Subgroups — #17053221

Order 4352.

638 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	M₁₆×Dic₃₄	C₂	1	2	yes	maximal
2	C₂×#11762939	C₂	1	1	yes	maximal, characteristic
3	#14529559	C₂	1	2	yes	maximal
4	#14529558	C₂	1	4	yes	maximal
5	#14528957	C₂	1	2	yes	maximal
6	#14528956	C₂	1	1	yes	maximal, characteristic
7	#14528692	C₂	1	2	yes	maximal
8	C₄×C₁₇×#108	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₃₄	C₂²	1	1	yes	characteristic
10	#11762939	—	2	2	no	—
11	#11762939	C₄	1	4	yes	—
12	#11762939	C₄	1	4	yes	—
13	#11762937	—	2	4	no	—
14	C₈×Dic₃₄	—	2	2	no	—
15	C₂×#10587490	C₂²	1	2	yes	—
16	C₂×#10587490	C₂²	1	2	yes	—
17	C₂×#10587493	C₂²	1	4	yes	—
18	#11762833	C₂²	1	2	yes	—
19	#11762832	C₂²	1	4	yes	—
20	C₂×C₄⋊#3622	C₂²	1	1	yes	characteristic
21	C₂×#10587485	C₂²	1	2	yes	—
22	M₁₆×Dic₁₇	C₂²	1	4	yes	—
23	C₂×#10587486	C₂²	1	2	yes	—
24	#11762573	C₂²	1	4	yes	—
25	C₂×C₁₇×#108	C₂²	1	1	yes	characteristic
26	C₂×C₁₇×#108	C₂²	1	1	yes	characteristic
27	C₄×M₁₆×C₁₇	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
29	C₁₇×#341	C₂²	1	2	yes	—
30	C₂²×Dic₃₄	Q₈	1	1	yes	characteristic
31	C₂×M₁₆×C₁₇	C₂³	1	2	yes	—
32	C₂×M₁₆×C₁₇	C₂³	1	2	yes	—
33	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
36	C₂×C₄⋊Dic₁₇	Q₈	1	1	yes	characteristic
37	C₂×#63225	Q₈	1	2	yes	—
38	C₂×C₄×Dic₁₇	C₂³	1	2	yes	—
39	C₂×#63223	C₂³	1	4	yes	—
40	C₂²×#3622	C₂³	1	2	yes	—
41	C₄×Dic₃₄	C₂×C₄	1	2	yes	—
42	C₄×Dic₃₄	C₂×C₄	1	2	yes	—
43	C₄×Dic₃₄	D₄	1	2	yes	—
44	C₄×Dic₃₄	D₄	1	2	yes	—
45	C₁₇×#108	—	2	2	no	—
46	C₁₇×#108	Q₈	1	4	yes	complemented
47	C₁₇×#97	—	2	2	no	—
48	C₄×C₈×C₁₇	—	2	2	no	—
49	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
50	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
51	#10587493	—	4	4	no	—
52	#10587490	—	2	4	no	—
53	#10587490	—	4	2	no	—
54	#10587486	—	2	4	no	—
55	#10587486	C₂×C₄	1	8	yes	—
56	#10587485	—	2	4	no	—
57	#10587485	—	2	4	no	—
58	#10587485	C₂×C₄	1	8	yes	—
59	C₈×Dic₁₇	—	2	4	no	—
60	C₄⋊#3622	—	2	2	no	—
61	C₄⋊#3622	—	2	2	no	—
62	C₄⋊#3622	C₂×C₄	1	2	yes	—
63	C₄⋊#3622	C₂×C₄	1	2	yes	—
64	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
67	C₂²×Dic₁₇	C₂×Q₈	1	2	yes	—
68	C₂²×Dic₁₇	C₂×Q₈	1	2	yes	—
69	C₂×Dic₃₄	—	4	2	no	—
70	C₂×Dic₃₄	C₄⋊C₄	1	2	yes	—
71	C₂×Dic₃₄	C₄⋊C₄	1	2	yes	—
72	M₁₆×C₁₇	—	2	2	no	—
73	M₁₆×C₁₇	—	2	2	no	—
74	M₁₆×C₁₇	—	2	2	no	—
75	M₁₆×C₁₇	C₂×Q₈	1	4	yes	—
76	C₂×C₈×C₁₇	—	2	2	no	—
77	C₂×C₈×C₁₇	—	2	1	no	—
78	C₂×C₈×C₁₇	—	2	1	no	—
79	C₂×C₈×C₁₇	—	2	2	no	—
80	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
83	C₂×C₈×C₁₇	#41	1	2	yes	—
84	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
88	C₄⋊Dic₁₇	C₄⋊C₄	1	2	yes	—
89	C₄⋊Dic₁₇	C₄⋊C₄	1	2	yes	—
90	#63225	C₄⋊C₄	1	4	yes	—
91	#63225	C₄⋊C₄	1	4	yes	—
92	C₄×Dic₁₇	C₂²×C₄	1	4	yes	—
93	C₄×Dic₁₇	C₂×D₄	1	4	yes	—
94	#63223	—	2	4	no	—
95	#63223	—	2	4	no	—
96	C₂×#3622	—	2	2	no	—
97	C₂×#3622	—	2	2	no	—
98	C₂×#3622	—	2	4	no	—
99	C₂×#3622	C₂²×C₄	1	4	yes	—
100	#22733	—	17	1	no	maximal, Sylow
101	C₂³×C₁₇	C₂²×Q₈	1	1	yes	characteristic
102	C₂×C₄×C₁₇	—	2	2	no	—
