Subgroups — #16925480

Order 4352.

438 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊#3622	C₂	1	1	yes	maximal, characteristic
2	#14528530	C₂	1	1	yes	maximal, characteristic
3	C₂×#11762531	C₂	1	1	yes	maximal, characteristic
4	C₂×#11762530	C₂	1	1	yes	maximal, characteristic
5	#14526611	C₂	1	1	yes	maximal, characteristic
6	#14526611	C₂	1	1	yes	maximal, characteristic
7	C₄×C₁₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊#3622	C₄	1	2	yes	—
9	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
10	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
11	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
12	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
13	#11762564	C₄	1	2	yes	—
14	#11762564	C₄	1	2	yes	—
15	#11762564	C₄	1	2	yes	—
16	#11762531	—	2	2	no	—
17	#11762530	—	2	2	no	—
18	C₂×C₁₇×C₄⋊C₈	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
21	C₄³×C₁₇	C₂²	1	1	yes	characteristic
22	C₁₇×#336	C₄	1	2	yes	—
23	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
32	C₂²×#3622	C₂×C₄	1	2	yes	—
33	C₂²×#3622	C₂×C₄	1	2	yes	—
34	C₂²×#3622	C₂×C₄	1	2	yes	—
35	C₂²×#3622	C₂×C₄	1	2	yes	—
36	C₁₇×C₄⋊C₈	—	2	4	no	—
37	C₄×C₈×C₁₇	—	2	1	no	—
38	C₄×C₈×C₁₇	—	2	1	no	—
39	C₄×C₈×C₁₇	—	2	1	no	—
40	C₄×C₈×C₁₇	—	2	1	no	—
41	C₄⋊#3622	—	2	4	no	—
42	C₄×#3622	—	2	1	no	—
43	C₄×#3622	—	2	1	no	—
44	C₄×#3622	—	2	1	no	—
45	C₄×#3622	—	2	1	no	—
46	C₄×#3622	—	2	1	no	—
47	C₄×#3622	—	2	1	no	—
48	C₄×#3622	D₄	1	2	yes	—
49	C₄×#3622	Q₈	1	2	yes	—
50	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
51	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
52	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
53	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
54	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₇	—	2	2	no	—
58	C₂×C₈×C₁₇	—	2	2	no	—
59	C₂×C₈×C₁₇	—	2	2	no	—
60	C₂×C₈×C₁₇	—	2	2	no	—
61	C₂×C₈×C₁₇	—	2	2	no	—
62	C₂×C₈×C₁₇	M₁₆	1	4	yes	—
63	C₄²×C₁₇	—	2	2	no	—
64	C₄²×C₁₇	—	2	2	no	—
65	C₄²×C₁₇	—	2	2	no	—
66	C₄²×C₁₇	—	2	2	no	—
67	C₄²×C₁₇	—	2	2	no	—
68	C₄²×C₁₇	M₁₆	1	4	yes	—
69	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
71	C₄²×C₁₇	#41	1	1	yes	characteristic
72	C₄²×C₁₇	#41	1	1	yes	characteristic
73	C₂×#3622	—	2	2	no	—
74	C₂×#3622	—	2	2	no	—
75	C₂×#3622	—	2	2	no	—
76	C₂×#3622	—	2	2	no	—
77	C₂×#3622	—	2	2	no	—
78	C₂×#3622	—	2	2	no	—
79	C₂×#3622	—	2	2	no	—
80	C₂×#3622	—	2	2	no	—
81	C₂×#3622	—	2	2	no	—
82	C₂×#3622	—	2	2	no	—
83	C₂×#3622	—	2	2	no	—
84	C₂×#3622	C₄⋊C₄	1	4	yes	—
85	#8265	—	17	1	no	maximal, Sylow
86	C₂³×C₁₇	C₂×C₄²	1	1	yes	characteristic
87	C₂×C₄×C₁₇	—	2	1	no	—
88	C₂×C₄×C₁₇	—	2	1	no	—
89	C₂×C₄×C₁₇	—	2	1	no	—
90	C₂×C₄×C₁₇	—	2	1	no	—
91	C₂×C₄×C₁₇	—	2	1	no	—
92	C₂×C₄×C₁₇	—	2	1	no	—
93	C₂×C₄×C₁₇	—	2	1	no	—
94	C₂×C₄×C₁₇	—	2	1	no	—
95	C₂×C₄×C₁₇	—	2	1	no	—
96	C₂×C₄×C₁₇	—	2	1	no	—
97	C₂×C₄×C₁₇	#97	1	2	yes	—
98	C₂×C₄×C₁₇	#97	1	2	yes	—
99	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
