Subgroups — #16928131

Order 4352.

322 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14528921	C₂	1	2	yes	maximal
2	#14527543	C₂	1	1	yes	maximal, characteristic
3	#14527539	C₂	1	2	yes	maximal
4	C₂×C₈×#3622	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#1567	C₂	1	1	yes	maximal, characteristic
6	#11762833	C₂²	1	2	yes	—
7	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
8	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
9	#11762573	—	2	4	no	—
10	#11762562	—	2	2	no	—
11	#11762559	—	2	4	no	—
12	C₈×#3622	C₄	1	1	yes	characteristic
13	C₈×#3622	C₄	1	1	yes	characteristic
14	C₈×#3622	C₄	1	1	yes	characteristic
15	C₈×#3622	C₄	1	1	yes	characteristic
16	C₁₇×#423	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
18	C₁₇×#335	—	2	2	no	—
19	C₂×M₁₆×C₁₇	—	2	2	no	—
20	C₂²×C₈×C₁₇	Q₈	1	2	yes	—
21	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
22	C₂×#63223	—	2	2	no	—
23	C₂²×#3622	D₄	1	2	yes	—
24	C₂²×#3622	Q₈	1	2	yes	—
25	C₁₇×C₄⋊C₈	—	2	2	no	—
26	C₁₇×C₄⋊C₈	—	2	2	no	—
27	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₁₇	D₄	1	2	yes	—
30	C₄⋊#3622	—	2	2	no	—
31	C₄⋊#3622	—	2	2	no	—
32	C₄×#3622	C₂×C₄	1	1	yes	characteristic
33	C₄×#3622	C₂×C₄	1	1	yes	characteristic
34	C₄×#3622	C₂×C₄	1	1	yes	characteristic
35	C₄×#3622	C₂×C₄	1	1	yes	characteristic
36	C₄×#3622	D₄	1	2	yes	—
37	C₄×#3622	Q₈	1	2	yes	—
38	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
41	M₁₆×C₁₇	—	4	4	no	—
42	C₂×C₈×C₁₇	—	2	4	no	—
43	C₂×C₈×C₁₇	—	4	2	no	—
44	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
46	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
50	#63223	—	4	4	no	—
51	C₂×#3622	—	2	2	no	—
52	C₂×#3622	—	2	2	no	—
53	C₂×#3622	—	2	4	no	—
54	C₂×#3622	—	4	2	no	—
55	C₂×#3622	C₄⋊C₄	1	2	yes	—
56	C₂×#3622	C₄⋊C₄	1	2	yes	—
57	C₂×#3622	C₄⋊C₄	1	2	yes	—
58	C₂×#3622	C₄⋊C₄	1	2	yes	—
59	#8824	—	17	1	no	maximal, Sylow
60	C₂³×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
61	C₂×C₄×C₁₇	—	2	2	no	—
62	C₂×C₄×C₁₇	—	2	2	no	—
63	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
67	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₁₇	#135	1	2	yes	—
70	C₂×C₄×C₁₇	#135	1	2	yes	—
71	C₈×C₁₇	—	2	2	no	—
72	C₈×C₁₇	—	8	2	no	—
73	C₈×C₁₇	#108	1	4	yes	—
74	#3622	—	2	2	no	—
75	#3622	—	2	2	no	—
76	#3622	—	2	2	no	—
77	#3622	—	8	2	no	—
78	#3622	#108	1	4	yes	—
79	#1837	—	17	2	no	—
80	#1567	—	17	1	no	—
81	#1567	D₁₇	1	1	yes	characteristic
82	#1566	—	17	2	no	—
83	C₂×C₈²	—	17	1	no	—
84	C₂²×C₁₇	—	2	1	no	—
85	C₂²×C₁₇	—	2	1	no	—
86	C₂²×C₁₇	#384	1	1	yes	characteristic
87	C₂²×C₁₇	#499	1	2	yes	—
88	C₄×C₁₇	—	2	1	no	—
89	C₄×C₁₇	—	2	1	no	—
90	C₄×C₁₇	#427	1	2	yes	—
91	C₄×C₁₇	C₂×#108	1	2	yes	—
92	C₄×C₁₇	C₂×#108	1	2	yes	—
93	C₄×C₁₇	#495	1	1	yes	characteristic
94	C₄×C₁₇	#495	1	1	yes	characteristic
95	#423	—	17	2	no	—
96	#423	C₂×D₁₇	1	2	yes	—
97	C₂×C₄×C₈	—	17	1	no	—
98	C₂×C₄×C₈	—	17	1	no	—
99	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
100	#341	—	34	4	no	—
101	#335	—	2	2	no	—
102	#335	—	34	2	no	—
103	#334	—	34	4	no	—
104	C₈²	—	17	1	no	—
105	C₈²	—	17	1	no	—
106	C₈²	—	17	1	no	—
