Subgroups — #17981631

Order 4352.

446 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊#3633	C₂	1	2	yes	maximal
2	C₂×#11763899	C₂	1	1	yes	maximal, characteristic
3	C₂×#11763898	C₂	1	1	yes	maximal, characteristic
4	#14545790	C₂	1	2	yes	maximal
5	#97×#595	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#3633	C₂²	1	2	yes	—
7	C₂×C₄⋊#3633	C₂²	1	2	yes	—
8	C₂×C₄×#3633	—	2	2	no	—
9	(C₂×#3633)⋊C₄	—	2	2	no	—
10	#11763899	C₄	1	4	yes	complemented
11	#11763898	C₄	1	4	yes	complemented
12	#11763768	C₄	1	1	yes	characteristic
13	C₂×#10587532	C₄	1	1	yes	characteristic
14	C₂×#10587532	C₄	1	1	yes	characteristic
15	C₂×C₈×#595	C₂²	1	1	yes	characteristic
16	C₂×C₈×#595	C₂²	1	1	yes	characteristic
17	C₄²×#595	C₂²	1	1	yes	characteristic
18	#97×D₁₇	C₄	1	1	yes	characteristic
19	C₂²×#3633	—	2	2	no	—
20	C₂²×#3633	—	2	2	no	—
21	C₂×C₄×#595	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×#595	D₄	1	2	yes	—
23	C₂×C₄×#595	Q₈	1	2	yes	—
24	C₂×C₄×#595	C₂³	1	1	yes	characteristic
25	C₂×#63242	C₂×C₄	1	1	yes	characteristic
26	C₂×#63242	C₂×C₄	1	1	yes	characteristic
27	C₄⋊#3633	—	2	2	no	—
28	C₄⋊#3633	—	2	2	no	—
29	C₄⋊#3633	C₂×C₄	1	4	yes	—
30	C₄⋊#3633	C₂×C₄	1	4	yes	—
31	C₄×#3633	—	4	4	no	—
32	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
34	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
35	#10587542	—	2	2	no	—
36	#10587532	—	2	2	no	—
37	#10587532	C₈	1	4	yes	complemented
38	C₈×#595	—	2	1	no	—
39	C₈×#595	—	2	1	no	—
40	C₈×#595	C₂×C₄	1	2	yes	—
41	C₈×#595	C₂×C₄	1	2	yes	—
42	C₁₇×#97	C₈	1	1	yes	characteristic, complemented
43	#10587486	C₈	1	1	yes	characteristic, complemented
44	#10587486	C₈	1	1	yes	characteristic, complemented
45	#10587474	C₈	1	1	yes	characteristic, complemented
46	C₂²×#595	C₄⋊C₄	1	2	yes	—
47	C₂²×#595	C₂×D₄	1	1	yes	characteristic
48	C₂²×#595	C₂×Q₈	1	1	yes	characteristic
49	C₂×#3633	—	2	4	no	—
50	C₂×#3633	—	2	4	no	—
51	C₂×#3633	—	4	2	no	—
52	C₂×#3633	—	4	2	no	—
53	C₂×#3633	—	4	2	no	—
54	C₂×#3633	—	4	2	no	—
55	C₂×C₄×D₁₇	C₄⋊C₄	1	2	yes	—
56	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×#3624	—	2	1	no	—
58	C₂×#3624	—	2	1	no	—
59	C₄×#595	—	2	2	no	—
60	C₄×#595	—	2	2	no	—
61	C₄×#595	—	4	2	no	—
62	C₄×#595	C₂×C₈	1	2	yes	—
63	C₄×#595	M₁₆	1	2	yes	—
64	C₄×#595	C₂²×C₄	1	1	yes	characteristic
65	C₄×#595	C₂²×C₄	1	1	yes	characteristic
66	C₄×#595	#41	1	1	yes	characteristic
67	C₄×#595	#41	1	1	yes	characteristic
68	#63242	—	2	1	no	—
69	#63242	—	2	1	no	—
70	#63242	C₄²	1	2	yes	—
71	#63242	C₂×C₈	1	2	yes	—
72	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
73	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
74	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
75	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
76	C₄×Dic₁₇	M₁₆	1	2	yes	—
77	C₂×#3622	C₂×C₈	1	1	yes	characteristic
78	C₂×#3622	C₂×C₈	1	1	yes	characteristic
79	C₈×D₁₇	—	2	1	no	—
80	C₈×D₁₇	—	2	1	no	—
81	C₈×D₁₇	C₄²	1	2	yes	—