101	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
104	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
105	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
106	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
107	C₂×C₄×C₁₇	#131	1	2	yes	—
108	C₂×C₄×C₁₇	#131	1	2	yes	—
109	C₂×C₄×C₁₇	#131	1	2	yes	—
110	C₂×C₄×C₁₇	#131	1	2	yes	—
111	C₈×C₁₇	—	2	4	no	—
112	C₈×C₁₇	—	4	2	no	—
113	#3622	—	2	4	no	—
114	#3622	—	4	2	no	—
115	#3622	—	4	2	no	—
116	#3622	—	4	2	no	—
117	C₄×C₄⋊C₈	—	17	1	no	—
118	C₄×C₄⋊C₈	D₁₇	1	1	yes	characteristic
119	#1744	—	17	1	no	—
120	C₂×#322	—	17	1	no	—
121	C₂×#322	—	17	1	no	—
122	#1273	—	17	1	no	—
123	#1273	—	17	1	no	—
124	C₂²×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
125	C₂²×C₁₇	C₂×#97	1	1	yes	characteristic
126	C₂²×C₁₇	C₄×M₁₆	1	1	yes	characteristic
127	C₂²×C₁₇	#405	1	1	yes	characteristic
128	C₂²×C₁₇	#405	1	1	yes	characteristic
129	C₂²×C₁₇	#405	1	1	yes	characteristic
130	C₂²×C₁₇	#405	1	1	yes	characteristic
131	C₄×C₁₇	—	2	2	no	—
132	C₄×C₁₇	—	2	2	no	—
133	C₄×C₁₇	—	4	1	no	—
134	C₄×C₁₇	—	4	1	no	—
135	C₄×C₁₇	#424	1	2	yes	—
136	C₄×C₁₇	#433	1	2	yes	—
137	C₄×C₁₇	#435	1	2	yes	—
138	C₄×C₁₇	#436	1	2	yes	—
139	C₄×C₁₇	#446	1	2	yes	—
140	C₄×C₁₇	#446	1	2	yes	—
141	C₂×C₄⋊C₈	—	17	2	no	—
142	C₂×C₄⋊C₈	Dic₁₇	1	2	yes	—
143	C₂×C₄×C₈	—	17	1	no	—
144	C₂×C₄×C₈	—	17	1	no	—
145	C₂×C₄×C₈	—	17	1	no	—
146	C₂×C₄×C₈	—	17	1	no	—
147	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
148	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
149	C₄³	C₂×D₁₇	1	1	yes	characteristic
150	#336	—	17	2	no	—
151	#336	—	17	2	no	—
152	#336	—	17	2	no	—
153	#336	Dic₁₇	1	2	yes	—
154	#322	—	34	2	no	—
155	#322	—	34	2	no	—
156	C₂×C₁₇	#1451	1	1	yes	characteristic
157	C₂×C₁₇	#1452	1	1	yes	characteristic
158	C₂×C₁₇	#1453	1	1	yes	characteristic
159	C₂×C₁₇	#1454	1	1	yes	characteristic
160	C₂×C₁₇	#1768	1	1	yes	characteristic
161	C₂×C₁₇	#1770	1	1	yes	characteristic
162	C₂×C₁₇	#1771	1	1	yes	characteristic
163	C₂²×C₈	—	17	2	no	—
164	C₂²×C₈	—	17	2	no	—
165	C₂²×C₈	—	17	2	no	—
166	C₂²×C₈	—	17	2	no	—
167	C₂²×C₈	C₂×Dic₁₇	1	2	yes	—
168	C₂²×C₈	C₂×Dic₁₇	1	2	yes	—
169	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
170	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
171	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
172	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
173	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
174	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
175	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
176	C₄⋊C₈	—	2	4	no	—
177	C₄⋊C₈	—	34	4	no	—
178	C₄×C₈	—	2	1	no	—
179	C₄×C₈	—	2	1	no	—
180	C₄×C₈	—	2	1	no	—
181	C₄×C₈	—	2	1	no	—
182	C₄×C₈	—	17	2	no	—
183	C₄×C₈	—	17	2	no	—
184	C₄×C₈	—	34	1	no	—
185	C₄×C₈	—	34	1	no	—
186	C₄×C₈	—	34	1	no	—
187	C₄×C₈	—	34	1	no	—
188	C₄×C₈	—	34	1	no	—
189	C₄×C₈	—	34	1	no	—
190	C₁₇	#8265	1	1	yes	characteristic, complemented, Sylow
