Subgroups — #17952326

Order 4352.

446 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊#3624	C₂	1	2	yes	maximal
2	C₂×#11763672	C₂	1	1	yes	maximal, characteristic
3	C₂×#11763670	C₂	1	1	yes	maximal, characteristic
4	#14543614	C₂	1	2	yes	maximal
5	#97×Dic₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#3624	C₂²	1	2	yes	—
7	C₂×C₄⋊#3624	C₂²	1	2	yes	—
8	C₂×C₄×#3624	—	2	2	no	—
9	#11763675	—	2	2	no	—
10	#11763672	C₄	1	4	yes	complemented
11	#11763670	C₄	1	4	yes	complemented
12	C₂×#10587474	C₄	1	1	yes	characteristic
13	C₂×#10587486	C₄	1	1	yes	characteristic
14	C₂×#10587486	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
17	C₄²×Dic₁₇	C₂²	1	1	yes	characteristic
18	C₂×C₁₇×#97	C₄	1	1	yes	characteristic
19	C₂²×#3624	—	2	2	no	—
20	C₂²×#3624	—	2	2	no	—
21	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₁₇	D₄	1	2	yes	—
26	C₂×C₄×Dic₁₇	Q₈	1	2	yes	—
27	C₂×C₄×Dic₁₇	C₂³	1	1	yes	characteristic
28	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
29	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
30	C₄⋊#3624	—	2	2	no	—
31	C₄⋊#3624	—	2	2	no	—
32	C₄⋊#3624	C₂×C₄	1	4	yes	—
33	C₄⋊#3624	C₂×C₄	1	4	yes	—
34	C₄×#3624	—	4	4	no	—
35	C₁₇×#97	C₈	1	4	yes	complemented
36	#10587486	—	2	2	no	—
37	#10587486	C₈	1	4	yes	complemented
38	C₈×Dic₁₇	—	2	1	no	—
39	C₈×Dic₁₇	—	2	1	no	—
40	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
41	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
42	#10587474	—	2	2	no	—
43	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
44	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
45	C₂²×Dic₁₇	C₄⋊C₄	1	2	yes	—
46	C₂²×Dic₁₇	C₂×D₄	1	1	yes	characteristic
47	C₂²×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
48	C₂×#3624	—	2	4	no	—
49	C₂×#3624	—	2	4	no	—
50	C₂×#3624	—	4	2	no	—
51	C₂×#3624	—	4	2	no	—
52	C₂×#3624	—	4	2	no	—
53	C₂×#3624	—	4	2	no	—
54	C₂×C₈×C₁₇	—	2	1	no	—
55	C₂×C₈×C₁₇	—	2	1	no	—
56	C₂×C₈×C₁₇	C₄²	1	2	yes	—
57	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
58	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
59	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
60	C₄²×C₁₇	C₂×C₈	1	2	yes	—
61	C₄²×C₁₇	M₁₆	1	2	yes	—
62	C₄×Dic₁₇	—	2	2	no	—
63	C₄×Dic₁₇	—	2	2	no	—
64	C₄×Dic₁₇	—	4	2	no	—
65	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
66	C₄×Dic₁₇	M₁₆	1	2	yes	—
67	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
68	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
69	C₄×Dic₁₇	#41	1	1	yes	characteristic
70	C₄×Dic₁₇	#41	1	1	yes	characteristic
71	C₂×#3622	—	2	1	no	—
72	C₂×#3622	—	2	1	no	—
73	C₂×#3622	—	2	1	no	—
74	C₂×#3622	—	2	1	no	—
75	C₂×#3622	C₄²	1	2	yes	—
76	C₂×#3622	C₂×C₈	1	2	yes	—
77	#8275	—	17	1	no	maximal, Sylow
78	C₂³×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₇	—	2	2	no	—
