Subgroups — #17953760

Order 4352.

422 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14544716	C₂	1	2	yes	maximal
2	#14543614	C₂	1	4	yes	maximal
3	C₄×C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#3624	—	2	2	no	—
5	C₂×C₄⋊#3624	—	2	2	no	—
6	C₂×C₄⋊#3624	C₂²	1	4	yes	—
7	C₂×C₄×#3622	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₁₇	C₄	1	2	yes	—
9	C₂×C₈×Dic₁₇	C₂²	1	2	yes	—
10	C₄²×Dic₁₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₇	C₄	1	1	yes	characteristic
12	C₂²×#3624	—	2	4	no	—
13	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×Dic₁₇	D₄	1	1	yes	characteristic
17	C₂×C₄×Dic₁₇	D₄	1	1	yes	characteristic
18	C₂×C₄×Dic₁₇	Q₈	1	1	yes	characteristic
19	C₂×C₄×Dic₁₇	Q₈	1	1	yes	characteristic
20	C₂×C₄×Dic₁₇	C₂³	1	1	yes	characteristic
21	C₂²×#3622	C₂×C₄	1	2	yes	—
22	C₄⋊#3624	—	2	8	no	—
23	C₄⋊#3624	—	4	4	no	—
24	C₄⋊#3624	—	4	4	no	—
25	C₄×C₈×C₁₇	C₈	1	4	yes	complemented
26	C₈×Dic₁₇	—	2	4	no	—
27	C₈×Dic₁₇	—	2	4	no	—
28	C₄×#3622	—	2	2	no	—
29	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
32	C₂²×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
33	C₂²×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×Dic₁₇	C₂×D₄	1	1	yes	characteristic
35	C₂²×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
36	C₂×#3624	—	4	4	no	—
37	C₂×#3624	—	4	4	no	—
38	C₂×#3624	—	4	4	no	—
39	C₂×C₈×C₁₇	—	2	2	no	—
40	C₂×C₈×C₁₇	—	2	2	no	—
41	C₂×C₈×C₁₇	C₂×C₈	1	4	yes	—
42	C₄²×C₁₇	C₂×C₈	1	2	yes	—
43	C₄²×C₁₇	M₁₆	1	2	yes	—
44	C₄×Dic₁₇	—	2	2	no	—
45	C₄×Dic₁₇	—	2	2	no	—
46	C₄×Dic₁₇	—	2	2	no	—
47	C₄×Dic₁₇	D₈	1	2	yes	—
48	C₄×Dic₁₇	SD₁₆	1	4	yes	—
49	C₄×Dic₁₇	Q₁₆	1	2	yes	—
50	C₄×Dic₁₇	#41	1	4	yes	—
51	C₂×#3622	—	2	2	no	—
52	C₂×#3622	—	2	2	no	—
53	C₂×#3622	M₁₆	1	4	yes	—
54	#8784	—	17	1	no	maximal, Sylow
55	C₂³×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
57	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
58	C₂×C₄×C₁₇	#106	1	2	yes	—
59	C₂×C₄×C₁₇	#107	1	2	yes	—
60	C₂×C₄×C₁₇	#108	1	2	yes	—
61	C₂×C₄×C₁₇	#108	1	2	yes	—
62	C₂×C₄×C₁₇	#117	1	2	yes	—
63	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
65	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
66	C₂×Dic₁₇	—	2	1	no	—
67	C₂×Dic₁₇	—	2	1	no	—
68	C₂×Dic₁₇	—	2	1	no	—
69	C₂×Dic₁₇	—	2	1	no	—
70	C₂×Dic₁₇	—	2	1	no	—
71	C₂×Dic₁₇	—	2	1	no	—
