Subgroups — #14538735

Order 2176.

418 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763375	C₂	1	1	yes	maximal, characteristic
2	#11763290	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763136	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763134	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763131	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763130	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763127	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763126	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763124	C₂	1	1	yes	maximal, characteristic, complemented
10	#11762850	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762846	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762812	C₂	1	1	yes	maximal, characteristic, complemented
13	#102×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762774	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#480	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₄×D₁₇	C₂²	1	1	yes	characteristic
17	C₂×#63252	C₂²	1	1	yes	characteristic
18	C₁₇×#121	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic, complemented
20	#10587626	—	2	1	no	—
21	#10587624	C₂²	1	1	yes	characteristic
22	#10587623	—	2	1	no	—
23	#10587616	C₂²	1	1	yes	characteristic
24	C₄×C₂²⋊D₁₇	—	2	1	no	—
25	C₄×C₂²⋊D₁₇	—	2	1	no	—
26	#10587612	C₂²	1	1	yes	characteristic, complemented
27	C₂×#63218	C₂²	1	1	yes	characteristic
28	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
29	#10587581	C₂²	1	1	yes	characteristic
30	#10587579	—	2	1	no	—
31	#10587578	C₂²	1	1	yes	characteristic
32	#10587578	C₂²	1	1	yes	characteristic
33	#10587578	C₂²	1	1	yes	characteristic, complemented
34	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic, complemented
36	#10587573	—	2	1	no	—
37	#10587571	—	2	1	no	—
38	#10587568	—	2	1	no	—
39	C₁₇×#107	C₂²	1	1	yes	characteristic, complemented
40	C₁₇×#107	C₂²	1	1	yes	characteristic, complemented
41	C₁₇×#102	C₂²	1	1	yes	characteristic
42	C₁₇×#102	C₂²	1	1	yes	characteristic
43	C₁₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic, complemented
44	#10587503	C₂²	1	1	yes	characteristic
45	#10587503	C₂²	1	1	yes	characteristic
46	C₂²⋊#3622	C₂²	1	1	yes	characteristic, complemented
47	#10587492	C₂²	1	1	yes	characteristic
48	#10587492	C₂²	1	1	yes	characteristic
49	#10587491	C₂²	1	1	yes	characteristic, complemented
50	#10587491	C₂²	1	1	yes	characteristic, complemented
51	#10587489	C₂²	1	1	yes	characteristic, complemented
52	#10587488	C₂²	1	1	yes	characteristic, complemented
53	#10587480	C₂²	1	1	yes	characteristic
54	#10587480	C₂²	1	1	yes	characteristic
55	#10587479	C₂²	1	1	yes	characteristic, complemented
56	#10587479	C₂²	1	1	yes	characteristic, complemented
57	#10587479	C₂²	1	1	yes	characteristic, complemented
58	#10587478	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₁₇	—	2	1	no	—
60	C₂×D₄×C₁₇	—	2	1	no	—
61	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₇	—	2	1	no	—
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	—	2	1	no	—
67	C₂×C₂²⋊D₁₇	D₄	1	1	yes	characteristic
68	C₂×C₂²⋊D₁₇	D₄	1	1	yes	characteristic
69	D₄×D₁₇	—	2	2	no	—
70	D₄×D₁₇	—	2	2	no	—
71	#63252	—	2	2	no	—
72	#63252	—	2	2	no	—
73	C₂×D₆₈	—	2	1	no	—
74	C₂×D₆₈	D₄	1	1	yes	characteristic
75	C₂×D₆₈	C₂³	1	1	yes	characteristic
76	C₂×C₄×D₁₇	—	2	1	no	—
77	C₂×C₄×D₁₇	—	2	1	no	—
78	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
79	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
80	C₂×Dic₃₄	D₄	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
85	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
86	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
87	C₂²⋊C₄×C₁₇	—	2	1	no	—