191	C₂²×C₄	C₄×Dic₁₇	1	1	yes	characteristic
192	C₂²×C₄	C₄×Dic₁₇	1	1	yes	characteristic
193	C₂²×C₄	C₄×Dic₁₇	1	1	yes	characteristic
194	C₂²×C₄	C₄×Dic₁₇	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
197	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	17	4	no	—
204	C₂×C₈	—	34	2	no	—
205	C₂×C₈	—	34	2	no	—
206	C₂×C₈	—	34	2	no	—
207	C₂×C₈	—	34	2	no	—
208	C₂×C₈	—	34	2	no	—
209	C₂×C₈	—	34	2	no	—
210	C₂×C₈	—	34	2	no	—
211	C₂×C₈	—	34	2	no	—
212	C₂×C₈	—	34	2	no	—
213	C₂×C₈	—	34	2	no	—
214	C₂×C₈	—	34	2	no	—
215	C₂×C₈	#63223	1	4	yes	—
216	C₄²	—	2	2	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	2	2	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	2	2	no	—
221	C₄²	#63218	1	4	yes	—
222	C₄²	D₄×D₁₇	1	1	yes	characteristic
223	C₄²	#63254	1	1	yes	characteristic
224	C₄²	Q₈×D₁₇	1	1	yes	characteristic
225	C₄²	#63256	1	1	yes	characteristic
226	C₂³	C₂×C₄×Dic₁₇	1	1	yes	characteristic
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	#10587486	1	2	yes	—
238	C₂×C₄	#10587486	1	2	yes	—
239	C₂×C₄	#10587573	1	1	yes	characteristic
240	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
241	C₂×C₄	#10587578	1	1	yes	characteristic
242	C₂×C₄	#10587579	1	1	yes	characteristic
243	C₂×C₄	C₂×#63218	1	2	yes	—
244	C₂×C₄	#10587587	1	2	yes	—
245	C₂×C₄	#10587593	1	2	yes	—
246	C₂×C₄	#10587593	1	2	yes	—
247	C₂×C₄	C₂×#63223	1	2	yes	—
248	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
249	C₂×C₄	Q₈×Dic₁₇	1	1	yes	characteristic
250	C₂×C₄	#10587634	1	2	yes	—
251	C₈	—	2	4	no	—
252	C₈	—	4	2	no	—
253	C₈	—	34	4	no	—
254	C₈	—	68	2	no	—
255	C₈	—	68	2	no	—
256	C₈	—	68	2	no	—
257	C₂²	C₄⋊C₄×Dic₁₇	1	1	yes	characteristic
258	C₂²	C₂×#10587486	1	1	yes	characteristic
259	C₂²	C₄×#63223	1	1	yes	characteristic
260	C₂²	#11762764	1	1	yes	characteristic
261	C₂²	#11762765	1	1	yes	characteristic
262	C₂²	#11762767	1	1	yes	characteristic
263	C₂²	#11762767	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	4	1	no	—
267	C₄	—	4	1	no	—
268	C₄	#11762834	1	2	yes	—
269	C₄	#11762869	1	2	yes	—
270	C₄	#11762881	1	2	yes	—
271	C₄	#11762886	1	2	yes	—
272	C₄	#11762936	1	2	yes	—
273	C₄	#11762940	1	2	yes	—
274	C₂	#14527073	1	1	yes	characteristic
275	C₂	#14527077	1	1	yes	characteristic
276	C₂	#14527081	1	1	yes	characteristic
277	C₂	#14527084	1	1	yes	characteristic
278	C₂	#14528643	1	1	yes	characteristic
279	C₂	#14528654	1	1	yes	characteristic
280	C₂	#14528659	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₁₇
commutator, socle, prime radical	C₂³×C₁₇	C₂×C₄²
Fitting	C₄×C₁₇×C₄⋊C₈	C₂
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	C₄×C₄⋊C₈	D₁₇
hypocenter	C₁₇	#8265
LCS	G ▷ C₂³×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄² ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
190. C₁₇	#8265	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#3622	2
#14528530	2
C₂×#11762531	2
C₂×#11762530	2
#14526611	2
C₄×C₁₇×C₄⋊C₈	2
#8265	17