Subgroups — #16950772

Order 4352.

314 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11762638	C₂	1	4	yes	maximal
2	C₂×C₄×#63214	C₂	1	1	yes	maximal, characteristic
3	#14528956	C₂	1	1	yes	maximal, characteristic
4	C₁₇×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂²×#63214	C₂²	1	2	yes	—
6	C₂×#10587490	—	2	1	no	—
7	C₂×#10587490	C₂²	1	2	yes	—
8	#11762833	—	2	1	no	—
9	#11762638	—	2	8	no	—
10	C₄×#63214	C₄	1	2	yes	—
11	C₄×#63214	C₄	1	2	yes	—
12	C₂×C₁₇×#108	—	2	1	no	—
13	C₂×C₁₇×#108	C₂²	1	2	yes	—
14	C₁₇×#423	—	2	1	no	—
15	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₁₇	—	2	2	no	—
17	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
19	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
20	C₂×#63223	—	2	2	no	—
21	C₁₇×#108	—	2	4	no	—
22	C₁₇×#108	—	4	2	no	—
23	C₁₇×C₄⋊C₈	—	4	2	no	—
24	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₁₇	Q₈	1	1	yes	characteristic
27	C₄×C₈×C₁₇	Q₈	1	1	yes	characteristic
28	#10587490	—	2	4	no	—
29	#10587490	—	4	2	no	—
30	C₂×#63214	C₂×C₄	1	4	yes	—
31	C₂×#63214	D₄	1	4	yes	—
32	C₂×#63214	Q₈	1	4	yes	—
33	C₄⋊#3622	—	4	2	no	—
34	C₂²×C₄×C₁₇	Q₁₆	1	2	yes	—
35	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
36	M₁₆×C₁₇	—	4	2	no	—
37	M₁₆×C₁₇	—	4	2	no	—
38	C₂×C₈×C₁₇	—	4	2	no	—
39	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₇	D₈	1	2	yes	—
42	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
43	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
45	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
47	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₁₇	D₈	1	2	yes	—
50	#63223	—	4	2	no	—
51	#63223	—	4	2	no	—
52	C₂×#3622	—	4	2	no	—
53	#63214	C₄⋊C₄	1	8	yes	—
54	#12909	—	17	1	no	maximal, Sylow
55	C₂³×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
56	C₂×C₄×C₁₇	—	2	4	no	—
57	C₂×C₄×C₁₇	#107	1	2	yes	—
58	C₂×C₄×C₁₇	#107	1	2	yes	—
59	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
61	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
64	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
65	C₈×C₁₇	—	8	2	no	—
66	C₈×C₁₇	#107	1	2	yes	—
67	C₈×C₁₇	#107	1	2	yes	—
68	C₈×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
69	C₈×C₁₇	C₄⋊Q₈	1	2	yes	—
70	#3622	—	8	2	no	—
71	C₂×#368	—	17	4	no	—
72	C₂×C₄×C₁₆	—	17	1	no	—
73	#1848	—	17	1	no	—
74	#1848	D₁₇	1	1	yes	characteristic
75	C₂²×C₁₇	—	2	2	no	—
76	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
77	C₂²×C₁₇	#494	1	1	yes	characteristic
78	C₂²×C₁₇	#500	1	1	yes	characteristic
79	C₄×C₁₇	—	2	2	no	—
80	C₄×C₁₇	#368	1	4	yes	complemented
81	C₄×C₁₇	#384	1	1	yes	characteristic
82	C₄×C₁₇	C₂×#107	1	1	yes	characteristic
83	C₄×C₁₇	#493	1	1	yes	characteristic
84	C₄×C₁₇	#500	1	1	yes	characteristic
85	C₂²×C₁₆	—	17	2	no	—
86	C₂×#108	—	2	1	no	—
87	C₂×#108	—	17	2	no	—
88	C₂×#108	—	34	1	no	—
89	C₂×#108	C₂×D₁₇	1	2	yes	—
90	#423	—	2	1	no	—
91	#423	—	34	1	no	—
92	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
93	#368	—	34	8	no	—
94	C₄×C₁₆	—	17	2	no	—
