Subgroups — #17188902

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	#14538504	C₂	1	2	yes	maximal, complemented
3	#14530914	C₂	1	4	yes	maximal, complemented
4	#14530909	C₂	1	2	yes	maximal, complemented
5	#14530904	C₂	1	2	yes	maximal, complemented
6	C₂×#11762639	C₂	1	1	yes	maximal, characteristic, complemented
7	#14530534	C₂	1	2	yes	maximal, complemented
8	C₂×C₁₇×#504	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763607	—	2	2	no	—
10	C₂×#10587590	C₂²	1	2	yes	—
11	#11763241	—	2	1	no	—
12	#11763241	—	2	1	no	—
13	#11763241	—	2	1	no	—
14	#11763241	—	2	1	no	—
15	C₂×Dic₁₃₆	C₂²	1	1	yes	characteristic, complemented
16	C₂×#10587471	C₂²	1	2	yes	complemented
17	C₂×D₂₇₂	C₂²	1	1	yes	characteristic, complemented
18	#11763084	C₂²	1	4	yes	—
19	#11763079	C₂²	1	4	yes	—
20	#11763072	—	2	4	no	—
21	C₂×#10587490	C₂²	1	2	yes	—
22	#11762791	—	2	2	no	—
23	#11762639	C₂²	1	2	yes	complemented
24	#11762639	C₂²	1	2	yes	complemented
25	#11762626	C₂²	1	4	yes	complemented
26	#11762623	C₂²	1	4	yes	complemented
27	C₂×C₁₇×#131	C₂²	1	1	yes	characteristic
28	C₁₇×#504	—	2	1	no	—
29	C₁₇×#504	—	2	1	no	—
30	C₁₇×#504	C₂²	1	2	yes	—
31	C₁₇×#504	C₂²	1	2	yes	—
32	C₂×C₁₇×M₃₂	—	2	1	no	—
33	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
34	C₂²×C₁₆×C₁₇	—	2	1	no	—
35	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
36	C₂×#41×C₁₇	D₄	1	1	yes	characteristic
37	C₂×#63252	—	2	2	no	—
38	C₁₇×#131	—	2	1	no	—
39	C₁₇×#131	—	2	1	no	—
40	C₁₇×#131	D₄	1	2	yes	—
41	C₁₇×#131	C₂³	1	2	yes	—
42	C₂×M₁₆×C₁₇	—	2	1	no	—
43	C₂×M₁₆×C₁₇	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₁₇	—	2	1	no	—
45	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
46	#10587637	—	2	4	no	—
47	#10587636	—	4	4	no	—
48	#10587635	—	2	4	no	—
49	#10587628	—	4	2	no	—
50	#10587618	—	4	2	no	—
51	C₂×#63223	—	2	2	no	—
52	C₂×Dic₆₈	C₂³	1	2	yes	—
53	#10587590	—	2	2	no	—
54	#10587590	—	2	2	no	—
55	#10587590	—	2	2	no	—
56	#10587590	—	2	2	no	—
57	C₂×D₁₃₆	C₂³	1	2	yes	—
58	C₂×#63219	—	2	2	no	—
59	C₁₇×M₃₂	—	2	2	no	—
60	C₁₇×M₃₂	—	2	2	no	—
61	C₁₇×M₃₂	—	2	1	no	—
62	C₁₇×M₃₂	C₂³	1	2	yes	—
63	C₂×C₁₆×C₁₇	—	2	2	no	—
64	C₂×C₁₆×C₁₇	—	2	2	no	—
65	C₂×C₁₆×C₁₇	—	2	1	no	—
66	C₂×C₁₆×C₁₇	D₄	1	1	yes	characteristic
67	C₂×C₁₆×C₁₇	D₄	1	1	yes	characteristic
68	C₂×C₁₆×C₁₇	D₄	1	1	yes	characteristic
69	C₂×C₁₆×C₁₇	D₄	1	1	yes	characteristic
70	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
71	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
72	#10587495	—	2	4	no	—
73	#10587494	—	2	4	no	—
74	#10587490	—	2	2	no	—
75	#10587490	C₂³	1	4	yes	—
76	Dic₁₃₆	—	2	1	no	—
77	Dic₁₃₆	—	2	1	no	—
78	#10587471	—	2	2	no	—
79	#10587471	—	2	2	no	—