80	C₂×C₄×C₁₇	—	2	2	no	—
81	C₂×C₄×C₁₇	C₄⋊C₈	1	4	yes	—
82	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
83	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
84	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
87	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
88	C₂×Dic₁₇	—	2	2	no	—
89	C₂×Dic₁₇	—	2	2	no	—
90	C₂×Dic₁₇	—	2	2	no	—
91	C₂×Dic₁₇	C₄⋊C₈	1	4	yes	—
92	C₂×Dic₁₇	C₄×D₄	1	1	yes	characteristic
93	C₂×Dic₁₇	C₄×D₄	1	1	yes	characteristic
94	C₂×Dic₁₇	C₄×Q₈	1	1	yes	characteristic
95	C₂×Dic₁₇	C₄×Q₈	1	1	yes	characteristic
96	C₂×Dic₁₇	AGL(1,ℤ/8)	1	2	yes	—
97	C₂×Dic₁₇	#137	1	2	yes	—
98	#3624	—	4	4	no	—
99	#3624	—	4	4	no	—
100	C₈×C₁₇	—	2	2	no	—
101	C₈×C₁₇	C₄×C₈	1	4	yes	complemented
102	#3622	—	2	2	no	—
103	#3622	—	2	2	no	—
104	C₄×C₄⋊C₈	—	17	2	no	—
105	C₄×#97	—	17	1	no	—
106	C₂×#335	—	17	1	no	—
107	C₂×#334	—	17	1	no	—
108	#1277	—	17	2	no	—
109	C₂²×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
110	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
111	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
112	C₂²×C₁₇	#428	1	2	yes	—
113	C₂²×C₁₇	#430	1	2	yes	—
114	C₄×C₁₇	—	2	4	no	—
115	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
116	C₄×C₁₇	C₄×M₁₆	1	1	yes	characteristic
117	C₄×C₁₇	#431	1	1	yes	characteristic
118	C₄×C₁₇	#431	1	1	yes	characteristic
119	Dic₁₇	—	2	4	no	—
120	Dic₁₇	—	2	1	no	—
121	Dic₁₇	—	2	1	no	—
122	Dic₁₇	#440	1	2	yes	—
123	Dic₁₇	#441	1	1	yes	characteristic
124	Dic₁₇	#442	1	1	yes	characteristic
125	C₂×C₄⋊C₈	—	17	2	no	—
126	C₂×C₄⋊C₈	—	17	2	no	—
127	C₂×#97	—	17	1	no	—
128	C₂×#97	—	17	1	no	—
129	C₂×#97	—	17	1	no	—
130	C₂×#97	#595	1	1	yes	characteristic
131	C₂×C₄×C₈	—	17	1	no	—
132	C₂×C₄×C₈	—	17	1	no	—
133	C₂×C₄×C₈	—	34	2	no	—
134	C₄³	—	17	1	no	—
135	#336	—	34	2	no	—
136	#335	—	17	4	no	—
137	#334	—	17	4	no	—
138	C₂×C₁₇	#1565	1	2	yes	—
139	C₂×C₁₇	#1565	1	2	yes	—
140	C₂×C₁₇	C₄×C₄⋊C₈	1	1	yes	characteristic
141	C₂×C₁₇	#1776	1	1	yes	characteristic
142	C₂×C₁₇	#1778	1	1	yes	characteristic
143	C₂²×C₈	—	17	1	no	—
144	C₂²×C₈	—	17	1	no	—
145	C₂²×C₈	—	34	2	no	—
146	C₂²×C₈	—	34	2	no	—
147	C₂²×C₈	C₂×#595	1	1	yes	characteristic
148	C₂²×C₈	C₂×#595	1	1	yes	characteristic
149	C₂×C₄²	—	17	1	no	—
150	C₂×C₄²	—	17	1	no	—
151	C₂×C₄²	—	17	2	no	—
152	C₂×C₄²	—	17	2	no	—
153	C₂×C₄²	C₂×#595	1	1	yes	characteristic
154	C₄⋊C₈	—	17	4	no	—
155	C₄⋊C₈	—	17	4	no	—
156	C₄⋊C₈	—	34	2	no	—
157	C₄⋊C₈	—	34	2	no	—
158	#97	—	17	4	no	—
159	#97	—	34	2	no	—
160	#97	—	34	2	no	—
161	#97	#3624	1	4	yes	complemented
162	C₄×C₈	—	17	2	no	—
163	C₄×C₈	—	17	2	no	—
164	C₄×C₈	—	34	1	no	—
165	C₄×C₈	—	34	1	no	—
166	C₄×C₈	—	68	4	no	—
167	C₁₇	#8275	1	1	yes	characteristic, complemented, Sylow
168	C₂²×C₄	—	17	1	no	—
169	C₂²×C₄	—	17	2	no	—
170	C₂²×C₄	—	17	1	no	—
171	C₂²×C₄	C₄⋊#595	1	2	yes	—
