Subgroups — #17961370

Order 4352.

450 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14545116	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄⋊#63216	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11763707	C₂	1	2	yes	maximal
4	#14543838	C₂	1	2	yes	maximal, complemented
5	C₄×#10587634	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587538	C₂²	1	2	yes	—
7	C₂²×#63216	C₂²	1	2	yes	—
8	C₄⋊#63216	—	2	1	no	—
9	C₄⋊#63216	—	2	1	no	—
10	C₄⋊#63216	—	2	1	no	—
11	C₄⋊#63216	—	2	1	no	—
12	#11763707	—	2	4	no	—
13	C₂×#10587634	C₄	1	1	yes	characteristic
14	C₂×#10587634	C₂²	1	1	yes	characteristic
15	#11762765	—	2	1	no	—
16	#11762765	—	2	1	no	—
17	#11762765	C₄	1	1	yes	characteristic
18	#11762765	C₂²	1	1	yes	characteristic
19	C₄×#63223	—	2	1	no	—
20	C₄×#63223	C₄	1	1	yes	characteristic
21	C₂×C₄×#3622	—	2	1	no	—
22	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
23	C₄×#41×C₁₇	C₄	1	1	yes	characteristic
24	C₂×#41×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₄×Q₈×C₁₇	C₈	1	1	yes	characteristic
26	C₄×Q₈×C₁₇	C₈	1	1	yes	characteristic
27	C₄×D₄×C₁₇	—	2	1	no	—
28	C₄×D₄×C₁₇	—	2	1	no	—
29	C₄×D₄×C₁₇	C₈	1	1	yes	characteristic
30	C₄×D₄×C₁₇	C₈	1	1	yes	characteristic
31	C₁₇×#117	—	2	1	no	—
32	C₁₇×#117	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₇	—	2	1	no	—
34	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
35	#10587634	—	2	2	no	—
36	#10587634	—	2	1	no	—
37	#10587634	—	2	1	no	—
38	#10587634	—	2	2	no	—
39	#10587634	D₄	1	2	yes	—
40	#10587634	Q₈	1	2	yes	—
41	C₂×#63223	—	2	1	no	—
42	C₂×#63223	—	2	1	no	—
43	C₂×#63223	C₂×C₄	1	1	yes	characteristic
44	C₂×#63223	C₂×C₄	1	1	yes	characteristic
45	C₂²×#3622	—	2	1	no	—
46	C₂²×#3622	—	2	1	no	—
47	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
48	C₂²×#3622	C₂³	1	1	yes	characteristic
49	#10587538	—	2	2	no	—
50	#10587538	—	2	2	no	—
51	C₂×#63216	—	2	2	no	—
52	C₂×#63216	—	2	2	no	—
53	C₂×#63216	—	2	2	no	—
54	C₂×#63216	—	2	2	no	—
55	#10587474	—	2	1	no	—
56	#10587474	—	2	1	no	—
57	#10587474	—	2	1	no	—
58	C₄×#3622	—	2	1	no	—
59	C₄×#3622	—	2	1	no	—
60	C₄×#3622	—	2	1	no	—
61	C₄×#3622	D₄	1	1	yes	characteristic
62	C₄×#3622	D₄	1	1	yes	characteristic
63	C₄×#3622	D₄	1	1	yes	characteristic
64	C₄×#3622	D₄	1	1	yes	characteristic
65	#41×C₁₇	—	2	1	no	—
66	#41×C₁₇	—	2	1	no	—
67	#41×C₁₇	C₄⋊C₄	1	2	yes	—
68	#41×C₁₇	M₁₆	1	2	yes	—
69	C₂×Q₈×C₁₇	C₂×C₈	1	1	yes	characteristic
70	C₂×D₄×C₁₇	—	2	1	no	—
71	C₂×D₄×C₁₇	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₁₇	—	2	1	no	—
73	C₂²×C₄×C₁₇	—	2	1	no	—
74	C₂²×C₄×C₁₇	—	2	1	no	—
75	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
76	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
77	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
78	C₄⋊C₄×C₁₇	—	2	1	no	—
79	C₄⋊C₄×C₁₇	—	2	1	no	—
