Subgroups — #17953810

Order 4352.

462 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14544717	C₂	1	2	yes	maximal
2	C₂×#11763671	C₂	1	2	yes	maximal
3	C₂×#11763670	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587544	—	2	4	no	—
6	C₂×C₄⋊#3624	C₂²	1	4	yes	—
7	#11763671	—	2	4	no	—
8	#11763670	—	2	4	no	—
9	C₂×C₄×#3622	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₁₇	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₁₇	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
14	C₄²×Dic₁₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₇	C₄	1	1	yes	characteristic
16	C₂²×#3624	—	2	4	no	—
17	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₁₇	D₄	1	2	yes	—
22	C₂×C₄×Dic₁₇	Q₈	1	2	yes	—
23	C₂×C₄×Dic₁₇	C₂³	1	1	yes	characteristic
24	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
25	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
26	#10587544	—	4	8	no	—
27	C₄⋊#3624	—	2	4	no	—
28	C₄⋊#3624	—	2	4	no	—
29	C₄×C₈×C₁₇	—	2	2	no	—
30	C₈×Dic₁₇	—	2	2	no	—
31	C₈×Dic₁₇	C₈	1	4	yes	complemented
32	C₈×Dic₁₇	D₄	1	2	yes	—
33	C₈×Dic₁₇	D₄	1	2	yes	—
34	C₈×Dic₁₇	Q₈	1	2	yes	—
35	C₈×Dic₁₇	Q₈	1	2	yes	—
36	C₄×#3622	—	2	2	no	—
37	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
38	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
39	C₂²×Dic₁₇	C₄⋊C₄	1	2	yes	—
40	C₂²×Dic₁₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
42	C₂×#3624	—	4	4	no	—
43	C₂×#3624	—	4	8	no	—
44	C₂×C₈×C₁₇	—	2	1	no	—
45	C₂×C₈×C₁₇	—	2	1	no	—
46	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
47	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
49	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
50	C₄²×C₁₇	—	2	1	no	—
51	C₄²×C₁₇	—	2	1	no	—
52	C₄×Dic₁₇	—	2	4	no	—
53	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
54	C₄×Dic₁₇	M₁₆	1	2	yes	—
55	C₄×Dic₁₇	SD₁₆	1	8	yes	—
56	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
57	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
58	C₄×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
59	C₄×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
60	C₂×#3622	—	2	1	no	—
61	C₂×#3622	—	2	1	no	—
62	C₂×#3622	C₄⋊C₄	1	2	yes	—
63	C₂×#3622	C₄⋊C₄	1	2	yes	—
64	C₂×#3622	C₂×C₈	1	2	yes	—
65	C₂×#3622	M₁₆	1	2	yes	—
66	#8805	—	17	1	no	maximal, Sylow
67	C₂³×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₇	—	2	2	no	—
69	C₂×C₄×C₁₇	—	2	2	no	—
70	C₂×C₄×C₁₇	#106	1	4	yes	—
71	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
75	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
77	C₂×Dic₁₇	—	2	2	no	—
78	C₂×Dic₁₇	C₄⋊C₈	1	4	yes	—
79	C₂×Dic₁₇	#106	1	4	yes	—
80	C₂×Dic₁₇	#108	1	4	yes	—
81	C₂×Dic₁₇	#127	1	1	yes	characteristic
82	C₂×Dic₁₇	C₄⋊Q₈	1	1	yes	characteristic
83	C₂×Dic₁₇	C₄⋊Q₈	1	1	yes	characteristic
84	C₂×Dic₁₇	C₄⋊Q₈	1	1	yes	characteristic
85	C₂×Dic₁₇	C₂×SD₁₆	1	4	yes	—
86	#3624	—	8	4	no	—
87	C₈×C₁₇	—	2	2	no	—
88	C₈×C₁₇	C₄⋊C₈	1	4	yes	complemented
89	#3622	—	2	2	no	—
90	#3622	C₄⋊C₈	1	4	yes	complemented
91	#1842	—	17	2	no	—
92	C₄²×C₈	—	17	1	no	—
93	C₂×#334	—	17	2	no	—
94	C₂×#334	—	17	2	no	—
95	C₂²×C₁₇	#384	1	1	yes	characteristic
96	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
97	C₂²×C₁₇	#423	1	1	yes	characteristic
98	C₂²×C₁₇	C₂×#106	1	2	yes	—
99	C₂²×C₁₇	C₂×#108	1	2	yes	—
100	C₄×C₁₇	—	2	4	no	—
101	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
102	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
103	C₄×C₁₇	#423	1	1	yes	characteristic
104	C₄×C₁₇	#423	1	1	yes	characteristic
105	Dic₁₇	—	2	1	no	—
106	Dic₁₇	—	2	1	no	—
107	Dic₁₇	#334	1	8	yes	complemented
