Subgroups — #17982474

Order 4352.

398 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14545970	C₂	1	1	yes	maximal, characteristic
2	C₂×#11763902	C₂	1	2	yes	maximal
3	C₄×#10587665	C₂	1	1	yes	maximal, characteristic
4	#14545855	C₂	1	2	yes	maximal
5	#14544577	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587665	C₂²	1	1	yes	characteristic
7	C₂×#10587665	C₂²	1	2	yes	—
8	C₂×#10587665	C₂²	1	1	yes	characteristic
9	#11763912	C₄	1	2	yes	complemented
10	C₄⋊#63241	C₄	1	2	yes	complemented
11	#11763902	—	2	4	no	—
12	C₄×#63241	C₄	1	2	yes	complemented
13	C₄×#63241	C₄	1	2	yes	complemented
14	C₄×#63242	C₂²	1	1	yes	characteristic
15	C₂×C₈×#595	C₂²	1	2	yes	—
16	C₄×C₄⋊#595	C₄	1	1	yes	characteristic
17	#11763674	C₄	1	2	yes	—
18	C₄⋊C₈×D₁₇	C₄	1	1	yes	characteristic
19	C₂×C₄⋊#595	C₈	1	2	yes	—
20	C₂×C₄×#595	C₂×C₄	1	2	yes	—
21	C₂×#63242	C₂×C₄	1	1	yes	characteristic
22	C₂×#63242	C₂³	1	1	yes	characteristic
23	C₂×#63241	C₂×C₄	1	2	yes	—
24	C₂×#63241	C₂×C₄	1	4	yes	—
25	C₂×#63241	C₂×C₄	1	2	yes	—
26	#10587665	—	2	2	no	—
27	#10587665	—	2	1	no	—
28	#10587665	—	2	1	no	—
29	#10587665	—	2	2	no	—
30	#10587665	D₄	1	2	yes	—
31	#10587665	Q₈	1	2	yes	—
32	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
33	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
34	(C₂×#595)⋊C₄	C₈	1	2	yes	—
35	C₄×#3624	C₂×C₄	1	1	yes	characteristic
36	C₄×#3624	C₂×C₄	1	1	yes	characteristic
37	C₄×#3624	C₂×C₄	1	1	yes	characteristic
38	C₄×#3624	C₂×C₄	1	1	yes	characteristic
39	C₈×#595	—	2	2	no	—
40	C₈×#595	—	2	2	no	—
41	C₁₇×C₄⋊C₈	C₈	1	1	yes	characteristic
42	#10587485	C₈	1	2	yes	—
43	C₄⋊#3622	C₈	1	1	yes	characteristic
44	C₂²×#595	C₂×C₈	1	4	yes	—
45	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
46	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
47	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×#3624	C₄²	1	1	yes	characteristic
49	C₂×#3624	C₄²	1	1	yes	characteristic
50	C₂×#3624	C₂²×C₄	1	1	yes	characteristic
51	C₂×#3624	C₂²×C₄	1	1	yes	characteristic
52	C₄⋊#595	C₁₆	1	8	yes	complemented
53	C₄×#595	—	2	2	no	—
54	C₄×#595	—	2	2	no	—
55	#63242	—	2	1	no	—
56	#63242	—	2	1	no	—
57	#63242	C₂×D₄	1	1	yes	characteristic
58	#63242	C₂×Q₈	1	1	yes	characteristic
59	#63242	#41	1	1	yes	characteristic
60	#63242	#41	1	1	yes	characteristic
61	#63241	—	2	2	no	—
62	#63241	—	2	4	no	—
63	#63241	C₄⋊C₄	1	4	yes	complemented
64	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
65	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
66	C₄×Dic₁₇	C₄²	1	1	yes	characteristic
67	C₄×Dic₁₇	C₄²	1	1	yes	characteristic
68	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
69	C₂×#3622	C₂×C₈	1	2	yes	—
