Subgroups — #14538967

Order 2176.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763428	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763281	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763148	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763132	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763100	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763089	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763021	C₂	1	1	yes	maximal, characteristic, complemented
8	#11762958	C₂	1	1	yes	maximal, characteristic, complemented
9	#11762941	C₂	1	1	yes	maximal, characteristic
10	#11762921	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762912	C₂	1	1	yes	maximal, characteristic, complemented
12	SD₁₆×Dic₁₇	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762889	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762869	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#463	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₄×D₁₇	C₂²	1	1	yes	characteristic
17	C₂×SD₁₆×C₁₇	C₂²	1	1	yes	characteristic
18	C₂×D₈×C₁₇	C₂²	1	1	yes	characteristic
19	C₁₇×#124	C₂²	1	1	yes	characteristic, complemented
20	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
21	#10587632	C₂²	1	1	yes	characteristic
22	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic, complemented
23	C₂×#63230	C₂²	1	1	yes	characteristic
24	#10587624	C₂²	1	1	yes	characteristic
25	D₄×Dic₁₇	C₂²	1	1	yes	characteristic
26	D₄×Dic₁₇	C₂²	1	1	yes	characteristic, complemented
27	C₂×#63228	C₂²	1	1	yes	characteristic
28	C₄×C₂²⋊D₁₇	—	2	1	no	—
29	C₂×#63227	—	2	1	no	—
30	#10587579	C₂²	1	1	yes	characteristic
31	#10587578	C₂²	1	1	yes	characteristic
32	#10587572	—	2	1	no	—
33	#10587571	—	2	1	no	—
34	#10587567	—	2	1	no	—
35	C₂²⋊C₄×D₁₇	—	2	1	no	—
36	#10587565	—	2	1	no	—
37	#10587564	—	2	1	no	—
38	C₄×D₆₈	C₂²	1	1	yes	characteristic
39	#10587558	C₂²	1	1	yes	characteristic
40	#10587555	C₂²	1	1	yes	characteristic, complemented
41	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
42	C₁₇×#103	C₂²	1	1	yes	characteristic, complemented
43	C₁₇×#102	C₂²	1	1	yes	characteristic, complemented
44	#10587506	C₂²	1	1	yes	characteristic, complemented
45	#10587503	C₂²	1	1	yes	characteristic
46	#10587503	C₂²	1	1	yes	characteristic
47	#10587503	C₂²	1	1	yes	characteristic, complemented
48	#10587492	C₂²	1	1	yes	characteristic
49	#10587492	C₂²	1	1	yes	characteristic
50	#10587491	C₂²	1	1	yes	characteristic
51	#10587491	C₂²	1	1	yes	characteristic
52	#10587488	C₂²	1	1	yes	characteristic, complemented
53	#10587488	C₂²	1	1	yes	characteristic, complemented
54	#10587486	C₂²	1	1	yes	characteristic, complemented
55	C₈×Dic₁₇	C₂²	1	1	yes	characteristic, complemented
56	#10587478	C₂²	1	1	yes	characteristic, complemented
57	#10587478	C₂²	1	1	yes	characteristic, complemented
58	C₄⋊#3622	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₁₇	—	2	1	no	—
60	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₇	—	2	1	no	—
64	C₂×C₂²⋊D₁₇	—	2	1	no	—
65	C₂×C₂²⋊D₁₇	—	2	1	no	—
66	C₂²×Dic₁₇	—	2	1	no	—
67	C₂²×Dic₁₇	—	2	1	no	—
68	D₄×D₁₇	—	2	2	no	—
69	D₄×D₁₇	—	2	2	no	—
70	C₂×D₆₈	C₂³	1	1	yes	characteristic
71	C₂×C₄×D₁₇	—	2	1	no	—
72	C₂×C₄×D₁₇	—	2	1	no	—
73	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
74	C₂×Dic₃₄	—	2	1	no	—
75	SD₁₆×C₁₇	—	2	2	no	—
76	D₈×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	C₂²⋊C₄×C₁₇	—	2	1	no	—
81	C₂²⋊C₄×C₁₇	—	2	1	no	—
82	C₂²⋊C₄×C₁₇	—	2	1	no	—
83	C₄²×C₁₇	C₂³	1	1	yes	characteristic
84	C₂²⋊Dic₁₇	—	2	1	no	—
85	C₂²⋊Dic₁₇	—	2	1	no	—
86	C₂²⋊Dic₁₇	—	2	1	no	—
87	#63230	—	2	2	no	—
