Subgroups — #16950713

Order 4352.

282 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#63214	C₂	1	1	yes	maximal, characteristic
2	#14528956	C₂	1	1	yes	maximal, characteristic
3	#14526877	C₂	1	4	yes	maximal
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	C₄×#63214	C₄	1	2	yes	—
10	C₄×#63214	C₄	1	2	yes	—
11	C₂×C₁₇×#108	—	2	1	no	—
12	C₂×C₁₇×#108	C₂²	1	2	yes	—
13	C₁₇×#423	—	2	1	no	—
14	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
15	C₂×M₁₆×C₁₇	—	2	2	no	—
16	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
19	C₂×#63223	—	2	2	no	—
20	C₁₇×#108	—	2	4	no	—
21	C₁₇×#108	—	4	2	no	—
22	C₁₇×C₄⋊C₈	—	4	2	no	—
23	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₇	Q₈	1	1	yes	characteristic
26	C₄×C₈×C₁₇	Q₈	1	1	yes	characteristic
27	#10587490	—	2	4	no	—
28	#10587490	—	4	2	no	—
29	C₂×#63214	—	2	2	no	—
30	C₂×#63214	—	2	2	no	—
31	C₂×#63214	C₂×C₄	1	4	yes	—
32	C₄⋊#3622	—	4	2	no	—
33	C₂²×C₄×C₁₇	Q₁₆	1	2	yes	—
34	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
35	M₁₆×C₁₇	—	4	2	no	—
36	M₁₆×C₁₇	—	4	2	no	—
37	C₂×C₈×C₁₇	—	4	2	no	—
38	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
41	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
42	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₁₇	#41	1	2	yes	—
45	C₂×C₈×C₁₇	#41	1	2	yes	—
46	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₁₇	D₈	1	2	yes	—
49	#63223	—	4	2	no	—
50	#63223	—	4	2	no	—
51	C₂×#3622	—	4	2	no	—
52	#63214	—	2	4	no	—
53	#12891	—	17	1	no	maximal, Sylow
54	C₂³×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
55	C₂×C₄×C₁₇	—	2	4	no	—
56	C₂×C₄×C₁₇	#107	1	2	yes	—
57	C₂×C₄×C₁₇	#107	1	2	yes	—
58	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
60	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
61	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
62	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
64	C₈×C₁₇	—	8	2	no	—
65	C₈×C₁₇	#106	1	2	yes	—
66	C₈×C₁₇	#106	1	2	yes	—
67	C₈×C₁₇	#117	1	2	yes	—
68	C₈×C₁₇	#125	1	2	yes	—
69	#3622	—	8	2	no	—
70	C₂×C₄×C₁₆	—	17	1	no	—
71	#1848	—	17	1	no	—
72	#1848	D₁₇	1	1	yes	characteristic
73	#1381	—	17	4	no	—
74	C₂²×C₁₇	—	2	2	no	—
75	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
76	C₂²×C₁₇	#487	1	1	yes	characteristic
77	C₂²×C₁₇	#487	1	1	yes	characteristic
78	C₄×C₁₇	—	2	2	no	—
79	C₄×C₁₇	#368	1	4	yes	—
80	C₄×C₁₇	#382	1	1	yes	characteristic
81	C₄×C₁₇	C₂×#106	1	1	yes	characteristic
82	C₄×C₁₇	#486	1	1	yes	characteristic
83	C₄×C₁₇	#486	1	1	yes	characteristic
84	C₂²×C₁₆	—	17	2	no	—
85	C₂×#108	—	2	1	no	—
86	C₂×#108	—	17	2	no	—
87	C₂×#108	—	34	1	no	—
88	C₂×#108	C₂×D₁₇	1	2	yes	—
89	#423	—	2	1	no	—
90	#423	—	34	1	no	—
91	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
92	C₄×C₁₆	—	17	2	no	—
93	C₄×C₁₆	—	17	2	no	—
94	C₂×C₁₇	—	2	1	no	—
95	C₂×C₁₇	—	2	1	no	—
96	C₂×C₁₇	#1834	1	1	yes	characteristic
97	C₂×C₁₇	C₂×#368	1	2	yes	—
98	C₂×M₁₆	—	2	2	no	—
99	C₂×M₁₆	—	34	2	no	—
100	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
101	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
102	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
103	C₂×C₁₆	—	17	4	no	—
104	C₂×C₁₆	—	34	2	no	—
105	C₂×C₁₆	—	34	2	no	—
106	#108	—	2	4	no	—
107	#108	—	4	2	no	—
108	#108	—	34	4	no	—
109	#108	—	68	2	no	—
110	C₄⋊C₈	—	4	2	no	—
111	C₄⋊C₈	—	68	2	no	—
112	C₄×C₈	Dic₃₄	1	1	yes	characteristic
113	C₄×C₈	Dic₃₄	1	1	yes	characteristic
114	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
115	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
116	C₁₇	#12891	1	1	yes	characteristic, complemented, Sylow
117	C₂²×C₄	#63231	1	2	yes	—
118	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
119	M₁₆	—	4	2	no	—
120	M₁₆	—	4	2	no	—
121	M₁₆	—	68	2	no	—
122	M₁₆	—	68	2	no	—
123	C₂×C₈	—	4	2	no	—
124	C₂×C₈	—	68	2	no	—
125	C₂×C₈	#63225	1	1	yes	characteristic
126	C₂×C₈	#63225	1	1	yes	characteristic
127	C₂×C₈	#63229	1	2	yes	—
128	C₂×C₈	#63230	1	2	yes	—
129	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
130	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
131	C₂×C₈	#63254	1	2	yes	—
132	C₂×C₈	#63256	1	2	yes	—
133	C₄²	#63225	1	1	yes	characteristic
134	C₄²	#63225	1	1	yes	characteristic
135	C₄²	#63228	1	2	yes	—
136	C₁₆	—	34	4	no	—
137	C₂³	C₂×#63231	1	1	yes	characteristic
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	#10587478	1	2	yes	—
140	C₂×C₄	#10587478	1	2	yes	—
141	C₂×C₄	C₂×#63225	1	1	yes	characteristic
142	C₂×C₄	C₂×#63228	1	1	yes	characteristic
143	C₂×C₄	C₂×#63229	1	1	yes	characteristic
144	C₂×C₄	#10587622	1	1	yes	characteristic
145	C₂×C₄	C₂×#63230	1	1	yes	characteristic
146	C₂×C₄	#10587633	1	1	yes	characteristic
147	C₈	—	8	2	no	—
148	C₈	—	136	2	no	—
149	C₈	#10587479	1	2	yes	—
150	C₈	#10587479	1	2	yes	—
151	C₈	#10587576	1	2	yes	—
152	C₈	#10587578	1	2	yes	—
153	C₂²	—	2	2	no	—
154	C₂²	C₂×#10587478	1	1	yes	characteristic
155	C₂²	#11763159	1	1	yes	characteristic
156	C₂²	#11763161	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	#11762638	1	4	yes	—
159	C₄	#11762674	1	1	yes	characteristic
160	C₄	C₂×#10587479	1	1	yes	characteristic
161	C₄	#11763155	1	1	yes	characteristic
162	C₄	#11763157	1	1	yes	characteristic
163	C₂	—	2	1	no	—
164	C₂	—	2	1	no	—
165	C₂	#14528911	1	1	yes	characteristic
166	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	#12891
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
116. C₁₇	#12891	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#63214	2
#14528956	2
#14526877	2
C₁₇×#1848	2
#12891	17