Subgroups — C₂²×#11762562 #17003293

Order 4352.

942 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊#3622	C₂	1	2	yes	maximal
2	C₂×#11762562	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₄×C₈×C₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×#3622	—	2	2	no	—
5	C₂×C₄⋊#3622	—	2	6	no	—
6	C₂×C₄⋊#3622	C₂²	1	12	yes	—
7	#11762562	C₂²	1	16	yes	direct product factor
8	C₂³×C₈×C₁₇	C₄	1	1	yes	characteristic
9	C₂³×C₈×C₁₇	C₄	1	1	yes	characteristic
10	C₂²×C₄²×C₁₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₇	C₄	1	6	yes	—
12	C₂×C₄×C₈×C₁₇	C₂²	1	6	yes	—
13	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
14	C₂³×C₄×C₁₇	D₄	1	1	yes	characteristic
15	C₂³×C₄×C₁₇	Q₈	1	1	yes	characteristic
16	C₂²×C₈×C₁₇	—	2	4	no	—
17	C₂²×C₈×C₁₇	C₈	1	8	yes	complemented
18	C₂²×C₈×C₁₇	C₂×C₄	1	6	yes	—
19	C₂²×C₈×C₁₇	C₂×C₄	1	6	yes	—
20	C₂×C₄²×C₁₇	C₂×C₄	1	3	yes	—
21	C₂×C₄²×C₁₇	D₄	1	3	yes	—
22	C₂×C₄²×C₁₇	Q₈	1	3	yes	—
23	C₂×C₄²×C₁₇	C₂³	1	3	yes	—
24	C₂²×#3622	—	2	12	no	—
25	C₂²×#3622	—	4	8	no	—
26	C₄×C₈×C₁₇	C₂×C₄	1	12	yes	—
27	C₄×C₈×C₁₇	C₂³	1	4	yes	—
28	C₄⋊#3622	—	2	12	no	—
29	C₄⋊#3622	C₂³	1	8	yes	—
30	C₂⁴×C₁₇	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₇	—	2	4	no	—
32	C₂²×C₄×C₁₇	C₄⋊C₄	1	3	yes	—
33	C₂²×C₄×C₁₇	C₄⋊C₄	1	3	yes	—
34	C₂²×C₄×C₁₇	C₄⋊C₄	1	3	yes	—
35	C₂²×C₄×C₁₇	C₂×C₈	1	4	yes	—
36	C₂²×C₄×C₁₇	M₁₆	1	4	yes	—
37	C₂²×C₄×C₁₇	D₈	1	4	yes	—
38	C₂²×C₄×C₁₇	Q₁₆	1	4	yes	—
39	C₂²×C₄×C₁₇	C₂²×C₄	1	3	yes	—
40	C₂²×C₄×C₁₇	C₂×D₄	1	3	yes	—
41	C₂²×C₄×C₁₇	C₂×Q₈	1	3	yes	—
42	C₂×C₈×C₁₇	—	2	12	no	—
43	C₂×C₈×C₁₇	C₂×C₈	1	24	yes	complemented
44	C₂×C₈×C₁₇	C₂²×C₄	1	4	yes	—
45	C₂×C₈×C₁₇	C₂²×C₄	1	4	yes	—
46	C₄²×C₁₇	C₄⋊C₄	1	6	yes	—
47	C₄²×C₁₇	C₂²×C₄	1	3	yes	—
48	C₄²×C₁₇	C₂×D₄	1	3	yes	—
49	C₄²×C₁₇	C₂×Q₈	1	3	yes	—
50	C₄²×C₁₇	C₂⁴	1	1	yes	characteristic
51	C₂×#3622	—	2	8	no	—
52	C₂×#3622	—	4	24	no	—
53	C₂²×#335	—	17	1	no	maximal, Sylow
54	C₂³×C₁₇	C₄⋊C₈	1	4	yes	—
55	C₂³×C₁₇	#107	1	4	yes	—
56	C₂³×C₁₇	#108	1	4	yes	—
57	C₂³×C₁₇	C₂×C₄⋊C₄	1	3	yes	—
58	C₂×C₄×C₁₇	—	2	6	no	—
59	C₂×C₄×C₁₇	—	2	6	no	—
60	C₂×C₄×C₁₇	C₄⋊C₈	1	12	yes	—
61	C₂×C₄×C₁₇	#107	1	12	yes	—
62	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	3	yes	—
63	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	3	yes	—
64	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	3	yes	—
65	C₂×C₄×C₁₇	C₂²×C₈	1	6	yes	—
66	C₂×C₄×C₁₇	C₂×M₁₆	1	6	yes	—