88	#63228	—	2	2	no	—
89	#63227	—	2	1	no	—
90	#63227	—	2	1	no	—
91	#63227	—	2	1	no	—
92	#63227	—	2	1	no	—
93	#63227	D₄	1	1	yes	characteristic
94	#63227	D₄	1	1	yes	characteristic
95	C₄⋊Dic₁₇	—	2	1	no	—
96	C₄⋊Dic₁₇	—	2	1	no	—
97	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
98	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
99	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
100	#63225	—	2	1	no	—
101	#63225	D₄	1	1	yes	characteristic
102	#63225	D₄	1	1	yes	characteristic
103	C₄×Dic₁₇	—	2	1	no	—
104	C₄×Dic₁₇	—	2	1	no	—
105	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
106	C₂×#3622	C₂³	1	1	yes	characteristic
107	C₂×#3622	C₂³	1	1	yes	characteristic
108	C₂³×C₁₇	—	2	1	no	—
109	C₂³×C₁₇	—	2	1	no	—
110	C₂²×D₁₇	—	2	1	no	—
111	C₂²×D₁₇	—	2	4	no	—
112	C₂²×D₁₇	—	8	1	no	—
113	C₂²×D₁₇	—	8	1	no	—
114	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
115	Q₈×C₁₇	—	2	1	no	—
116	Q₈×C₁₇	—	2	1	no	—
117	D₄×C₁₇	—	2	1	no	—
118	D₄×C₁₇	—	2	1	no	—
119	D₄×C₁₇	—	2	1	no	—
120	D₄×C₁₇	#41	1	2	yes	—
121	C₂×C₄×C₁₇	—	2	1	no	—
122	C₂×C₄×C₁₇	—	2	1	no	—
123	C₂×C₄×C₁₇	—	8	1	no	—
124	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
127	C₂²⋊D₁₇	—	2	4	no	—
128	C₂²⋊D₁₇	—	8	1	no	—
129	C₂×Dic₁₇	—	2	1	no	—
130	C₂×Dic₁₇	—	2	1	no	—
131	C₂×Dic₁₇	—	2	1	no	—
132	C₂×Dic₁₇	—	8	1	no	—
133	C₂×Dic₁₇	—	8	1	no	—
134	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
135	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
137	D₆₈	—	2	1	no	—
138	D₆₈	#41	1	2	yes	—
139	C₄×D₁₇	—	2	1	no	—
140	C₄×D₁₇	—	2	1	no	—
141	C₄×D₁₇	—	8	1	no	—
142	C₄×D₁₇	—	8	1	no	—
143	Dic₃₄	—	8	1	no	—
144	C₈×C₁₇	—	2	1	no	—
145	C₈×C₁₇	—	2	1	no	—
146	#3622	—	2	1	no	—
147	#3622	—	2	1	no	—
148	#3118	—	17	1	no	maximal, Sylow
149	C₂²×C₁₇	—	2	2	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	—	2	1	no	—
157	C₂×D₁₇	—	2	2	no	—
158	C₂×D₁₇	—	2	1	no	—
159	C₂×D₁₇	—	2	1	no	—
160	C₂×D₁₇	—	4	1	no	—
161	C₂×D₁₇	—	4	1	no	—
162	C₂×D₁₇	—	8	4	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	C₂×#41	1	1	yes	characteristic
169	Dic₁₇	—	2	1	no	—
170	Dic₁₇	—	2	1	no	—
171	Dic₁₇	—	2	1	no	—
172	Dic₁₇	—	4	1	no	—
173	Dic₁₇	—	4	1	no	—
174	Dic₁₇	—	4	1	no	—
175	#542	—	17	1	no	—
176	#518	—	17	1	no	—
177	#485	—	17	1	no	—
178	#482	—	17	1	no	—
179	#476	—	17	1	no	—
180	#474	—	17	1	no	—
181	#463	—	17	1	no	—
182	#463	D₁₇	1	1	yes	characteristic, complemented
183	#450	—	17	1	no	—
184	#447	—	17	1	no	—
185	#442	—	17	1	no	—
186	#440	—	17	1	no	—
187	C₄×SD₁₆	—	17	1	no	—
188	C₄×D₈	—	17	1	no	—
189	#433	—	17	1	no	—
190	C₂×C₁₇	—	4	2	no	—
191	C₂×C₁₇	—	8	1	no	—
192	C₂×C₁₇	#525	1	1	yes	characteristic
193	C₂×C₁₇	#576	1	1	yes	characteristic
194	C₂×C₁₇	#577	1	1	yes	characteristic
195	D₁₇	—	4	1	no	—
196	D₁₇	—	4	2	no	—
197	C₂²×D₄	—	17	1	no	—
198	C₂×SD₁₆	—	17	1	no	—
199	C₂×SD₁₆	C₂×D₁₇	1	1	yes	characteristic
200	C₂×D₈	—	17	1	no	—
201	C₂×D₈	C₂×D₁₇	1	1	yes	characteristic
202	C₄⋊Q₈	—	17	1	no	—
203	#126	—	34	1	no	—
204	#124	—	34	1	no	—
205	#124	C₂×D₁₇	1	1	yes	characteristic, complemented
206	#123	—	34	1	no	—
207	C₂²⋊Q₈	—	17	1	no	—
208	C₂²⋊Q₈	—	34	1	no	—
209	#121	—	17	1	no	—
210	C₄×Q₈	—	17	1	no	—
211	C₄×D₄	—	17	1	no	—
212	C₄×D₄	—	17	1	no	—
213	C₄×D₄	—	17	1	no	—
214	C₄×D₄	—	17	1	no	—
215	C₄×D₄	—	34	1	no	—
216	C₄×D₄	—	34	1	no	—
217	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
218	#117	—	17	1	no	—
219	#117	—	17	1	no	—
220	C₂×C₂²⋊C₄	—	34	1	no	—
221	C₂×C₂²⋊C₄	—	34	1	no	—
222	#107	—	17	1	no	—
223	#107	—	17	1	no	—
224	#106	—	17	1	no	—
225	#106	—	17	1	no	—
226	C₄⋊C₈	—	17	1	no	—
227	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic, complemented
