Subgroups — #17054027

Order 4352.

710 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#10587590	C₂	1	1	yes	maximal, characteristic
2	#14537449	C₂	1	2	yes	maximal, complemented
3	#14529677	C₂	1	2	yes	maximal
4	#14529433	C₂	1	4	yes	maximal, complemented
5	#14528956	C₂	1	1	yes	maximal, characteristic, complemented
6	#14528805	C₂	1	2	yes	maximal, complemented
7	#14528668	C₂	1	2	yes	maximal, complemented
8	C₁₇×#1770	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#10587590	C₂²	1	1	yes	characteristic
10	C₄×#63252	C₂²	1	2	yes	—
11	C₄×Dic₆₈	C₄	1	1	yes	characteristic, complemented
12	C₄×Dic₆₈	C₄	1	1	yes	characteristic, complemented
13	C₄×#63219	C₄	1	2	yes	complemented
14	C₄×#63219	C₄	1	2	yes	complemented
15	C₄×D₁₃₆	C₄	1	1	yes	characteristic, complemented
16	C₄×D₁₃₆	C₄	1	1	yes	characteristic, complemented
17	#11762869	—	2	4	no	—
18	#11762865	—	2	2	no	—
19	#11762861	—	2	2	no	—
20	C₂×#10587490	C₂²	1	2	yes	—
21	C₂×#10587490	C₂²	1	2	yes	—
22	#11762847	C₂²	1	1	yes	characteristic
23	#11762833	C₂²	1	2	yes	—
24	C₂×#10587476	C₂²	1	4	yes	—
25	#11762806	C₂²	1	2	yes	—
26	#11762775	C₂²	1	4	yes	—
27	#11762764	C₂²	1	4	yes	—
28	#11762569	C₂²	1	4	yes	—
29	C₂×C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₁₇×#97	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
33	C₁₇×#336	C₂²	1	2	yes	—
34	C₂×#63252	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₁₇	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
39	C₂×C₄²×C₁₇	C₂³	1	2	yes	—
40	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
41	C₄×C₂²⋊D₁₇	—	2	2	no	—
42	C₄×C₂²⋊D₁₇	—	2	2	no	—
43	#10587612	C₂³	1	2	yes	—
44	C₂×#63223	C₂³	1	4	yes	—
45	C₂×Dic₆₈	C₂×C₄	1	1	yes	characteristic
46	#10587590	—	2	2	no	—
47	#10587590	—	2	2	no	—
48	C₂×D₁₃₆	C₂×C₄	1	1	yes	characteristic
49	C₂×#63219	C₂×C₄	1	2	yes	—
50	C₄×D₆₈	C₂×C₄	1	2	yes	—
51	C₄×D₆₈	C₂×C₄	1	2	yes	—
52	#10587558	—	2	2	no	—
53	C₄²×D₁₇	—	2	2	no	—
54	C₄×Dic₃₄	C₂×C₄	1	2	yes	—
55	C₄×Dic₃₄	C₂×C₄	1	2	yes	—
56	C₁₇×C₄⋊C₈	D₄	1	2	yes	—
57	C₁₇×C₄⋊C₈	D₄	1	2	yes	—
58	C₁₇×#97	—	2	4	no	—
59	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
63	C₁₇×#95	—	2	1	no	—
64	#10587492	C₂×C₄	1	2	yes	—
65	#10587492	C₂×C₄	1	2	yes	—
66	#10587491	—	2	4	no	—
67	#10587491	—	2	4	no	—
68	#10587490	—	2	4	no	—
69	#10587490	—	4	2	no	—
70	#10587489	C₂×C₄	1	1	yes	characteristic
71	#10587489	C₂×C₄	1	1	yes	characteristic
72	#10587488	C₂×C₄	1	2	yes	—
73	#10587487	C₂×C₄	1	2	yes	—
74	#10587487	C₂×C₄	1	2	yes	—
75	#10587486	—	2	4	no	—
76	C₈×Dic₁₇	—	2	4	no	—
77	#10587476	—	4	4	no	—
78	C₄⋊#3622	—	2	2	no	—
79	C₄⋊#3622	—	2	2	no	—
80	C₂²×C₄×C₁₇	—	2	2	no	—
81	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
82	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
83	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
84	C₂×C₂²⋊D₁₇	—	2	2	no	—
85	#63252	—	2	2	no	—
86	#63252	—	2	2	no	—
87	#63252	—	2	2	no	—
88	#63252	—	2	2	no	—
89	C₂×D₆₈	C₂²×C₄	1	2	yes	—
90	C₂×C₄×D₁₇	—	2	2	no	—
91	C₂×C₄×D₁₇	—	2	2	no	—
