Subgroups — #16928127

Order 4352.

354 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14528955	C₂	1	2	yes	maximal
2	#14527543	C₂	1	1	yes	maximal, characteristic
3	#14527542	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	C₂×#10587493	—	2	4	no	—
7	#11762833	C₂²	1	2	yes	—
8	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
9	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
10	#11762562	—	2	2	no	—
11	#11762561	—	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	#10587493	—	4	8	no	—
31	C₄⋊#3622	—	2	2	no	—
32	C₄⋊#3622	—	2	2	no	—
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	C₂×C₄	1	1	yes	characteristic
37	C₄×#3622	D₄	1	2	yes	—
38	C₄×#3622	Q₈	1	2	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	—	4	4	no	—
43	C₂×C₈×C₁₇	—	2	4	no	—
44	C₂×C₈×C₁₇	—	4	2	no	—
45	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
46	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
47	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
48	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
50	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
51	#63223	—	4	4	no	—
52	C₂×#3622	—	2	4	no	—
53	C₂×#3622	—	4	2	no	—
54	C₂×#3622	C₄⋊C₄	1	2	yes	—
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	D₈	1	4	yes	—
59	C₂×#3622	Q₁₆	1	4	yes	—
60	#8823	—	17	1	no	maximal, Sylow
61	C₂³×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₁₇	—	2	2	no	—
63	C₂×C₄×C₁₇	—	2	2	no	—
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	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
68	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
71	C₂×C₄×C₁₇	C₂×Q₁₆	1	2	yes	—
72	C₈×C₁₇	—	2	2	no	—
73	C₈×C₁₇	—	8	2	no	—
74	C₈×C₁₇	#108	1	4	yes	complemented
75	#3622	—	2	2	no	—
76	#3622	—	8	2	no	—
77	#3622	#107	1	4	yes	—
78	#3622	#107	1	4	yes	—
79	#3622	#108	1	4	yes	complemented
80	#1848	—	17	2	no	—
81	#1567	—	17	2	no	—
82	#1567	—	17	1	no	—
83	#1567	D₁₇	1	1	yes	characteristic
84	C₂×C₈²	—	17	1	no	—
85	C₂²×C₁₇	—	2	1	no	—
86	C₂²×C₁₇	—	2	1	no	—
87	C₂²×C₁₇	#384	1	1	yes	characteristic
88	C₂²×C₁₇	#500	1	2	yes	—
89	C₄×C₁₇	—	2	1	no	—
90	C₄×C₁₇	—	2	1	no	—
91	C₄×C₁₇	C₂×#107	1	2	yes	—
92	C₄×C₁₇	C₂×#108	1	2	yes	—
93	C₄×C₁₇	C₂×#108	1	2	yes	—
94	C₄×C₁₇	#493	1	1	yes	characteristic
95	C₄×C₁₇	#494	1	1	yes	characteristic
96	C₂×#108	—	34	4	no	—
97	#423	—	17	2	no	—
98	#423	C₂×D₁₇	1	2	yes	—
99	C₂×C₄×C₈	—	17	1	no	—
100	C₂×C₄×C₈	—	17	1	no	—
101	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
102	#335	—	2	2	no	—
103	#335	—	34	4	no	—
104	#335	—	34	2	no	—
105	C₈²	—	17	1	no	—
106	C₈²	—	17	1	no	—
107	C₈²	—	17	1	no	—
108	C₈²	—	17	1	no	—
109	C₂×C₁₇	—	2	2	no	—
110	C₂×C₁₇	#1845	1	1	yes	characteristic
111	C₂×C₁₇	#1848	1	1	yes	characteristic
112	C₂×C₁₇	#1848	1	1	yes	characteristic
113	C₂×M₁₆	—	2	2	no	—
114	C₂×M₁₆	—	34	2	no	—
115	C₂²×C₈	—	17	2	no	—
116	C₂²×C₈	—	17	2	no	—
117	C₂²×C₈	Dic₃₄	1	2	yes	—
118	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
119	#108	—	68	8	no	—
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	—	2	2	no	—
122	C₄⋊C₈	—	34	2	no	—
123	C₄⋊C₈	—	34	2	no	—
124	C₄×C₈	—	17	1	no	—
125	C₄×C₈	—	17	1	no	—
126	C₄×C₈	—	17	1	no	—
127	C₄×C₈	—	17	1	no	—
128	C₄×C₈	—	17	2	no	—
129	C₄×C₈	—	17	2	no	—
130	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
131	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
132	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
133	C₁₇	#8823	1	1	yes	characteristic, complemented, Sylow
134	C₂²×C₄	C₂×Dic₃₄	1	1	yes	characteristic
135	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
136	C₂²×C₄	Q₈×D₁₇	1	1	yes	characteristic
137	M₁₆	—	4	4	no	—
138	M₁₆	—	68	4	no	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	17	2	no	—
142	C₂×C₈	—	17	2	no	—
143	C₂×C₈	—	17	2	no	—
144	C₂×C₈	—	17	4	no	—
145	C₂×C₈	—	17	4	no	—
146	C₂×C₈	—	17	2	no	—
147	C₂×C₈	—	34	4	no	—
148	C₂×C₈	—	68	2	no	—
149	C₂×C₈	#63225	1	2	yes	—
150	C₂×C₈	#63225	1	2	yes	—
151	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
152	C₄²	D₄×D₁₇	1	1	yes	characteristic
153	C₄²	Q₈×D₁₇	1	1	yes	characteristic
154	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
155	C₂³	#10587574	1	1	yes	characteristic
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	#10587574	1	1	yes	characteristic
159	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
160	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
161	C₂×C₄	D₈×D₁₇	1	2	yes	—
162	C₂×C₄	Q₁₆×D₁₇	1	2	yes	—
163	C₂×C₄	C₂×#63225	1	1	yes	characteristic
164	C₂×C₄	#10587626	1	1	yes	characteristic
165	C₂×C₄	#10587630	1	1	yes	characteristic
166	C₈	—	2	2	no	—
167	C₈	—	8	2	no	—
168	C₈	—	17	4	no	—
169	C₈	—	17	4	no	—
170	C₈	—	17	4	no	—
171	C₈	—	34	2	no	—
172	C₈	—	136	2	no	—
173	C₈	#10587493	1	4	yes	complemented
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	#11762678	1	1	yes	characteristic
177	C₂²	#11763213	1	2	yes	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	#107×D₁₇	1	2	yes	—
181	C₄	#108×D₁₇	1	2	yes	—
182	C₄	C₂×#10587493	1	2	yes	—
183	C₄	#11763184	1	1	yes	characteristic
184	C₄	#11763187	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	#14528942	1	1	yes	characteristic
187	C₂	#14528954	1	1	yes	characteristic
188	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	#8823
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
74. C₈×C₁₇	#108	semidirect
79. #3622	#108	semidirect
133. C₁₇	#8823	semidirect
173. C₈	#10587493	semidirect

Maximal subgroups

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

subgroup	index
#14528955	2
#14527543	2
#14527542	2
C₂×C₈×#3622	2
C₁₇×#1567	2
#8823	17