Subgroups — #16945952

Order 4352.

236 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14530583	C₂	1	1	yes	maximal, characteristic
2	#14530540	C₂	1	1	yes	maximal, characteristic
3	#14526854	C₂	1	2	yes	maximal, complemented
4	#14526848	C₂	1	2	yes	maximal, complemented
5	C₈×M₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587483	—	2	1	no	—
7	C₄⋊#63214	—	2	1	no	—
8	C₄⋊#63214	C₂²	1	1	yes	characteristic
9	C₄⋊#63214	C₂²	1	1	yes	characteristic
10	C₄⋊#63214	C₂²	1	2	yes	—
11	C₂²⋊#63214	—	2	1	no	—
12	C₄×#63214	—	2	1	no	—
13	C₁₇×#431	C₄	1	1	yes	characteristic
14	C₄×M₁₆×C₁₇	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
16	C₁₇×#322	C₄	1	2	yes	—
17	C₈²×C₁₇	C₂²	1	2	yes	—
18	C₂×M₁₆×C₁₇	—	2	1	no	—
19	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₇	Q₈	1	1	yes	characteristic
21	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₁₇×C₄⋊C₈	C₈	1	2	yes	—
23	C₁₇×C₂²⋊C₈	C₈	1	2	yes	—
24	C₁₇×#97	—	2	1	no	—
25	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
29	C₄×C₈×C₁₇	Q₈	1	1	yes	characteristic
30	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
31	#10587483	—	4	2	no	—
32	C₂×#63214	—	2	2	no	—
33	C₂×#63214	—	2	1	no	—
34	C₂×#63214	—	2	1	no	—
35	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
38	M₁₆×C₁₇	—	2	4	no	—
39	C₂×C₈×C₁₇	—	2	2	no	—
40	C₂×C₈×C₁₇	—	2	1	no	—
41	C₂×C₈×C₁₇	—	2	1	no	—
42	C₂×C₈×C₁₇	—	4	1	no	—
43	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₇	C₂×C₈	1	4	yes	—
46	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
48	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
50	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
51	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
52	#63214	—	4	1	no	—
53	#63214	—	4	2	no	—
54	#63214	—	4	1	no	—
55	#12018	—	17	1	no	maximal, Sylow
56	C₂³×C₁₇	C₄⋊C₈	1	1	yes	characteristic
57	C₂×C₄×C₁₇	—	2	2	no	—
58	C₂×C₄×C₁₇	—	2	1	no	—
59	C₂×C₄×C₁₇	—	2	1	no	—
60	C₂×C₄×C₁₇	—	2	2	no	—
61	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
67	C₈×C₁₇	—	2	4	no	—
68	C₈×C₁₇	—	2	1	no	—
69	C₈×C₁₇	—	2	1	no	—
70	C₈×C₁₇	—	2	1	no	—
71	C₈×C₁₇	—	2	4	no	—
72	C₈×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
73	C₈×C₁₇	#137	1	1	yes	characteristic
74	#2162	—	17	1	no	—
75	#2150	—	17	1	no	—
76	C₈×M₁₆	D₁₇	1	1	yes	characteristic
77	#1370	—	17	2	no	—
78	#1367	—	17	2	no	—
79	C₂²×C₁₇	—	2	1	no	—
80	C₂²×C₁₇	—	2	1	no	—
81	C₂²×C₁₇	—	2	1	no	—
82	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
83	C₄×C₁₇	—	2	1	no	—
84	C₄×C₁₇	—	2	1	no	—
85	C₄×C₁₇	—	2	1	no	—
86	C₄×C₁₇	#364	1	2	yes	—
87	C₄×C₁₇	#428	1	1	yes	characteristic
88	C₄×C₁₇	#430	1	1	yes	characteristic
89	C₄×C₁₇	#504	1	2	yes	—
90	C₂×M₃₂	—	34	1	no	—
91	#431	Dic₁₇	1	1	yes	characteristic
92	C₄×M₁₆	Dic₁₇	1	1	yes	characteristic
93	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
94	C₄⋊C₁₆	—	17	1	no	—
95	C₄⋊C₁₆	—	17	2	no	—
96	C₄⋊C₁₆	—	17	1	no	—
97	C₄⋊C₁₆	—	34	1	no	—
98	C₂²⋊C₁₆	—	34	1	no	—
99	C₄×C₁₆	—	34	1	no	—
100	#322	Dic₁₇	1	2	yes	—
101	C₈²	C₂×D₁₇	1	2	yes	—