80	D₂₇₂	—	2	1	no	—
81	D₂₇₂	—	2	1	no	—
82	#41×C₁₇	—	2	1	no	—
83	#41×C₁₇	—	2	1	no	—
84	#41×C₁₇	D₈	1	2	yes	—
85	#41×C₁₇	C₂×D₄	1	2	yes	—
86	C₂×Q₈×C₁₇	D₈	1	1	yes	characteristic
87	C₂×D₄×C₁₇	—	2	1	no	—
88	C₂×D₄×C₁₇	D₈	1	1	yes	characteristic
89	C₂²×C₄×C₁₇	—	2	1	no	—
90	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₂²⋊D₁₇	—	4	2	no	—
92	#63252	—	4	2	no	—
93	#63252	—	4	2	no	—
94	#63252	—	4	2	no	—
95	#63252	—	4	2	no	—
96	C₂×D₆₈	—	2	2	no	—
97	C₂×C₄×D₁₇	—	4	2	no	—
98	C₂×Dic₃₄	—	2	2	no	—
99	M₁₆×C₁₇	—	2	1	no	—
100	M₁₆×C₁₇	—	2	2	no	—
101	M₁₆×C₁₇	—	2	2	no	—
102	M₁₆×C₁₇	C₂×D₄	1	2	yes	—
103	C₂×C₈×C₁₇	—	2	1	no	—
104	C₂×C₈×C₁₇	—	2	2	no	—
105	C₂×C₈×C₁₇	—	2	2	no	—
106	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
110	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
111	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
112	#63231	—	4	4	no	—
113	#63230	—	4	4	no	—
114	#63229	—	4	4	no	—
115	#63228	—	4	4	no	—
116	#63223	—	2	4	no	—
117	#63223	—	4	2	no	—
118	C₂×#3622	—	4	2	no	—
119	Dic₆₈	—	2	2	no	—
120	Dic₆₈	—	2	2	no	—
121	D₁₃₆	—	2	2	no	—
122	D₁₃₆	—	2	2	no	—
123	#63219	—	4	2	no	—
124	#63219	—	4	2	no	—
125	C₁₆×C₁₇	—	2	2	no	—
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	C₁₆×C₁₇	C₂×D₄	1	1	yes	characteristic
130	#33617	—	17	1	no	maximal, Sylow
131	C₂³×C₁₇	—	2	1	no	—
132	C₂³×C₁₇	#135	1	1	yes	characteristic
133	C₂²×D₁₇	—	4	2	no	—
134	Q₈×C₁₇	—	2	1	no	—
135	Q₈×C₁₇	C₂×D₈	1	2	yes	—
136	D₄×C₁₇	—	2	2	no	—
137	D₄×C₁₇	—	2	2	no	—
138	D₄×C₁₇	—	2	1	no	—
139	D₄×C₁₇	C₂×D₈	1	2	yes	—
140	C₂×C₄×C₁₇	—	2	1	no	—
141	C₂×C₄×C₁₇	—	2	2	no	—
142	C₂×C₄×C₁₇	—	2	2	no	—
143	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
144	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
145	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
148	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
149	C₂²⋊D₁₇	—	8	2	no	—
150	C₂²⋊D₁₇	—	8	2	no	—
151	C₂×Dic₁₇	—	4	2	no	—
152	D₆₈	—	4	2	no	—
153	D₆₈	—	4	2	no	—
154	C₄×D₁₇	—	8	2	no	—
155	C₄×D₁₇	—	8	2	no	—
156	Dic₃₄	—	4	2	no	—
157	Dic₃₄	—	4	2	no	—
158	C₈×C₁₇	—	2	2	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	—	4	4	no	—
164	C₂×#508	—	17	1	no	—
165	C₂×#504	D₁₇	1	1	yes	characteristic, complemented
166	#3067	—	17	2	no	—
167	#2223	—	17	4	no	—
168	#2222	—	17	2	no	—
169	#2221	—	17	2	no	—
170	C₂×#368	—	17	1	no	—
171	#2148	—	17	2	no	—
172	C₂²×C₁₇	—	2	2	no	—
173	C₂²×C₁₇	—	4	1	no	—
174	C₂²×C₁₇	—	4	2	no	—
175	C₂²×C₁₇	#466	1	1	yes	characteristic
176	C₂²×C₁₇	#466	1	1	yes	characteristic
177	C₂²×C₁₇	C₂×#135	1	1	yes	characteristic
178	C₂×D₁₇	—	8	2	no	—
179	C₂×D₁₇	—	16	2	no	—
180	C₄×C₁₇	—	2	2	no	—
181	C₄×C₁₇	#466	1	1	yes	characteristic
