Subgroups — #16945913

Order 4352.

242 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14527532	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762562	C₂	1	1	yes	maximal, characteristic
3	#14526858	C₂	1	2	yes	maximal, complemented
4	#14526856	C₂	1	2	yes	maximal, complemented
5	C₄×C₁₇×M₃₂	C₂	1	1	yes	maximal, characteristic
6	#11762862	—	2	1	no	—
7	C₂×C₄⋊#3622	—	2	1	no	—
8	#11762562	—	2	1	no	—
9	#11762562	C₂²	1	2	yes	—
10	#11762562	C₂²	1	1	yes	characteristic
11	#11762562	C₂²	1	1	yes	characteristic
12	#11762560	—	2	1	no	—
13	C₂×C₁₇×M₃₂	C₄	1	1	yes	characteristic
14	C₂×C₁₇×M₃₂	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
16	C₁₇×#346	C₄	1	2	yes	—
17	C₄×C₁₆×C₁₇	C₂²	1	2	yes	—
18	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₇	Q₈	1	1	yes	characteristic
20	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
21	C₂²×#3622	—	4	1	no	—
22	C₁₇×M₃₂	—	2	2	no	—
23	C₁₇×M₃₂	C₈	1	4	yes	complemented
24	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
25	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
28	C₄×C₈×C₁₇	Q₈	1	1	yes	characteristic
29	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
30	C₂²⋊#3622	—	4	1	no	—
31	C₄⋊#3622	—	2	1	no	—
32	C₄⋊#3622	—	2	1	no	—
33	C₄⋊#3622	—	2	2	no	—
34	C₄⋊#3622	—	4	1	no	—
35	C₄×#3622	—	4	1	no	—
36	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
38	C₂²×C₄×C₁₇	Q₁₆	1	1	yes	characteristic
39	C₂×C₈×C₁₇	—	2	2	no	—
40	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
43	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
44	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
46	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₁₇	D₈	1	1	yes	characteristic
48	C₄²×C₁₇	Q₁₆	1	1	yes	characteristic
49	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
50	C₂×#3622	—	4	1	no	—
51	C₂×#3622	—	4	1	no	—
52	C₂×#3622	—	4	2	no	—
53	C₂×#3622	—	8	2	no	—
54	C₁₆×C₁₇	—	2	2	no	—
55	C₁₆×C₁₇	C₂×C₈	1	4	yes	complemented
56	#12009	—	17	1	no	maximal, Sylow
57	C₂³×C₁₇	#107	1	1	yes	characteristic
58	C₂×C₄×C₁₇	—	2	1	no	—
59	C₂×C₄×C₁₇	—	2	1	no	—
60	C₂×C₄×C₁₇	—	4	1	no	—
61	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
62	C₂×C₄×C₁₇	#107	1	1	yes	characteristic
63	C₂×C₄×C₁₇	#107	1	1	yes	characteristic
64	C₂×C₄×C₁₇	#107	1	1	yes	characteristic
65	C₂×C₄×C₁₇	#108	1	2	yes	—
66	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
69	C₈×C₁₇	—	2	1	no	—
70	C₈×C₁₇	—	2	1	no	—
71	C₈×C₁₇	C₄⋊C₈	1	2	yes	—
72	C₈×C₁₇	C₂²×C₈	1	1	yes	characteristic
73	C₈×C₁₇	C₂×M₁₆	1	1	yes	characteristic
74	#3622	—	8	1	no	—
75	#3622	—	8	2	no	—
76	#3622	—	8	1	no	—
77	C₄×M₃₂	D₁₇	1	1	yes	characteristic
78	#1564	—	17	1	no	—
79	C₂×#335	—	17	1	no	—
80	#1372	—	17	2	no	—
81	#1371	—	17	2	no	—
82	C₂²×C₁₇	—	2	1	no	—
83	C₂²×C₁₇	—	2	1	no	—
84	C₂²×C₁₇	#335	1	2	yes	—
85	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
86	C₄×C₁₇	—	2	1	no	—
87	C₄×C₁₇	—	2	1	no	—
88	C₄×C₁₇	—	2	1	no	—
89	C₄×C₁₇	#335	1	2	yes	—
90	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
91	C₄×C₁₇	C₂×#108	1	1	yes	characteristic
92	C₄×C₁₇	#509	1	1	yes	characteristic
93	C₄×C₁₇	#510	1	1	yes	characteristic
94	C₂×M₃₂	Dic₁₇	1	1	yes	characteristic
95	C₂×M₃₂	Dic₁₇	1	1	yes	characteristic
96	#431	—	34	1	no	—
97	C₂×C₄⋊C₈	—	34	1	no	—
98	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
99	#346	Dic₁₇	1	2	yes	—
100	C₄×C₁₆	C₂×D₁₇	1	2	yes	—
101	#335	—	17	2	no	—
102	#335	—	17	1	no	—
103	#335	—	17	1	no	—