70	C₈×D₁₇	—	2	2	no	—
71	C₈×D₁₇	—	2	2	no	—
72	C₁₆×C₄⋊C₄	—	17	1	no	maximal, Sylow
73	C₂²×D₁₇	C₂²×C₈	1	1	yes	characteristic
74	C₂×#595	—	2	4	no	—
75	C₂×#595	—	2	4	no	—
76	C₂×#595	C₂×C₁₆	1	8	yes	—
77	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
78	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
79	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
80	C₂×Dic₁₇	C₄×C₈	1	1	yes	characteristic
81	C₂×Dic₁₇	C₄×C₈	1	1	yes	characteristic
82	C₂×Dic₁₇	C₂×C₄²	1	1	yes	characteristic
83	C₄×D₁₇	—	2	2	no	—
84	C₄×D₁₇	C₂×C₁₆	1	4	yes	—
85	C₄×D₁₇	C₄×D₄	1	1	yes	characteristic
86	C₄×D₁₇	C₄×Q₈	1	1	yes	characteristic
87	C₄×D₁₇	#131	1	1	yes	characteristic
88	C₄×D₁₇	#131	1	1	yes	characteristic
89	#3624	—	2	1	no	—
90	#3624	—	2	1	no	—
91	#3624	C₂×C₄⋊C₄	1	1	yes	characteristic
92	#3624	#117	1	1	yes	characteristic
93	#3624	C₄×D₄	1	1	yes	characteristic
94	#3624	C₄×Q₈	1	1	yes	characteristic
95	C₈×C₁₇	—	2	2	no	—
96	#3622	—	2	2	no	—
97	C₂×C₄⋊C₁₆	—	17	1	no	—
98	C₂×C₄×C₁₆	—	17	1	no	—
99	C₂×C₄×C₁₆	—	17	2	no	—
100	C₈×C₄⋊C₄	—	17	1	no	—
101	#1374	—	17	2	no	—
102	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
103	C₂×D₁₇	C₈×D₄	1	1	yes	characteristic
104	C₂×D₁₇	C₈×Q₈	1	1	yes	characteristic
105	C₂×D₁₇	C₂²×C₁₆	1	2	yes	—
106	C₂×D₁₇	#504	1	2	yes	—
107	#595	—	2	8	no	—
108	C₄×C₁₇	—	2	1	no	—
109	C₄×C₁₇	C₄×C₁₆	1	2	yes	complemented
110	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
111	C₄×C₁₇	C₈×Q₈	1	1	yes	characteristic
112	Dic₁₇	—	2	1	no	—
113	Dic₁₇	C₄×C₁₆	1	2	yes	complemented
114	Dic₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
115	Dic₁₇	#405	1	1	yes	characteristic
116	C₂²×C₁₆	—	17	1	no	—
117	C₂²×C₁₆	—	17	1	no	—
118	C₂²×C₁₆	—	17	2	no	—
119	C₂×C₄⋊C₈	—	17	1	no	—
120	C₂×C₄×C₈	—	17	1	no	—
121	C₂×C₄×C₈	—	17	2	no	—
122	C₄×C₄⋊C₄	—	17	1	no	—
123	C₄⋊C₁₆	—	17	2	no	—
124	C₄⋊C₁₆	—	17	2	no	—
125	C₄×C₁₆	—	17	2	no	—
126	C₄×C₁₆	—	17	2	no	—
127	C₄×C₁₆	—	34	4	no	—
128	#336	—	17	2	no	—
129	C₂×C₁₇	C₈×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₁₇	C₂×C₄×C₁₆	1	1	yes	characteristic
131	C₂×C₁₇	#2108	1	1	yes	characteristic
132	D₁₇	D₄×C₁₆	1	2	yes	—
133	D₁₇	Q₈×C₁₆	1	2	yes	—
134	C₂²×C₈	—	17	1	no	—
135	C₂²×C₈	—	17	1	no	—
136	C₂²×C₈	—	17	2	no	—
137	C₂×C₄⋊C₄	—	17	2	no	—
138	C₂×C₄²	—	17	1	no	—
139	C₂×C₄²	—	17	2	no	—
140	C₂×C₁₆	—	17	2	no	—
141	C₂×C₁₆	—	17	2	no	—
142	C₂×C₁₆	—	17	2	no	—
143	C₂×C₁₆	—	17	2	no	—
144	C₂×C₁₆	—	17	4	no	—
145	C₂×C₁₆	—	34	2	no	—
146	C₂×C₁₆	—	34	1	no	—
147	C₂×C₁₆	—	34	1	no	—
148	C₂×C₁₆	—	34	2	no	—
149	C₄⋊C₈	—	17	2	no	—
150	C₄⋊C₈	—	17	1	no	—
151	C₄⋊C₈	#3633	1	1	yes	characteristic
152	C₄×C₈	—	17	1	no	—
153	C₄×C₈	—	17	1	no	—
154	C₄×C₈	—	17	1	no	—
