Subgroups — #16944136

Order 4352.

334 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14537718	C₂	1	1	yes	maximal, characteristic
2	#14530540	C₂	1	1	yes	maximal, characteristic
3	#14526764	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
5	#11763366	C₄	1	1	yes	characteristic
6	C₂×#10587483	—	2	1	no	—
7	#11762939	—	2	1	no	—
8	#11762939	C₄	1	2	yes	—
9	C₈×Dic₃₄	—	2	1	no	—
10	C₈×Dic₃₄	C₂²	1	2	yes	—
11	#11762835	—	2	1	no	—
12	#11762833	C₄	1	1	yes	characteristic
13	#11762764	—	2	1	no	—
14	C₄⋊#63214	—	2	1	no	—
15	C₄⋊#63214	C₂²	1	2	yes	—
16	C₂²×C₁₆×C₁₇	—	2	1	no	—
17	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
18	C₄×C₁₆×C₁₇	—	2	1	no	—
19	C₄×C₁₆×C₁₇	C₂²	1	2	yes	—
20	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
23	#10587612	—	2	1	no	—
24	#10587610	—	2	1	no	—
25	#10587610	C₈	1	2	yes	—
26	C₂×#63223	—	2	1	no	—
27	#10587556	—	2	1	no	—
28	#10587555	C₈	1	1	yes	characteristic
29	#10587555	C₈	1	1	yes	characteristic
30	C₄×Dic₃₄	—	2	1	no	—
31	C₄×Dic₃₄	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₁₇	—	2	4	no	—
33	C₂×C₁₆×C₁₇	—	2	2	no	—
34	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
36	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
37	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
38	#10587486	—	2	2	no	—
39	#10587485	—	2	2	no	—
40	#10587485	—	2	2	no	—
41	C₈×Dic₁₇	—	2	2	no	—
42	#10587483	—	4	2	no	—
43	C₂×#63214	—	2	2	no	—
44	C₄⋊#3622	—	2	1	no	—
45	C₄⋊#3622	C₂×C₄	1	2	yes	—
46	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
49	C₂×Dic₃₄	—	2	1	no	—
50	C₂×Dic₃₄	C₂×C₈	1	2	yes	—
51	C₂×C₈×C₁₇	—	2	2	no	—
52	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
55	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
56	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
60	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
61	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²⋊Dic₁₇	—	2	2	no	—
63	C₄⋊Dic₁₇	—	2	1	no	—
64	C₄⋊Dic₁₇	—	2	1	no	—
65	C₄⋊Dic₁₇	—	2	1	no	—
66	C₄⋊Dic₁₇	C₂×C₈	1	2	yes	—
67	#63225	—	2	2	no	—
68	#63225	—	2	2	no	—
69	C₄×Dic₁₇	—	2	2	no	—
70	#63223	—	4	2	no	—
71	C₂×#3622	—	2	2	no	—
72	C₁₆×C₁₇	—	2	4	no	—
73	#63214	—	4	2	no	—
74	#11608	—	17	1	no	maximal, Sylow
75	C₂³×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
76	C₂×C₄×C₁₇	—	2	2	no	—
77	C₂×C₄×C₁₇	—	2	2	no	—
78	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
79	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
80	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
81	C₂×C₄×C₁₇	#103	1	2	yes	—
82	C₂×C₄×C₁₇	#104	1	2	yes	—
83	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
86	C₂×Dic₁₇	—	2	2	no	—
87	C₂×Dic₁₇	—	2	2	no	—
88	Dic₃₄	—	2	4	no	—
89	Dic₃₄	—	4	2	no	—
90	C₈×C₁₇	—	2	1	no	—
91	C₈×C₁₇	—	2	1	no	—
92	C₈×C₁₇	#103	1	2	yes	—
93	C₈×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
94	C₈×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
95	#3622	—	4	2	no	—
96	#2960	—	17	1	no	—
97	#2150	—	17	1	no	—
98	C₂×C₄×C₁₆	D₁₇	1	1	yes	characteristic
99	Q₈⋊C₁₆	—	17	4	no	—
100	C₂²×C₁₇	—	2	1	no	—
101	C₂²×C₁₇	—	2	1	no	—
102	C₂²×C₁₇	Q₈⋊C₈	1	2	yes	—
103	C₂²×C₁₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
104	C₄×C₁₇	—	2	1	no	—
105	C₄×C₁₇	—	2	1	no	—
106	C₄×C₁₇	Q₈⋊C₈	1	2	yes	—
107	C₄×C₁₇	#350	1	2	yes	—
108	C₄×C₁₇	C₂×#103	1	1	yes	characteristic
109	C₄×C₁₇	C₂×#104	1	1	yes	characteristic
110	C₄×C₁₇	#504	1	2	yes	—
111	Dic₁₇	—	4	2	no	—
112	Dic₁₇	—	4	2	no	—
113	#533	—	17	1	no	—
114	C₂×M₃₂	—	34	1	no	—
115	C₂²×C₁₆	—	2	1	no	—
116	#446	—	17	2	no	—
117	#446	—	34	1	no	—
118	C₈×Q₈	—	17	2	no	—
119	C₈×Q₈	—	34	1	no	—
120	#424	—	34	1	no	—
121	#423	—	17	1	no	—
122	#405	—	34	1	no	—
123	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
