Subgroups — #17188904

Order 4352.

642 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763241	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11763225	C₂	1	1	yes	maximal, characteristic, complemented
3	#14538505	C₂	1	2	yes	maximal, complemented
4	#14530915	C₂	1	4	yes	maximal, complemented
5	#14530910	C₂	1	2	yes	maximal, complemented
6	#14530905	C₂	1	2	yes	maximal, complemented
7	#14530531	C₂	1	2	yes	maximal, complemented
8	C₂×C₁₇×#368	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763609	—	2	2	no	—
10	C₂×#10587590	C₂²	1	2	yes	—
11	C₂×#10587587	C₂²	1	1	yes	characteristic
12	#11763241	—	2	1	no	—
13	#11763241	—	2	1	no	—
14	#11763241	—	2	1	no	—
15	#11763241	—	2	1	no	—
16	C₂×Dic₁₃₆	C₂²	1	1	yes	characteristic, complemented
17	C₂×#10587471	C₂²	1	2	yes	complemented
18	C₂×D₂₇₂	C₂²	1	1	yes	characteristic, complemented
19	#11763225	—	2	1	no	—
20	#11763225	—	2	1	no	—
21	#11763225	C₂²	1	2	yes	—
22	#11763225	C₂²	1	2	yes	—
23	C₂×#10587483	C₂²	1	1	yes	characteristic
24	C₂×#10587469	—	2	1	no	—
25	C₂×C₁₆×D₁₇	—	2	1	no	—
26	#11763085	C₂²	1	4	yes	—
27	#11763080	C₂²	1	4	yes	—
28	#11763073	—	2	4	no	—
29	#11762791	—	2	2	no	—
30	#11762624	C₂²	1	4	yes	complemented
31	#11762621	C₂²	1	4	yes	complemented
32	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
33	C₂×C₁₇×#108	C₂²	1	2	yes	—
34	C₁₇×#368	C₂²	1	2	yes	complemented
35	C₁₇×#368	C₂²	1	2	yes	complemented
36	C₂×#63252	—	2	2	no	—
37	C₂×#63252	D₄	1	1	yes	characteristic
38	C₂×M₁₆×C₁₇	—	2	2	no	—
39	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
40	C₂×#63223	D₄	1	1	yes	characteristic
41	#10587595	—	2	4	no	—
42	#10587595	—	4	2	no	—
43	#10587594	—	2	4	no	—
44	#10587594	—	4	2	no	—
45	C₂×Dic₆₈	C₂³	1	2	yes	—
46	#10587590	—	2	2	no	—
47	#10587590	—	2	2	no	—
48	#10587590	—	2	2	no	—
49	#10587590	—	2	2	no	—
50	#10587590	—	4	4	no	—
51	C₂×D₁₃₆	C₂³	1	2	yes	—
52	C₂×#63219	—	2	2	no	—
53	#10587587	—	2	1	no	—
54	#10587587	—	2	1	no	—
55	#10587587	D₄	1	2	yes	—
56	#10587587	C₂³	1	2	yes	—
57	C₂×#63218	—	2	1	no	—
58	C₂×C₈×D₁₇	—	2	1	no	—
59	C₂×C₁₆×C₁₇	D₄	1	1	yes	characteristic
60	C₂×C₁₆×C₁₇	D₄	1	1	yes	characteristic
61	C₂×C₁₆×C₁₇	D₄	1	1	yes	characteristic
62	C₂×C₁₆×C₁₇	D₄	1	1	yes	characteristic
63	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
64	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
