Subgroups — #14540747

Order 2176.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763463	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763455	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763169	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763140	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763128	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763100	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763062	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763038	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763025	C₂	1	1	yes	maximal, characteristic, complemented
10	#11762956	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762945	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762922	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762811	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#10587492	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#435	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#63254	—	2	1	no	—
17	C₂×#63252	C₂²	1	1	yes	characteristic
18	C₂²×D₆₈	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic, complemented
20	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
21	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
22	#10587623	—	2	1	no	—
23	#10587616	C₂²	1	1	yes	characteristic
24	#10587616	C₂²	1	1	yes	characteristic
25	#10587616	C₂²	1	1	yes	characteristic, complemented
26	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic, complemented
28	#10587610	C₂²	1	1	yes	characteristic
29	C₂×D₁₃₆	C₂²	1	1	yes	characteristic
30	C₂×D₁₃₆	C₂²	1	1	yes	characteristic
31	C₂×#63219	C₂²	1	1	yes	characteristic, complemented
32	#10587583	—	2	1	no	—
33	#10587580	—	2	1	no	—
34	#10587572	—	2	1	no	—
35	#10587572	—	2	1	no	—
36	#10587570	—	2	1	no	—
37	C₂²⋊C₄×D₁₇	—	2	1	no	—
38	#10587561	C₂²	1	1	yes	characteristic
39	C₄×D₆₈	C₂²	1	1	yes	characteristic
40	C₄×D₆₈	C₂²	1	1	yes	characteristic, complemented
41	#10587555	C₂²	1	1	yes	characteristic, complemented
42	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
43	C₁₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
44	C₁₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic, complemented
45	C₁₇×#97	C₂²	1	1	yes	characteristic, complemented
46	#10587492	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	#10587492	C₂²	1	1	yes	characteristic
52	#10587492	C₂²	1	1	yes	characteristic, complemented
53	#10587488	C₂²	1	1	yes	characteristic, complemented
54	#10587488	C₂²	1	1	yes	characteristic, complemented
55	#10587488	C₂²	1	1	yes	characteristic, complemented
56	#10587487	C₂²	1	1	yes	characteristic, complemented
57	#10587487	C₂²	1	1	yes	characteristic, complemented
58	#10587487	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₁₇	—	2	1	no	—
60	C₂×D₄×C₁₇	D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₇	—	2	1	no	—
64	C₂×C₂²⋊D₁₇	—	2	1	no	—
65	C₂×C₂²⋊D₁₇	—	2	1	no	—
66	C₂×C₂²⋊D₁₇	—	2	1	no	—
67	C₂²×Dic₁₇	—	2	1	no	—
68	C₂²×Dic₁₇	—	2	1	no	—
69	#63254	—	4	4	no	—
70	#63252	—	2	2	no	—
71	#63252	—	4	1	no	—
72	C₂×D₆₈	—	2	2	no	—
73	C₂×D₆₈	—	2	1	no	—
74	C₂×D₆₈	—	4	1	no	—
75	C₂×D₆₈	C₂³	1	1	yes	characteristic
76	C₂×D₆₈	C₂³	1	1	yes	characteristic
77	C₂×D₆₈	C₂³	1	1	yes	characteristic
78	C₂×C₄×D₁₇	—	2	1	no	—
79	C₂×C₄×D₁₇	—	2	1	no	—
80	C₂×C₄×D₁₇	—	2	1	no	—
81	C₂×Dic₃₄	—	2	1	no	—
82	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
83	M₁₆×C₁₇	—	4	1	no	—
84	C₂×C₈×C₁₇	—	4	1	no	—
85	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
86	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
87	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
88	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
89	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
90	C₂²⋊C₄×C₁₇	D₄	1	1	yes	characteristic
91	C₂²⋊C₄×C₁₇	D₄	1	1	yes	characteristic
92	C₄²×C₁₇	C₂³	1	1	yes	characteristic
93	C₂²⋊Dic₁₇	—	2	1	no	—
