Subgroups — #16925513

Order 4352.

438 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14528637	C₂	1	2	yes	maximal
2	C₂×#11762560	C₂	1	2	yes	maximal
3	#14526619	C₂	1	2	yes	maximal
4	C₄²×C₈×C₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#3622	C₂²	1	4	yes	—
6	C₂×#10587474	—	2	2	no	—
7	#11762564	—	2	2	no	—
8	#11762560	C₄	1	8	yes	complemented
9	C₂×C₄×C₈×C₁₇	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₁₇	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
13	C₄³×C₁₇	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
16	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
19	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₇	Q₈	1	2	yes	—
21	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
22	C₂²×#3622	—	2	4	no	—
23	C₄×C₈×C₁₇	—	2	4	no	—
24	C₄×C₈×C₁₇	—	2	4	no	—
25	C₄×C₈×C₁₇	—	2	2	no	—
26	C₄×C₈×C₁₇	C₂×C₄	1	4	yes	—
27	C₄⋊#3622	—	2	4	no	—
28	C₄⋊#3622	C₂×C₄	1	8	yes	—
29	#10587474	—	4	4	no	—
30	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
31	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
32	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₈×C₁₇	—	2	2	no	—
36	C₂×C₈×C₁₇	—	2	2	no	—
37	C₂×C₈×C₁₇	—	2	2	no	—
38	C₂×C₈×C₁₇	C₄²	1	2	yes	—
39	C₂×C₈×C₁₇	C₄²	1	2	yes	—
40	C₂×C₈×C₁₇	M₁₆	1	4	yes	—
41	C₂×C₈×C₁₇	M₁₆	1	4	yes	—
42	C₄²×C₁₇	—	2	4	no	—
43	C₄²×C₁₇	—	2	2	no	—
44	C₄²×C₁₇	—	2	2	no	—
45	C₄²×C₁₇	—	2	2	no	—
46	C₄²×C₁₇	SD₁₆	1	4	yes	—
47	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
48	C₄²×C₁₇	#41	1	2	yes	—
49	C₂×#3622	—	2	8	no	—
50	C₂×#3622	—	4	4	no	—
51	C₂×#3622	—	4	4	no	—
52	#8273	—	17	1	no	maximal, Sylow
53	C₂³×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₇	—	2	2	no	—
55	C₂×C₄×C₁₇	—	2	2	no	—
56	C₂×C₄×C₁₇	—	2	2	no	—
57	C₂×C₄×C₁₇	—	2	2	no	—
58	C₂×C₄×C₁₇	—	2	2	no	—
59	C₂×C₄×C₁₇	#106	1	4	yes	—
60	C₂×C₄×C₁₇	#108	1	4	yes	—
61	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
62	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
63	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
64	C₂×C₄×C₁₇	C₄×Q₈	1	2	yes	—
65	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
66	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
67	C₂×C₄×C₁₇	C₂×SD₁₆	1	2	yes	—
68	C₂×C₄×C₁₇	#135	1	2	yes	—
69	C₈×C₁₇	—	2	2	no	—
70	C₈×C₁₇	—	2	4	no	—
71	C₈×C₁₇	#97	1	4	yes	complemented
72	#3622	—	4	8	no	—
73	#1767	—	17	2	no	—
74	C₄²×C₈	D₁₇	1	1	yes	characteristic
75	C₂×#334	—	17	2	no	—
76	#1277	—	17	2	no	—
77	C₂²×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₁₇	#423	1	2	yes	—
79	C₂²×C₁₇	C₂×#106	1	1	yes	characteristic
80	C₂²×C₁₇	#427	1	1	yes	characteristic
81	C₂²×C₁₇	C₂×#108	1	2	yes	—
82	C₄×C₁₇	—	2	4	no	—
83	C₄×C₁₇	—	2	4	no	—
84	C₄×C₁₇	—	2	2	no	—
85	C₄×C₁₇	C₂×#97	1	1	yes	characteristic
86	C₄×C₁₇	C₄×M₁₆	1	1	yes	characteristic
87	C₄×C₁₇	#432	1	2	yes	—
88	C₄×C₁₇	C₄×SD₁₆	1	4	yes	—
89	C₂×C₄⋊C₈	—	17	4	no	—
90	C₂×#97	—	34	2	no	—
91	C₂×C₄×C₈	Dic₁₇	1	2	yes	—
92	C₂×C₄×C₈	Dic₁₇	1	2	yes	—
93	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
94	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