72	C₂×Dic₁₇	#106	1	2	yes	—
73	C₂×Dic₁₇	#107	1	2	yes	—
74	C₂×Dic₁₇	#124	1	2	yes	—
75	C₂×Dic₁₇	#125	1	2	yes	—
76	C₂×Dic₁₇	C₂×D₈	1	1	yes	characteristic
77	C₂×Dic₁₇	C₂×SD₁₆	1	2	yes	—
78	C₂×Dic₁₇	C₂×Q₁₆	1	1	yes	characteristic
79	#3624	—	8	4	no	—
80	C₈×C₁₇	—	2	4	no	—
81	#3622	—	2	4	no	—
82	#1842	—	17	2	no	—
83	C₄²×C₈	—	17	1	no	—
84	#1277	—	17	4	no	—
85	C₂²×C₁₇	#382	1	1	yes	characteristic
86	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
87	C₂²×C₁₇	#423	1	1	yes	characteristic
88	C₂²×C₁₇	C₂×#106	1	1	yes	characteristic
89	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
90	C₂²×C₁₇	C₂×#108	1	1	yes	characteristic
91	C₂²×C₁₇	C₂×#108	1	1	yes	characteristic
92	C₄×C₁₇	#334	1	4	yes	—
93	C₄×C₁₇	#335	1	4	yes	—
94	C₄×C₁₇	#431	1	2	yes	—
95	C₄×C₁₇	#432	1	2	yes	—
96	Dic₁₇	—	2	2	no	—
97	Dic₁₇	—	2	2	no	—
98	Dic₁₇	—	2	2	no	—
99	Dic₁₇	#486	1	2	yes	—
100	Dic₁₇	#487	1	2	yes	—
101	C₂×C₄⋊C₈	—	17	4	no	—
102	C₂×C₄⋊C₈	—	34	2	no	—
103	C₂×C₄⋊C₈	—	34	2	no	—
104	C₂×C₄×C₈	—	17	2	no	—
105	C₂×C₄×C₈	—	17	2	no	—
106	C₂×C₄×C₈	—	17	1	no	—
107	C₂×C₄×C₈	#595	1	1	yes	characteristic
108	C₄³	—	17	1	no	—
109	C₂×C₁₇	C₂×#334	1	1	yes	characteristic
110	C₂×C₁₇	C₂×#335	1	1	yes	characteristic
111	C₂×C₁₇	#1566	1	1	yes	characteristic
112	C₂×C₁₇	#1567	1	1	yes	characteristic
113	C₂×C₁₇	#1831	1	1	yes	characteristic
114	C₂×C₁₇	#1834	1	1	yes	characteristic
115	C₂×C₁₇	#1837	1	1	yes	characteristic
116	C₂²×C₈	—	17	2	no	—
117	C₂²×C₈	—	34	4	no	—
118	C₂²×C₈	C₂×#595	1	2	yes	—
119	C₂×C₄²	—	17	1	no	—
120	C₂×C₄²	—	17	1	no	—
121	C₂×C₄²	—	17	1	no	—
122	C₂×C₄²	—	17	1	no	—
123	C₂×C₄²	—	17	1	no	—
124	C₂×C₄²	—	17	1	no	—
125	C₂×C₄²	C₂×#595	1	1	yes	characteristic
126	C₄⋊C₈	—	34	8	no	—
127	C₄⋊C₈	—	68	4	no	—
128	C₄⋊C₈	—	68	4	no	—
129	C₄×C₈	—	34	4	no	—
130	C₄×C₈	—	34	4	no	—
131	C₄×C₈	—	34	2	no	—
132	C₄×C₈	#3624	1	4	yes	complemented
133	C₁₇	#8784	1	1	yes	characteristic, complemented, Sylow
134	C₂²×C₄	—	17	1	no	—
135	C₂²×C₄	—	17	1	no	—
136	C₂²×C₄	—	17	1	no	—
137	C₂²×C₄	—	17	1	no	—
138	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
139	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
140	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	17	4	no	—
144	C₂×C₈	—	34	2	no	—
145	C₂×C₈	—	34	2	no	—
146	C₂×C₈	—	68	4	no	—
147	C₂×C₈	—	68	4	no	—
148	C₂×C₈	—	68	4	no	—
149	C₂×C₈	C₂×#3624	1	4	yes	—