103	C₂×C₄×C₁₇	—	4	1	no	—
104	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
111	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
112	C₂×C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
113	C₂×C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
114	C₂×Dic₁₇	—	4	2	no	—
115	C₂×Dic₁₇	—	4	2	no	—
116	C₂×Dic₁₇	C₂×C₄⋊C₄	1	4	yes	—
117	C₂×Dic₁₇	C₂×C₄⋊C₄	1	4	yes	—
118	Dic₃₄	—	2	4	no	—
119	Dic₃₄	—	4	2	no	—
120	C₈×C₁₇	—	2	2	no	—
121	C₈×C₁₇	—	2	4	no	—
122	C₈×C₁₇	—	4	2	no	—
123	C₈×C₁₇	C₄×Q₈	1	4	yes	—
124	#3622	—	2	4	no	—
125	#3622	—	4	4	no	—
126	Q₈×M₁₆	—	17	2	no	—
127	C₂×#446	—	17	1	no	—
128	#1950	—	17	2	no	—
129	#1950	—	17	4	no	—
130	#1849	—	17	2	no	—
131	#1848	—	17	1	no	—
132	#1778	—	17	2	no	—
133	C₄×#108	D₁₇	1	1	yes	characteristic
134	C₂²×C₁₇	—	2	1	no	—
135	C₂²×C₁₇	—	2	1	no	—
136	C₂²×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
137	C₂²×C₁₇	#557	1	1	yes	characteristic
138	C₂²×C₁₇	Q₈²	1	1	yes	characteristic
139	C₄×C₁₇	—	2	1	no	—
140	C₄×C₁₇	—	2	1	no	—
141	C₄×C₁₇	—	2	1	no	—
142	C₄×C₁₇	#430	1	2	yes	—
143	C₄×C₁₇	C₂×C₄×Q₈	1	1	yes	characteristic
144	C₄×C₁₇	#520	1	1	yes	characteristic
145	C₄×C₁₇	D₄×Q₈	1	1	yes	characteristic
146	C₄×C₁₇	#550	1	1	yes	characteristic
147	Dic₁₇	—	2	4	no	—
148	Dic₁₇	—	2	4	no	—
149	C₂×C₄×Q₈	—	17	1	no	—
150	#446	—	17	4	no	—
151	#446	—	17	4	no	—
152	#446	—	34	2	no	—
153	#446	—	34	4	no	—
154	C₈×Q₈	—	34	2	no	—
155	C₂×#108	—	17	2	no	—
156	C₂×#108	—	17	4	no	—
157	C₂×#108	—	17	2	no	—
158	C₂×#108	C₂×D₁₇	1	1	yes	characteristic
159	C₂×#108	C₂×D₁₇	1	1	yes	characteristic
160	#423	—	17	4	no	—
161	#423	—	17	2	no	—
162	C₂×C₄⋊C₈	—	17	2	no	—
163	C₂×C₄⋊C₈	—	17	1	no	—
164	C₄×M₁₆	—	17	4	no	—
165	C₄×M₁₆	C₂×D₁₇	1	2	yes	—
166	C₂×#97	—	17	2	no	—
167	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
168	#341	—	17	4	no	—
169	#341	C₂×D₁₇	1	2	yes	—
170	C₂×C₁₇	—	2	1	no	—
171	C₂×C₁₇	—	2	1	no	—
172	C₂×C₁₇	Q₈×C₄⋊C₄	1	1	yes	characteristic
173	C₂×C₁₇	C₂×#430	1	1	yes	characteristic
174	C₂×C₁₇	#2916	1	1	yes	characteristic
175	C₂²×Q₈	—	17	1	no	—
176	C₂×M₁₆	—	17	4	no	—
177	C₂×M₁₆	C₂²×D₁₇	1	2	yes	—
178	C₂×M₁₆	C₂²×D₁₇	1	2	yes	—
179	C₂²×C₈	—	17	2	no	—
180	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
181	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
182	C₄×Q₈	—	17	2	no	—
183	C₄×Q₈	—	17	2	no	—
184	C₄×Q₈	—	17	2	no	—
185	C₄×Q₈	—	17	2	no	—
186	C₂×C₄⋊C₄	—	17	2	no	—
187	C₂×C₄⋊C₄	—	17	1	no	—
188	C₂×C₄²	—	17	2	no	—
189	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
190	#108	—	2	2	no	—
191	#108	—	34	4	no	—
192	#108	—	68	2	no	—
193	#108	—	68	4	no	—
194	#108	Dic₃₄	1	4	yes	complemented
195	C₄⋊C₈	—	17	8	no	—
196	C₄⋊C₈	—	17	2	no	—
197	C₄⋊C₈	—	17	2	no	—
198	C₄⋊C₈	—	34	4	no	—
199	C₄⋊C₈	—	34	4	no	—
200	C₄⋊C₈	—	34	2	no	—
201	C₄⋊C₈	—	34	2	no	—
202	#97	—	2	2	no	—
203	#97	—	17	8	no	—
204	#97	—	34	4	no	—
205	C₄×C₈	—	2	2	no	—
206	C₄×C₈	—	34	4	no	—
207	C₄×C₈	C₄×D₁₇	1	2	yes	—
208	C₄×C₈	C₄×D₁₇	1	2	yes	—
209	C₁₇	#22733	1	1	yes	characteristic, complemented, Sylow
210	C₂×Q₈	—	17	2	no	—