107	C₈²	—	17	1	no	—
108	C₂×C₁₇	—	2	2	no	—
109	C₂×C₁₇	#1846	1	1	yes	characteristic
110	C₂×C₁₇	#1848	1	1	yes	characteristic
111	C₂×C₁₇	#1848	1	1	yes	characteristic
112	C₂×M₁₆	—	2	2	no	—
113	C₂×M₁₆	—	34	2	no	—
114	C₂²×C₈	—	17	2	no	—
115	C₂²×C₈	—	17	2	no	—
116	C₂²×C₈	Dic₃₄	1	2	yes	—
117	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
118	C₄⋊C₈	—	2	2	no	—
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	34	2	no	—
121	C₄⋊C₈	—	34	2	no	—
122	C₄×C₈	—	17	1	no	—
123	C₄×C₈	—	17	1	no	—
124	C₄×C₈	—	17	1	no	—
125	C₄×C₈	—	17	1	no	—
126	C₄×C₈	—	17	2	no	—
127	C₄×C₈	—	17	2	no	—
128	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
129	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
130	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
131	C₁₇	#8824	1	1	yes	characteristic, complemented, Sylow
132	C₂²×C₄	C₂×Dic₃₄	1	1	yes	characteristic
133	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
134	C₂²×C₄	Q₈×D₁₇	1	1	yes	characteristic
135	M₁₆	—	4	4	no	—
136	M₁₆	—	68	4	no	—
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	17	2	no	—
140	C₂×C₈	—	17	2	no	—
141	C₂×C₈	—	17	2	no	—
142	C₂×C₈	—	17	2	no	—
143	C₂×C₈	—	34	4	no	—
144	C₂×C₈	—	34	2	no	—
145	C₂×C₈	—	34	2	no	—
146	C₂×C₈	—	68	2	no	—
147	C₂×C₈	#63225	1	2	yes	—
148	C₂×C₈	#63225	1	2	yes	—
149	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
150	C₄²	D₄×D₁₇	1	1	yes	characteristic
151	C₄²	Q₈×D₁₇	1	1	yes	characteristic
152	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
153	C₂³	#10587574	1	1	yes	characteristic
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	#10587574	1	1	yes	characteristic
157	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
158	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
159	C₂×C₄	#10587598	1	2	yes	—
160	C₂×C₄	#10587605	1	2	yes	—
161	C₂×C₄	C₂×#63225	1	1	yes	characteristic
162	C₂×C₄	#10587626	1	1	yes	characteristic
163	C₂×C₄	#10587630	1	1	yes	characteristic
164	C₈	—	2	2	no	—
165	C₈	—	8	2	no	—
166	C₈	—	17	4	no	—
167	C₈	—	34	2	no	—
168	C₈	—	34	2	no	—
169	C₈	—	34	2	no	—
170	C₈	—	136	2	no	—
171	C₈	#10587493	1	4	yes	—
172	C₂²	—	2	1	no	—
173	C₂²	—	2	1	no	—
174	C₂²	#11762678	1	1	yes	characteristic
175	C₂²	#11763210	1	2	yes	—
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	#11762846	1	2	yes	—
179	C₄	#108×D₁₇	1	2	yes	—
180	C₄	C₂×#10587493	1	2	yes	—
181	C₄	#11763192	1	1	yes	characteristic
182	C₄	#11763194	1	1	yes	characteristic
183	C₂	—	2	2	no	—
184	C₂	#14528947	1	1	yes	characteristic
185	C₂	#14528954	1	1	yes	characteristic
186	C₂	#14528955	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10587626
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1567	C₂
socle	C₂²×C₁₇	#384
prime radical	C₂³×C₁₇	C₄⋊Q₈
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#1567	D₁₇
hypocenter	C₁₇	#8824
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
131. C₁₇	#8824	semidirect

Maximal subgroups

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

subgroup	index
#14528921	2
#14527543	2
#14527539	2
C₂×C₈×#3622	2
C₁₇×#1567	2
#8824	17