Subgroups — C₂²×#11762634 #17187344

Order 4352.

782 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#63214	C₂	1	1	yes	maximal, characteristic
2	C₂²×#10587489	C₂	1	1	yes	maximal, characteristic
3	C₂×#11762634	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₁₇×#107	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₁₇	—	2	1	no	—
6	C₂²×#63214	C₄	1	8	yes	complemented
7	C₂²×#63214	C₂²	1	6	yes	—
8	C₂×#10587489	—	2	3	no	—
9	C₂×#10587489	C₂²	1	6	yes	—
10	#11762634	C₂²	1	16	yes	direct product factor
11	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₁₇	—	2	1	no	—
13	C₂×C₁₇×#107	—	2	3	no	—
14	C₂×C₁₇×#107	C₂²	1	6	yes	—
15	C₂³×C₄×C₁₇	—	4	1	no	—
16	C₂³×C₄×C₁₇	D₄	1	1	yes	characteristic
17	C₂³×Dic₁₇	—	4	1	no	—
18	C₂²×C₈×C₁₇	C₂×C₄	1	4	yes	—
19	C₂²×C₈×C₁₇	D₄	1	3	yes	—
20	C₂²×C₈×C₁₇	Q₈	1	4	yes	—
21	C₂²×C₈×C₁₇	C₂³	1	3	yes	—
22	C₂×C₄⋊C₄×C₁₇	—	2	6	no	—
23	C₂×C₄⋊C₄×C₁₇	—	4	3	no	—
24	C₂×C₄⋊Dic₁₇	—	2	6	no	—
25	C₂×C₄⋊Dic₁₇	—	4	3	no	—
26	C₁₇×#107	—	2	6	no	—
27	C₁₇×#107	C₂³	1	4	yes	—
28	#10587489	—	2	6	no	—
29	#10587489	C₂³	1	4	yes	—
30	C₂×#63214	C₂×C₄	1	24	yes	complemented
31	C₂×#63214	C₂³	1	4	yes	—
32	C₂⁴×C₁₇	D₈	1	1	yes	characteristic
33	C₂²×C₄×C₁₇	—	4	6	no	—
34	C₂²×C₄×C₁₇	—	8	4	no	—
35	C₂²×C₄×C₁₇	C₄⋊C₄	1	4	yes	—
36	C₂²×C₄×C₁₇	D₈	1	3	yes	—
37	C₂²×C₄×C₁₇	Q₁₆	1	4	yes	—
38	C₂²×C₄×C₁₇	C₂×D₄	1	3	yes	—
39	C₂²×Dic₁₇	—	4	6	no	—
40	C₂²×Dic₁₇	—	8	4	no	—
41	C₂×C₈×C₁₇	C₄⋊C₄	1	12	yes	—
42	C₂×C₈×C₁₇	C₂²×C₄	1	6	yes	—
43	C₂×C₈×C₁₇	C₂×D₄	1	3	yes	—
44	C₂×C₈×C₁₇	C₂×Q₈	1	6	yes	—
45	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
46	C₄⋊C₄×C₁₇	—	2	4	no	—
47	C₄⋊C₄×C₁₇	—	4	6	no	—
48	C₄⋊Dic₁₇	—	2	4	no	—
49	C₄⋊Dic₁₇	—	4	6	no	—
50	#63214	C₂²×C₄	1	8	yes	complemented
51	C₂²×#366	—	17	1	no	maximal, Sylow
52	C₂³×C₁₇	#107	1	4	yes	—
53	C₂³×C₁₇	D₁₆	1	4	yes	—
54	C₂³×C₁₇	Q₃₂	1	4	yes	—
55	C₂³×C₁₇	C₂×D₈	1	3	yes	—
56	C₂×C₄×C₁₇	—	4	4	no	—
57	C₂×C₄×C₁₇	—	8	12	no	—
58	C₂×C₄×C₁₇	#107	1	12	yes	—
59	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	6	yes	—
60	C₂×C₄×C₁₇	C₂×D₈	1	3	yes	—
61	C₂×C₄×C₁₇	C₂×Q₁₆	1	6	yes	—
62	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
63	C₂×Dic₁₇	—	4	4	no	—
64	C₂×Dic₁₇	—	8	12	no	—
65	C₈×C₁₇	C₂×C₄⋊C₄	1	6	yes	—
66	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
67	C₈×C₁₇	C₂²×Q₈	1	1	yes	characteristic
68	C₂³×C₁₆	—	17	1	no	—
69	C₂²×#107	—	17	1	no	—
70	C₂²×#107	D₁₇	1	1	yes	characteristic
71	C₂×#366	—	17	12	no	—
72	C₂²×C₁₇	#366	1	16	yes	complemented
73	C₂²×C₁₇	C₂×#107	1	6	yes	—
74	C₂²×C₁₇	C₂×D₁₆	1	6	yes	—
75	C₂²×C₁₇	C₂×Q₃₂	1	6	yes	—
76	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
77	C₄×C₁₇	—	8	4	no	—
78	C₄×C₁₇	C₂×#107	1	6	yes	—
79	C₄×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
80	C₄×C₁₇	C₂²×Q₁₆	1	1	yes	characteristic
81	Dic₁₇	—	8	4	no	—
82	C₂³×C₈	C₂×D₁₇	1	1	yes	characteristic
83	C₂²×C₄⋊C₄	—	2	1	no	—
84	C₂²×C₄⋊C₄	—	34	1	no	—
85	C₂²×C₁₆	—	17	8	no	—
86	C₂²×C₁₆	—	17	6	no	—
87	C₂×#107	—	2	3	no	—
88	C₂×#107	—	17	6	no	—
89	C₂×#107	—	34	3	no	—
90	C₂×#107	C₂×D₁₇	1	6	yes	—
91	#366	—	17	16	no	—
92	C₂×C₁₇	C₂×#366	1	12	yes	complemented