182	C₄×C₁₇	#466	1	1	yes	characteristic
183	C₄×C₁₇	C₂×#121	1	1	yes	characteristic
184	C₄×C₁₇	C₂²×D₈	1	1	yes	characteristic
185	Dic₁₇	—	8	2	no	—
186	#575	—	34	2	no	—
187	C₂×#135	—	17	2	no	—
188	C₂×#131	C₂×D₁₇	1	1	yes	characteristic
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	—	2	1	no	—
197	#504	—	2	1	no	—
198	#504	C₂×D₁₇	1	2	yes	—
199	#504	C₂×D₁₇	1	2	yes	—
200	C₂×M₃₂	—	2	1	no	—
201	C₂×M₃₂	C₂×D₁₇	1	1	yes	characteristic
202	C₂²×C₁₆	—	2	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	—	17	2	no	—
208	#413	—	34	2	no	—
209	#368	—	17	2	no	—
210	#368	—	17	2	no	—
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	—	4	2	no	—
217	C₂×C₁₇	C₂×#466	1	1	yes	characteristic
218	D₁₇	—	16	2	no	—
219	C₂×#41	—	34	2	no	—
220	C₂×#41	D₆₈	1	1	yes	characteristic
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	—	2	1	no	—
234	#131	—	2	1	no	—
235	#131	D₆₈	1	2	yes	—
236	#131	C₂²×D₁₇	1	2	yes	—
237	C₂×M₁₆	—	2	1	no	—
238	C₂×M₁₆	—	34	2	no	—
239	C₂×M₁₆	D₆₈	1	1	yes	characteristic
240	C₂²×C₈	—	2	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	—	2	1	no	—
249	M₃₂	—	2	2	no	—
250	M₃₂	—	2	2	no	—
251	M₃₂	C₂²×D₁₇	1	2	yes	—
252	C₂×C₁₆	—	2	1	no	—
253	C₂×C₁₆	—	2	2	no	—
254	C₂×C₁₆	—	2	2	no	—
255	C₂×C₁₆	C₂²⋊D₁₇	1	1	yes	characteristic
256	C₂×C₁₆	C₂²⋊D₁₇	1	1	yes	characteristic
257	C₂×C₁₆	C₂²⋊D₁₇	1	1	yes	characteristic
258	C₂×C₁₆	C₂²⋊D₁₇	1	1	yes	characteristic
259	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
260	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
261	#108	—	17	4	no	—
262	#108	—	34	2	no	—
263	#101	—	34	4	no	—
264	#100	—	34	4	no	—
265	C₁₇	#33617	1	1	yes	characteristic, complemented, Sylow
266	#41	—	2	1	no	—
267	#41	—	2	1	no	—
268	#41	—	68	2	no	—
269	#41	—	68	2	no	—
270	#41	—	68	2	no	—
271	#41	—	68	2	no	—
272	#41	D₁₃₆	1	2	yes	—
273	#41	C₂×D₆₈	1	2	yes	—
274	C₂×Q₈	—	34	2	no	—
275	C₂×Q₈	D₁₃₆	1	1	yes	characteristic
276	C₂×D₄	—	2	1	no	—
277	C₂×D₄	—	34	2	no	—
278	C₂×D₄	—	68	2	no	—
279	C₂×D₄	D₁₃₆	1	1	yes	characteristic
280	C₂²×C₄	—	2	1	no	—
281	C₂²×C₄	—	68	2	no	—
282	C₂²×C₄	C₂×D₆₈	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	1	no	—
294	M₁₆	—	2	2	no	—
295	M₁₆	—	2	2	no	—
296	M₁₆	—	34	4	no	—
297	M₁₆	—	68	2	no	—
298	M₁₆	C₂×D₆₈	1	2	yes	—
299	C₂×C₈	—	2	1	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	68	2	no	—
303	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
304	C₂×C₈	#63252	1	1	yes	characteristic
305	C₂×C₈	#63252	1	1	yes	characteristic
306	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
307	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
308	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
309	C₁₆	—	2	2	no	—
