Subgroups — #16933023

Order 4352.

298 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14529559	C₂	1	2	yes	maximal
2	#14527540	C₂	1	1	yes	maximal, characteristic
3	C₂×#11762560	C₂	1	1	yes	maximal, characteristic
4	#14526680	C₂	1	2	yes	maximal
5	C₈×M₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587493	—	2	2	no	—
7	#11762833	C₂²	1	2	yes	—
8	C₂×C₄⋊#3622	C₂²	1	2	yes	—
9	#11762573	—	2	2	no	—
10	#11762560	—	2	1	no	—
11	#11762560	—	2	1	no	—
12	#11762560	C₄	1	2	yes	—
13	#11762560	C₄	1	2	yes	—
14	C₁₇×#431	C₂²	1	1	yes	characteristic
15	C₄×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
17	C₁₇×#322	—	2	1	no	—
18	C₈²×C₁₇	—	2	1	no	—
19	C₂×M₁₆×C₁₇	D₄	1	2	yes	—
20	C₂²×C₈×C₁₇	Q₈	1	1	yes	characteristic
21	C₂²×C₈×C₁₇	Q₈	1	1	yes	characteristic
22	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
23	C₂×#63223	—	2	2	no	—
24	C₂²×#3622	—	2	2	no	—
25	C₁₇×C₄⋊C₈	—	2	1	no	—
26	C₁₇×C₂²⋊C₈	Q₈	1	2	yes	—
27	C₁₇×#97	—	2	1	no	—
28	C₄×C₈×C₁₇	—	2	1	no	—
29	C₄×C₈×C₁₇	—	2	1	no	—
30	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
33	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
34	#10587493	—	4	4	no	—
35	C₄⋊#3622	—	2	2	no	—
36	C₄⋊#3622	—	2	2	no	—
37	C₄⋊#3622	—	2	2	no	—
38	C₄⋊#3622	C₂×C₄	1	4	yes	—
39	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
42	M₁₆×C₁₇	—	2	2	no	—
43	M₁₆×C₁₇	SD₁₆	1	4	yes	—
44	C₂×C₈×C₁₇	—	2	2	no	—
45	C₂×C₈×C₁₇	—	2	2	no	—
46	C₂×C₈×C₁₇	—	2	2	no	—
47	C₂×C₈×C₁₇	—	4	1	no	—
48	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₁₇	#41	1	1	yes	characteristic
55	C₄²×C₁₇	#41	1	1	yes	characteristic
56	#63223	—	4	2	no	—
57	#63223	—	4	2	no	—
58	C₂×#3622	—	2	4	no	—
59	C₂×#3622	—	4	2	no	—
60	C₂×#3622	—	4	2	no	—
61	C₂×#3622	—	4	2	no	—
62	#9623	—	17	1	no	maximal, Sylow
63	C₂³×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₁₇	—	2	1	no	—
65	C₂×C₄×C₁₇	—	2	1	no	—
66	C₂×C₄×C₁₇	—	4	1	no	—
67	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
71	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₁₇	#125	1	1	yes	characteristic
73	C₂×C₄×C₁₇	C₂×SD₁₆	1	2	yes	—
74	C₂×C₄×C₁₇	#135	1	2	yes	—
75	C₈×C₁₇	—	2	1	no	—
76	C₈×C₁₇	—	2	1	no	—
77	C₈×C₁₇	—	2	1	no	—
78	C₈×C₁₇	—	2	4	no	—
79	C₈×C₁₇	—	4	2	no	—
80	C₈×C₁₇	#108	1	2	yes	complemented
81	#3622	—	4	4	no	—
82	#3622	—	8	2	no	—
83	#1950	—	17	2	no	—
84	#1566	—	17	1	no	—
85	C₂×#334	—	17	1	no	—
86	C₈×M₁₆	D₁₇	1	1	yes	characteristic
87	#1301	—	17	2	no	—
88	C₂²×C₁₇	—	2	1	no	—
89	C₂²×C₁₇	—	2	1	no	—
90	C₂²×C₁₇	#389	1	1	yes	characteristic
91	C₂²×C₁₇	#498	1	1	yes	characteristic
92	C₂²×C₁₇	#499	1	1	yes	characteristic
93	C₄×C₁₇	—	2	1	no	—
94	C₄×C₁₇	—	2	1	no	—
95	C₄×C₁₇	—	2	1	no	—
96	C₄×C₁₇	C₂×#108	1	1	yes	characteristic
97	C₄×C₁₇	#430	1	1	yes	characteristic
98	C₄×C₁₇	C₄×SD₁₆	1	2	yes	—
99	C₄×C₁₇	#465	1	2	yes	—
100	#431	C₂×D₁₇	1	1	yes	characteristic
101	C₂×#108	—	34	2	no	—
102	#423	—	17	2	no	—
103	C₂×C₄⋊C₈	—	17	2	no	—
104	C₄×M₁₆	C₂×D₁₇	1	1	yes	characteristic
105	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
106	#341	—	34	2	no	—
107	#334	—	17	2	no	—
108	#334	—	17	2	no	—
109	#334	—	34	1	no	—
110	#334	—	34	1	no	—
111	#322	—	2	1	no	—
112	C₈²	—	2	1	no	—
113	C₂×C₁₇	—	2	1	no	—
114	C₂×C₁₇	—	2	1	no	—
115	C₂×C₁₇	#1941	1	1	yes	characteristic
116	C₂×C₁₇	#1950	1	1	yes	characteristic