92	C₂×C₄×D₁₇	—	2	2	no	—
93	C₂×Dic₃₄	C₂²×C₄	1	2	yes	—
94	C₂×C₈×C₁₇	—	2	2	no	—
95	C₂×C₈×C₁₇	—	2	1	no	—
96	C₂×C₈×C₁₇	—	2	1	no	—
97	C₂×C₈×C₁₇	—	2	2	no	—
98	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₁₇	C₂×D₄	1	4	yes	—
103	C₂×C₈×C₁₇	#41	1	2	yes	—
104	C₂×C₈×C₁₇	#41	1	2	yes	—
105	C₄⋊C₄×C₁₇	—	4	2	no	—
106	C₄²×C₁₇	—	2	2	no	—
107	C₄²×C₁₇	—	2	2	no	—
108	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
112	C₂²⋊Dic₁₇	—	2	2	no	—
113	#63227	—	2	2	no	—
114	#63227	—	2	2	no	—
115	C₄⋊Dic₁₇	C₂²×C₄	1	2	yes	—
116	C₄⋊Dic₁₇	C₂²×C₄	1	2	yes	—
117	#63225	—	2	2	no	—
118	#63225	—	2	2	no	—
119	C₄×Dic₁₇	—	2	2	no	—
120	C₄×Dic₁₇	—	2	2	no	—
121	C₄×Dic₁₇	—	2	2	no	—
122	#63223	—	2	4	no	—
123	#63223	—	2	4	no	—
124	C₂×#3622	—	2	4	no	—
125	Dic₆₈	—	2	2	no	—
126	D₁₃₆	—	2	2	no	—
127	#63219	—	2	4	no	—
128	#22803	—	17	1	no	maximal, Sylow
129	C₂³×C₁₇	C₂×#41	1	1	yes	characteristic
130	C₂²×D₁₇	—	2	2	no	—
131	C₂×C₄×C₁₇	—	2	2	no	—
132	C₂×C₄×C₁₇	—	4	1	no	—
133	C₂×C₄×C₁₇	—	4	2	no	—
134	C₂×C₄×C₁₇	—	4	2	no	—
135	C₂×C₄×C₁₇	—	4	2	no	—
136	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
141	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
143	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
144	C₂×C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
145	C₂×C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
146	C₂²⋊D₁₇	—	4	2	no	—
147	C₂²⋊D₁₇	—	4	2	no	—
148	C₂×Dic₁₇	—	2	2	no	—
149	C₂×Dic₁₇	—	2	2	no	—
150	C₂×Dic₁₇	—	2	2	no	—
151	D₆₈	—	2	2	no	—
152	D₆₈	—	2	2	no	—
153	C₄×D₁₇	—	4	2	no	—
154	C₄×D₁₇	—	4	2	no	—
155	C₄×D₁₇	—	4	4	no	—
156	C₄×D₁₇	—	8	2	no	—
157	Dic₃₄	—	2	2	no	—
158	Dic₃₄	—	2	2	no	—
159	C₈×C₁₇	—	2	1	no	—
160	C₈×C₁₇	—	2	1	no	—
161	C₈×C₁₇	—	2	4	no	—
162	C₈×C₁₇	#117	1	2	yes	—
163	C₈×C₁₇	#117	1	2	yes	—
164	#3622	—	4	4	no	—
165	C₄×#135	—	17	1	no	—
166	#2925	—	17	2	no	—
167	#1975	—	17	2	no	—
168	#1929	—	17	4	no	—
169	#1848	—	17	1	no	—
170	#1806	—	17	2	no	—
171	#1773	—	17	2	no	—
172	#1770	D₁₇	1	1	yes	characteristic, complemented
173	C₂²×C₁₇	—	2	1	no	—
174	C₂²×C₁₇	—	2	1	no	—
175	C₂²×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
176	C₂²×C₁₇	#555	1	1	yes	characteristic
177	C₂²×C₁₇	#556	1	1	yes	characteristic
178	C₂×D₁₇	—	4	2	no	—
179	C₂×D₁₇	—	4	2	no	—
180	C₂×D₁₇	—	4	2	no	—
181	C₄×C₁₇	—	2	1	no	—
182	C₄×C₁₇	—	2	1	no	—
183	C₄×C₁₇	—	2	1	no	—
184	C₄×C₁₇	—	8	2	no	—
185	C₄×C₁₇	#444	1	2	yes	—
186	C₄×C₁₇	C₂×#117	1	1	yes	characteristic
187	C₄×C₁₇	#520	1	1	yes	characteristic
188	C₄×C₁₇	#547	1	1	yes	characteristic
189	C₄×C₁₇	#550	1	1	yes	characteristic
190	Dic₁₇	—	4	2	no	—
191	Dic₁₇	—	4	2	no	—
192	Dic₁₇	—	4	2	no	—
193	C₂×#135	—	17	1	no	—
194	C₄×#41	—	17	2	no	—
195	C₄×Q₁₆	—	17	1	no	—
196	C₄×Q₁₆	—	17	1	no	—
197	C₄×SD₁₆	—	17	2	no	—
198	C₄×SD₁₆	—	17	2	no	—
199	C₄×D₈	—	17	1	no	—
200	C₄×D₈	—	17	1	no	—
201	#433	—	34	4	no	—
202	#432	—	34	2	no	—
203	#431	—	34	2	no	—
204	C₂×#108	—	17	2	no	—
205	C₂×#108	—	17	2	no	—
206	#427	—	17	1	no	—
207	#423	—	17	2	no	—
208	C₂×C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