80	C₄⋊C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
81	C₄⋊C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
82	C₂²⋊C₄×C₁₇	—	2	1	no	—
83	C₂²⋊C₄×C₁₇	—	2	1	no	—
84	C₂²⋊C₄×C₁₇	—	2	1	no	—
85	C₄²×C₁₇	—	2	1	no	—
86	C₄²×C₁₇	—	2	1	no	—
87	C₄²×C₁₇	—	2	1	no	—
88	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
89	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
90	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
91	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
92	#63223	—	2	4	no	—
93	#63223	—	2	1	no	—
94	#63223	—	2	1	no	—
95	#63223	—	2	2	no	—
96	#63223	—	2	2	no	—
97	#63223	C₄⋊C₄	1	2	yes	—
98	#63223	M₁₆	1	2	yes	—
99	C₂×#3622	—	2	4	no	—
100	C₂×#3622	—	2	1	no	—
101	C₂×#3622	—	2	1	no	—
102	C₂×#3622	—	2	1	no	—
103	C₂×#3622	—	2	1	no	—
104	C₂×#3622	—	2	2	no	—
105	C₂×#3622	—	2	2	no	—
106	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
107	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
108	C₂×#3622	C₂×D₄	1	1	yes	characteristic
109	C₂×#3622	C₂×D₄	1	1	yes	characteristic
110	C₂×#3622	C₂×D₄	1	1	yes	characteristic
111	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
112	C₂×#3622	#41	1	1	yes	characteristic
113	C₂×#3622	#41	1	1	yes	characteristic
114	#63216	—	4	2	no	—
115	#63216	—	4	2	no	—
116	#12787	—	17	1	no	maximal, Sylow
117	C₂³×C₁₇	—	2	1	no	—
118	C₂³×C₁₇	C₂×M₁₆	1	1	yes	characteristic
119	Q₈×C₁₇	—	2	1	no	—
120	Q₈×C₁₇	C₄⋊C₈	1	2	yes	—
121	D₄×C₁₇	—	2	2	no	—
122	D₄×C₁₇	—	2	2	no	—
123	D₄×C₁₇	—	2	1	no	—
124	D₄×C₁₇	C₄⋊C₈	1	2	yes	—
125	C₂×C₄×C₁₇	—	2	1	no	—
126	C₂×C₄×C₁₇	—	2	2	no	—
127	C₂×C₄×C₁₇	—	2	2	no	—
128	C₂×C₄×C₁₇	—	2	2	no	—
129	C₂×C₄×C₁₇	—	2	1	no	—
130	C₂×C₄×C₁₇	—	2	1	no	—
131	C₂×C₄×C₁₇	—	4	1	no	—
132	C₂×C₄×C₁₇	—	4	2	no	—
133	C₂×C₄×C₁₇	—	4	2	no	—
134	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
135	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
136	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
137	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
138	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
141	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
142	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
143	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
144	#3622	—	2	2	no	—
145	#3622	—	2	1	no	—
146	#3622	—	2	2	no	—
147	#3622	—	2	1	no	—
148	#3622	#121	1	1	yes	characteristic
149	#3622	#121	1	1	yes	characteristic
150	#3622	C₂²⋊Q₈	1	1	yes	characteristic
151	#3622	C₂²⋊Q₈	1	1	yes	characteristic
152	C₄×#131	—	17	1	no	—
153	#2150	—	17	1	no	—
154	C₂×C₄⋊C₁₆	—	17	1	no	—
155	C₂×#350	—	17	2	no	—
156	#1379	—	17	2	no	—
157	C₂²×C₁₇	—	2	1	no	—
158	C₂²×C₁₇	—	2	1	no	—
159	C₂²×C₁₇	—	2	2	no	—
160	C₂²×C₁₇	—	4	1	no	—
161	C₂²×C₁₇	—	4	1	no	—
162	C₂²×C₁₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
163	C₂²×C₁₇	#423	1	1	yes	characteristic
164	C₂²×C₁₇	#432	1	1	yes	characteristic