228	#103	—	17	1	no	—
229	#103	C₂×D₁₇	1	1	yes	characteristic, complemented
230	#102	—	17	1	no	—
231	#102	—	17	1	no	—
232	#102	—	17	1	no	—
233	#102	—	17	1	no	—
234	#102	—	17	1	no	—
235	#102	C₂×D₁₇	1	1	yes	characteristic, complemented
236	C₂²⋊C₈	—	17	1	no	—
237	C₂²⋊C₈	—	17	1	no	—
238	#97	—	17	1	no	—
239	C₄×C₈	—	17	1	no	—
240	C₁₇	#3118	1	1	yes	characteristic, complemented, Sylow
241	C₂⁴	—	34	1	no	—
242	C₂×Q₈	—	34	1	no	—
243	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
244	C₂×D₄	—	17	1	no	—
245	C₂×D₄	—	34	1	no	—
246	C₂×D₄	—	34	2	no	—
247	C₂×D₄	—	34	2	no	—
248	C₂×D₄	—	34	1	no	—
249	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
250	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
251	C₂²×C₄	—	2	1	no	—
252	C₂²×C₄	—	17	1	no	—
253	C₂²×C₄	—	34	1	no	—
254	C₂²×C₄	—	34	1	no	—
255	C₂²×C₄	—	34	1	no	—
256	C₂²×C₄	—	34	1	no	—
257	SD₁₆	—	2	2	no	—
258	SD₁₆	—	34	2	no	—
259	D₈	—	2	2	no	—
260	D₈	—	34	2	no	—
261	C₂×C₈	—	17	1	no	—
262	C₂×C₈	—	17	1	no	—
263	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
264	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
265	C₄⋊C₄	—	17	1	no	—
266	C₄⋊C₄	—	17	1	no	—
267	C₄⋊C₄	—	17	1	no	—
268	C₄⋊C₄	—	17	1	no	—
269	C₄⋊C₄	—	17	1	no	—
270	C₄⋊C₄	—	34	1	no	—
271	C₄⋊C₄	—	34	1	no	—
272	C₄⋊C₄	—	34	1	no	—
273	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
274	C₂²⋊C₄	—	2	1	no	—
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	2	1	no	—
277	C₂²⋊C₄	—	17	1	no	—
278	C₂²⋊C₄	—	17	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	C₂²⋊C₄	—	34	1	no	—
284	C₂²⋊C₄	—	34	1	no	—
285	C₂²⋊C₄	—	34	1	no	—
286	C₄²	—	17	1	no	—
287	C₄²	—	34	1	no	—
288	C₄²	—	34	1	no	—
289	C₄²	C₂²×D₁₇	1	1	yes	characteristic
290	C₂³	—	2	1	no	—
291	C₂³	—	2	1	no	—
292	C₂³	—	17	1	no	—
293	C₂³	—	34	4	no	—
294	C₂³	—	34	1	no	—
295	C₂³	—	136	1	no	—
296	C₂³	—	136	1	no	—
297	Q₈	—	2	1	no	—
298	Q₈	—	2	1	no	—
299	Q₈	—	136	1	no	—
300	D₄	—	2	1	no	—
301	D₄	—	2	1	no	—
302	D₄	—	2	1	no	—
303	D₄	—	17	2	no	—
304	D₄	—	34	4	no	—
305	D₄	—	34	1	no	—
306	D₄	—	136	1	no	—
307	D₄	#63254	1	2	yes	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	8	1	no	—
311	C₂×C₄	—	17	1	no	—
312	C₂×C₄	—	17	1	no	—
313	C₂×C₄	—	17	1	no	—
314	C₂×C₄	—	34	1	no	—
315	C₂×C₄	—	34	1	no	—
316	C₂×C₄	—	34	1	no	—
317	C₂×C₄	—	34	1	no	—
318	C₂×C₄	—	34	1	no	—
319	C₂×C₄	—	136	1	no	—
320	C₂×C₄	—	136	1	no	—
321	C₂×C₄	—	136	1	no	—
322	C₂×C₄	—	136	1	no	—
323	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
324	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
325	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
326	C₈	—	2	1	no	—
327	C₈	—	2	1	no	—
328	C₈	—	34	1	no	—
329	C₈	—	34	1	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	4	1	no	—
334	C₂²	—	4	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	34	1	no	—
337	C₂²	—	34	2	no	—
338	C₂²	—	34	1	no	—
339	C₂²	—	34	1	no	—
340	C₂²	—	68	1	no	—
341	C₂²	—	68	1	no	—
342	C₂²	—	136	4	no	—
343	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	4	1	no	—
347	C₄	—	4	1	no	—
348	C₄	—	34	1	no	—
349	C₄	—	34	1	no	—
350	C₄	—	34	1	no	—
351	C₄	—	68	1	no	—
352	C₄	—	68	1	no	—
353	C₄	—	68	1	no	—
354	C₄	C₂×#63254	1	1	yes	characteristic
355	C₄	#41×D₁₇	1	1	yes	characteristic
356	C₂	—	4	2	no	—
357	C₂	—	8	1	no	—
358	C₂	—	68	1	no	—
359	C₂	—	68	2	no	—
360	C₂	#11763319	1	1	yes	characteristic
361	C₂	#11763613	1	1	yes	characteristic
362	C₂	#11763616	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×#463	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#11762941	C₂
hypercenter	#463	D₁₇
hypocenter	C₁₇	#3118
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #463