95	C₄³	C₂×D₁₇	1	1	yes	characteristic
96	#336	—	34	2	no	—
97	#334	—	17	8	no	—
98	C₂×C₁₇	#1566	1	2	yes	—
99	C₂×C₁₇	#1568	1	2	yes	—
100	C₂×C₁₇	#1767	1	1	yes	characteristic
101	C₂×C₁₇	C₄×#106	1	1	yes	characteristic
102	C₂×C₁₇	C₄×#108	1	1	yes	characteristic
103	C₂²×C₈	—	34	4	no	—
104	C₂²×C₈	C₂×Dic₁₇	1	2	yes	—
105	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
106	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
107	C₂×C₄²	Dic₃₄	1	2	yes	—
108	C₂×C₄²	D₆₈	1	1	yes	characteristic
109	C₂×C₄²	D₆₈	1	1	yes	characteristic
110	C₂×C₄²	C₂×Dic₁₇	1	2	yes	—
111	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
112	C₄⋊C₈	—	17	8	no	—
113	C₄⋊C₈	—	34	4	no	—
114	#97	—	68	4	no	—
115	C₄×C₈	—	2	4	no	—
116	C₄×C₈	—	2	4	no	—
117	C₄×C₈	—	2	2	no	—
118	C₄×C₈	C₄×D₁₇	1	4	yes	—
119	C₁₇	#8273	1	1	yes	characteristic, complemented, Sylow
120	C₂²×C₄	C₄⋊Dic₁₇	1	2	yes	—
121	C₂²×C₄	C₄⋊Dic₁₇	1	2	yes	—
122	C₂²×C₄	C₂×Dic₃₄	1	1	yes	characteristic
123	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
124	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	34	8	no	—
129	C₂×C₈	—	68	4	no	—
130	C₂×C₈	—	68	4	no	—
131	C₂×C₈	#63223	1	4	yes	—
132	C₂×C₈	#63223	1	4	yes	—
133	C₂×C₈	C₄×Dic₁₇	1	2	yes	—
134	C₂×C₈	C₄×Dic₁₇	1	2	yes	—
135	C₄²	—	2	4	no	—
136	C₄²	—	2	2	no	—
137	C₄²	—	2	2	no	—
138	C₄²	—	2	2	no	—
139	C₄²	#63219	1	4	yes	—
140	C₄²	C₂×C₄×D₁₇	1	2	yes	—
141	C₄²	#63252	1	2	yes	—
142	C₂³	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	#10587488	1	4	yes	—
149	C₂×C₄	#10587490	1	4	yes	—
150	C₂×C₄	C₄×Dic₃₄	1	2	yes	—
151	C₂×C₄	C₄×D₆₈	1	2	yes	—
152	C₂×C₄	C₂×#63219	1	2	yes	—
153	C₂×C₄	#10587590	1	2	yes	—
154	C₂×C₄	C₂×#63223	1	2	yes	—
155	C₂×C₄	C₂×#63223	1	2	yes	—
156	C₂×C₄	C₂×C₄×Dic₁₇	1	1	yes	characteristic
157	C₂×C₄	#10587612	1	1	yes	characteristic
158	C₈	—	2	2	no	—
159	C₈	—	2	4	no	—
160	C₈	—	68	8	no	—
161	C₈	#10587474	1	4	yes	complemented
162	C₂²	C₄×C₄⋊Dic₁₇	1	1	yes	characteristic
163	C₂²	#11762833	1	2	yes	—
164	C₂²	C₂×#10587488	1	1	yes	characteristic
165	C₂²	#11762847	1	1	yes	characteristic
166	C₂²	C₂×#10587490	1	2	yes	—
167	C₄	—	2	4	no	—
168	C₄	—	2	4	no	—
169	C₄	—	2	2	no	—
170	C₄	C₂×#10587474	1	1	yes	characteristic
171	C₄	C₄×#63223	1	1	yes	characteristic
172	C₄	#11762866	1	2	yes	—
173	C₄	C₄×#63219	1	4	yes	—
174	C₂	#14527540	1	2	yes	—
175	C₂	#14527547	1	2	yes	—
176	C₂	#14528637	1	1	yes	characteristic
177	C₂	C₄×#10587488	1	1	yes	characteristic
178	C₂	C₄×#10587490	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₆₈
commutator	C₂×C₄×C₁₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₄²×C₈×C₁₇	C₂
socle, prime radical	C₂³×C₁₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	C₄²×C₈	D₁₇
hypocenter	C₁₇	#8273
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
8. #11762560	C₄	semidirect
71. C₈×C₁₇	#97	semidirect
119. C₁₇	#8273	semidirect
161. C₈	#10587474	semidirect

Maximal subgroups

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

subgroup	index
#14528637	2
C₂×#11762560	2
#14526619	2
C₄²×C₈×C₁₇	2
#8273	17