88	C₂²⋊Dic₁₇	—	2	1	no	—
89	C₂²⋊Dic₁₇	—	2	1	no	—
90	#63227	—	2	1	no	—
91	#63227	—	2	1	no	—
92	#63227	—	2	1	no	—
93	#63227	—	2	1	no	—
94	#63227	—	2	1	no	—
95	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
96	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
97	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
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	C₂×#3622	C₂³	1	1	yes	characteristic
105	C₂×#3622	C₂³	1	1	yes	characteristic
106	#63218	—	2	2	no	—
107	C₈×D₁₇	—	2	2	no	—
108	C₂³×C₁₇	—	2	1	no	—
109	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
110	C₂²×D₁₇	—	2	1	no	—
111	C₂²×D₁₇	—	4	2	no	—
112	C₂²×D₁₇	—	8	1	no	—
113	C₂²×D₁₇	—	8	1	no	—
114	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
115	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
116	D₄×C₁₇	—	2	1	no	—
117	D₄×C₁₇	—	2	1	no	—
118	D₄×C₁₇	—	8	1	no	—
119	C₂×C₄×C₁₇	—	2	1	no	—
120	C₂×C₄×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	C₂⁴	1	1	yes	characteristic
127	C₂²⋊D₁₇	—	2	2	no	—
128	C₂²⋊D₁₇	—	2	2	no	—
129	C₂²⋊D₁₇	—	4	2	no	—
130	C₂²⋊D₁₇	—	8	1	no	—
131	C₂²⋊D₁₇	—	8	1	no	—
132	C₂×Dic₁₇	—	2	1	no	—
133	C₂×Dic₁₇	—	2	1	no	—
134	C₂×Dic₁₇	—	2	1	no	—
135	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
137	D₆₈	—	2	2	no	—
138	D₆₈	—	2	1	no	—
139	D₆₈	—	2	1	no	—
140	D₆₈	—	8	1	no	—
141	C₄×D₁₇	—	2	1	no	—
142	C₄×D₁₇	—	2	1	no	—
143	C₄×D₁₇	—	8	1	no	—
144	C₄×D₁₇	—	8	1	no	—
145	C₄×D₁₇	#41	1	2	yes	—
146	C₄×D₁₇	#41	1	2	yes	—
147	Dic₃₄	—	2	2	no	—
148	C₈×C₁₇	—	2	1	no	—
149	C₈×C₁₇	—	2	1	no	—
150	#3622	—	2	1	no	—
151	#3622	—	2	1	no	—
152	#3096	—	17	1	no	maximal, Sylow
153	C₂²×C₁₇	—	2	1	no	—
154	C₂²×C₁₇	—	2	1	no	—
155	C₂²×C₁₇	—	2	1	no	—
156	C₂²×C₁₇	—	4	1	no	—
157	C₂²×C₁₇	—	4	1	no	—
158	C₂²×C₁₇	—	4	1	no	—
159	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
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	—	2	1	no	—
165	C₂×D₁₇	—	4	1	no	—
166	C₂×D₁₇	—	4	1	no	—
167	C₂×D₁₇	—	4	1	no	—
168	C₂×D₁₇	—	8	4	no	—
169	C₂×D₁₇	#123	1	2	yes	—
170	C₄×C₁₇	—	2	1	no	—
171	C₄×C₁₇	—	4	1	no	—
172	C₄×C₁₇	—	4	1	no	—
173	C₄×C₁₇	—	4	1	no	—
174	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
175	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
176	Dic₁₇	—	2	1	no	—
177	Dic₁₇	—	4	1	no	—
178	Dic₁₇	—	4	1	no	—
179	Dic₁₇	—	4	1	no	—
180	Dic₁₇	#123	1	2	yes	—
181	#535	—	17	1	no	—
182	#520	—	17	1	no	—
183	#482	—	17	1	no	—
184	#482	—	17	1	no	—
185	#482	—	17	1	no	—
186	#482	—	17	1	no	—
187	#481	—	17	1	no	—
188	#481	—	17	1	no	—
189	#480	—	17	1	no	—
190	#480	D₁₇	1	1	yes	characteristic, complemented
191	#428	—	17	1	no	—
192	#427	—	17	1	no	—
193	#418	—	17	1	no	—
194	C₂×#102	—	17	1	no	—
195	#408	—	17	1	no	—
196	C₂×C₁₇	—	4	1	no	—
197	C₂×C₁₇	—	8	1	no	—
198	C₂×C₁₇	C₂×#123	1	1	yes	characteristic
199	C₂×C₁₇	#576	1	1	yes	characteristic
200	C₂×C₁₇	#577	1	1	yes	characteristic
201	D₁₇	—	2	2	no	—
202	D₁₇	—	4	1	no	—
203	D₁₇	—	8	1	no	—
204	C₂×#41	—	17	1	no	—
205	C₂²×D₄	—	17	1	no	—
206	C₂×M₁₆	—	17	1	no	—
207	C₂²×C₈	—	17	1	no	—
208	#127	—	34	1	no	—
209	#124	—	34	1	no	—
210	#121	—	17	1	no	—
211	#121	—	17	1	no	—
212	#121	—	17	1	no	—
213	#121	C₂×D₁₇	1	1	yes	characteristic
214	#120	—	34	1	no	—
215	#120	—	34	1	no	—
216	C₄×Q₈	—	34	1	no	—
217	C₄×D₄	—	34	1	no	—
218	C₄×D₄	—	34	1	no	—
219	C₄×D₄	—	34	1	no	—
220	#117	—	17	1	no	—
221	#117	—	17	1	no	—
222	#117	—	17	1	no	—
223	#117	—	17	1	no	—
224	C₂×C₄⋊C₄	—	17	1	no	—
225	C₂×C₄⋊C₄	—	17	1	no	—
226	C₂×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic, complemented
227	#107	—	17	1	no	—
228	#107	—	17	1	no	—
229	#107	C₂×D₁₇	1	1	yes	characteristic, complemented
230	#107	C₂×D₁₇	1	1	yes	characteristic, complemented
231	#106	—	17	1	no	—