150	C₄²	—	17	4	no	—
151	C₄²	—	17	4	no	—
152	C₄²	—	17	2	no	—
153	C₄²	—	17	2	no	—
154	C₄²	—	34	2	no	—
155	C₄²	—	34	2	no	—
156	C₄²	—	34	2	no	—
157	C₄²	#63243	1	2	yes	—
158	C₄²	C₂×#3624	1	2	yes	—
159	C₂³	C₂×C₄⋊#595	1	1	yes	characteristic
160	C₂×C₄	—	17	2	no	—
161	C₂×C₄	—	17	2	no	—
162	C₂×C₄	—	17	1	no	—
163	C₂×C₄	—	17	1	no	—
164	C₂×C₄	—	17	2	no	—
165	C₂×C₄	—	17	2	no	—
166	C₂×C₄	—	17	2	no	—
167	C₂×C₄	—	34	1	no	—
168	C₂×C₄	—	34	1	no	—
169	C₂×C₄	—	34	1	no	—
170	C₂×C₄	—	34	1	no	—
171	C₂×C₄	—	34	1	no	—
172	C₂×C₄	—	34	1	no	—
173	C₂×C₄	#10587533	1	2	yes	—
174	C₂×C₄	#10587534	1	2	yes	—
175	C₂×C₄	#10587535	1	2	yes	—
176	C₂×C₄	#10587536	1	2	yes	—
177	C₂×C₄	C₄⋊#3624	1	2	yes	—
178	C₂×C₄	C₄⋊#3624	1	2	yes	—
179	C₂×C₄	C₂×#63243	1	1	yes	characteristic
180	C₂×C₄	D₁₇⋊M₁₆	1	2	yes	—
181	C₂×C₄	#10587677	1	2	yes	—
182	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
183	C₈	—	2	4	no	—
184	C₈	—	34	4	no	—
185	C₈	—	136	4	no	—
186	C₂²	#11763753	1	1	yes	characteristic
187	C₂²	C₂×C₄⋊#3624	1	1	yes	characteristic
188	C₂²	#11763814	1	1	yes	characteristic
189	C₂²	C₂×#10587533	1	1	yes	characteristic
190	C₂²	C₂×#10587534	1	1	yes	characteristic
191	C₂²	C₂×#10587535	1	1	yes	characteristic
192	C₂²	C₂×#10587536	1	1	yes	characteristic
193	C₄	—	17	2	no	—
194	C₄	—	17	2	no	—
195	C₄	—	34	2	no	—
196	C₄	—	34	2	no	—
197	C₄	—	34	2	no	—
198	C₄	#11763670	1	4	yes	—
199	C₄	#11763672	1	4	yes	—
200	C₄	#11763825	1	2	yes	—
201	C₄	#11763831	1	2	yes	—
202	C₂	C₂×#11763670	1	1	yes	characteristic
203	C₂	C₂×#11763672	1	1	yes	characteristic
204	C₂	#14544340	1	1	yes	characteristic
205	C₂	#14544343	1	1	yes	characteristic
206	C₂	#14544699	1	1	yes	characteristic
207	C₂	#14544705	1	1	yes	characteristic
208	C₂	#14544708	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#595
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#595
Fitting	C₂×C₄×C₈×C₁₇	C₄
socle, prime radical	C₂³×C₁₇	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₁₇	C₂³
hypercenter	C₂×C₄×C₈	#595
hypocenter	C₁₇	#8784
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
25. C₄×C₈×C₁₇	C₈	semidirect
132. C₄×C₈	#3624	semidirect
133. C₁₇	#8784	semidirect

Maximal subgroups

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

subgroup	index
#14544716	2
#14543614	2
C₄×C₈×Dic₁₇	2
#8784	17