124	C₄⋊C₁₆	—	17	2	no	—
125	C₄⋊C₁₆	—	34	1	no	—
126	C₄×C₁₆	—	2	1	no	—
127	C₄×C₁₆	C₂×D₁₇	1	2	yes	—
128	C₂×C₁₇	—	2	2	no	—
129	C₂×C₁₇	C₂×Q₈⋊C₈	1	1	yes	characteristic
130	C₂×C₁₇	#2113	1	1	yes	characteristic
131	C₂×C₁₇	C₂×#350	1	1	yes	characteristic
132	C₂×M₁₆	—	34	1	no	—
133	C₂²×C₈	D₆₈	1	1	yes	characteristic
134	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
135	C₄⋊Q₈	—	17	1	no	—
136	C₄⋊Q₈	—	17	1	no	—
137	#125	—	34	1	no	—
138	C₂²⋊Q₈	—	17	2	no	—
139	C₂²⋊Q₈	—	34	1	no	—
140	C₄×Q₈	—	17	2	no	—
141	C₄×Q₈	—	34	1	no	—
142	#117	—	34	1	no	—
143	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
144	M₃₂	—	68	2	no	—
145	C₂×C₁₆	—	2	4	no	—
146	C₂×C₁₆	—	2	2	no	—
147	C₂×C₁₆	—	34	2	no	—
148	C₄⋊C₈	—	17	2	no	—
149	C₄⋊C₈	—	34	2	no	—
150	C₄⋊C₈	—	34	2	no	—
151	C₄⋊C₈	—	34	1	no	—
152	#97	—	34	2	no	—
153	C₄×C₈	—	34	2	no	—
154	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
155	C₄×C₈	D₆₈	1	1	yes	characteristic
156	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
157	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
158	C₁₇	#11608	1	1	yes	characteristic, complemented, Sylow
159	C₂×Q₈	—	17	2	no	—
160	C₂×Q₈	—	34	1	no	—
161	C₂²×C₄	C₈×D₁₇	1	1	yes	characteristic
162	C₂²×C₄	#63218	1	1	yes	characteristic
163	C₂²×C₄	#63227	1	1	yes	characteristic
164	M₁₆	—	68	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	34	2	no	—
167	C₂×C₈	#63219	1	2	yes	—
168	C₂×C₈	Dic₆₈	1	2	yes	—
169	C₂×C₈	#63227	1	1	yes	characteristic
170	C₂×C₈	#63227	1	1	yes	characteristic
171	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
172	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
173	C₄⋊C₄	—	17	2	no	—
174	C₄⋊C₄	—	34	1	no	—
175	C₄⋊C₄	—	34	1	no	—
176	C₄⋊C₄	—	34	2	no	—
177	C₄⋊C₄	—	34	1	no	—
178	C₄⋊C₄	—	34	2	no	—
179	C₂²⋊C₄	—	34	2	no	—
180	C₄²	—	34	2	no	—
181	C₄²	C₈×D₁₇	1	1	yes	characteristic
182	C₄²	#63218	1	1	yes	characteristic
183	C₄²	#63227	1	1	yes	characteristic
184	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
185	C₁₆	—	2	4	no	—
186	C₁₆	—	68	2	no	—
187	C₂³	#10587491	1	1	yes	characteristic
188	Q₈	—	34	4	no	—
189	Q₈	—	68	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	34	2	no	—
193	C₂×C₄	—	34	2	no	—
194	C₂×C₄	#10587476	1	2	yes	—
195	C₂×C₄	#10587487	1	2	yes	—
196	C₂×C₄	#10587491	1	1	yes	characteristic
197	C₂×C₄	#10587491	1	1	yes	characteristic
198	C₂×C₄	#10587491	1	1	yes	characteristic
199	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
200	C₂×C₄	C₂×#63218	1	1	yes	characteristic
201	C₂×C₄	C₂×#63227	1	1	yes	characteristic
202	C₈	—	2	1	no	—
203	C₈	—	2	1	no	—
204	C₈	—	68	2	no	—
205	C₈	#10587487	1	2	yes	—
206	C₈	C₂×#63219	1	1	yes	characteristic
207	C₈	C₂×Dic₆₈	1	1	yes	characteristic
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	#11762541	1	2	yes	—
211	C₂²	C₂×#10587491	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	68	2	no	—
215	C₄	—	68	2	no	—
216	C₄	#11762541	1	2	yes	—
217	C₄	#11762592	1	2	yes	—
218	C₄	C₂×#10587487	1	1	yes	characteristic
219	C₄	C₂×#10587476	1	1	yes	characteristic
220	C₄	#11763225	1	2	yes	—
221	C₂	—	2	2	no	—
222	C₂	C₂×#11762541	1	1	yes	characteristic
223	C₂	#14530397	1	1	yes	characteristic
224	C₂	C₂×#11762592	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₆₈
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₂×C₄×C₁₆×C₁₇	C₂
socle	C₂²×C₁₇	C₂×C₂²⋊C₈
prime radical	C₂³×C₁₇	C₂²⋊C₈
cosocle	C₄×C₈×C₁₇	C₂³
hypercenter	C₂×C₄×C₁₆	D₁₇
hypocenter	C₁₇	#11608
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
3. #14526764	C₂	semidirect
158. C₁₇	#11608	semidirect

Maximal subgroups

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

subgroup	index
#14537718	2
#14530540	2
#14526764	2
C₂×C₄×C₁₆×C₁₇	2
#11608	17