172	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	17	2	no	—
176	C₂×C₈	—	17	2	no	—
177	C₂×C₈	—	34	1	no	—
178	C₂×C₈	—	34	1	no	—
179	C₂×C₈	—	34	1	no	—
180	C₂×C₈	—	34	1	no	—
181	C₂×C₈	—	34	4	no	—
182	C₂×C₈	—	34	4	no	—
183	C₂×C₈	—	68	2	no	—
184	C₂×C₈	—	68	2	no	—
185	C₂×C₈	—	68	2	no	—
186	C₂×C₈	—	68	2	no	—
187	C₂×C₈	#63242	1	2	yes	—
188	C₂×C₈	C₄×#595	1	2	yes	—
189	C₂×C₈	C₂×#3624	1	2	yes	—
190	C₂×C₈	C₂×#3624	1	2	yes	—
191	C₄²	—	17	2	no	—
192	C₄²	—	17	2	no	—
193	C₄²	—	17	1	no	—
194	C₄²	—	17	1	no	—
195	C₄²	—	17	1	no	—
196	C₄²	—	17	1	no	—
197	C₄²	—	34	2	no	—
198	C₄²	—	34	2	no	—
199	C₄²	—	68	2	no	—
200	C₄²	#63243	1	2	yes	—
201	C₄²	C₂×#3624	1	2	yes	—
202	C₂³	C₂×C₄⋊#595	1	1	yes	characteristic
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	17	1	no	—
206	C₂×C₄	—	17	1	no	—
207	C₂×C₄	—	17	2	no	—
208	C₂×C₄	—	17	2	no	—
209	C₂×C₄	—	17	1	no	—
210	C₂×C₄	—	17	1	no	—
211	C₂×C₄	—	17	4	no	—
212	C₂×C₄	—	34	2	no	—
213	C₂×C₄	—	34	2	no	—
214	C₂×C₄	—	34	2	no	—
215	C₂×C₄	C₄⋊#3624	1	4	yes	—
216	C₂×C₄	C₂×#63242	1	1	yes	characteristic
217	C₂×C₄	C₂×#63243	1	1	yes	characteristic
218	C₂×C₄	D₁₇⋊M₁₆	1	1	yes	characteristic
219	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
220	C₂×C₄	#10587677	1	1	yes	characteristic
221	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
222	C₈	—	2	2	no	—
223	C₈	—	34	2	no	—
224	C₈	—	34	2	no	—
225	C₈	—	68	4	no	—
226	C₈	—	68	4	no	—
227	C₈	C₄×#3624	1	4	yes	complemented
228	C₂²	C₄×C₄⋊#595	1	1	yes	characteristic
229	C₂²	C₂×C₄⋊#3624	1	1	yes	characteristic
230	C₂²	#11763810	1	1	yes	characteristic
231	C₂²	#11763820	1	2	yes	—
232	C₂²	#11763824	1	2	yes	—
233	C₄	—	2	4	no	—
234	C₄	—	17	1	no	—
235	C₄	—	17	2	no	—
236	C₄	—	17	1	no	—
237	C₄	—	34	4	no	—
238	C₄	—	34	1	no	—
239	C₄	—	34	1	no	—
240	C₄	C₂×C₄×#3624	1	1	yes	characteristic
241	C₄	C₄×#63243	1	1	yes	characteristic
242	C₄	#11763825	1	1	yes	characteristic
243	C₄	#11763828	1	1	yes	characteristic
244	C₂	#14544335	1	2	yes	—
245	C₂	#14544336	1	2	yes	—
246	C₂	C₄×C₄⋊#3624	1	1	yes	characteristic
247	C₂	#14544598	1	1	yes	characteristic
248	C₂	#14544604	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#595
commutator	C₄×C₁₇	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#595
Fitting	C₂×C₁₇×#97	C₄
socle, prime radical	C₂³×C₁₇	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₁₇	C₂³
hypercenter	C₂×#97	#595
hypocenter	C₁₇	#8275
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
10. #11763672	C₄	semidirect
11. #11763670	C₄	semidirect
35. C₁₇×#97	C₈	semidirect
37. #10587486	C₈	semidirect
101. C₈×C₁₇	C₄×C₈	semidirect
161. #97	#3624	semidirect
167. C₁₇	#8275	semidirect
227. C₈	C₄×#3624	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#3624	2
C₂×#11763672	2
C₂×#11763670	2
#14543614	2
#97×Dic₁₇	2
#8275	17