232	#106	—	17	1	no	—
233	#106	—	17	1	no	—
234	#106	—	17	1	no	—
235	#102	—	17	1	no	—
236	#102	—	17	1	no	—
237	#102	—	17	1	no	—
238	#102	—	17	1	no	—
239	#102	—	17	1	no	—
240	#102	—	17	1	no	—
241	#102	C₂×D₁₇	1	1	yes	characteristic
242	#102	C₂×D₁₇	1	1	yes	characteristic
243	C₂²⋊C₈	—	17	1	no	—
244	C₂²⋊C₈	—	17	1	no	—
245	C₂²⋊C₈	—	17	1	no	—
246	C₂²⋊C₈	C₂×D₁₇	1	1	yes	characteristic, complemented
247	C₁₇	#3096	1	1	yes	characteristic, complemented, Sylow
248	C₂⁴	—	34	1	no	—
249	#41	—	34	2	no	—
250	#41	—	34	2	no	—
251	C₂×Q₈	—	17	1	no	—
252	C₂×D₄	—	2	1	no	—
253	C₂×D₄	—	17	1	no	—
254	C₂×D₄	—	17	1	no	—
255	C₂×D₄	—	17	1	no	—
256	C₂×D₄	—	17	1	no	—
257	C₂×D₄	—	34	1	no	—
258	C₂×D₄	—	34	2	no	—
259	C₂×D₄	—	34	2	no	—
260	C₂×D₄	—	34	1	no	—
261	C₂×D₄	—	34	1	no	—
262	C₂×D₄	—	34	1	no	—
263	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
264	C₂²×C₄	—	2	1	no	—
265	C₂²×C₄	—	17	1	no	—
266	C₂²×C₄	—	17	1	no	—
267	C₂²×C₄	—	34	1	no	—
268	C₂²×C₄	—	34	1	no	—
269	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
270	M₁₆	—	34	2	no	—
271	C₂×C₈	—	17	1	no	—
272	C₂×C₈	—	17	1	no	—
273	C₂×C₈	—	34	2	no	—
274	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
275	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	17	1	no	—
278	C₄⋊C₄	—	17	1	no	—
279	C₄⋊C₄	—	17	1	no	—
280	C₄⋊C₄	—	34	1	no	—
281	C₄⋊C₄	—	34	1	no	—
282	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
283	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
284	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
285	C₂²⋊C₄	—	2	1	no	—
286	C₂²⋊C₄	—	34	1	no	—
287	C₂²⋊C₄	—	34	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	C₄²	—	34	1	no	—
294	C₄²	—	34	1	no	—
295	C₄²	—	34	1	no	—
296	C₄²	—	34	1	no	—
297	C₂³	—	2	1	no	—
298	C₂³	—	17	1	no	—
299	C₂³	—	17	1	no	—
300	C₂³	—	34	1	no	—
301	C₂³	—	68	2	no	—
302	C₂³	—	136	1	no	—
303	C₂³	—	136	1	no	—
304	C₂³	D₄×D₁₇	1	1	yes	characteristic
305	Q₈	—	34	2	no	—
306	D₄	—	2	1	no	—
307	D₄	—	2	1	no	—
308	D₄	—	8	1	no	—
309	D₄	—	34	2	no	—
310	D₄	—	34	2	no	—
311	D₄	—	34	2	no	—
312	D₄	—	34	1	no	—
313	D₄	—	34	1	no	—
314	D₄	—	68	2	no	—
315	D₄	—	136	1	no	—
316	D₄	—	136	1	no	—
317	D₄	—	136	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	8	1	no	—
324	C₂×C₄	—	17	1	no	—
325	C₂×C₄	—	17	2	no	—
326	C₂×C₄	—	17	2	no	—
327	C₂×C₄	—	17	1	no	—
328	C₂×C₄	—	34	1	no	—
329	C₂×C₄	—	34	1	no	—
330	C₂×C₄	—	34	1	no	—
331	C₂×C₄	—	34	1	no	—
332	C₂×C₄	—	34	1	no	—
333	C₂×C₄	—	136	1	no	—
334	C₂×C₄	—	136	1	no	—
335	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
336	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
337	C₈	—	2	1	no	—
338	C₈	—	2	1	no	—
339	C₈	—	34	1	no	—
340	C₈	—	34	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	4	1	no	—
345	C₂²	—	4	1	no	—
346	C₂²	—	4	1	no	—
347	C₂²	—	17	2	no	—
348	C₂²	—	34	1	no	—
349	C₂²	—	34	1	no	—
350	C₂²	—	34	1	no	—
351	C₂²	—	34	1	no	—
352	C₂²	—	34	1	no	—
353	C₂²	—	68	1	no	—
354	C₂²	—	68	1	no	—
355	C₂²	—	68	1	no	—
356	C₂²	—	136	4	no	—
357	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
358	C₄	—	2	1	no	—
359	C₄	—	4	1	no	—
360	C₄	—	4	1	no	—
361	C₄	—	4	1	no	—
362	C₄	—	17	2	no	—
363	C₄	—	34	1	no	—
364	C₄	—	68	1	no	—
365	C₄	—	68	1	no	—
366	C₄	—	68	1	no	—
367	C₄	#41×D₁₇	1	1	yes	characteristic
368	C₄	#41×D₁₇	1	1	yes	characteristic
369	C₂	—	4	1	no	—
370	C₂	—	8	1	no	—
371	C₂	—	34	2	no	—
372	C₂	—	68	1	no	—
373	C₂	—	136	1	no	—
374	C₂	#123×D₁₇	1	1	yes	characteristic
375	C₂	#11763610	1	1	yes	characteristic
376	C₂	#11763618	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×#480	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#11763375	C₂
hypercenter	#480	D₁₇
hypocenter	C₁₇	#3096
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #480