310	C₁₆	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
311	C₁₆	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
312	C₁₆	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
313	C₁₆	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
314	C₂³	—	2	1	no	—
315	C₂³	—	68	2	no	—
316	C₂³	#10587590	1	1	yes	characteristic
317	Q₈	—	2	1	no	—
318	Q₈	—	68	2	no	—
319	Q₈	—	68	2	no	—
320	Q₈	C₂×D₁₃₆	1	2	yes	—
321	D₄	—	2	2	no	—
322	D₄	—	2	2	no	—
323	D₄	—	2	1	no	—
324	D₄	—	68	2	no	—
325	D₄	—	68	2	no	—
326	D₄	—	136	2	no	—
327	D₄	—	136	2	no	—
328	D₄	C₂×D₁₃₆	1	2	yes	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	68	2	no	—
333	C₂×C₄	—	136	2	no	—
334	C₂×C₄	—	136	2	no	—
335	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
336	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
337	C₂×C₄	#10587590	1	1	yes	characteristic
338	C₂×C₄	#10587616	1	1	yes	characteristic
339	C₂×C₄	#10587616	1	1	yes	characteristic
340	C₂×C₄	C₂²×D₆₈	1	1	yes	characteristic
341	C₈	—	2	2	no	—
342	C₈	—	68	4	no	—
343	C₈	#10587616	1	1	yes	characteristic
344	C₈	#10587616	1	1	yes	characteristic
345	C₈	C₂×#63252	1	1	yes	characteristic
346	C₈	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
347	C₂²	—	2	2	no	—
348	C₂²	—	4	1	no	—
349	C₂²	—	4	2	no	—
350	C₂²	—	272	2	no	—
351	C₂²	—	136	2	no	—
352	C₂²	#11763045	1	1	yes	characteristic
353	C₂²	#11763045	1	1	yes	characteristic
354	C₂²	C₂×#10587590	1	1	yes	characteristic
355	C₄	—	2	2	no	—
356	C₄	—	136	2	no	—
357	C₄	#11763045	1	1	yes	characteristic
358	C₄	#11763045	1	1	yes	characteristic
359	C₄	C₂×#10587616	1	1	yes	characteristic
360	C₄	C₂²×D₁₃₆	1	1	yes	characteristic
361	C₂	—	2	1	no	—
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	—	4	2	no	—
365	C₂	—	272	2	no	—
366	C₂	C₂×#11763045	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#11763045
commutator, cosocle	C₂×C₈×C₁₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂×C₁₇×#504	C₂
socle	C₂×C₁₇	C₂×#466
hypercenter	C₂×#504	D₁₇
hypocenter	C₁₇	#33617
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂×#11763241	C₂	semidirect
2. #14538504	C₂	semidirect
3. #14530914	C₂	semidirect
4. #14530909	C₂	semidirect
5. #14530904	C₂	semidirect
6. C₂×#11762639	C₂	semidirect
7. #14530534	C₂	semidirect
8. C₂×C₁₇×#504	C₂	semidirect
15. C₂×Dic₁₃₆	C₂²	semidirect
16. C₂×#10587471	C₂²	semidirect
17. C₂×D₂₇₂	C₂²	semidirect
23. #11762639	C₂²	semidirect
24. #11762639	C₂²	semidirect
25. #11762626	C₂²	semidirect
26. #11762623	C₂²	semidirect
165. C₂×#504	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
#14538504	2
#14530914	2
#14530909	2
#14530904	2
C₂×#11762639	2
#14530534	2
C₂×C₁₇×#504	2
#33617	17