117	C₂×C₁₇	#1953	1	1	yes	characteristic
118	C₂×M₁₆	—	34	2	no	—
119	C₂×M₁₆	D₆₈	1	2	yes	—
120	C₂²×C₈	—	34	2	no	—
121	C₂²×C₈	Dic₃₄	1	1	yes	characteristic
122	C₂²×C₈	Dic₃₄	1	1	yes	characteristic
123	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
124	#108	—	68	4	no	—
125	C₄⋊C₈	—	2	1	no	—
126	C₄⋊C₈	—	17	4	no	—
127	C₄⋊C₈	—	34	2	no	—
128	C₄⋊C₈	—	34	2	no	—
129	C₄⋊C₈	—	34	2	no	—
130	C₂²⋊C₈	Dic₃₄	1	2	yes	—
131	#97	—	2	1	no	—
132	C₄×C₈	—	2	1	no	—
133	C₄×C₈	—	2	1	no	—
134	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
135	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
136	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
137	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
138	C₁₇	#9623	1	1	yes	characteristic, complemented, Sylow
139	C₂²×C₄	C₂×Dic₃₄	1	1	yes	characteristic
140	C₂²×C₄	C₂×Dic₃₄	1	1	yes	characteristic
141	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
142	M₁₆	—	2	2	no	—
143	M₁₆	—	68	2	no	—
144	M₁₆	—	68	2	no	—
145	M₁₆	#63219	1	4	yes	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	4	1	no	—
150	C₂×C₈	—	34	4	no	—
151	C₂×C₈	—	68	2	no	—
152	C₂×C₈	—	68	2	no	—
153	C₂×C₈	—	68	2	no	—
154	C₂×C₈	#63225	1	1	yes	characteristic
155	C₂×C₈	#63225	1	1	yes	characteristic
156	C₂×C₈	#63225	1	1	yes	characteristic
157	C₂×C₈	#63225	1	1	yes	characteristic
158	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
159	C₄²	#63252	1	1	yes	characteristic
160	C₄²	#63252	1	1	yes	characteristic
161	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
162	C₂³	#10587555	1	1	yes	characteristic
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	4	1	no	—
166	C₂×C₄	C₄×Dic₃₄	1	1	yes	characteristic
167	C₂×C₄	#10587556	1	1	yes	characteristic
168	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
169	C₂×C₄	C₂×#63219	1	2	yes	—
170	C₂×C₄	#10587590	1	2	yes	—
171	C₂×C₄	C₂×#63225	1	1	yes	characteristic
172	C₂×C₄	#10587610	1	1	yes	characteristic
173	C₂×C₄	#10587616	1	1	yes	characteristic
174	C₈	—	2	4	no	—
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	2	1	no	—
178	C₈	—	4	2	no	—
179	C₈	—	68	4	no	—
180	C₈	—	136	2	no	—
181	C₈	#10587493	1	2	yes	complemented
182	C₂²	—	2	1	no	—
183	C₂²	—	2	1	no	—
184	C₂²	#11762707	1	1	yes	characteristic
185	C₂²	#11763207	1	1	yes	characteristic
186	C₂²	#11763211	1	1	yes	characteristic
187	C₄	—	2	1	no	—
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	C₂×#10587493	1	1	yes	characteristic
191	C₄	#11762857	1	1	yes	characteristic
192	C₄	C₄×#63219	1	2	yes	—
193	C₄	#11763038	1	2	yes	—
194	C₂	—	2	1	no	—
195	C₂	—	2	1	no	—
196	C₂	#14529512	1	1	yes	characteristic
197	C₂	#14529559	1	1	yes	characteristic
198	C₂	#14529574	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10587616
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₈×M₁₆×C₁₇	C₂
socle	C₂²×C₁₇	#389
prime radical	C₂³×C₁₇	C₄⋊Q₈
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	C₈×M₁₆	D₁₇
hypocenter	C₁₇	#9623
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
80. C₈×C₁₇	#108	semidirect
138. C₁₇	#9623	semidirect
181. C₈	#10587493	semidirect

Maximal subgroups

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

subgroup	index
#14529559	2
#14527540	2
C₂×#11762560	2
#14526680	2
C₈×M₁₆×C₁₇	2
#9623	17