65	C₁₇×#108	—	2	2	no	—
66	C₁₇×#108	C₂³	1	4	yes	—
67	#10587495	—	2	4	no	—
68	#10587494	—	2	4	no	—
69	#10587483	—	2	1	no	—
70	#10587483	C₂³	1	2	yes	—
71	C₂×#63214	—	2	1	no	—
72	Dic₁₃₆	—	2	1	no	—
73	Dic₁₃₆	—	2	1	no	—
74	#10587471	—	2	2	no	—
75	#10587471	—	2	2	no	—
76	D₂₇₂	—	2	1	no	—
77	D₂₇₂	—	2	1	no	—
78	#10587469	—	2	2	no	—
79	#10587469	—	2	2	no	—
80	C₁₆×D₁₇	—	2	2	no	—
81	C₁₆×D₁₇	—	2	2	no	—
82	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
83	C₂×C₂²⋊D₁₇	—	2	1	no	—
84	C₂×C₂²⋊D₁₇	—	4	2	no	—
85	#63252	—	2	1	no	—
86	#63252	—	2	1	no	—
87	#63252	—	4	2	no	—
88	#63252	—	4	2	no	—
89	#63252	—	4	2	no	—
90	#63252	—	4	2	no	—
91	#63252	D₈	1	2	yes	—
92	#63252	C₂×D₄	1	2	yes	—
93	C₂×D₆₈	—	2	2	no	—
94	C₂×D₆₈	D₈	1	1	yes	characteristic
95	C₂×C₄×D₁₇	—	2	1	no	—
96	C₂×C₄×D₁₇	—	4	2	no	—
97	C₂×Dic₃₄	—	2	2	no	—
98	C₂×Dic₃₄	D₈	1	1	yes	characteristic
99	M₁₆×C₁₇	—	2	4	no	—
100	M₁₆×C₁₇	—	4	2	no	—
101	C₂×C₈×C₁₇	—	4	2	no	—
102	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
106	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
107	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
108	#63223	—	2	1	no	—
109	#63223	C₂×D₄	1	2	yes	—
110	C₂×#3622	—	2	1	no	—
111	Dic₆₈	—	2	2	no	—
112	Dic₆₈	—	2	2	no	—
113	Dic₆₈	—	4	4	no	—
114	D₁₃₆	—	2	2	no	—
115	D₁₃₆	—	2	2	no	—
116	D₁₃₆	—	4	4	no	—
117	#63219	—	4	4	no	—
118	#63219	—	4	4	no	—
119	#63219	—	4	2	no	—
120	#63219	—	4	2	no	—
121	#63218	—	2	2	no	—
122	#63218	—	2	2	no	—
123	C₈×D₁₇	—	2	2	no	—
124	C₈×D₁₇	—	2	2	no	—
125	C₁₆×C₁₇	C₂×D₄	1	1	yes	characteristic
126	C₁₆×C₁₇	C₂×D₄	1	1	yes	characteristic
127	C₁₆×C₁₇	C₂×D₄	1	1	yes	characteristic
128	C₁₆×C₁₇	C₂×D₄	1	1	yes	characteristic
129	#63214	—	2	2	no	—
130	#33617	—	17	1	no	maximal, Sylow
131	C₂³×C₁₇	#135	1	1	yes	characteristic
132	C₂²×D₁₇	—	2	1	no	—
133	C₂²×D₁₇	—	4	2	no	—
134	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
135	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
136	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
139	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
140	C₂²⋊D₁₇	—	2	2	no	—
141	C₂²⋊D₁₇	—	2	2	no	—
142	C₂²⋊D₁₇	—	8	2	no	—
143	C₂²⋊D₁₇	—	8	2	no	—
144	C₂×Dic₁₇	—	2	1	no	—
145	C₂×Dic₁₇	—	4	2	no	—
146	D₆₈	—	2	1	no	—
147	D₆₈	—	4	2	no	—
148	D₆₈	—	4	2	no	—
149	D₆₈	C₂×D₈	1	2	yes	—
150	C₄×D₁₇	—	2	2	no	—
151	C₄×D₁₇	—	2	2	no	—
152	C₄×D₁₇	—	8	2	no	—
