Subgroups — #17953818

Order 4352.

422 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14544701	C₂	1	2	yes	maximal
2	C₂×#11763672	C₂	1	2	yes	maximal
3	C₂×#11763671	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#3624	C₂²	1	4	yes	—
6	#11763675	—	2	4	no	—
7	#11763672	—	2	4	no	—
8	#11763671	—	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	C₄⋊#3624	—	2	4	no	—
27	C₄⋊#3624	—	2	4	no	—
28	C₄×C₈×C₁₇	—	2	2	no	—
29	C₈×Dic₁₇	—	2	2	no	—
30	C₈×Dic₁₇	C₈	1	4	yes	complemented
31	C₈×Dic₁₇	D₄	1	2	yes	—
32	C₈×Dic₁₇	D₄	1	2	yes	—
33	C₈×Dic₁₇	Q₈	1	2	yes	—
34	C₈×Dic₁₇	Q₈	1	2	yes	—
35	C₄×#3622	—	2	2	no	—
36	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
37	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
38	C₂²×Dic₁₇	C₄⋊C₄	1	2	yes	—
39	C₂²×Dic₁₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
41	C₂×#3624	—	4	4	no	—
42	C₂×#3624	—	4	8	no	—
43	C₂×C₈×C₁₇	—	2	1	no	—
44	C₂×C₈×C₁₇	—	2	1	no	—
45	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
46	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
47	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
48	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
49	C₄²×C₁₇	—	2	1	no	—
50	C₄²×C₁₇	—	2	1	no	—
51	C₄×Dic₁₇	—	2	4	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	C₂×D₄	1	1	yes	characteristic
56	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
57	C₄×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
58	C₄×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
59	C₂×#3622	—	2	1	no	—
60	C₂×#3622	—	2	1	no	—
61	C₂×#3622	C₄⋊C₄	1	2	yes	—
62	C₂×#3622	C₄⋊C₄	1	2	yes	—
63	C₂×#3622	C₂×C₈	1	2	yes	—
64	C₂×#3622	M₁₆	1	2	yes	—
65	#8807	—	17	1	no	maximal, Sylow
66	C₂³×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₇	—	2	2	no	—
68	C₂×C₄×C₁₇	—	2	2	no	—
69	C₂×C₄×C₁₇	#108	1	4	yes	—
70	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₁₇	C₂×M₁₆	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×Dic₁₇	—	2	2	no	—
77	C₂×Dic₁₇	—	2	2	no	—
78	C₂×Dic₁₇	—	2	2	no	—
79	C₂×Dic₁₇	C₄⋊C₈	1	4	yes	—
80	C₂×Dic₁₇	#127	1	1	yes	characteristic
81	C₂×Dic₁₇	C₄⋊Q₈	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	#135	1	4	yes	—
85	#3624	—	8	4	no	—
86	C₈×C₁₇	—	2	2	no	—
87	C₈×C₁₇	C₄⋊C₈	1	4	yes	—
88	#3622	—	2	2	no	—
89	#3622	C₄⋊C₈	1	4	yes	—
90	#1831	—	17	2	no	—
91	C₄²×C₈	—	17	1	no	—
92	C₂×#335	—	17	2	no	—
93	C₂×#334	—	17	2	no	—
94	C₂²×C₁₇	#384	1	1	yes	characteristic
95	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
96	C₂²×C₁₇	#423	1	1	yes	characteristic
97	C₂²×C₁₇	#427	1	2	yes	—
98	C₂²×C₁₇	C₂×#108	1	2	yes	—
99	C₄×C₁₇	—	2	4	no	—
100	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
101	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
102	C₄×C₁₇	#423	1	1	yes	characteristic
103	C₄×C₁₇	#423	1	1	yes	characteristic
104	Dic₁₇	—	2	4	no	—
105	Dic₁₇	—	2	1	no	—
106	Dic₁₇	—	2	1	no	—
107	Dic₁₇	#495	1	2	yes	—
108	Dic₁₇	#499	1	2	yes	—
109	C₂×C₄⋊C₈	—	17	4	no	—
110	C₂×C₄×C₈	—	17	1	no	—
111	C₂×C₄×C₈	—	17	1	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	#595	1	1	yes	characteristic
116	C₄³	—	17	1	no	—
117	#336	—	34	4	no	—