95	C₄×C₁₆	—	17	2	no	—
96	C₂×C₁₇	—	2	1	no	—
97	C₂×C₁₇	—	2	1	no	—
98	C₂×C₁₇	#1845	1	1	yes	characteristic
99	C₂×C₁₇	C₂×#368	1	2	yes	—
100	C₂×M₁₆	—	2	2	no	—
101	C₂×M₁₆	—	34	2	no	—
102	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
104	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
105	C₂×C₁₆	—	17	4	no	—
106	C₂×C₁₆	—	17	4	no	—
107	C₂×C₁₆	—	17	4	no	—
108	#108	—	2	4	no	—
109	#108	—	4	2	no	—
110	#108	—	34	4	no	—
111	#108	—	68	2	no	—
112	C₄⋊C₈	—	4	2	no	—
113	C₄⋊C₈	—	68	2	no	—
114	C₄×C₈	Dic₃₄	1	1	yes	characteristic
115	C₄×C₈	Dic₃₄	1	1	yes	characteristic
116	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
117	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
118	C₁₇	#12909	1	1	yes	characteristic, complemented, Sylow
119	C₂²×C₄	#63231	1	2	yes	—
120	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
121	M₁₆	—	4	2	no	—
122	M₁₆	—	4	2	no	—
123	M₁₆	—	68	2	no	—
124	M₁₆	—	68	2	no	—
125	C₂×C₈	—	4	2	no	—
126	C₂×C₈	—	68	2	no	—
127	C₂×C₈	#63225	1	1	yes	characteristic
128	C₂×C₈	#63225	1	1	yes	characteristic
129	C₂×C₈	#63228	1	2	yes	—
130	C₂×C₈	#63231	1	2	yes	—
131	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
132	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
133	C₂×C₈	D₄×D₁₇	1	2	yes	—
134	C₂×C₈	Q₈×D₁₇	1	2	yes	—
135	C₄²	#63225	1	1	yes	characteristic
136	C₄²	#63225	1	1	yes	characteristic
137	C₄²	#63228	1	2	yes	—
138	C₁₆	—	17	8	no	—
139	C₂³	C₂×#63231	1	1	yes	characteristic
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	#10587478	1	2	yes	—
142	C₂×C₄	#10587478	1	2	yes	—
143	C₂×C₄	C₂×#63225	1	1	yes	characteristic
144	C₂×C₄	C₂×#63228	1	1	yes	characteristic
145	C₂×C₄	C₂×#63228	1	1	yes	characteristic
146	C₂×C₄	#10587626	1	1	yes	characteristic
147	C₂×C₄	C₂×#63231	1	1	yes	characteristic
148	C₂×C₄	#10587630	1	1	yes	characteristic
149	C₈	—	8	2	no	—
150	C₈	—	136	2	no	—
151	C₈	#10587478	1	2	yes	—
152	C₈	#10587478	1	2	yes	—
153	C₈	#10587574	1	2	yes	—
154	C₈	C₄⋊C₄×D₁₇	1	2	yes	—
155	C₂²	—	2	2	no	—
156	C₂²	C₂×#10587478	1	1	yes	characteristic
157	C₂²	#11763188	1	1	yes	characteristic
158	C₂²	#11763212	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	#11762638	1	4	yes	complemented
161	C₄	#11762678	1	1	yes	characteristic
162	C₄	C₂×#10587478	1	1	yes	characteristic
163	C₄	#11763185	1	1	yes	characteristic
164	C₄	#11763212	1	1	yes	characteristic
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	#14528943	1	1	yes	characteristic
168	C₂	C₂×#11762638	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#63228
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#1848	C₂
socle	C₂²×C₁₇	C₂×#107
prime radical	C₂³×C₁₇	C₂×Q₁₆
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	#1848	D₁₇
hypocenter	C₁₇	#12909
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
80. C₄×C₁₇	#368	semidirect
118. C₁₇	#12909	semidirect
160. C₄	#11762638	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11762638	2
C₂×C₄×#63214	2
#14528956	2
C₁₇×#1848	2
#12909	17