209	C₂×#104	—	17	4	no	—
210	#416	—	17	2	no	—
211	#408	—	17	4	no	—
212	#405	—	17	4	no	—
213	C₂×#97	C₂×D₁₇	1	2	yes	—
214	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
215	C₄×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
216	#339	—	17	4	no	—
217	#336	C₂×D₁₇	1	2	yes	—
218	C₂×C₁₇	—	2	1	no	—
219	C₂×C₁₇	—	2	1	no	—
220	C₂×C₁₇	#2354	1	1	yes	characteristic
221	C₂×C₁₇	C₂×#444	1	1	yes	characteristic
222	C₂×C₁₇	#2957	1	1	yes	characteristic
223	D₁₇	—	8	2	no	—
224	C₂×#41	—	17	2	no	—
225	#135	—	34	2	no	—
226	#135	—	34	2	no	—
227	C₂×Q₁₆	—	17	1	no	—
228	C₂×SD₁₆	—	17	2	no	—
229	C₂×D₈	—	17	1	no	—
230	C₂×M₁₆	—	17	4	no	—
231	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
232	C₂²×C₈	C₂²×D₁₇	1	2	yes	—
233	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
234	C₄×Q₈	—	17	2	no	—
235	C₄×Q₈	—	17	2	no	—
236	C₄×D₄	—	17	2	no	—
237	C₄×D₄	—	17	2	no	—
238	C₄×D₄	—	34	2	no	—
239	C₄×D₄	—	34	2	no	—
240	#117	—	17	2	no	—
241	#117	—	34	2	no	—
242	C₂×C₄⋊C₄	—	2	1	no	—
243	C₂×C₄²	—	34	2	no	—
244	C₂×C₄²	C₂²×D₁₇	1	2	yes	—
245	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
246	#108	—	34	4	no	—
247	#108	—	68	2	no	—
248	#107	—	17	1	no	—
249	#107	—	17	1	no	—
250	#106	—	17	2	no	—
251	C₄⋊C₈	—	34	2	no	—
252	C₄⋊C₈	—	34	2	no	—
253	C₄⋊C₈	D₆₈	1	2	yes	—
254	C₄⋊C₈	D₆₈	1	2	yes	—
255	#104	—	68	4	no	—
256	#103	—	17	2	no	—
257	#103	—	17	2	no	—
258	#102	—	17	2	no	—
259	#102	—	17	2	no	—
260	C₂²⋊C₈	—	34	4	no	—
261	C₂²⋊C₈	—	34	4	no	—
262	#97	—	2	4	no	—
263	#97	—	34	4	no	—
264	C₄×C₈	—	34	4	no	—
265	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
266	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
267	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
268	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
269	#95	—	2	1	no	—
270	C₁₇	#22803	1	1	yes	characteristic, complemented, Sylow
271	#41	—	34	2	no	—
272	#41	—	34	2	no	—
273	#41	—	34	2	no	—
274	#41	—	34	2	no	—
275	C₂×Q₈	—	17	2	no	—
276	C₂×D₄	—	17	2	no	—
277	C₂×D₄	—	34	2	no	—
278	C₂²×C₄	—	2	2	no	—
279	C₂²×C₄	—	34	2	no	—
280	C₂²×C₄	—	34	2	no	—
281	C₂²×C₄	—	34	2	no	—
282	C₂²×C₄	#63252	1	1	yes	characteristic
283	C₂²×C₄	#63252	1	1	yes	characteristic
284	C₂²×C₄	C₂³×D₁₇	1	1	yes	characteristic
285	Q₁₆	—	34	2	no	—
286	SD₁₆	—	34	4	no	—
287	D₈	—	34	2	no	—
288	M₁₆	—	34	4	no	—
289	M₁₆	—	34	4	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	34	4	no	—
295	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
296	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
297	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
298	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
299	C₂×C₈	C₂×D₆₈	1	4	yes	—
300	C₂×C₈	#63254	1	2	yes	—
301	C₂×C₈	#63256	1	2	yes	—
302	C₄⋊C₄	—	4	2	no	—
303	C₄⋊C₄	—	17	2	no	—
304	C₄⋊C₄	—	17	2	no	—
305	C₄⋊C₄	—	34	2	no	—
306	C₄⋊C₄	—	34	2	no	—
307	C₂²⋊C₄	—	34	2	no	—
308	C₂²⋊C₄	—	34	2	no	—
309	C₂²⋊C₄	—	34	2	no	—
310	C₄²	—	2	2	no	—
311	C₄²	—	2	2	no	—
312	C₄²	—	34	2	no	—
313	C₄²	—	34	2	no	—