94	#63227	—	2	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	#63227	—	2	1	no	—
101	C₄⋊Dic₁₇	—	2	1	no	—
102	C₄⋊Dic₁₇	—	2	1	no	—
103	C₄⋊Dic₁₇	—	2	1	no	—
104	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
105	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
106	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
107	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
108	#63225	—	2	1	no	—
109	D₁₃₆	—	2	2	no	—
110	D₁₃₆	—	4	1	no	—
111	#63219	—	4	1	no	—
112	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
113	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
114	C₂²×D₁₇	—	2	1	no	—
115	C₂²×D₁₇	—	2	1	no	—
116	C₂²×D₁₇	—	2	1	no	—
117	C₂²×D₁₇	—	4	2	no	—
118	C₂²×D₁₇	—	4	2	no	—
119	C₂²×D₁₇	—	8	1	no	—
120	D₄×C₁₇	—	2	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	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
130	C₂²⋊D₁₇	—	4	2	no	—
131	C₂²⋊D₁₇	—	4	2	no	—
132	C₂²⋊D₁₇	—	4	2	no	—
133	C₂²⋊D₁₇	—	8	1	no	—
134	C₂×Dic₁₇	—	2	1	no	—
135	C₂×Dic₁₇	—	2	1	no	—
136	C₂×Dic₁₇	—	2	1	no	—
137	C₂×Dic₁₇	—	2	1	no	—
138	C₂×Dic₁₇	—	2	1	no	—
139	C₂×Dic₁₇	—	4	2	no	—
140	C₂×Dic₁₇	—	4	2	no	—
141	D₆₈	—	2	1	no	—
142	D₆₈	—	2	1	no	—
143	D₆₈	—	2	1	no	—
144	D₆₈	—	2	1	no	—
145	D₆₈	—	2	1	no	—
146	D₆₈	—	4	2	no	—
147	D₆₈	#41	1	2	yes	complemented
148	C₄×D₁₇	—	4	2	no	—
149	C₄×D₁₇	—	4	2	no	—
150	C₄×D₁₇	—	8	1	no	—
151	Dic₃₄	—	2	1	no	—
152	Dic₃₄	—	2	1	no	—
153	Dic₃₄	—	4	2	no	—
154	C₈×C₁₇	—	2	1	no	—
155	C₈×C₁₇	—	2	1	no	—
156	C₈×C₁₇	—	2	1	no	—
157	C₈×C₁₇	—	2	1	no	—
158	#3306	—	17	1	no	maximal, Sylow
159	C₂²×C₁₇	—	2	1	no	—
160	C₂²×C₁₇	—	2	1	no	—
161	C₂²×C₁₇	—	2	1	no	—
162	C₂²×C₁₇	—	2	1	no	—
163	C₂²×C₁₇	—	2	1	no	—
164	C₂²×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
165	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
166	C₂×D₁₇	—	2	2	no	—
167	C₂×D₁₇	—	4	1	no	—
168	C₂×D₁₇	—	4	1	no	—
169	C₂×D₁₇	—	4	1	no	—
170	C₂×D₁₇	—	4	1	no	—
171	C₂×D₁₇	—	4	1	no	—
172	C₂×D₁₇	—	4	1	no	—
173	C₂×D₁₇	—	4	1	no	—
174	C₂×D₁₇	—	4	1	no	—
175	C₂×D₁₇	—	8	4	no	—
176	C₄×C₁₇	—	2	1	no	—
177	C₄×C₁₇	—	2	1	no	—
178	C₄×C₁₇	—	2	1	no	—
179	C₄×C₁₇	—	2	1	no	—
180	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
181	C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
182	Dic₁₇	—	4	1	no	—
183	Dic₁₇	—	4	1	no	—
184	Dic₁₇	—	4	1	no	—
185	Dic₁₇	—	4	1	no	—
186	Dic₁₇	—	4	1	no	—
187	#547	—	17	1	no	—
188	#546	—	17	1	no	—
189	#489	—	17	1	no	—
190	#483	—	17	1	no	—
191	#481	—	17	1	no	—
192	#476	—	17	1	no	—
193	#469	—	17	1	no	—
194	#465	—	17	1	no	—
195	#463	—	17	1	no	—
196	#449	—	17	1	no	—
197	#447	—	17	1	no	—
198	#442	—	17	1	no	—
199	#435	D₁₇	1	1	yes	characteristic, complemented
200	#417	—	17	1	no	—
201	C₂×#102	—	17	1	no	—
202	C₂×C₁₇	—	2	2	no	—
203	C₂×C₁₇	—	4	1	no	—
204	C₂×C₁₇	#546	1	1	yes	characteristic
205	C₂×C₁₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
206	C₂×C₁₇	#577	1	1	yes	characteristic
207	D₁₇	—	4	2	no	—
208	D₁₇	—	8	1	no	—
209	D₁₇	—	8	1	no	—
210	C₂×#41	—	17	1	no	—
211	C₂×#41	—	34	1	no	—
212	C₂²×D₄	—	17	1	no	—
213	C₂×SD₁₆	—	17	1	no	—
214	C₂×D₈	—	17	1	no	—
215	C₂×D₈	—	17	1	no	—
216	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic, complemented
217	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
218	C₄⋊Q₈	—	17	1	no	—
219	#124	—	17	1	no	—
220	#123	—	34	1	no	—
221	#123	—	34	1	no	—
222	#123	—	34	1	no	—
223	C₂²⋊Q₈	—	17	1	no	—
224	C₂²⋊Q₈	—	34	1	no	—
225	#121	—	17	1	no	—
226	#121	—	17	1	no	—
227	#121	—	17	1	no	—
228	#121	—	34	1	no	—
229	#120	—	34	1	no	—
230	C₄×D₄	—	17	1	no	—
231	C₄×D₄	—	17	1	no	—
232	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
233	C₂×C₄⋊C₄	—	17	1	no	—
234	C₂×C₄⋊C₄	—	17	1	no	—
235	C₂×C₂²⋊C₄	—	34	1	no	—
236	#106	—	17	1	no	—
237	#106	—	17	1	no	—
238	#106	—	17	1	no	—
239	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic, complemented
240	#103	—	17	1	no	—
241	#103	—	17	1	no	—
242	#103	—	17	1	no	—
243	#102	—	17	1	no	—
244	#102	—	17	1	no	—
245	#102	—	17	1	no	—