153	C₄×D₁₇	—	8	2	no	—
154	Dic₃₄	—	2	1	no	—
155	Dic₃₄	—	4	2	no	—
156	Dic₃₄	—	4	2	no	—
157	Dic₃₄	C₂×D₈	1	2	yes	—
158	C₈×C₁₇	—	4	4	no	—
159	C₈×C₁₇	#121	1	1	yes	characteristic
160	C₈×C₁₇	#121	1	1	yes	characteristic
161	C₈×C₁₇	C₂²×D₄	1	1	yes	characteristic
162	C₈×C₁₇	C₂×#41	1	1	yes	characteristic
163	#3622	—	2	2	no	—
164	C₂×#508	—	17	1	no	—
165	C₂×#504	—	17	1	no	—
166	#3067	—	17	2	no	—
167	#2223	—	17	4	no	—
168	#2222	—	17	2	no	—
169	#2221	—	17	2	no	—
170	C₂×#368	D₁₇	1	1	yes	characteristic, complemented
171	#2148	—	17	2	no	—
172	C₂²×C₁₇	—	4	1	no	—
173	C₂²×C₁₇	#466	1	1	yes	characteristic
174	C₂²×C₁₇	#466	1	1	yes	characteristic
175	C₂²×C₁₇	C₂×#135	1	1	yes	characteristic
176	C₂×D₁₇	—	2	2	no	—
177	C₂×D₁₇	—	4	2	no	—
178	C₂×D₁₇	—	8	2	no	—
179	C₂×D₁₇	—	16	2	no	—
180	C₄×C₁₇	#466	1	1	yes	characteristic
181	C₄×C₁₇	#466	1	1	yes	characteristic
182	C₄×C₁₇	C₂×#121	1	1	yes	characteristic
183	C₄×C₁₇	C₂²×D₈	1	1	yes	characteristic
184	Dic₁₇	—	2	2	no	—
185	Dic₁₇	—	8	2	no	—
186	#575	—	34	2	no	—
187	C₂×#135	—	17	2	no	—
188	C₂×#131	—	17	1	no	—
189	#508	—	34	1	no	—
190	#508	—	34	1	no	—
191	#508	—	34	1	no	—
192	#508	—	34	1	no	—
193	C₂×Q₃₂	—	17	1	no	—
194	C₂×SD₃₂	—	17	2	no	—
195	C₂×D₁₆	—	17	1	no	—
196	#504	—	17	2	no	—
197	#504	—	17	2	no	—
198	#504	—	34	1	no	—
199	#504	—	34	1	no	—
200	C₂×M₃₂	—	17	1	no	—
201	C₂×M₃₂	—	34	1	no	—
202	C₂²×C₁₆	—	34	1	no	—
203	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
204	#473	—	17	4	no	—
205	#472	—	17	4	no	—
206	#471	—	34	4	no	—
207	C₂×#108	C₂×D₁₇	1	2	yes	—
208	#413	—	34	2	no	—
209	#368	C₂×D₁₇	1	2	yes	complemented
210	#368	C₂×D₁₇	1	2	yes	complemented
211	#362	—	17	4	no	—
212	#361	—	17	4	no	—
213	C₂×C₁₇	—	2	1	no	—
214	C₂×C₁₇	—	2	1	no	—
215	C₂×C₁₇	—	2	1	no	—
216	C₂×C₁₇	C₂×#466	1	1	yes	characteristic
217	D₁₇	—	4	2	no	—
218	D₁₇	—	16	2	no	—
219	C₂×#41	—	17	1	no	—
220	C₂×#41	—	34	2	no	—
221	#137	—	34	4	no	—
222	#137	—	68	2	no	—
223	AGL(1,ℤ/8)	—	34	4	no	—
224	AGL(1,ℤ/8)	—	68	2	no	—
225	#135	—	34	2	no	—
226	#135	—	34	2	no	—
227	#135	—	34	2	no	—
228	#135	—	34	2	no	—
229	#135	—	68	4	no	—
230	C₂×Q₁₆	—	17	2	no	—
231	C₂×SD₁₆	—	34	2	no	—
232	C₂×D₈	—	17	2	no	—
233	#131	—	17	2	no	—
234	#131	—	17	2	no	—
235	#131	—	34	1	no	—
236	#131	—	34	1	no	—
237	C₂×M₁₆	—	2	2	no	—
238	C₂×M₁₆	—	17	1	no	—
239	C₂×M₁₆	—	34	1	no	—
240	C₂²×C₈	—	34	1	no	—
