Subgroups — #16928151

Order 4352.

556 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊#3622	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊#3622	C₂	1	1	yes	maximal, characteristic
3	C₂×#11762561	C₂	1	2	yes	maximal, complemented
4	#14526618	C₂	1	2	yes	maximal, complemented
5	C₁₇×#1797	C₂	1	1	yes	maximal, characteristic
6	C₂³×#3622	—	2	1	no	—
7	C₂×C₄⋊#3622	—	2	1	no	—
8	C₂×C₄⋊#3622	—	2	4	no	—
9	C₂×C₄⋊#3622	C₂²	1	2	yes	—
10	C₂×C₂²⋊#3622	C₄	1	1	yes	characteristic
11	C₂×C₂²⋊#3622	C₄	1	1	yes	characteristic
12	C₂×C₄×#3622	C₂²	1	2	yes	—
13	#11762564	C₄	1	2	yes	—
14	#11762561	—	2	4	no	—
15	C₂²×C₄²×C₁₇	C₂²	1	1	yes	characteristic
16	C₂×C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₁₇×C₂²⋊C₈	—	2	1	no	—
19	C₁₇×#336	—	2	1	no	—
20	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₇	D₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₇	Q₈	1	1	yes	characteristic
23	C₂²×C₈×C₁₇	—	2	1	no	—
24	C₂²×C₈×C₁₇	—	2	1	no	—
25	C₂×C₄²×C₁₇	—	2	2	no	—
26	C₂×C₄²×C₁₇	—	2	2	no	—
27	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₇	Q₈	1	1	yes	characteristic
30	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
31	C₂²×#3622	—	2	4	no	—
32	C₂²×#3622	—	2	2	no	—
33	C₂²×#3622	—	4	2	no	—
34	C₂²×#3622	—	4	2	no	—
35	C₂²×#3622	C₂×C₄	1	2	yes	—
36	C₂²×#3622	C₂×C₄	1	2	yes	—
37	C₁₇×C₄⋊C₈	—	2	2	no	—
38	C₁₇×C₄⋊C₈	—	2	2	no	—
39	C₁₇×C₂²⋊C₈	—	4	2	no	—
40	C₂²⋊#3622	—	2	2	no	—
41	C₂²⋊#3622	C₈	1	4	yes	complemented
42	C₄⋊#3622	—	2	4	no	—
43	C₄⋊#3622	—	2	2	no	—
44	C₄⋊#3622	D₄	1	4	yes	—
45	C₄×#3622	—	2	2	no	—
46	C₄×#3622	—	2	2	no	—
47	C₂⁴×C₁₇	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₇	—	2	2	no	—
49	C₂²×C₄×C₁₇	—	2	1	no	—
50	C₂²×C₄×C₁₇	—	2	1	no	—
51	C₂²×C₄×C₁₇	—	2	2	no	—
52	C₂²×C₄×C₁₇	—	2	1	no	—
53	C₂²×C₄×C₁₇	—	2	1	no	—
54	C₂²×C₄×C₁₇	—	2	2	no	—
55	C₂²×C₄×C₁₇	—	2	2	no	—
56	C₂²×C₄×C₁₇	—	4	1	no	—
57	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₁₇	M₁₆	1	2	yes	—
62	C₂²×C₄×C₁₇	D₈	1	2	yes	—
63	C₂²×C₄×C₁₇	Q₁₆	1	2	yes	—
64	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₈×C₁₇	—	4	1	no	—
68	C₂×C₈×C₁₇	—	4	1	no	—
69	C₂×C₈×C₁₇	—	4	1	no	—
70	C₂×C₈×C₁₇	—	4	1	no	—
71	C₂×C₈×C₁₇	—	4	1	no	—
72	C₂×C₈×C₁₇	—	4	1	no	—
73	C₄²×C₁₇	—	2	2	no	—
74	C₄²×C₁₇	—	2	1	no	—
75	C₄²×C₁₇	—	2	2	no	—
76	C₄²×C₁₇	—	2	1	no	—
77	C₄²×C₁₇	C₂×D₄	1	2	yes	—
78	C₄²×C₁₇	#41	1	2	yes	—
79	C₂×#3622	—	2	2	no	—
80	C₂×#3622	—	2	2	no	—
81	C₂×#3622	—	2	2	no	—
82	C₂×#3622	—	2	4	no	—
83	C₂×#3622	—	2	2	no	—
84	C₂×#3622	—	2	2	no	—