246	#102	—	17	1	no	—
247	#102	—	17	1	no	—
248	#102	—	17	1	no	—
249	#102	—	17	1	no	—
250	C₂²⋊C₈	C₂×D₁₇	1	1	yes	characteristic
251	C₂²⋊C₈	C₂×D₁₇	1	1	yes	characteristic, complemented
252	#97	C₂×D₁₇	1	1	yes	characteristic, complemented
253	C₁₇	#3306	1	1	yes	characteristic, complemented, Sylow
254	C₂⁴	—	34	1	no	—
255	#41	—	34	2	no	—
256	#41	—	68	1	no	—
257	#41	—	68	4	no	—
258	C₂×Q₈	—	17	1	no	—
259	C₂×Q₈	—	34	1	no	—
260	C₂×D₄	—	17	1	no	—
261	C₂×D₄	—	17	1	no	—
262	C₂×D₄	—	17	1	no	—
263	C₂×D₄	—	34	2	no	—
264	C₂×D₄	—	34	1	no	—
265	C₂×D₄	—	34	1	no	—
266	C₂×D₄	—	34	1	no	—
267	C₂×D₄	—	34	1	no	—
268	C₂×D₄	—	34	1	no	—
269	C₂×D₄	—	68	1	no	—
270	C₂×D₄	D₆₈	1	1	yes	characteristic
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	C₂²×C₄	—	34	1	no	—
276	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
277	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
278	SD₁₆	—	68	1	no	—
279	D₈	—	34	2	no	—
280	D₈	—	68	1	no	—
281	M₁₆	—	4	1	no	—
282	C₂×C₈	—	4	1	no	—
283	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
284	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
285	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
286	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
287	C₄⋊C₄	—	17	1	no	—
288	C₄⋊C₄	—	17	1	no	—
289	C₄⋊C₄	—	17	1	no	—
290	C₄⋊C₄	—	17	1	no	—
291	C₄⋊C₄	—	34	1	no	—
292	C₄⋊C₄	—	34	1	no	—
293	C₄⋊C₄	—	34	1	no	—
294	C₄⋊C₄	—	34	1	no	—
295	C₄⋊C₄	D₆₈	1	1	yes	characteristic
296	C₂²⋊C₄	—	34	1	no	—
297	C₂²⋊C₄	—	34	1	no	—
298	C₂²⋊C₄	—	34	1	no	—
299	C₂²⋊C₄	—	34	1	no	—
300	C₂²⋊C₄	—	34	1	no	—
301	C₂²⋊C₄	—	34	1	no	—
302	C₂²⋊C₄	—	34	1	no	—
303	C₂²⋊C₄	—	34	1	no	—
304	C₂²⋊C₄	D₆₈	1	1	yes	characteristic
305	C₂²⋊C₄	D₆₈	1	1	yes	characteristic
306	C₄²	C₂²×D₁₇	1	1	yes	characteristic
307	C₂³	—	34	1	no	—
308	C₂³	—	34	1	no	—
309	C₂³	—	34	1	no	—
310	C₂³	—	68	2	no	—
311	C₂³	—	68	2	no	—
312	C₂³	—	136	1	no	—
313	C₂³	C₂×D₆₈	1	1	yes	characteristic
314	C₂³	C₂×D₆₈	1	1	yes	characteristic
315	Q₈	—	34	1	no	—
316	Q₈	—	34	1	no	—
317	Q₈	—	68	2	no	—
318	D₄	—	2	2	no	—
319	D₄	—	17	2	no	—
320	D₄	—	34	1	no	—
321	D₄	—	34	1	no	—
322	D₄	—	34	1	no	—
323	D₄	—	34	1	no	—
324	D₄	—	34	1	no	—
325	D₄	—	68	2	no	—
326	D₄	—	68	2	no	—
327	D₄	—	68	2	no	—
328	D₄	—	68	2	no	—
329	D₄	—	136	1	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	34	1	no	—
335	C₂×C₄	—	34	1	no	—
336	C₂×C₄	—	34	1	no	—
337	C₂×C₄	—	34	1	no	—
338	C₂×C₄	—	34	1	no	—
339	C₂×C₄	—	68	2	no	—
340	C₂×C₄	—	68	2	no	—
341	C₂×C₄	—	68	2	no	—
342	C₂×C₄	—	68	2	no	—
343	C₂×C₄	—	136	1	no	—
344	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
345	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
346	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
347	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
348	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
349	C₈	—	2	1	no	—
350	C₈	—	2	1	no	—
351	C₈	—	2	1	no	—
352	C₈	—	2	1	no	—
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	34	2	no	—
359	C₂²	—	68	1	no	—
360	C₂²	—	68	1	no	—
361	C₂²	—	68	1	no	—
362	C₂²	—	68	1	no	—
363	C₂²	—	68	1	no	—
364	C₂²	—	68	1	no	—
365	C₂²	—	68	1	no	—
366	C₂²	—	68	1	no	—
367	C₂²	—	136	4	no	—
368	C₂²	#10587594	1	2	yes	—
369	C₂²	C₂²×D₆₈	1	1	yes	characteristic
370	C₄	—	2	1	no	—
371	C₄	—	2	1	no	—
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	68	1	no	—
375	C₄	—	68	1	no	—
376	C₄	—	68	1	no	—
377	C₄	—	68	1	no	—
378	C₄	—	68	1	no	—
379	C₄	#10587691	1	1	yes	characteristic
380	C₄	#41×D₁₇	1	1	yes	characteristic
381	C₂	—	2	2	no	—
382	C₂	—	4	1	no	—
383	C₂	—	68	2	no	—
384	C₂	—	136	1	no	—
385	C₂	—	136	1	no	—
386	C₂	#11763454	1	1	yes	characteristic
387	C₂	C₂×#10587594	1	1	yes	characteristic
388	C₂	#11763621	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₆₈
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#435	C₂
socle	C₂²×C₁₇	C₂²×D₄
hypercenter	#435	D₁₇
hypocenter	C₁₇	#3306
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #435