102	C₂×C₁₇	—	4	1	no	—
103	C₂×C₁₇	#1565	1	1	yes	characteristic
104	C₂×C₁₇	#2151	1	1	yes	characteristic
105	C₂×C₁₇	C₂×#364	1	1	yes	characteristic
106	C₂×M₁₆	—	2	1	no	—
107	C₂²×C₈	Dic₃₄	1	1	yes	characteristic
108	C₂²×C₈	D₆₈	1	1	yes	characteristic
109	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
110	M₃₂	—	68	2	no	—
111	C₂×C₁₆	—	34	1	no	—
112	C₂×C₁₆	—	34	1	no	—
113	C₂×C₁₆	—	34	2	no	—
114	C₄⋊C₈	#3622	1	2	yes	—
115	C₂²⋊C₈	#3622	1	2	yes	—
116	#97	—	2	1	no	—
117	C₄×C₈	Dic₃₄	1	1	yes	characteristic
118	C₄×C₈	D₆₈	1	1	yes	characteristic
119	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
120	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
121	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
122	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
123	C₁₇	#12018	1	1	yes	characteristic, complemented, Sylow
124	C₂²×C₄	C₂×#3622	1	1	yes	characteristic
125	C₂²×C₄	#63223	1	1	yes	characteristic
126	C₂²×C₄	C₄⋊Dic₁₇	1	1	yes	characteristic
127	M₁₆	—	2	4	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	4	1	no	—
132	C₂×C₈	C₂×#3622	1	4	yes	—
133	C₂×C₈	C₄⋊Dic₁₇	1	1	yes	characteristic
134	C₂×C₈	C₄⋊Dic₁₇	1	1	yes	characteristic
135	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
136	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
137	C₄²	C₂×#3622	1	1	yes	characteristic
138	C₄²	#63223	1	1	yes	characteristic
139	C₄²	C₄⋊Dic₁₇	1	1	yes	characteristic
140	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
141	C₁₆	—	68	2	no	—
142	C₁₆	—	68	1	no	—
143	C₁₆	—	68	1	no	—
144	C₂³	C₄⋊#3622	1	1	yes	characteristic
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	C₄⋊#3622	1	1	yes	characteristic
150	C₂×C₄	C₄⋊#3622	1	1	yes	characteristic
151	C₂×C₄	C₄⋊#3622	1	1	yes	characteristic
152	C₂×C₄	C₂²×#3622	1	1	yes	characteristic
153	C₂×C₄	C₂×#63223	1	1	yes	characteristic
154	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
155	C₈	—	2	4	no	—
156	C₈	—	2	4	no	—
157	C₈	—	2	1	no	—
158	C₈	—	2	1	no	—
159	C₈	—	2	1	no	—
160	C₈	#10587594	1	1	yes	characteristic
161	C₈	#10587595	1	1	yes	characteristic
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	C₂×C₄⋊#3622	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	#11762630	1	2	yes	—
170	C₄	#11762851	1	1	yes	characteristic
171	C₄	#11762858	1	1	yes	characteristic
172	C₄	#11763226	1	2	yes	—
173	C₂	—	4	1	no	—
174	C₂	#14527536	1	1	yes	characteristic
175	C₂	#14530543	1	1	yes	characteristic
176	C₂	C₂×#11762630	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₆₈
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₈×M₁₆×C₁₇	C₂
socle	C₂²×C₁₇	C₂×C₄⋊C₈
prime radical	C₂³×C₁₇	C₄⋊C₈
cosocle	C₄×C₈×C₁₇	C₂³
hypercenter	C₈×M₁₆	D₁₇
hypocenter	C₁₇	#12018
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
3. #14526854	C₂	semidirect
4. #14526848	C₂	semidirect
123. C₁₇	#12018	semidirect

Maximal subgroups

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

subgroup	index
#14530583	2
#14530540	2
#14526854	2
#14526848	2
C₈×M₁₆×C₁₇	2
#12018	17