Subgroups — #17960118

Order 4352.

246 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	#14545141	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#63216	C₂	1	1	yes	maximal, characteristic
3	#14543813	C₂	1	2	yes	maximal, complemented
4	#14543806	C₂	1	2	yes	maximal, complemented
5	M₁₆×#3622	C₂	1	1	yes	maximal, characteristic
6	C₂²×#63216	—	2	1	no	—
7	C₄⋊#63216	—	2	1	no	—
8	C₄⋊#63216	C₂²	1	1	yes	characteristic
9	C₄⋊#63216	C₂²	1	2	yes	—
10	C₄⋊#63216	C₂²	1	1	yes	characteristic
11	C₂²⋊#63216	—	2	1	no	—
12	C₄×#63216	—	2	1	no	—
13	#11762862	C₄	1	1	yes	characteristic
14	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
15	#11762530	C₄	1	2	yes	—
16	C₈×#3622	C₂²	1	2	yes	—
17	C₄×M₁₆×C₁₇	C₄	1	1	yes	characteristic
18	C₂×M₁₆×C₁₇	C₈	1	1	yes	characteristic
19	C₂×M₁₆×C₁₇	C₈	1	1	yes	characteristic
20	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×#3622	D₄	1	1	yes	characteristic
22	C₂²×#3622	Q₈	1	1	yes	characteristic
23	C₂×#63216	—	2	2	no	—
24	C₂×#63216	—	2	1	no	—
25	C₂×#63216	—	2	1	no	—
26	C₂×#63216	—	4	2	no	—
27	C₁₇×#97	C₈	1	2	yes	—
28	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
29	C₂²⋊#3622	—	2	1	no	—
30	C₄⋊#3622	—	2	1	no	—
31	C₄×#3622	C₂×C₄	1	2	yes	—
32	C₄×#3622	C₂×C₄	1	1	yes	characteristic
33	C₄×#3622	D₄	1	1	yes	characteristic
34	C₄×#3622	Q₈	1	1	yes	characteristic
35	C₄×#3622	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
39	M₁₆×C₁₇	—	2	2	no	—
40	M₁₆×C₁₇	C₁₆	1	4	yes	complemented
41	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
42	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
43	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
44	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
45	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
46	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×#3622	—	2	1	no	—
48	C₂×#3622	—	2	1	no	—
49	C₂×#3622	—	2	2	no	—
50	C₂×#3622	—	4	1	no	—
51	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
52	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
53	C₂×#3622	C₂×D₄	1	1	yes	characteristic
54	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
55	#63216	—	4	1	no	—
56	#63216	—	4	2	no	—
57	#63216	—	4	1	no	—
58	#12015	—	17	1	no	maximal, Sylow
59	C₂³×C₁₇	C₄⋊C₈	1	1	yes	characteristic
60	C₂×C₄×C₁₇	—	2	1	no	—
61	C₂×C₄×C₁₇	—	2	1	no	—
62	C₂×C₄×C₁₇	—	4	1	no	—
63	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
65	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
66	C₂×C₄×C₁₇	C₂×C₁₆	1	2	yes	—
67	C₂×C₄×C₁₇	M₃₂	1	2	yes	—
68	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
71	C₈×C₁₇	—	2	2	no	—
72	C₈×C₁₇	C₂×C₁₆	1	4	yes	complemented
73	#3622	—	2	1	no	—
74	#3622	—	2	1	no	—
75	#3622	—	2	1	no	—
76	#3622	—	4	2	no	—
77	#3622	AGL(1,ℤ/8)	1	1	yes	characteristic
78	#3622	#137	1	1	yes	characteristic
79	#2162	—	17	1	no	—
80	C₂×C₄⋊C₁₆	—	17	1	no	—
81	C₈×M₁₆	—	17	1	no	—
82	#1370	—	17	2	no	—
83	#1367	—	17	2	no	—
84	C₂²×C₁₇	—	2	1	no	—
85	C₂²×C₁₇	—	2	1	no	—
86	C₂²×C₁₇	C₄⋊C₁₆	1	2	yes	—
87	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
88	C₄×C₁₇	—	2	1	no	—
89	C₄×C₁₇	—	2	1	no	—
90	C₄×C₁₇	—	2	1	no	—
91	C₄×C₁₇	C₄⋊C₁₆	1	2	yes	—
92	C₄×C₁₇	#428	1	1	yes	characteristic
93	C₄×C₁₇	#430	1	1	yes	characteristic
94	C₄×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
95	C₄×C₁₇	C₂×M₃₂	1	1	yes	characteristic
96	C₂²×C₁₆	—	34	1	no	—