241	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
242	Q₃₂	—	34	1	no	—
243	Q₃₂	—	34	1	no	—
244	SD₃₂	—	34	2	no	—
245	SD₃₂	—	34	2	no	—
246	D₁₆	—	34	1	no	—
247	D₁₆	—	34	1	no	—
248	M₃₂	—	17	2	no	—
249	M₃₂	—	34	1	no	—
250	M₃₂	—	34	2	no	—
251	M₃₂	—	34	2	no	—
252	C₂×C₁₆	—	34	1	no	—
253	C₂×C₁₆	—	34	2	no	—
254	C₂×C₁₆	—	34	2	no	—
255	C₂×C₁₆	D₆₈	1	1	yes	characteristic
256	C₂×C₁₆	D₆₈	1	1	yes	characteristic
257	C₂×C₁₆	D₆₈	1	1	yes	characteristic
258	C₂×C₁₆	D₆₈	1	1	yes	characteristic
259	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
260	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
261	#108	—	2	2	no	—
262	#108	C₂²×D₁₇	1	4	yes	—
263	#101	—	34	4	no	—
264	#100	—	34	4	no	—
265	C₁₇	#33617	1	1	yes	characteristic, complemented, Sylow
266	#41	—	17	2	no	—
267	#41	—	17	2	no	—
268	#41	—	34	1	no	—
269	#41	—	34	1	no	—
270	#41	—	68	2	no	—
271	#41	—	68	2	no	—
272	#41	—	68	2	no	—
273	#41	—	68	2	no	—
274	C₂×Q₈	—	17	1	no	—
275	C₂×Q₈	—	34	2	no	—
276	C₂×D₄	—	17	1	no	—
277	C₂×D₄	—	34	1	no	—
278	C₂×D₄	—	34	2	no	—
279	C₂×D₄	—	68	2	no	—
280	C₂²×C₄	—	34	1	no	—
281	C₂²×C₄	—	68	2	no	—
282	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
283	Q₁₆	—	34	2	no	—
284	Q₁₆	—	34	2	no	—
285	Q₁₆	—	68	4	no	—
286	SD₁₆	—	68	4	no	—
287	SD₁₆	—	68	4	no	—
288	SD₁₆	—	68	2	no	—
289	SD₁₆	—	68	2	no	—
290	D₈	—	34	2	no	—
291	D₈	—	34	2	no	—
292	D₈	—	68	4	no	—
293	M₁₆	—	2	4	no	—
294	M₁₆	—	4	2	no	—
295	M₁₆	—	17	2	no	—
296	M₁₆	—	34	1	no	—
297	M₁₆	—	34	2	no	—
298	M₁₆	—	34	2	no	—
299	C₂×C₈	—	4	2	no	—
300	C₂×C₈	—	34	1	no	—
301	C₂×C₈	—	34	2	no	—
302	C₂×C₈	—	34	2	no	—
303	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
304	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
305	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
306	C₂×C₈	#63256	1	1	yes	characteristic
307	C₂×C₈	#63256	1	1	yes	characteristic
308	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
309	C₁₆	—	34	2	no	—
310	C₁₆	C₂×D₆₈	1	1	yes	characteristic
311	C₁₆	C₂×D₆₈	1	1	yes	characteristic
312	C₁₆	C₂×D₆₈	1	1	yes	characteristic
313	C₁₆	C₂×D₆₈	1	1	yes	characteristic
314	C₂³	—	34	1	no	—
315	C₂³	—	68	2	no	—
316	C₂³	#10587605	1	1	yes	characteristic
317	Q₈	—	17	2	no	—
318	Q₈	—	34	1	no	—
319	Q₈	—	68	2	no	—
320	Q₈	—	68	2	no	—
321	D₄	—	17	2	no	—
322	D₄	—	34	2	no	—
323	D₄	—	34	2	no	—
324	D₄	—	34	1	no	—
325	D₄	—	68	2	no	—
326	D₄	—	68	2	no	—