165	C₄×C₁₇	—	2	2	no	—
166	C₄×C₁₇	—	2	1	no	—
167	C₄×C₁₇	—	2	1	no	—
168	C₄×C₁₇	—	2	1	no	—
169	C₄×C₁₇	—	4	1	no	—
170	C₄×C₁₇	—	4	1	no	—
171	C₄×C₁₇	#350	1	2	yes	complemented
172	C₄×C₁₇	#380	1	1	yes	characteristic
173	C₄×C₁₇	#407	1	1	yes	characteristic
174	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
175	C₄×C₁₇	#431	1	1	yes	characteristic
176	C₂×#131	—	17	1	no	—
177	C₂×#131	—	17	1	no	—
178	C₄×#41	#595	1	1	yes	characteristic
179	C₂×M₃₂	—	17	2	no	—
180	C₂²×C₁₆	—	17	2	no	—
181	#405	—	17	1	no	—
182	#405	—	17	1	no	—
183	#405	—	34	1	no	—
184	#405	—	34	1	no	—
185	C₄×M₁₆	—	17	1	no	—
186	C₄×M₁₆	—	34	1	no	—
187	C₂×C₄×C₈	—	17	1	no	—
188	C₂×C₄×C₈	—	34	1	no	—
189	C₄⋊C₁₆	—	34	1	no	—
190	C₄⋊C₁₆	—	34	1	no	—
191	C₄⋊C₁₆	—	34	1	no	—
192	C₄⋊C₁₆	—	34	1	no	—
193	#350	—	34	4	no	—
194	C₂×C₁₇	—	2	1	no	—
195	C₂×C₁₇	—	2	1	no	—
196	C₂×C₁₇	—	4	2	no	—
197	C₂×C₁₇	#1797	1	1	yes	characteristic
198	C₂×C₁₇	C₂×#350	1	1	yes	characteristic
199	C₂×C₁₇	#2117	1	1	yes	characteristic
200	C₂×#41	C₂×#595	1	1	yes	characteristic
201	#131	—	17	2	no	—
202	#131	—	17	2	no	—
203	#131	—	34	1	no	—
204	#131	—	34	1	no	—
205	#131	—	34	2	no	—
206	#131	—	34	2	no	—
207	C₂×M₁₆	—	17	1	no	—
208	C₂×M₁₆	—	17	1	no	—
209	C₂×M₁₆	—	34	1	no	—
210	C₂×M₁₆	—	34	1	no	—
211	C₂²×C₈	—	17	1	no	—
212	C₂²×C₈	—	17	1	no	—
213	C₂²×C₈	—	34	1	no	—
214	C₂²×C₈	—	34	1	no	—
215	C₄×Q₈	#3624	1	1	yes	characteristic
216	C₄×Q₈	#3624	1	1	yes	characteristic
217	C₄×D₄	—	2	1	no	—
218	C₄×D₄	—	2	1	no	—
219	C₄×D₄	#3624	1	1	yes	characteristic
220	C₄×D₄	#3624	1	1	yes	characteristic
221	#117	—	2	1	no	—
222	#117	C₂×#595	1	1	yes	characteristic
223	C₂×C₄²	—	2	1	no	—
224	C₂×C₄²	C₂×#595	1	1	yes	characteristic
225	M₃₂	—	34	2	no	—
226	M₃₂	—	34	2	no	—
227	C₂×C₁₆	—	34	2	no	—
228	C₂×C₁₆	—	34	2	no	—
229	C₂×C₁₆	—	34	2	no	—
230	C₂×C₁₆	—	34	2	no	—
231	#97	—	34	1	no	—
232	#97	—	34	1	no	—
233	#97	—	34	1	no	—
234	C₄×C₈	—	17	1	no	—
235	C₄×C₈	—	17	1	no	—
236	C₄×C₈	—	17	1	no	—
237	C₄×C₈	—	17	1	no	—
238	C₄×C₈	—	34	1	no	—
239	C₄×C₈	—	34	1	no	—
240	C₄×C₈	—	34	1	no	—
241	C₁₇	#12787	1	1	yes	characteristic, complemented, Sylow
242	#41	—	2	1	no	—
243	#41	—	2	1	no	—
244	#41	#63243	1	2	yes	—
245	#41	C₄⋊#595	1	2	yes	—
246	C₂×Q₈	C₂×#3624	1	1	yes	characteristic
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	C₂×#3624	1	1	yes	characteristic
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	—	2	1	no	—
252	C₂²×C₄	#63247	1	1	yes	characteristic
253	C₂²×C₄	#63247	1	1	yes	characteristic
254	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
255	M₁₆	—	17	2	no	—
256	M₁₆	—	17	2	no	—
257	M₁₆	—	34	2	no	—
258	M₁₆	—	34	2	no	—
259	M₁₆	—	34	4	no	—
260	M₁₆	—	34	1	no	—
261	M₁₆	—	34	1	no	—
262	C₂×C₈	—	17	1	no	—
263	C₂×C₈	—	17	1	no	—
264	C₂×C₈	—	17	1	no	—
265	C₂×C₈	—	17	1	no	—
266	C₂×C₈	—	17	1	no	—
267	C₂×C₈	—	17	1	no	—