67	C₂×C₄×C₁₇	C₂×D₈	1	6	yes	—
68	C₂×C₄×C₁₇	C₂×Q₁₆	1	6	yes	—
69	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
71	C₂×C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
72	C₈×C₁₇	—	2	4	no	—
73	C₈×C₁₇	C₂²×C₈	1	8	yes	complemented
74	#3622	—	4	8	no	—
75	C₂²×C₄⋊C₈	—	17	2	no	—
76	C₂²×C₄×C₈	D₁₇	1	1	yes	characteristic
77	C₂×#335	—	17	12	no	—
78	C₂²×C₁₇	#335	1	16	yes	complemented
79	C₂²×C₁₇	C₂×C₄⋊C₈	1	6	yes	—
80	C₂²×C₁₇	C₂×#107	1	6	yes	—
81	C₂²×C₁₇	C₂×#108	1	6	yes	—
82	C₂²×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
83	C₄×C₁₇	—	2	3	no	—
84	C₄×C₁₇	—	2	1	no	—
85	C₄×C₁₇	C₂×C₄⋊C₈	1	6	yes	—
86	C₄×C₁₇	C₂×#107	1	6	yes	—
87	C₄×C₁₇	C₂³×C₈	1	1	yes	characteristic
88	C₄×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
89	C₄×C₁₇	C₂²×D₈	1	1	yes	characteristic
90	C₄×C₁₇	C₂²×Q₁₆	1	1	yes	characteristic
91	C₂³×C₈	—	34	2	no	—
92	C₂³×C₈	Dic₁₇	1	1	yes	characteristic
93	C₂³×C₈	Dic₁₇	1	1	yes	characteristic
94	C₂²×C₄²	C₂×D₁₇	1	1	yes	characteristic
95	C₂×C₄⋊C₈	—	17	12	no	—
96	C₂×C₄⋊C₈	—	34	6	no	—
97	C₂×C₄×C₈	Dic₁₇	1	6	yes	—
98	C₂×C₄×C₈	C₂×D₁₇	1	6	yes	—
99	#335	—	17	16	no	—
100	C₂×C₁₇	C₂×#335	1	12	yes	complemented
101	C₂×C₁₇	C₂²×C₄⋊C₈	1	1	yes	characteristic
102	C₂×C₁₇	C₂²×#107	1	1	yes	characteristic
103	C₂×C₁₇	C₂²×#108	1	1	yes	characteristic
104	C₂³×C₄	Dic₃₄	1	1	yes	characteristic
105	C₂³×C₄	D₆₈	1	1	yes	characteristic
106	C₂³×C₄	C₂×Dic₁₇	1	1	yes	characteristic
107	C₂²×C₈	—	2	4	no	—
108	C₂²×C₈	—	34	12	no	—
109	C₂²×C₈	—	68	8	no	—
110	C₂²×C₈	#3622	1	8	yes	complemented
111	C₂²×C₈	C₂×Dic₁₇	1	6	yes	—
112	C₂²×C₈	C₂×Dic₁₇	1	6	yes	—
113	C₂×C₄²	Dic₃₄	1	3	yes	—
114	C₂×C₄²	D₆₈	1	3	yes	—
115	C₂×C₄²	C₂×Dic₁₇	1	3	yes	—
116	C₂×C₄²	C₂²×D₁₇	1	3	yes	—
117	C₄⋊C₈	—	17	8	no	—
118	C₄⋊C₈	—	34	12	no	—
119	C₄×C₈	C₂×Dic₁₇	1	12	yes	—
120	C₄×C₈	C₂²×D₁₇	1	4	yes	—
121	C₁₇	C₂²×#335	1	1	yes	characteristic, complemented, Sylow
122	C₂⁴	C₄⋊Dic₁₇	1	1	yes	characteristic
123	C₂²×C₄	—	2	4	no	—
124	C₂²×C₄	D₁₃₆	1	4	yes	—
125	C₂²×C₄	Dic₆₈	1	4	yes	—
126	C₂²×C₄	C₂×#3622	1	4	yes	—
127	C₂²×C₄	#63223	1	4	yes	—
128	C₂²×C₄	C₄⋊Dic₁₇	1	3	yes	—
129	C₂²×C₄	C₄⋊Dic₁₇	1	3	yes	—
130	C₂²×C₄	C₄⋊Dic₁₇	1	3	yes	—
131	C₂²×C₄	C₂×Dic₃₄	1	3	yes	—
132	C₂²×C₄	C₂×D₆₈	1	3	yes	—
133	C₂²×C₄	C₂²×Dic₁₇	1	3	yes	—
134	C₂×C₈	—	2	12	no	—
135	C₂×C₈	—	34	8	no	—
136	C₂×C₈	—	68	24	no	—
137	C₂×C₈	C₂×#3622	1	24	yes	complemented
138	C₂×C₈	C₂²×Dic₁₇	1	4	yes	—