314	C₄²	—	34	2	no	—
315	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
316	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
317	C₄²	C₂×D₆₈	1	1	yes	characteristic
318	C₄²	C₂×D₆₈	1	1	yes	characteristic
319	C₂³	—	34	2	no	—
320	C₂³	C₂×#63252	1	1	yes	characteristic
321	Q₈	—	34	2	no	—
322	Q₈	—	34	2	no	—
323	D₄	—	34	2	no	—
324	D₄	—	34	2	no	—
325	D₄	—	68	2	no	—
326	D₄	—	68	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	34	2	no	—
333	C₂×C₄	—	34	2	no	—
334	C₂×C₄	—	34	2	no	—
335	C₂×C₄	—	68	2	no	—
336	C₂×C₄	—	68	2	no	—
337	C₂×C₄	—	68	4	no	—
338	C₂×C₄	—	136	2	no	—
339	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
340	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
341	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
342	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
343	C₂×C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
344	C₂×C₄	C₂²×D₆₈	1	1	yes	characteristic
345	C₂×C₄	C₂×#63254	1	1	yes	characteristic
346	C₂×C₄	C₂×#63256	1	1	yes	characteristic
347	C₂×C₄	#10587696	1	1	yes	characteristic
348	C₂×C₄	#10587697	1	1	yes	characteristic
349	C₈	—	2	4	no	—
350	C₈	—	2	1	no	—
351	C₈	—	2	1	no	—
352	C₈	—	68	4	no	—
353	C₈	#10587578	1	2	yes	—
354	C₈	#10587578	1	2	yes	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	68	2	no	—
358	C₂²	—	68	2	no	—
359	C₂²	—	68	2	no	—
360	C₂²	C₂×C₄×D₆₈	1	1	yes	characteristic
361	C₂²	#11763524	1	1	yes	characteristic
362	C₂²	#11763532	1	1	yes	characteristic
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	8	2	no	—
367	C₄	—	68	2	no	—
368	C₄	—	68	2	no	—
369	C₄	—	68	2	no	—
370	C₄	#11762932	1	2	yes	—
371	C₄	C₂×#10587578	1	1	yes	characteristic
372	C₄	#11763289	1	1	yes	characteristic
373	C₄	#11763463	1	1	yes	characteristic
374	C₄	#11763481	1	1	yes	characteristic
375	C₂	—	2	1	no	—
376	C₂	—	2	1	no	—
377	C₂	—	136	2	no	—
378	C₂	#14531731	1	1	yes	characteristic
379	C₂	C₂×#11762932	1	1	yes	characteristic
380	C₂	#14537702	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₆₈
commutator	C₂×C₄×C₁₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₇
Fitting	C₁₇×#1770	C₂
socle	C₂²×C₁₇	C₂×C₄×D₄
prime radical	C₂×#10587590	C₂²
cosocle	C₂²×C₄×C₁₇	C₂⁴
hypercenter	#1770	D₁₇
hypocenter	C₁₇	#22803
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈ ◁ #1770

Complements

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

N	H ≅ G/N	type
2. #14537449	C₂	semidirect
4. #14529433	C₂	semidirect
5. #14528956	C₂	semidirect
6. #14528805	C₂	semidirect
7. #14528668	C₂	semidirect
8. C₁₇×#1770	C₂	semidirect
11. C₄×Dic₆₈	C₄	semidirect
12. C₄×Dic₆₈	C₄	semidirect
13. C₄×#63219	C₄	semidirect
14. C₄×#63219	C₄	semidirect
15. C₄×D₁₃₆	C₄	semidirect
16. C₄×D₁₃₆	C₄	semidirect
172. #1770	D₁₇	semidirect
270. C₁₇	#22803	semidirect

Maximal subgroups

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

subgroup	index
C₄×#10587590	2
#14537449	2
#14529677	2
#14529433	2
#14528956	2
#14528805	2
#14528668	2
C₁₇×#1770	2
#22803	17