104	#335	—	34	1	no	—
105	#334	—	34	1	no	—
106	C₂×C₁₇	—	2	2	no	—
107	C₂×C₁₇	C₂×#335	1	1	yes	characteristic
108	C₂×C₁₇	#2159	1	1	yes	characteristic
109	C₂×C₁₇	#2161	1	1	yes	characteristic
110	C₂²×C₈	—	68	1	no	—
111	C₂²×C₈	Dic₃₄	1	1	yes	characteristic
112	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
113	C₂×C₄²	D₆₈	1	1	yes	characteristic
114	M₃₂	—	2	2	no	—
115	M₃₂	#3622	1	4	yes	complemented
116	C₂×C₁₆	C₂×Dic₁₇	1	2	yes	—
117	C₂×C₁₆	C₂×Dic₁₇	1	2	yes	—
118	C₄⋊C₈	—	34	1	no	—
119	C₄⋊C₈	—	34	1	no	—
120	C₄⋊C₈	—	34	2	no	—
121	C₄⋊C₈	—	68	1	no	—
122	C₂²⋊C₈	—	68	1	no	—
123	C₄×C₈	—	68	1	no	—
124	C₄×C₈	Dic₃₄	1	1	yes	characteristic
125	C₄×C₈	D₆₈	1	1	yes	characteristic
126	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
127	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
128	C₁₇	#12009	1	1	yes	characteristic, complemented, Sylow
129	C₂²×C₄	D₁₃₆	1	1	yes	characteristic
130	C₂²×C₄	Dic₆₈	1	1	yes	characteristic
131	C₂²×C₄	C₄⋊Dic₁₇	1	1	yes	characteristic
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	68	1	no	—
134	C₂×C₈	—	68	1	no	—
135	C₂×C₈	—	68	2	no	—
136	C₂×C₈	—	136	2	no	—
137	C₂×C₈	C₂×#3622	1	2	yes	—
138	C₂×C₈	#63223	1	2	yes	—
139	C₂×C₈	C₄⋊Dic₁₇	1	1	yes	characteristic
140	C₂×C₈	C₄⋊Dic₁₇	1	1	yes	characteristic
141	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
142	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
143	C₄²	D₁₃₆	1	1	yes	characteristic
144	C₄²	Dic₆₈	1	1	yes	characteristic
145	C₄²	C₄⋊Dic₁₇	1	1	yes	characteristic
146	C₄²	C₂×D₆₈	1	1	yes	characteristic
147	C₁₆	—	2	2	no	—
148	C₁₆	C₂×#3622	1	4	yes	complemented
149	C₂³	#10587489	1	1	yes	characteristic
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	C₄⋊#3622	1	2	yes	—
154	C₂×C₄	#10587489	1	1	yes	characteristic
155	C₂×C₄	#10587489	1	1	yes	characteristic
156	C₂×C₄	#10587489	1	1	yes	characteristic
157	C₂×C₄	#10587490	1	2	yes	—
158	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
159	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
160	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
161	C₈	—	2	1	no	—
162	C₈	—	2	1	no	—
163	C₈	—	136	2	no	—
164	C₈	—	136	1	no	—
165	C₈	—	136	1	no	—
166	C₈	C₄⋊#3622	1	2	yes	—
167	C₈	C₂²×#3622	1	1	yes	characteristic
168	C₈	C₂×#63223	1	1	yes	characteristic
169	C₂²	—	2	1	no	—
170	C₂²	—	2	1	no	—
171	C₂²	#11762562	1	2	yes	—
172	C₂²	C₂×#10587489	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	#11762562	1	2	yes	—
177	C₄	C₂×C₄⋊#3622	1	1	yes	characteristic
178	C₄	C₂×#10587490	1	1	yes	characteristic
179	C₄	#11763247	1	1	yes	characteristic
180	C₄	#11763252	1	1	yes	characteristic
181	C₂	—	2	2	no	—
182	C₂	C₂×#11762562	1	1	yes	characteristic
183	C₂	#14530572	1	1	yes	characteristic
184	C₂	#14530579	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₃₆
commutator	C₈×C₁₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₄×C₁₇×M₃₂	C₂
socle	C₂²×C₁₇	C₂×#107
prime radical	C₂³×C₁₇	#107
cosocle	C₄×C₈×C₁₇	C₂³
hypercenter	C₄×M₃₂	D₁₇
hypocenter	C₁₇	#12009
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #14526858	C₂	semidirect
4. #14526856	C₂	semidirect
23. C₁₇×M₃₂	C₈	semidirect
55. C₁₆×C₁₇	C₂×C₈	semidirect
115. M₃₂	#3622	semidirect
128. C₁₇	#12009	semidirect
148. C₁₆	C₂×#3622	semidirect

Maximal subgroups

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

subgroup	index
#14527532	2
C₂×#11762562	2
#14526858	2
#14526856	2
C₄×C₁₇×M₃₂	2
#12009	17