139	C₂×C₈	C₂²×Dic₁₇	1	4	yes	—
140	C₄²	C₄⋊Dic₁₇	1	6	yes	—
141	C₄²	C₂×Dic₃₄	1	3	yes	—
142	C₄²	C₂×D₆₈	1	3	yes	—
143	C₄²	C₂²×Dic₁₇	1	3	yes	—
144	C₄²	C₂³×D₁₇	1	1	yes	characteristic
145	C₂³	C₄⋊#3622	1	4	yes	—
146	C₂³	#10587489	1	4	yes	—
147	C₂³	#10587490	1	4	yes	—
148	C₂³	C₂×C₄⋊Dic₁₇	1	3	yes	—
149	C₂×C₄	—	2	6	no	—
150	C₂×C₄	—	2	6	no	—
151	C₂×C₄	C₄⋊#3622	1	12	yes	—
152	C₂×C₄	#10587489	1	12	yes	—
153	C₂×C₄	C₂×D₁₃₆	1	6	yes	—
154	C₂×C₄	C₂×Dic₆₈	1	6	yes	—
155	C₂×C₄	C₂²×#3622	1	6	yes	—
156	C₂×C₄	C₂×#63223	1	6	yes	—
157	C₂×C₄	C₂×C₄⋊Dic₁₇	1	3	yes	—
158	C₂×C₄	C₂×C₄⋊Dic₁₇	1	3	yes	—
159	C₂×C₄	C₂×C₄⋊Dic₁₇	1	3	yes	—
160	C₂×C₄	C₂²×Dic₃₄	1	1	yes	characteristic
161	C₂×C₄	C₂²×D₆₈	1	1	yes	characteristic
162	C₂×C₄	C₂³×Dic₁₇	1	1	yes	characteristic
163	C₈	—	2	4	no	—
164	C₈	—	68	8	no	—
165	C₈	C₂²×#3622	1	8	yes	complemented
166	C₂²	#11762562	1	16	yes	direct product factor
167	C₂²	C₂×C₄⋊#3622	1	6	yes	—
168	C₂²	C₂×#10587489	1	6	yes	—
169	C₂²	C₂×#10587490	1	6	yes	—
170	C₂²	C₂²×C₄⋊Dic₁₇	1	1	yes	characteristic
171	C₄	—	2	1	no	—
172	C₄	—	2	3	no	—
173	C₄	C₂×C₄⋊#3622	1	6	yes	—
174	C₄	C₂×#10587489	1	6	yes	—
175	C₄	C₂³×#3622	1	1	yes	characteristic
176	C₄	C₂²×#63223	1	1	yes	characteristic
177	C₄	C₂²×D₁₃₆	1	1	yes	characteristic
178	C₄	C₂²×Dic₆₈	1	1	yes	characteristic
179	C₂	C₂×#11762562	1	12	yes	direct product factor
180	C₂	C₂²×C₄⋊#3622	1	1	yes	characteristic
181	C₂	C₂²×#10587489	1	1	yes	characteristic
182	C₂	C₂²×#10587490	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	D₆₈
commutator	C₄×C₁₇	C₂³×C₈
Frattini	C₄²	C₂³×D₁₇
Fitting	C₂²×C₄×C₈×C₁₇	C₂
socle, prime radical	C₂⁴×C₁₇	C₄⋊C₄
cosocle	C₄²×C₁₇	C₂⁴
hypercenter	C₂²×C₄×C₈	D₁₇
hypocenter	C₁₇	C₂²×#335
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#11762562	C₂	direct (N × H)
7. #11762562	C₂²	direct (N × H)
17. C₂²×C₈×C₁₇	C₈	semidirect
43. C₂×C₈×C₁₇	C₂×C₈	semidirect
73. C₈×C₁₇	C₂²×C₈	semidirect
78. C₂²×C₁₇	#335	semidirect
100. C₂×C₁₇	C₂×#335	semidirect
110. C₂²×C₈	#3622	semidirect
121. C₁₇	C₂²×#335	semidirect
137. C₂×C₈	C₂×#3622	semidirect
165. C₈	C₂²×#3622	semidirect
166. C₂²	#11762562	direct (N × H)
179. C₂	C₂×#11762562	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊#3622	2
C₂×#11762562	2
C₂²×C₄×C₈×C₁₇	2
C₂²×#335	17

Subgroups — C22×#11762562 #17003293

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11762562 #17003293