁₆	—	17	1	no	—
192	C₂×SD₁₆	—	17	1	no	—
193	C₂×SD₁₆	—	17	1	no	—
194	C₂×D₈	—	17	1	no	—
195	#126	—	34	1	no	—
196	#125	—	17	1	no	—
197	#124	—	17	1	no	—
198	#123	—	34	1	no	—
199	C₂²⋊Q₈	—	17	1	no	—
200	#121	—	17	1	no	—
201	C₄×Q₈	—	17	1	no	—
202	C₄×Q₈	—	17	1	no	—
203	C₄×Q₈	—	17	1	no	—
204	C₄×D₄	—	17	1	no	—
205	C₄×D₄	—	17	1	no	—
206	C₄×D₄	—	17	1	no	—
207	C₄×D₄	—	34	1	no	—
208	C₄×D₄	—	34	1	no	—
209	C₄×D₄	—	34	1	no	—
210	#117	—	17	1	no	—
211	#117	—	34	1	no	—
212	#117	—	34	1	no	—
213	C₂×C₄²	—	34	1	no	—
214	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
215	#107	—	17	1	no	—
216	#107	—	17	1	no	—
217	#106	—	17	1	no	—
218	#106	—	17	1	no	—
219	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
220	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
221	#103	—	17	1	no	—
222	#103	—	17	1	no	—
223	#103	—	17	1	no	—
224	#103	—	17	1	no	—
225	#102	—	17	1	no	—
226	#102	—	17	1	no	—
227	#102	—	17	1	no	—
228	#102	—	17	1	no	—
229	C₂²⋊C₈	C₂×D₁₇	1	1	yes	characteristic
230	C₂²⋊C₈	C₂×D₁₇	1	1	yes	characteristic
231	#97	C₂×D₁₇	1	1	yes	characteristic
232	#97	C₂×D₁₇	1	1	yes	characteristic
233	C₁₇	#3111	1	1	yes	characteristic, complemented, Sylow
234	#41	—	34	2	no	—
235	#41	—	34	2	no	—
236	C₂×Q₈	—	17	1	no	—
237	C₂×Q₈	—	17	1	no	—
238	C₂×D₄	—	17	1	no	—
239	C₂×D₄	—	17	1	no	—
240	C₂×D₄	—	34	1	no	—
241	C₂×D₄	—	34	1	no	—
242	C₂²×C₄	—	34	1	no	—
243	C₂²×C₄	—	34	1	no	—
244	C₂²×C₄	—	34	1	no	—
245	C₂²×C₄	—	34	1	no	—
246	C₂²×C₄	D₆₈	1	1	yes	characteristic
247	C₂²×C₄	D₆₈	1	1	yes	characteristic
248	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
249	Q₁₆	—	34	2	no	—
250	SD₁₆	—	34	2	no	—
251	SD₁₆	—	34	2	no	—
252	D₈	—	34	2	no	—
253	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
254	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
255	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
256	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
257	C₄⋊C₄	—	17	1	no	—
258	C₄⋊C₄	—	17	1	no	—
259	C₄⋊C₄	—	17	1	no	—
260	C₄⋊C₄	—	17	1	no	—
261	C₄⋊C₄	—	34	1	no	—
262	C₄⋊C₄	—	34	1	no	—
263	C₄⋊C₄	—	34	1	no	—
264	C₄⋊C₄	—	34	1	no	—
265	C₄⋊C₄	—	34	1	no	—
266	C₄⋊C₄	—	34	1	no	—
267	C₂²⋊C₄	—	34	1	no	—
268	C₂²⋊C₄	—	34	1	no	—
269	C₂²⋊C₄	—	34	1	no	—
270	C₂²⋊C₄	—	34	1	no	—
271	C₂²⋊C₄	—	34	1	no	—
272	C₂²⋊C₄	—	34	1	no	—
273	C₂²⋊C₄	—	34	1	no	—
274	C₂²⋊C₄	—	34	1	no	—
275	C₄²	—	34	1	no	—
276	C₄²	—	34	1	no	—
277	C₄²	—	34	1	no	—
278	C₄²	—	34	1	no	—
279	C₄²	D₆₈	1	1	yes	characteristic
280	C₄²	D₆₈	1	1	yes	characteristic
281	C₄²	C₂²×D₁₇	1	1	yes	characteristic
282	C₄²	C₂²×D₁₇	1	1	yes	characteristic
283	C₂³	—	34	1	no	—
284	C₂³	—	34	1	no	—
285	C₂³	C₂×D₆₈	1	1	yes	characteristic
286	Q₈	—	17	2	no	—
287	Q₈	—	34	1	no	—
288	Q₈	—	34	1	no	—
289	Q₈	—	34	1	no	—
290	D₄	—	17	2	no	—
291	D₄	—	34	4	no	—
292	D₄	—	34	1	no	—
293	D₄	—	34	1	no	—
294	D₄	—	34	1	no	—
295	D₄	—	68	2	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	34	4	no	—
301	C₂×C₄	—	34	1	no	—
302	C₂×C₄	—	34	1	no	—
303	C₂×C₄	—	34	1	no	—
304	C₂×C₄	—	34	1	no	—
305	C₂×C₄	—	34	1	no	—
306	C₂×C₄	—	34	1	no	—
307	C₂×C₄	—	68	2	no	—
308	C₂×C₄	—	68	2	no	—
309	C₂×C₄	—	68	2	no	—
310	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
311	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
312	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
313	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
314	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
315	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	34	2	no	—
324	C₂²	—	68	1	no	—
325	C₂²	—	68	1	no	—
326	C₂²	—	68	1	no	—
327	C₂²	—	68	1	no	—
328	C₂²	—	68	1	no	—
329	C₂²	C₂²×D₆₈	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	34	2	no	—
336	C₄	—	68	1	no	—
337	C₄	—	68	1	no	—
338	C₄	—	68	1	no	—
339	C₄	—	68	1	no	—
340	C₄	—	68	1	no	—
341	C₄	#41×D₁₇	1	1	yes	characteristic
342	C₄	#41×D₁₇	1	1	yes	characteristic
343	C₂	—	4	1	no	—
344	C₂	—	68	2	no	—
345	C₂	—	136	1	no	—
346	C₂	#11763321	1	1	yes	characteristic
347	C₂	#11763609	1	1	yes	characteristic
348	C₂	#11763609	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₆₈
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#432	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#11762956	C₂
hypercenter	#432	D₁₇
hypocenter	C₁₇	#3111
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

Complements

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

N	H ≅ G/N	type
1. #11763343	C₂	semidirect
2. C₄×#63252	C₂	semidirect
3. #11763151	C₂	semidirect
4. #11763134	C₂	semidirect
5. #11763117	C₂	semidirect
6. #11763110	C₂	semidirect
7. #11763032	C₂	semidirect
8. #11763025	C₂	semidirect
9. #11762973	C₂	semidirect
11. #11762922	C₂	semidirect
12. #11762917	C₂	semidirect
13. #11762911	C₂	semidirect
14. #11762911	C₂	semidirect
15. C₁₇×#432	C₂	semidirect
183. #432	D₁₇	semidirect
233. C₁₇	#3111	semidirect

Maximal subgroups

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

subgroup	index
#11763343	2
C₄×#63252	2
#11763151	2
#11763134	2
#11763117	2
#11763110	2
#11763032	2
#11763025	2
#11762973	2
#11762956	2
#11762922	2
#11762917	2
#11762911	2
C₁₇×#432	2
#3111	17