97	#431	—	17	1	no	—
98	C₄×M₁₆	#595	1	1	yes	characteristic
99	C₂×C₄×C₈	—	17	1	no	—
100	C₄⋊C₁₆	—	17	1	no	—
101	C₄⋊C₁₆	—	17	2	no	—
102	C₄⋊C₁₆	—	17	1	no	—
103	C₄⋊C₁₆	—	34	1	no	—
104	C₂²⋊C₁₆	—	34	1	no	—
105	C₄×C₁₆	—	34	1	no	—
106	#322	—	17	2	no	—
107	C₈²	—	17	2	no	—
108	C₂×C₁₇	—	2	2	no	—
109	C₂×C₁₇	#1565	1	1	yes	characteristic
110	C₂×C₁₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
111	C₂×C₁₇	#2153	1	1	yes	characteristic
112	C₂×M₁₆	#3624	1	1	yes	characteristic
113	C₂×M₁₆	#3624	1	1	yes	characteristic
114	C₂²×C₈	—	17	1	no	—
115	C₂²×C₈	—	17	1	no	—
116	C₂×C₄²	C₂×#595	1	1	yes	characteristic
117	C₂×C₁₆	—	34	1	no	—
118	C₂×C₁₆	—	34	1	no	—
119	C₂×C₁₆	—	34	2	no	—
120	C₂×C₁₆	—	68	2	no	—
121	C₄⋊C₈	—	34	1	no	—
122	C₂²⋊C₈	—	34	1	no	—
123	#97	#3624	1	2	yes	—
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	C₂×#595	1	2	yes	—
130	C₁₇	#12015	1	1	yes	characteristic, complemented, Sylow
131	C₂²×C₄	#63243	1	1	yes	characteristic
132	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
133	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
134	M₁₆	—	2	2	no	—
135	M₁₆	#63216	1	4	yes	complemented
136	C₂×C₈	—	17	1	no	—
137	C₂×C₈	—	17	1	no	—
138	C₂×C₈	—	17	1	no	—
139	C₂×C₈	—	17	1	no	—
140	C₂×C₈	—	34	1	no	—
141	C₂×C₈	—	34	1	no	—
142	C₂×C₈	—	34	2	no	—
143	C₂×C₈	—	68	1	no	—
144	C₂×C₈	C₂×#3624	1	2	yes	—
145	C₂×C₈	C₂×#3624	1	2	yes	—
146	C₄²	#63243	1	1	yes	characteristic
147	C₄²	C₄⋊#595	1	1	yes	characteristic
148	C₄²	C₂×#3624	1	1	yes	characteristic
149	C₄²	C₂²×#595	1	1	yes	characteristic
150	C₁₆	—	68	2	no	—
151	C₁₆	—	68	1	no	—
152	C₁₆	—	68	1	no	—
153	C₂³	C₄⋊#3624	1	1	yes	characteristic
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	C₂×#63216	1	2	yes	—
158	C₂×C₄	#10587538	1	2	yes	—
159	C₂×C₄	C₄⋊#3624	1	1	yes	characteristic
160	C₂×C₄	C₄⋊#3624	1	1	yes	characteristic
161	C₂×C₄	C₄⋊#3624	1	1	yes	characteristic
162	C₂×C₄	C₂×#63243	1	1	yes	characteristic
163	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
164	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
165	C₈	—	2	2	no	—
166	C₈	—	17	1	no	—
167	C₈	—	17	1	no	—
168	C₈	—	34	1	no	—
169	C₈	—	34	1	no	—
170	C₈	—	34	1	no	—
171	C₈	—	68	2	no	—
172	C₈	C₂×#63216	1	4	yes	complemented
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	C₄⋊#63216	1	2	yes	—
176	C₂²	C₂×C₄⋊#3624	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	C₄⋊#63216	1	2	yes	—
181	C₄	#11763820	1	1	yes	characteristic
182	C₄	#11763824	1	1	yes	characteristic
183	C₄	C₂²×#63216	1	1	yes	characteristic
184	C₄	C₂×#10587538	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	#14544335	1	1	yes	characteristic
187	C₂	C₂×C₄⋊#63216	1	1	yes	characteristic
188	C₂	#14545125	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#595
commutator	C₄×C₁₇	C₂²×C₁₆
Frattini	C₄²	C₂²×#595
Fitting	C₄×M₁₆×C₁₇	C₄
socle	C₂²×C₁₇	C₂×C₄⋊C₈
prime radical	C₂³×C₁₇	C₄⋊C₈
cosocle	C₄×#3622	C₂³
hypercenter	C₄×M₁₆	#595
hypocenter	C₁₇	#12015
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #14543813	C₂	semidirect
4. #14543806	C₂	semidirect
40. M₁₆×C₁₇	C₁₆	semidirect
72. C₈×C₁₇	C₂×C₁₆	semidirect
130. C₁₇	#12015	semidirect
135. M₁₆	#63216	semidirect
172. C₈	C₂×#63216	semidirect

Maximal subgroups

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

subgroup	index
#14545141	2
C₂×C₄⋊#63216	2
#14543813	2
#14543806	2
M₁₆×#3622	2
#12015	17