85	C₂×#3622	—	4	8	no	—
86	C₂×#3622	—	4	2	no	—
87	C₂×#3622	—	4	2	no	—
88	C₂×#3622	C₂×C₈	1	4	yes	complemented
89	#8829	—	17	1	no	maximal, Sylow
90	C₂³×C₁₇	—	2	1	no	—
91	C₂³×C₁₇	—	2	1	no	—
92	C₂³×C₁₇	—	2	1	no	—
93	C₂³×C₁₇	—	2	1	no	—
94	C₂³×C₁₇	C₄⋊C₈	1	2	yes	—
95	C₂³×C₁₇	#107	1	2	yes	—
96	C₂³×C₁₇	#108	1	2	yes	—
97	C₂³×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₇	—	2	1	no	—
99	C₂×C₄×C₁₇	—	2	1	no	—
100	C₂×C₄×C₁₇	—	2	1	no	—
101	C₂×C₄×C₁₇	—	2	1	no	—
102	C₂×C₄×C₁₇	—	2	1	no	—
103	C₂×C₄×C₁₇	—	2	1	no	—
104	C₂×C₄×C₁₇	—	2	4	no	—
105	C₂×C₄×C₁₇	—	2	4	no	—
106	C₂×C₄×C₁₇	—	2	2	no	—
107	C₂×C₄×C₁₇	—	2	2	no	—
108	C₂×C₄×C₁₇	—	2	2	no	—
109	C₂×C₄×C₁₇	—	2	2	no	—
110	C₂×C₄×C₁₇	—	2	1	no	—
111	C₂×C₄×C₁₇	—	2	1	no	—
112	C₂×C₄×C₁₇	—	2	1	no	—
113	C₂×C₄×C₁₇	—	4	4	no	—
114	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
115	C₂×C₄×C₁₇	#107	1	2	yes	—
116	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₁₇	#120	1	1	yes	characteristic
118	C₂×C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₁₇	#123	1	1	yes	characteristic
120	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
122	C₂×C₄×C₁₇	#131	1	2	yes	—
123	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
125	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
127	C₈×C₁₇	—	8	1	no	—
128	C₈×C₁₇	—	8	1	no	—
129	#3622	—	2	4	no	—
130	#3622	—	4	2	no	—
131	#3622	—	4	4	no	—
132	C₂²×C₄⋊C₈	—	17	1	no	—
133	#1797	D₁₇	1	1	yes	characteristic
134	C₄×C₂²⋊C₈	—	17	1	no	—
135	C₂×#335	—	17	2	no	—
136	#1277	—	17	2	no	—
137	C₂²×C₁₇	—	2	2	no	—
138	C₂²×C₁₇	—	2	2	no	—
139	C₂²×C₁₇	—	2	2	no	—
140	C₂²×C₁₇	—	2	2	no	—
141	C₂²×C₁₇	—	2	2	no	—
142	C₂²×C₁₇	—	2	2	no	—
143	C₂²×C₁₇	#335	1	4	yes	complemented
144	C₂²×C₁₇	#385	1	1	yes	characteristic
145	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
146	C₂²×C₁₇	#424	1	1	yes	characteristic
147	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
148	C₂²×C₁₇	#428	1	1	yes	characteristic
149	C₂²×C₁₇	C₂×#108	1	1	yes	characteristic
150	C₂²×C₁₇	#430	1	1	yes	characteristic
151	C₄×C₁₇	—	2	2	no	—
152	C₄×C₁₇	—	2	2	no	—
153	C₄×C₁₇	—	2	2	no	—
154	C₄×C₁₇	—	4	1	no	—
155	C₄×C₁₇	C₈×D₄	1	2	yes	—
156	C₄×C₁₇	#435	1	2	yes	—
157	C₄×C₁₇	#447	1	1	yes	characteristic
158	C₄×C₁₇	#451	1	1	yes	characteristic
159	C₄×C₁₇	#480	1	1	yes	characteristic
160	C₄×C₁₇	#484	1	1	yes	characteristic
161	C₂³×C₈	—	34	1	no	—
162	C₂²×C₄²	C₂×D₁₇	1	1	yes	characteristic
163	C₂×C₄⋊C₈	—	17	2	no	—
164	C₂×C₄⋊C₈	—	34	4	no	—