93	C₂×C₁₇	C₂²×#107	1	1	yes	characteristic
94	C₂×C₁₇	C₂²×D₁₆	1	1	yes	characteristic
95	C₂×C₁₇	C₂²×Q₃₂	1	1	yes	characteristic
96	C₂³×C₄	—	4	1	no	—
97	C₂³×C₄	—	68	1	no	—
98	C₂³×C₄	C₂²⋊D₁₇	1	1	yes	characteristic
99	C₂²×C₈	Dic₃₄	1	4	yes	—
100	C₂²×C₈	C₄×D₁₇	1	4	yes	—
101	C₂²×C₈	C₂²⋊D₁₇	1	3	yes	—
102	C₂²×C₈	C₂²×D₁₇	1	3	yes	—
103	C₂×C₄⋊C₄	—	2	6	no	—
104	C₂×C₄⋊C₄	—	4	3	no	—
105	C₂×C₄⋊C₄	—	34	6	no	—
106	C₂×C₄⋊C₄	—	68	3	no	—
107	C₂×C₁₆	—	17	24	no	—
108	C₂×C₁₆	—	17	4	no	—
109	#107	—	2	6	no	—
110	#107	—	17	4	no	—
111	#107	—	34	6	no	—
112	#107	C₂²×D₁₇	1	4	yes	—
113	C₁₇	C₂²×#366	1	1	yes	characteristic, complemented, Sylow
114	C₂⁴	#63228	1	1	yes	characteristic
115	C₂²×C₄	—	4	6	no	—
116	C₂²×C₄	—	8	4	no	—
117	C₂²×C₄	—	68	6	no	—
118	C₂²×C₄	—	136	4	no	—
119	C₂²×C₄	#63225	1	4	yes	—
120	C₂²×C₄	#63228	1	3	yes	—
121	C₂²×C₄	#63231	1	4	yes	—
122	C₂²×C₄	C₂×C₂²⋊D₁₇	1	3	yes	—
123	C₂×C₈	#63225	1	12	yes	—
124	C₂×C₈	C₂×Dic₃₄	1	6	yes	—
125	C₂×C₈	C₂×C₄×D₁₇	1	6	yes	—
126	C₂×C₈	C₂×C₂²⋊D₁₇	1	3	yes	—
127	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
128	C₄⋊C₄	—	2	4	no	—
129	C₄⋊C₄	—	4	6	no	—
130	C₄⋊C₄	—	34	4	no	—
131	C₄⋊C₄	—	68	6	no	—
132	C₁₆	—	17	8	no	—
133	C₂³	#10587478	1	4	yes	—
134	C₂³	#10587497	1	4	yes	—
135	C₂³	#10587500	1	4	yes	—
136	C₂³	C₂×#63228	1	3	yes	—
137	C₂×C₄	—	4	4	no	—
138	C₂×C₄	—	8	12	no	—
139	C₂×C₄	—	68	4	no	—
140	C₂×C₄	—	136	12	no	—
141	C₂×C₄	#10587478	1	12	yes	—
142	C₂×C₄	C₂×#63225	1	6	yes	—
143	C₂×C₄	C₂×#63228	1	3	yes	—
144	C₂×C₄	C₂×#63231	1	6	yes	—
145	C₂×C₄	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
146	C₈	C₂×#63225	1	6	yes	—
147	C₈	C₂²×Dic₃₄	1	1	yes	characteristic
148	C₈	C₂²×C₄×D₁₇	1	1	yes	characteristic
149	C₂²	#11762634	1	16	yes	direct product factor
150	C₂²	C₂×#10587478	1	6	yes	—
151	C₂²	C₂×#10587497	1	6	yes	—
152	C₂²	C₂×#10587500	1	6	yes	—
153	C₂²	C₂²×#63228	1	1	yes	characteristic
154	C₄	—	8	4	no	—
155	C₄	—	136	4	no	—
156	C₄	C₂×#10587478	1	6	yes	—
157	C₄	C₂²×#63225	1	1	yes	characteristic
158	C₄	C₂²×#63231	1	1	yes	characteristic
159	C₂	C₂×#11762634	1	12	yes	direct product factor
160	C₂	C₂²×#10587478	1	1	yes	characteristic
161	C₂	C₂²×#10587497	1	1	yes	characteristic
162	C₂	C₂²×#10587500	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	#63228
commutator	C₈×C₁₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂²×C₁₇×#107	C₂
socle, prime radical	C₂⁴×C₁₇	D₈
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	C₂²×#107	D₁₇
hypocenter	C₁₇	C₂²×#366
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
3. C₂×#11762634	C₂	direct (N × H)
6. C₂²×#63214	C₄	semidirect
10. #11762634	C₂²	direct (N × H)
30. C₂×#63214	C₂×C₄	semidirect
50. #63214	C₂²×C₄	semidirect
72. C₂²×C₁₇	#366	semidirect
92. C₂×C₁₇	C₂×#366	semidirect
113. C₁₇	C₂²×#366	semidirect
149. C₂²	#11762634	direct (N × H)
159. C₂	C₂×#11762634	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#63214	2
C₂²×#10587489	2
C₂×#11762634	2
C₂²×C₁₇×#107	2
C₂²×#366	17

Subgroups — C22×#11762634 #17187344

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11762634 #17187344