155	C₄×C₈	—	17	1	no	—
156	C₄×C₈	—	34	2	no	—
157	C₄×C₈	—	34	2	no	—
158	#95	—	17	2	no	—
159	C₁₇	C₁₆×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
160	C₂²×C₄	—	17	1	no	—
161	C₂²×C₄	—	17	1	no	—
162	C₂²×C₄	—	17	4	no	—
163	C₂²×C₄	—	17	1	no	—
164	C₂×C₈	—	17	1	no	—
165	C₂×C₈	—	17	1	no	—
166	C₂×C₈	—	17	1	no	—
167	C₂×C₈	—	17	1	no	—
168	C₂×C₈	—	17	1	no	—
169	C₂×C₈	—	17	1	no	—
170	C₂×C₈	—	17	1	no	—
171	C₂×C₈	—	17	1	no	—
172	C₂×C₈	—	17	2	no	—
173	C₂×C₈	—	34	2	no	—
174	C₂×C₈	—	34	2	no	—
175	C₂×C₈	—	34	1	no	—
176	C₂×C₈	—	34	1	no	—
177	C₂×C₈	C₂×#3633	1	2	yes	—
178	C₄⋊C₄	—	17	8	no	—
179	C₄²	—	17	1	no	—
180	C₄²	—	17	1	no	—
181	C₄²	—	17	1	no	—
182	C₄²	—	34	2	no	—
183	C₄²	—	34	2	no	—
184	C₄²	C₂×#3633	1	1	yes	characteristic
185	C₁₆	—	17	4	no	—
186	C₁₆	—	34	4	no	—
187	C₁₆	—	34	2	no	—
188	C₂³	—	17	1	no	—
189	C₂×C₄	—	17	1	no	—
190	C₂×C₄	—	17	1	no	—
191	C₂×C₄	—	17	4	no	—
192	C₂×C₄	—	17	1	no	—
193	C₂×C₄	—	17	1	no	—
194	C₂×C₄	—	17	1	no	—
195	C₂×C₄	—	17	1	no	—
196	C₂×C₄	—	17	8	no	—
197	C₂×C₄	—	17	1	no	—
198	C₂×C₄	—	34	2	no	—
199	C₂×C₄	—	34	4	no	—
200	C₂×C₄	—	34	4	no	—
201	C₂×C₄	C₄×#3633	1	1	yes	characteristic
202	C₂×C₄	C₄×#3633	1	1	yes	characteristic
203	C₂×C₄	C₂²×#3633	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	—	17	1	no	—
206	C₈	—	17	1	no	—
207	C₈	—	17	1	no	—
208	C₈	—	17	1	no	—
209	C₈	—	34	1	no	—
210	C₈	—	34	1	no	—
211	C₈	—	34	2	no	—
212	C₂²	—	17	1	no	—
213	C₂²	—	17	1	no	—
214	C₂²	—	17	2	no	—
215	C₂²	—	17	2	no	—
216	C₂²	C₂×C₄×#3633	1	1	yes	characteristic
217	C₄	—	2	1	no	—
218	C₄	—	17	1	no	—
219	C₄	—	17	1	no	—
220	C₄	—	17	2	no	—
221	C₄	—	34	8	no	—
222	C₄	—	34	1	no	—
223	C₄	#11763902	1	2	yes	complemented
224	C₄	D₄×#3633	1	1	yes	characteristic
225	C₄	Q₈×#3633	1	1	yes	characteristic
226	C₂	—	17	2	no	—
227	C₂	—	17	2	no	—
228	C₂	C₄⋊C₄×#3633	1	1	yes	characteristic
229	C₂	C₂×#11763902	1	1	yes	characteristic
230	C₂	#14545947	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#3633
commutator	C₂×C₁₇	C₂×C₄×C₁₆
Fitting	C₁₇×C₄⋊C₈	C₈
socle	C₂²×C₁₇	C₂×C₄×C₈
prime radical	C₂²×D₁₇	C₂²×C₈
cosocle	C₂×#63242	C₂³
hypercenter	C₄⋊C₈	#3633
hypocenter	C₁₇	C₁₆×C₄⋊C₄
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
9. #11763912	C₄	semidirect
10. C₄⋊#63241	C₄	semidirect
12. C₄×#63241	C₄	semidirect
13. C₄×#63241	C₄	semidirect
52. C₄⋊#595	C₁₆	semidirect
63. #63241	C₄⋊C₄	semidirect
109. C₄×C₁₇	C₄×C₁₆	semidirect
113. Dic₁₇	C₄×C₁₆	semidirect
159. C₁₇	C₁₆×C₄⋊C₄	semidirect
223. C₄	#11763902	semidirect

Maximal subgroups

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

subgroup	index
#14545970	2
C₂×#11763902	2
C₄×#10587665	2
#14545855	2
#14544577	2
C₁₆×C₄⋊C₄	17