Complements

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

N	H ≅ G/N	type
1. #11763463	C₂	semidirect
2. #11763455	C₂	semidirect
3. #11763169	C₂	semidirect
4. #11763140	C₂	semidirect
5. #11763128	C₂	semidirect
6. #11763100	C₂	semidirect
7. #11763062	C₂	semidirect
8. #11763038	C₂	semidirect
9. #11763025	C₂	semidirect
10. #11762956	C₂	semidirect
11. #11762945	C₂	semidirect
12. #11762922	C₂	semidirect
13. #11762811	C₂	semidirect
14. C₂×#10587492	C₂	semidirect
15. C₁₇×#435	C₂	semidirect
19. C₂×M₁₆×C₁₇	C₂²	semidirect
25. #10587616	C₂²	semidirect
27. C₂×C₄⋊Dic₁₇	C₂²	semidirect
31. C₂×#63219	C₂²	semidirect
40. C₄×D₆₈	C₂²	semidirect
41. #10587555	C₂²	semidirect
42. C₁₇×C₄⋊C₈	C₂²	semidirect
44. C₁₇×C₂²⋊C₈	C₂²	semidirect
45. C₁₇×#97	C₂²	semidirect
52. #10587492	C₂²	semidirect
53. #10587488	C₂²	semidirect
54. #10587488	C₂²	semidirect
55. #10587488	C₂²	semidirect
56. #10587487	C₂²	semidirect
57. #10587487	C₂²	semidirect
58. #10587487	C₂²	semidirect
147. D₆₈	#41	semidirect
199. #435	D₁₇	semidirect
216. C₂×M₁₆	C₂×D₁₇	semidirect
239. C₄⋊C₈	C₂×D₁₇	semidirect
251. C₂²⋊C₈	C₂×D₁₇	semidirect
252. #97	C₂×D₁₇	semidirect
253. C₁₇	#3306	semidirect

Maximal subgroups

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

subgroup	index
#11763463	2
#11763455	2
#11763169	2
#11763140	2
#11763128	2
#11763100	2
#11763062	2
#11763038	2
#11763025	2
#11762956	2
#11762945	2
#11762922	2
#11762811	2
C₂×#10587492	2
C₁₇×#435	2
#3306	17