108	Dic₁₇	#492	1	2	yes	—
109	Dic₁₇	#498	1	2	yes	—
110	C₂×C₄⋊C₈	—	17	4	no	—
111	C₂×C₄⋊C₈	—	34	4	no	—
112	C₂×C₄×C₈	—	17	1	no	—
113	C₂×C₄×C₈	—	17	1	no	—
114	C₂×C₄×C₈	—	17	1	no	—
115	C₂×C₄×C₈	—	17	1	no	—
116	C₂×C₄×C₈	—	17	1	no	—
117	C₂×C₄×C₈	#595	1	1	yes	characteristic
118	C₄³	—	17	1	no	—
119	#334	—	34	4	no	—
120	#334	—	34	4	no	—
121	C₂×C₁₇	C₂×#334	1	2	yes	—
122	C₂×C₁₇	#1566	1	2	yes	—
123	C₂×C₁₇	#1842	1	1	yes	characteristic
124	C₂×C₁₇	#1844	1	1	yes	characteristic
125	C₂×C₁₇	#1848	1	1	yes	characteristic
126	C₂²×C₈	—	17	1	no	—
127	C₂²×C₈	—	17	1	no	—
128	C₂²×C₈	—	34	4	no	—
129	C₂²×C₈	C₂×#595	1	1	yes	characteristic
130	C₂²×C₈	C₂×#595	1	1	yes	characteristic
131	C₂×C₄²	—	17	2	no	—
132	C₂×C₄²	—	17	1	no	—
133	C₂×C₄²	—	17	1	no	—
134	C₂×C₄²	—	17	2	no	—
135	C₂×C₄²	C₂×#595	1	1	yes	characteristic
136	C₄⋊C₈	—	34	4	no	—
137	C₄⋊C₈	—	34	4	no	—
138	C₄⋊C₈	—	68	8	no	—
139	C₄×C₈	—	2	2	no	—
140	C₄×C₈	—	17	4	no	—
141	C₄×C₈	—	17	2	no	—
142	C₄×C₈	—	17	2	no	—
143	C₄×C₈	—	17	2	no	—
144	C₄×C₈	—	17	2	no	—
145	C₄×C₈	—	34	2	no	—
146	C₄×C₈	—	34	2	no	—
147	C₁₇	#8805	1	1	yes	characteristic, complemented, Sylow
148	C₂²×C₄	—	17	2	no	—
149	C₂²×C₄	—	17	1	no	—
150	C₂²×C₄	—	17	1	no	—
151	C₂²×C₄	C₄⋊#595	1	2	yes	—
152	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	17	2	no	—
156	C₂×C₈	—	17	2	no	—
157	C₂×C₈	—	17	2	no	—
158	C₂×C₈	—	17	2	no	—
159	C₂×C₈	—	34	1	no	—
160	C₂×C₈	—	34	1	no	—
161	C₂×C₈	—	68	4	no	—
162	C₂×C₈	—	68	8	no	—
163	C₂×C₈	#63242	1	2	yes	—
164	C₂×C₈	C₄⋊#595	1	2	yes	—
165	C₂×C₈	C₄⋊#595	1	2	yes	—
166	C₂×C₈	#63247	1	2	yes	—
167	C₄²	—	2	1	no	—
168	C₄²	—	2	1	no	—
169	C₄²	—	17	2	no	—
170	C₄²	—	17	2	no	—
171	C₄²	—	17	1	no	—
172	C₄²	—	17	1	no	—
173	C₄²	—	17	1	no	—
174	C₄²	—	17	1	no	—
175	C₄²	—	17	8	no	—
176	C₄²	—	34	4	no	—
177	C₂³	C₂×C₄⋊#595	1	1	yes	characteristic
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	17	1	no	—
181	C₂×C₄	—	17	1	no	—
182	C₂×C₄	—	17	4	no	—
183	C₂×C₄	—	17	4	no	—
184	C₂×C₄	—	17	4	no	—
185	C₂×C₄	—	17	4	no	—
186	C₂×C₄	—	17	1	no	—
187	C₂×C₄	—	17	1	no	—
188	C₂×C₄	—	34	2	no	—
189	C₂×C₄	#10587533	1	4	yes	—
190	C₂×C₄	C₂×#63242	1	1	yes	characteristic
191	C₂×C₄	D₁₇⋊M₁₆	1	1	yes	characteristic
192	C₂×C₄	D₁₇⋊M₁₆	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
194	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
195	C₂×C₄	C₂×#63247	1	1	yes	characteristic
196	C₈	—	2	2	no	—
197	C₈	—	17	4	no	—
198	C₈	—	34	2	no	—
199	C₈	—	136	4	no	—
200	C₈	#10587544	1	4	yes	complemented
201	C₂²	#11763755	1	1	yes	characteristic
202	C₂²	#11763810	1	1	yes	characteristic
203	C₂²	#11763812	1	1	yes	characteristic
204	C₂²	C₂×#10587533	1	2	yes	—
205	C₂²	D₁₇⋊#108	1	2	yes	—
206	C₄	—	2	4	no	—
207	C₄	—	17	2	no	—
208	C₄	—	17	2	no	—
209	C₄	—	17	8	no	—
210	C₄	—	34	1	no	—
211	C₄	—	34	1	no	—
212	C₄	#11763810	1	1	yes	characteristic
213	C₄	C₂×#10587544	1	1	yes	characteristic
214	C₄	#11763812	1	1	yes	characteristic
215	C₄	#11763814	1	1	yes	characteristic
216	C₂	#14544326	1	2	yes	—
217	C₂	#14544338	1	2	yes	—
218	C₂	#14544717	1	1	yes	characteristic
219	C₂	#14544719	1	1	yes	characteristic
220	C₂	#14544724	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	#8805
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
31. C₈×Dic₁₇	C₈	semidirect
88. C₈×C₁₇	C₄⋊C₈	semidirect
90. #3622	C₄⋊C₈	semidirect
107. Dic₁₇	#334	semidirect
147. C₁₇	#8805	semidirect
200. C₈	#10587544	semidirect

Maximal subgroups

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

subgroup	index
#14544717	2
C₂×#11763671	2
C₂×#11763670	2
C₄×C₈×Dic₁₇	2
#8805	17