165	C₂×C₄⋊C₈	—	34	1	no	—
166	C₂×C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
167	C₂×C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
168	C₂×C₂²⋊C₈	—	2	1	no	—
169	C₂×C₂²⋊C₈	—	17	1	no	—
170	C₂×C₂²⋊C₈	—	17	1	no	—
171	C₂×C₄×C₈	—	17	2	no	—
172	#336	—	2	1	no	—
173	#336	—	17	2	no	—
174	#335	—	34	4	no	—
175	C₂×C₁₇	—	2	4	no	—
176	C₂×C₁₇	C₂×#335	1	1	yes	characteristic
177	C₂×C₁₇	#1565	1	1	yes	characteristic
178	C₂×C₁₇	#1571	1	1	yes	characteristic
179	C₂×C₁₇	#1573	1	1	yes	characteristic
180	C₂×C₁₇	#1850	1	1	yes	characteristic
181	C₂×C₁₇	#1852	1	1	yes	characteristic
182	C₂×C₁₇	#1855	1	1	yes	characteristic
183	C₂³×C₄	Dic₃₄	1	1	yes	characteristic
184	C₂³×C₄	C₄×D₁₇	1	1	yes	characteristic
185	C₂³×C₄	C₂²⋊D₁₇	1	1	yes	characteristic
186	C₂²×C₈	—	2	1	no	—
187	C₂²×C₈	—	2	1	no	—
188	C₂²×C₈	—	17	2	no	—
189	C₂²×C₈	—	17	2	no	—
190	C₂²×C₈	—	34	4	no	—
191	C₂²×C₈	—	34	2	no	—
192	C₂²×C₈	—	68	2	no	—
193	C₂²×C₈	—	68	2	no	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	Dic₃₄	1	1	yes	characteristic
197	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
198	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
199	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
200	C₄⋊C₈	—	2	2	no	—
201	C₄⋊C₈	—	2	2	no	—
202	C₄⋊C₈	—	17	4	no	—
203	C₄⋊C₈	—	34	4	no	—
204	C₄⋊C₈	—	34	2	no	—
205	C₂²⋊C₈	—	4	2	no	—
206	C₂²⋊C₈	—	17	4	no	—
207	C₂²⋊C₈	—	34	2	no	—
208	C₄×C₈	—	34	2	no	—
209	C₄×C₈	—	34	2	no	—
210	C₁₇	#8829	1	1	yes	characteristic, complemented, Sylow
211	C₂⁴	#63225	1	1	yes	characteristic
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	4	1	no	—
221	C₂²×C₄	C₈×D₁₇	1	2	yes	—
222	C₂²×C₄	#63218	1	2	yes	—
223	C₂²×C₄	#63225	1	1	yes	characteristic
224	C₂²×C₄	#63225	1	1	yes	characteristic
225	C₂²×C₄	#63225	1	1	yes	characteristic
226	C₂²×C₄	#63228	1	2	yes	—
227	C₂²×C₄	#63231	1	2	yes	—
228	C₂²×C₄	C₂×Dic₃₄	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	4	1	no	—
234	C₂×C₈	—	4	1	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	—	17	4	no	—
238	C₂×C₈	—	34	2	no	—
239	C₂×C₈	—	34	2	no	—
240	C₂×C₈	—	34	2	no	—
241	C₂×C₈	—	34	2	no	—
242	C₂×C₈	—	34	2	no	—
243	C₂×C₈	—	34	4	no	—
244	C₂×C₈	—	68	8	no	—
245	C₂×C₈	—	68	2	no	—
246	C₂×C₈	—	68	2	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	1	no	—
250	C₄²	—	2	1	no	—
251	C₄²	D₄×D₁₇	1	2	yes	—
252	C₄²	#63254	1	2	yes	—
253	C₂³	—	2	1	no	—
254	C₂³	—	2	1	no	—
255	C₂³	—	2	1	no	—
256	C₂³	—	2	1	no	—
257	C₂³	#10587478	1	2	yes	—
258	C₂³	#10587485	1	2	yes	—