82	C₈×D₁₇	C₂×C₈	1	2	yes	—
83	#8275	—	17	1	no	maximal, Sylow
84	C₂²×D₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×#595	—	2	2	no	—
86	C₂×#595	—	2	2	no	—
87	C₂×#595	—	2	2	no	—
88	C₂×#595	C₄⋊C₈	1	4	yes	—
89	C₂×#595	C₄×D₄	1	1	yes	characteristic
90	C₂×#595	C₄×D₄	1	1	yes	characteristic
91	C₂×#595	C₄×Q₈	1	1	yes	characteristic
92	C₂×#595	C₄×Q₈	1	1	yes	characteristic
93	C₂×#595	AGL(1,ℤ/8)	1	2	yes	—
94	C₂×#595	#137	1	2	yes	—
95	#3633	—	4	4	no	—
96	#3633	—	4	4	no	—
97	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
98	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
99	C₂×Dic₁₇	C₄⋊C₈	1	2	yes	—
100	C₂×Dic₁₇	C₂×M₁₆	1	1	yes	characteristic
101	C₄×D₁₇	—	2	2	no	—
102	C₄×D₁₇	—	2	2	no	—
103	C₄×D₁₇	C₂×C₄²	1	1	yes	characteristic
104	C₄×D₁₇	#117	1	1	yes	characteristic
105	C₄×D₁₇	C₂²×C₈	1	1	yes	characteristic
106	C₄×D₁₇	C₂×M₁₆	1	1	yes	characteristic
107	#3624	—	2	2	no	—
108	#3624	—	2	2	no	—
109	C₈×C₁₇	—	2	1	no	—
110	C₈×C₁₇	C₄×C₈	1	2	yes	complemented
111	#3622	—	2	1	no	—
112	#3622	C₄×C₈	1	2	yes	complemented
113	C₄×C₄⋊C₈	—	17	2	no	—
114	C₄×#97	—	17	1	no	—
115	C₂×#335	—	17	1	no	—
116	C₂×#334	—	17	1	no	—
117	#1277	—	17	2	no	—
118	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
119	C₂×D₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
120	C₂×D₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
121	C₂×D₁₇	#428	1	2	yes	—
122	C₂×D₁₇	#430	1	2	yes	—
123	#595	—	2	4	no	—
124	#595	—	2	1	no	—
125	#595	—	2	1	no	—
126	#595	#440	1	2	yes	—
127	#595	#441	1	1	yes	characteristic
128	#595	#442	1	1	yes	characteristic
129	C₄×C₁₇	—	2	2	no	—
130	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
131	C₄×C₁₇	#431	1	1	yes	characteristic
132	Dic₁₇	—	2	2	no	—
133	Dic₁₇	C₄×M₁₆	1	1	yes	characteristic
134	Dic₁₇	#431	1	1	yes	characteristic
135	C₂×C₄⋊C₈	—	17	2	no	—
136	C₂×C₄⋊C₈	—	17	2	no	—
137	C₂×#97	—	17	1	no	—
138	C₂×#97	—	17	1	no	—
139	C₂×#97	—	17	1	no	—
140	C₂×#97	—	17	1	no	—
141	C₂×C₄×C₈	—	17	1	no	—
142	C₂×C₄×C₈	—	17	1	no	—
143	C₂×C₄×C₈	—	34	2	no	—
144	C₄³	—	17	1	no	—
145	#336	—	34	2	no	—
146	#335	—	17	4	no	—
147	#334	—	17	4	no	—
148	C₂×C₁₇	#1565	1	2	yes	—
149	C₂×C₁₇	C₄×C₄⋊C₈	1	1	yes	characteristic
150	D₁₇	#1565	1	2	yes	—
151	D₁₇	#1776	1	1	yes	characteristic
152	D₁₇	#1778	1	1	yes	characteristic
153	C₂²×C₈	—	17	1	no	—
154	C₂²×C₈	—	17	1	no	—
155	C₂²×C₈	—	17	1	no	—
156	C₂²×C₈	—	17	1	no	—
157	C₂²×C₈	—	34	2	no	—
158	C₂²×C₈	—	34	2	no	—
159	C₂×C₄²	—	17	1	no	—
160	C₂×C₄²	—	17	1	no	—
161	C₂×C₄²	—	17	1	no	—
162	C₂×C₄²	—	17	2	no	—
163	C₂×C₄²	—	17	2	no	—
164	C₄⋊C₈	—	17	4	no	—
165	C₄⋊C₈	—	17	4	no	—
166	C₄⋊C₈	—	34	2	no	—
167	C₄⋊C₈	—	34	2	no	—
168	#97	—	17	4	no	—
169	#97	—	17	1	no	—
170	#97	—	17	1	no	—
171	#97	—	17	1	no	—
172	#97	—	34	2	no	—
173	#97	—	34	2	no	—
174	#97	#3633	1	1	yes	characteristic, complemented
175	C₄×C₈	—	17	2	no	—
176	C₄×C₈	—	17	2	no	—
177	C₄×C₈	—	34	1	no	—
178	C₄×C₈	—	34	1	no	—