327	D₄	—	136	2	no	—
328	D₄	—	136	2	no	—
329	C₂×C₄	—	34	1	no	—
330	C₂×C₄	—	34	2	no	—
331	C₂×C₄	—	34	2	no	—
332	C₂×C₄	—	68	2	no	—
333	C₂×C₄	—	136	2	no	—
334	C₂×C₄	—	136	2	no	—
335	C₂×C₄	#10587581	1	1	yes	characteristic
336	C₂×C₄	#10587581	1	1	yes	characteristic
337	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
338	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
339	C₂×C₄	#10587605	1	1	yes	characteristic
340	C₂×C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
341	C₈	—	4	4	no	—
342	C₈	—	34	2	no	—
343	C₈	#10587581	1	1	yes	characteristic
344	C₈	#10587581	1	1	yes	characteristic
345	C₈	C₂²×D₆₈	1	1	yes	characteristic
346	C₈	C₂×#63256	1	1	yes	characteristic
347	C₂²	—	4	1	no	—
348	C₂²	—	272	2	no	—
349	C₂²	—	34	2	no	—
350	C₂²	—	68	2	no	—
351	C₂²	—	136	2	no	—
352	C₂²	#11763046	1	1	yes	characteristic
353	C₂²	#11763046	1	1	yes	characteristic
354	C₂²	C₂×#10587605	1	1	yes	characteristic
355	C₄	—	34	2	no	—
356	C₄	—	136	2	no	—
357	C₄	#11763046	1	1	yes	characteristic
358	C₄	#11763046	1	1	yes	characteristic
359	C₄	C₂×#10587581	1	1	yes	characteristic
360	C₄	C₂×D₈×D₁₇	1	1	yes	characteristic
361	C₂	—	2	1	no	—
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	—	272	2	no	—
365	C₂	—	68	2	no	—
366	C₂	C₂×#11763046	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#11763046
commutator, cosocle	C₂×C₈×C₁₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂×C₁₇×#368	C₂
socle	C₂×C₁₇	C₂×#466
hypercenter	C₂×#368	D₁₇
hypocenter	C₁₇	#33617
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

Complements

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

N	H ≅ G/N	type
1. C₂×#11763241	C₂	semidirect
2. C₂×#11763225	C₂	semidirect
3. #14538505	C₂	semidirect
4. #14530915	C₂	semidirect
5. #14530910	C₂	semidirect
6. #14530905	C₂	semidirect
7. #14530531	C₂	semidirect
8. C₂×C₁₇×#368	C₂	semidirect
16. C₂×Dic₁₃₆	C₂²	semidirect
17. C₂×#10587471	C₂²	semidirect
18. C₂×D₂₇₂	C₂²	semidirect
30. #11762624	C₂²	semidirect
31. #11762621	C₂²	semidirect
34. C₁₇×#368	C₂²	semidirect
35. C₁₇×#368	C₂²	semidirect
170. C₂×#368	D₁₇	semidirect
209. #368	C₂×D₁₇	semidirect
210. #368	C₂×D₁₇	semidirect
265. C₁₇	#33617	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11763241	2
C₂×#11763225	2
#14538505	2
#14530915	2
#14530910	2
#14530905	2
#14530531	2
C₂×C₁₇×#368	2
#33617	17