259	C₂³	#10587493	1	2	yes	—
260	C₂³	C₂×#63225	1	1	yes	characteristic
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	#10587478	1	2	yes	—
278	C₂×C₄	#10587485	1	2	yes	—
279	C₂×C₄	#10587564	1	2	yes	—
280	C₂×C₄	#10587567	1	2	yes	—
281	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
282	C₂×C₄	C₂×#63218	1	1	yes	characteristic
283	C₂×C₄	#10587593	1	2	yes	—
284	C₂×C₄	C₂×#63228	1	1	yes	characteristic
285	C₂×C₄	#10587621	1	1	yes	characteristic
286	C₂×C₄	#10587623	1	1	yes	characteristic
287	C₂×C₄	C₂×#63231	1	1	yes	characteristic
288	C₂×C₄	#10587635	1	1	yes	characteristic
289	C₂×C₄	#10587637	1	1	yes	characteristic
290	C₈	—	8	1	no	—
291	C₈	—	8	1	no	—
292	C₈	—	34	4	no	—
293	C₈	—	68	2	no	—
294	C₈	—	68	4	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	#11762561	1	4	yes	complemented
302	C₂²	#11762681	1	1	yes	characteristic
303	C₂²	C₂×#10587485	1	1	yes	characteristic
304	C₂²	#11762837	1	1	yes	characteristic
305	C₂²	C₂×#10587478	1	1	yes	characteristic
306	C₂²	#11762852	1	1	yes	characteristic
307	C₂²	C₂×#10587493	1	1	yes	characteristic
308	C₂²	#11762857	1	1	yes	characteristic
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	2	2	no	—
312	C₄	—	4	1	no	—
313	C₄	#11762872	1	2	yes	—
314	C₄	#11762877	1	2	yes	—
315	C₄	#11762942	1	1	yes	characteristic
316	C₄	#11762964	1	1	yes	characteristic
317	C₄	#11763120	1	1	yes	characteristic
318	C₄	#11763142	1	1	yes	characteristic
319	C₂	—	2	4	no	—
320	C₂	C₂×#11762561	1	1	yes	characteristic
321	C₂	#14527537	1	1	yes	characteristic
322	C₂	#14527560	1	1	yes	characteristic
323	C₂	#14527570	1	1	yes	characteristic
324	C₂	#14528961	1	1	yes	characteristic
325	C₂	#14528971	1	1	yes	characteristic
326	C₂	#14528988	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₇
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1797	C₂
socle	C₂³×C₁₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₇	C₄⋊C₄
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#1797	D₁₇
hypocenter	C₁₇	#8829
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #1797

Complements

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

N	H ≅ G/N	type
3. C₂×#11762561	C₂	semidirect
4. #14526618	C₂	semidirect
41. C₂²⋊#3622	C₈	semidirect
88. C₂×#3622	C₂×C₈	semidirect
143. C₂²×C₁₇	#335	semidirect
210. C₁₇	#8829	semidirect
301. C₂²	#11762561	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊#3622	2
C₄×C₂²⋊#3622	2
C₂×#11762561	2
#14526618	2
C₁₇×#1797	2
#8829	17