179	C₄×C₈	—	68	4	no	—
180	C₁₇	#8275	1	1	yes	characteristic, complemented, Sylow
181	C₂²×C₄	—	17	1	no	—
182	C₂²×C₄	—	17	1	no	—
183	C₂²×C₄	—	17	2	no	—
184	C₂²×C₄	—	17	1	no	—
185	C₂²×C₄	—	17	2	no	—
186	C₂×C₈	—	17	1	no	—
187	C₂×C₈	—	17	2	no	—
188	C₂×C₈	—	17	2	no	—
189	C₂×C₈	—	17	2	no	—
190	C₂×C₈	—	17	2	no	—
191	C₂×C₈	—	17	1	no	—
192	C₂×C₈	—	34	1	no	—
193	C₂×C₈	—	34	1	no	—
194	C₂×C₈	—	34	1	no	—
195	C₂×C₈	—	34	1	no	—
196	C₂×C₈	—	34	1	no	—
197	C₂×C₈	—	34	1	no	—
198	C₂×C₈	—	34	4	no	—
199	C₂×C₈	—	34	4	no	—
200	C₂×C₈	—	68	2	no	—
201	C₂×C₈	—	68	2	no	—
202	C₂×C₈	—	68	2	no	—
203	C₂×C₈	—	68	2	no	—
204	C₂×C₈	C₂×#3633	1	1	yes	characteristic
205	C₂×C₈	C₂×#3633	1	1	yes	characteristic
206	C₄²	—	17	2	no	—
207	C₄²	—	17	2	no	—
208	C₄²	—	17	1	no	—
209	C₄²	—	17	1	no	—
210	C₄²	—	17	1	no	—
211	C₄²	—	17	1	no	—
212	C₄²	—	17	2	no	—
213	C₄²	—	17	1	no	—
214	C₄²	—	34	2	no	—
215	C₄²	—	34	2	no	—
216	C₄²	—	68	2	no	—
217	C₄²	C₂×#3633	1	1	yes	characteristic
218	C₂³	—	17	1	no	—
219	C₂×C₄	—	17	1	no	—
220	C₂×C₄	—	17	1	no	—
221	C₂×C₄	—	17	2	no	—
222	C₂×C₄	—	17	2	no	—
223	C₂×C₄	—	17	1	no	—
224	C₂×C₄	—	17	1	no	—
225	C₂×C₄	—	17	1	no	—
226	C₂×C₄	—	17	1	no	—
227	C₂×C₄	—	17	1	no	—
228	C₂×C₄	—	17	4	no	—
229	C₂×C₄	—	17	1	no	—
230	C₂×C₄	—	17	1	no	—
231	C₂×C₄	—	17	2	no	—
232	C₂×C₄	—	34	2	no	—
233	C₂×C₄	—	34	2	no	—
234	C₂×C₄	—	34	2	no	—
235	C₂×C₄	—	34	2	no	—
236	C₂×C₄	—	34	2	no	—
237	C₂×C₄	C₄⋊#3633	1	2	yes	—
238	C₂×C₄	C₂²×#3633	1	1	yes	characteristic
239	C₈	—	2	1	no	—
240	C₈	—	17	2	no	—
241	C₈	—	34	2	no	—
242	C₈	—	34	2	no	—
243	C₈	—	34	1	no	—
244	C₈	—	68	4	no	—
245	C₈	—	68	4	no	—
246	C₈	C₄×#3633	1	2	yes	complemented
247	C₂²	—	17	1	no	—
248	C₂²	—	17	1	no	—
249	C₂²	—	17	2	no	—
250	C₂²	—	17	2	no	—
251	C₂²	C₂×C₄⋊#3633	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	17	1	no	—
254	C₄	—	17	2	no	—
255	C₄	—	17	1	no	—
256	C₄	—	17	1	no	—
257	C₄	—	17	1	no	—
258	C₄	—	34	4	no	—
259	C₄	—	34	1	no	—
260	C₄	—	34	1	no	—
261	C₄	—	34	2	no	—
262	C₄	C₂×C₄×#3633	1	1	yes	characteristic
263	C₄	#11763922	1	1	yes	characteristic
264	C₂	—	17	1	no	—
265	C₂	—	17	2	no	—
266	C₂	—	17	1	no	—
267	C₂	#14545909	1	2	yes	—
268	C₂	C₄×C₄⋊#3633	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊#3633
commutator	C₄×C₁₇	C₂×C₄×C₈
Frattini	C₂×C₄	C₂²×#3633
Fitting	C₁₇×#97	C₈
socle	C₂²×C₁₇	C₂×C₄⋊C₈
prime radical	C₂²×D₁₇	C₂×C₄⋊C₄
cosocle	C₂×C₄×#595	C₂³
hypercenter	#97	#3633
hypocenter	C₁₇	#8275
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
10. #11763899	C₄	semidirect
11. #11763898	C₄	semidirect
37. #10587532	C₈	semidirect
42. C₁₇×#97	C₈	semidirect
43. #10587486	C₈	semidirect
44. #10587486	C₈	semidirect
45. #10587474	C₈	semidirect
110. C₈×C₁₇	C₄×C₈	semidirect
112. #3622	C₄×C₈	semidirect
174. #97	#3633	semidirect
180. C₁₇	#8275	semidirect
246. C₈	C₄×#3633	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#3633	2
C₂×#11763899	2
C₂×#11763898	2
#14545790	2
#97×#595	2
#8275	17