Complements

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

N	H ≅ G/N	type
1. #11763428	C₂	semidirect
2. #11763281	C₂	semidirect
3. #11763148	C₂	semidirect
4. #11763132	C₂	semidirect
5. #11763100	C₂	semidirect
6. #11763089	C₂	semidirect
7. #11763021	C₂	semidirect
8. #11762958	C₂	semidirect
10. #11762921	C₂	semidirect
11. #11762912	C₂	semidirect
12. SD₁₆×Dic₁₇	C₂	semidirect
13. #11762889	C₂	semidirect
14. #11762869	C₂	semidirect
15. C₁₇×#463	C₂	semidirect
19. C₁₇×#124	C₂²	semidirect
22. Q₈×Dic₁₇	C₂²	semidirect
26. D₄×Dic₁₇	C₂²	semidirect
40. #10587555	C₂²	semidirect
41. C₁₇×C₄⋊C₈	C₂²	semidirect
42. C₁₇×#103	C₂²	semidirect
43. C₁₇×#102	C₂²	semidirect
44. #10587506	C₂²	semidirect
47. #10587503	C₂²	semidirect
52. #10587488	C₂²	semidirect
53. #10587488	C₂²	semidirect
54. #10587486	C₂²	semidirect
55. C₈×Dic₁₇	C₂²	semidirect
56. #10587478	C₂²	semidirect
57. #10587478	C₂²	semidirect
58. C₄⋊#3622	C₂²	semidirect
182. #463	D₁₇	semidirect
205. #124	C₂×D₁₇	semidirect
227. C₄⋊C₈	C₂×D₁₇	semidirect
229. #103	C₂×D₁₇	semidirect
235. #102	C₂×D₁₇	semidirect
240. C₁₇	#3118	semidirect

Maximal subgroups

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

subgroup	index
#11763428	2
#11763281	2
#11763148	2
#11763132	2
#11763100	2
#11763089	2
#11763021	2
#11762958	2
#11762941	2
#11762921	2
#11762912	2
SD₁₆×Dic₁₇	2
#11762889	2
#11762869	2
C₁₇×#463	2
#3118	17