Complements

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

N	H ≅ G/N	type
2. #11763290	C₂	semidirect
3. #11763136	C₂	semidirect
4. #11763134	C₂	semidirect
5. #11763131	C₂	semidirect
6. #11763130	C₂	semidirect
7. #11763127	C₂	semidirect
8. #11763126	C₂	semidirect
9. #11763124	C₂	semidirect
10. #11762850	C₂	semidirect
11. #11762846	C₂	semidirect
12. #11762812	C₂	semidirect
13. #102×D₁₇	C₂	semidirect
14. #11762774	C₂	semidirect
15. C₁₇×#480	C₂	semidirect
19. C₂×C₄⋊C₄×C₁₇	C₂²	semidirect
26. #10587612	C₂²	semidirect
33. #10587578	C₂²	semidirect
35. C₄⋊C₄×D₁₇	C₂²	semidirect
39. C₁₇×#107	C₂²	semidirect
40. C₁₇×#107	C₂²	semidirect
43. C₁₇×C₂²⋊C₈	C₂²	semidirect
46. C₂²⋊#3622	C₂²	semidirect
49. #10587491	C₂²	semidirect
50. #10587491	C₂²	semidirect
51. #10587489	C₂²	semidirect
52. #10587488	C₂²	semidirect
55. #10587479	C₂²	semidirect
56. #10587479	C₂²	semidirect
57. #10587479	C₂²	semidirect
58. #10587478	C₂²	semidirect
190. #480	D₁₇	semidirect
226. C₂×C₄⋊C₄	C₂×D₁₇	semidirect
229. #107	C₂×D₁₇	semidirect
230. #107	C₂×D₁₇	semidirect
246. C₂²⋊C₈	C₂×D₁₇	semidirect
247. C₁₇	#3096	semidirect

Maximal subgroups

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

subgroup	index
#11763375	2
#11763290	2
#11763136	2
#11763134	2
#11763131	2
#11763130	2
#11763127	2
#11763126	2
#11763124	2
#11762850	2
#11762846	2
#11762812	2
#102×D₁₇	2
#11762774	2
C₁₇×#480	2
#3096	17