211	C₂×Q₈	—	17	2	no	—
212	C₂×Q₈	—	68	2	no	—
213	C₂²×C₄	—	17	2	no	—
214	C₂²×C₄	—	17	2	no	—
215	C₂²×C₄	Q₈×D₁₇	1	1	yes	characteristic
216	C₂²×C₄	Q₈×D₁₇	1	1	yes	characteristic
217	C₂²×C₄	C₂³×D₁₇	1	1	yes	characteristic
218	M₁₆	—	2	2	no	—
219	M₁₆	—	2	2	no	—
220	M₁₆	—	2	2	no	—
221	M₁₆	—	34	4	no	—
222	M₁₆	—	34	4	no	—
223	M₁₆	C₂×Dic₃₄	1	4	yes	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	17	4	no	—
229	C₂×C₈	—	34	2	no	—
230	C₂×C₈	—	34	2	no	—
231	C₂×C₈	—	34	4	no	—
232	C₂×C₈	C₂×Dic₃₄	1	2	yes	—
233	C₂×C₈	C₂×C₄×D₁₇	1	2	yes	—
234	C₂×C₈	C₂×C₄×D₁₇	1	2	yes	—
235	C₂×C₈	#63252	1	2	yes	—
236	C₄⋊C₄	—	17	4	no	—
237	C₄⋊C₄	—	17	4	no	—
238	C₄⋊C₄	—	17	2	no	—
239	C₄⋊C₄	—	17	2	no	—
240	C₄²	—	17	4	no	—
241	C₄²	—	17	4	no	—
242	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
243	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
244	C₄²	D₄×D₁₇	1	1	yes	characteristic
245	C₄²	D₄×D₁₇	1	1	yes	characteristic
246	C₂³	C₂×Q₈×D₁₇	1	1	yes	characteristic
247	Q₈	—	34	4	no	—
248	Q₈	—	68	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	17	4	no	—
252	C₂×C₄	—	17	4	no	—
253	C₂×C₄	—	68	2	no	—
254	C₂×C₄	—	68	2	no	—
255	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
256	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
257	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
258	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
259	C₂×C₄	C₂²×Dic₃₄	1	1	yes	characteristic
260	C₂×C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
261	C₂×C₄	C₂×#63252	1	1	yes	characteristic
262	C₂×C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
263	C₂×C₄	#10587696	1	1	yes	characteristic
264	C₂×C₄	#10587697	1	1	yes	characteristic
265	C₈	—	2	4	no	—
266	C₈	—	2	2	no	—
267	C₈	—	4	2	no	—
268	C₈	—	34	4	no	—
269	C₈	—	68	4	no	—
270	C₈	C₄×Dic₃₄	1	4	yes	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	C₂×C₄⋊C₄×D₁₇	1	1	yes	characteristic
274	C₂²	#11763537	1	1	yes	characteristic
275	C₂²	Q₈×Dic₃₄	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	34	4	no	—
280	C₄	—	34	4	no	—
281	C₄	#11762856	1	2	yes	—
282	C₄	C₂×C₄×Dic₃₄	1	1	yes	characteristic
283	C₄	#11763289	1	1	yes	characteristic
284	C₄	D₄×Dic₃₄	1	1	yes	characteristic
285	C₄	#11763480	1	1	yes	characteristic
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	C₄⋊C₄×Dic₃₄	1	1	yes	characteristic
289	C₂	C₂×#11762856	1	1	yes	characteristic
290	C₂	#14537392	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₇
commutator	C₂×C₄×C₁₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₇
Fitting	C₄×C₁₇×#108	C₂
socle	C₂²×C₁₇	C₂²×C₄⋊C₄
prime radical	C₂³×C₁₇	C₂²×Q₈
cosocle	C₂²×C₄×C₁₇	C₂⁴
hypercenter	C₄×#108	D₁₇
hypocenter	C₁₇	#22733
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
46. C₁₇×#108	Q₈	semidirect
194. #108	Dic₃₄	semidirect
209. C₁₇	#22733	semidirect

Maximal subgroups

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

subgroup	index
M₁₆×Dic₃₄	2
C₂×#11762939	2
#14529559	2
#14529558	2
#14528957	2
#14528956	2
#14528692	2
C₄×C₁₇×#108	2
#22733	17