268	C₂×C₈	—	17	1	no	—
269	C₂×C₈	—	17	1	no	—
270	C₂×C₈	—	34	2	no	—
271	C₂×C₈	—	34	2	no	—
272	C₂×C₈	—	34	4	no	—
273	C₂×C₈	—	34	1	no	—
274	C₂×C₈	—	34	1	no	—
275	C₂×C₈	—	34	1	no	—
276	C₂×C₈	—	34	1	no	—
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	C₂×#3624	1	1	yes	characteristic
280	C₄⋊C₄	C₂×#3624	1	1	yes	characteristic
281	C₂²⋊C₄	—	2	1	no	—
282	C₂²⋊C₄	—	2	1	no	—
283	C₂²⋊C₄	—	2	1	no	—
284	C₄²	—	2	1	no	—
285	C₄²	—	2	1	no	—
286	C₄²	—	2	1	no	—
287	C₄²	C₂×#3624	1	1	yes	characteristic
288	C₄²	C₂×#3624	1	1	yes	characteristic
289	C₄²	C₂²⋊#595	1	1	yes	characteristic
290	C₄²	C₂²⋊#595	1	1	yes	characteristic
291	C₁₆	—	68	2	no	—
292	C₁₆	—	68	2	no	—
293	C₂³	—	2	1	no	—
294	C₂³	C₂×#63247	1	1	yes	characteristic
295	Q₈	—	2	1	no	—
296	Q₈	C₄⋊#3624	1	2	yes	—
297	D₄	—	2	2	no	—
298	D₄	—	2	2	no	—
299	D₄	—	2	1	no	—
300	D₄	C₄⋊#3624	1	2	yes	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	C₂²⋊#3624	1	1	yes	characteristic
311	C₂×C₄	C₂²⋊#3624	1	1	yes	characteristic
312	C₂×C₄	C₂²⋊#3624	1	1	yes	characteristic
313	C₂×C₄	C₂²⋊#3624	1	1	yes	characteristic
314	C₂×C₄	C₂×#63243	1	1	yes	characteristic
315	C₂×C₄	D₁₇⋊M₁₆	1	1	yes	characteristic
316	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
317	C₂×C₄	#10587677	1	1	yes	characteristic
318	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
319	C₂×C₄	C₂×C₂²⋊#595	1	1	yes	characteristic
320	C₈	—	17	1	no	—
321	C₈	—	17	1	no	—
322	C₈	—	17	1	no	—
323	C₈	—	17	1	no	—
324	C₈	—	34	2	no	—
325	C₈	—	34	1	no	—
326	C₈	—	34	1	no	—
327	C₈	—	34	2	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	4	1	no	—
333	C₂²	C₂×C₂²⋊#3624	1	1	yes	characteristic
334	C₂²	#11763812	1	1	yes	characteristic
335	C₂²	#11763830	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	2	no	—
340	C₄	—	4	1	no	—
341	C₄	—	4	1	no	—
342	C₄	#11763707	1	2	yes	complemented
343	C₄	#11763751	1	1	yes	characteristic
344	C₄	#11763780	1	1	yes	characteristic
345	C₄	C₂×C₄⋊#3624	1	1	yes	characteristic
346	C₄	#11763825	1	1	yes	characteristic
347	C₂	—	2	1	no	—
348	C₂	—	2	1	no	—
349	C₂	—	4	2	no	—
350	C₂	#14544638	1	1	yes	characteristic
351	C₂	C₂×#11763707	1	1	yes	characteristic
352	C₂	#14545026	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊#595
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#595
Fitting	C₄×#41×C₁₇	C₄
socle	C₂²×C₁₇	C₂×C₂²⋊C₈
prime radical	C₂×#41×C₁₇	C₂×C₄
cosocle	C₂²×#3622	C₂³
hypercenter	C₄×#41	#595
hypocenter	C₁₇	#12787
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #14545116	C₂	semidirect
2. C₂×C₄⋊#63216	C₂	semidirect
4. #14543838	C₂	semidirect
171. C₄×C₁₇	#350	semidirect
241. C₁₇	#12787	semidirect
342. C₄	#11763707	semidirect

Maximal subgroups

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

subgroup	index
#14545116	2
C₂×C₄⋊#63216	2
C₂×#11763707	2
#14543838	2
C₄×#10587634	2
#12787	17