118	#335	—	34	4	no	—
119	#334	—	34	4	no	—
120	C₂×C₁₇	#1564	1	2	yes	—
121	C₂×C₁₇	#1568	1	2	yes	—
122	C₂×C₁₇	#1842	1	1	yes	characteristic
123	C₂×C₁₇	#1846	1	1	yes	characteristic
124	C₂×C₁₇	#1848	1	1	yes	characteristic
125	C₂²×C₈	—	17	1	no	—
126	C₂²×C₈	—	17	1	no	—
127	C₂²×C₈	—	34	4	no	—
128	C₂²×C₈	C₂×#595	1	1	yes	characteristic
129	C₂²×C₈	C₂×#595	1	1	yes	characteristic
130	C₂×C₄²	—	17	1	no	—
131	C₂×C₄²	—	17	1	no	—
132	C₂×C₄²	—	17	2	no	—
133	C₂×C₄²	—	17	2	no	—
134	C₂×C₄²	C₂×#595	1	1	yes	characteristic
135	C₄⋊C₈	—	34	4	no	—
136	C₄⋊C₈	—	34	4	no	—
137	C₄×C₈	—	2	2	no	—
138	C₄×C₈	—	17	4	no	—
139	C₄×C₈	—	17	2	no	—
140	C₄×C₈	—	17	2	no	—
141	C₄×C₈	—	17	2	no	—
142	C₄×C₈	—	17	2	no	—
143	C₄×C₈	—	34	2	no	—
144	C₄×C₈	—	34	2	no	—
145	C₁₇	#8807	1	1	yes	characteristic, complemented, Sylow
146	C₂²×C₄	—	17	1	no	—
147	C₂²×C₄	—	17	2	no	—
148	C₂²×C₄	—	17	1	no	—
149	C₂²×C₄	C₄⋊#595	1	2	yes	—
150	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	17	2	no	—
154	C₂×C₈	—	17	2	no	—
155	C₂×C₈	—	17	2	no	—
156	C₂×C₈	—	17	2	no	—
157	C₂×C₈	—	34	1	no	—
158	C₂×C₈	—	34	1	no	—
159	C₂×C₈	—	68	4	no	—
160	C₂×C₈	—	68	8	no	—
161	C₂×C₈	#63242	1	2	yes	—
162	C₂×C₈	C₄⋊#595	1	2	yes	—
163	C₂×C₈	C₄⋊#595	1	2	yes	—
164	C₂×C₈	#63247	1	2	yes	—
165	C₄²	—	2	1	no	—
166	C₄²	—	2	1	no	—
167	C₄²	—	17	2	no	—
168	C₄²	—	17	2	no	—
169	C₄²	—	17	1	no	—
170	C₄²	—	17	1	no	—
171	C₄²	—	17	1	no	—
172	C₄²	—	17	1	no	—
173	C₄²	—	34	4	no	—
174	C₄²	—	34	4	no	—
175	C₂³	C₂×C₄⋊#595	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	17	1	no	—
179	C₂×C₄	—	17	1	no	—
180	C₂×C₄	—	17	4	no	—
181	C₂×C₄	—	17	1	no	—
182	C₂×C₄	—	17	1	no	—
183	C₂×C₄	—	17	4	no	—
184	C₂×C₄	—	34	2	no	—
185	C₂×C₄	—	34	2	no	—
186	C₂×C₄	—	34	2	no	—
187	C₂×C₄	#10587536	1	4	yes	—
188	C₂×C₄	C₂×#63242	1	1	yes	characteristic
189	C₂×C₄	D₁₇⋊M₁₆	1	1	yes	characteristic
190	C₂×C₄	D₁₇⋊M₁₆	1	1	yes	characteristic
191	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
192	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
193	C₂×C₄	C₂×#63247	1	1	yes	characteristic
194	C₈	—	2	2	no	—
195	C₈	—	17	4	no	—
196	C₈	—	34	2	no	—
197	C₈	—	136	4	no	—
198	C₈	#10587544	1	4	yes	—
199	C₂²	#11763755	1	1	yes	characteristic
200	C₂²	#11763810	1	1	yes	characteristic
201	C₂²	#11763812	1	1	yes	characteristic
202	C₂²	#11763819	1	2	yes	—
203	C₂²	C₂×#10587536	1	2	yes	—
204	C₄	—	2	4	no	—
205	C₄	—	17	2	no	—
206	C₄	—	17	2	no	—
207	C₄	—	34	1	no	—
208	C₄	—	34	1	no	—
209	C₄	—	34	4	no	—
210	C₄	#11763810	1	1	yes	characteristic
211	C₄	C₂×#10587544	1	1	yes	characteristic
212	C₄	#11763812	1	1	yes	characteristic
213	C₄	#11763814	1	1	yes	characteristic
214	C₂	#14544334	1	2	yes	—
215	C₂	#14544346	1	2	yes	—
216	C₂	#14544717	1	1	yes	characteristic
217	C₂	#14544721	1	1	yes	characteristic
218	C₂	#14544725	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	#8807
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
30. C₈×Dic₁₇	C₈	semidirect
145. C₁₇	#8807	semidirect

Maximal subgroups

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

subgroup	index
#14544701	2
C₂×#11763672	2
C₂×#11763671	2
C₄×C₈×Dic₁₇	2
#8807	17