Subgroups — C₂²×#11763670 #17966304

Order 4352.

942 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	C₂²×C₄⋊#3624	C₂	1	2	yes	maximal
2	C₂×#11763670	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×#3624	—	2	2	no	—
5	C₂×C₄⋊#3624	—	2	6	no	—
6	C₂×C₄⋊#3624	C₂²	1	12	yes	—
7	#11763670	C₂²	1	16	yes	direct product factor
8	C₂³×#3622	C₄	1	1	yes	characteristic
9	C₂²×C₄×Dic₁₇	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₁₇	C₄	1	6	yes	—
11	C₂×C₈×Dic₁₇	C₂²	1	6	yes	—
12	C₂³×C₈×C₁₇	C₄	1	1	yes	characteristic
13	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
14	C₂³×Dic₁₇	D₄	1	1	yes	characteristic
15	C₂³×Dic₁₇	Q₈	1	1	yes	characteristic
16	C₂²×#3624	—	2	12	no	—
17	C₂²×#3624	—	4	8	no	—
18	C₂²×C₈×C₁₇	C₈	1	8	yes	complemented
19	C₂²×C₈×C₁₇	C₂×C₄	1	6	yes	—
20	C₂×C₄×Dic₁₇	C₂×C₄	1	3	yes	—
21	C₂×C₄×Dic₁₇	D₄	1	3	yes	—
22	C₂×C₄×Dic₁₇	Q₈	1	3	yes	—
23	C₂×C₄×Dic₁₇	C₂³	1	3	yes	—
24	C₂²×#3622	—	2	4	no	—
25	C₂²×#3622	C₂×C₄	1	6	yes	—
26	C₄⋊#3624	—	2	12	no	—
27	C₄⋊#3624	C₂³	1	8	yes	—
28	C₈×Dic₁₇	C₂×C₄	1	12	yes	—
29	C₈×Dic₁₇	C₂³	1	4	yes	—
30	C₂⁴×C₁₇	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₇	C₄⋊C₄	1	3	yes	—
32	C₂²×C₄×C₁₇	C₂×C₈	1	4	yes	—
33	C₂²×C₄×C₁₇	M₁₆	1	4	yes	—
34	C₂²×C₄×C₁₇	C₂²×C₄	1	3	yes	—
35	C₂²×Dic₁₇	—	2	4	no	—
36	C₂²×Dic₁₇	C₄⋊C₄	1	3	yes	—
37	C₂²×Dic₁₇	C₄⋊C₄	1	3	yes	—
38	C₂²×Dic₁₇	SD₁₆	1	8	yes	—
39	C₂²×Dic₁₇	C₂×D₄	1	3	yes	—
40	C₂²×Dic₁₇	C₂×Q₈	1	3	yes	—
41	C₂×#3624	—	2	8	no	—
42	C₂×#3624	—	4	24	no	—
43	C₂×C₈×C₁₇	C₂×C₈	1	24	yes	complemented
44	C₂×C₈×C₁₇	C₂²×C₄	1	4	yes	—
45	C₄×Dic₁₇	C₄⋊C₄	1	6	yes	—
46	C₄×Dic₁₇	C₂²×C₄	1	3	yes	—
47	C₄×Dic₁₇	C₂×D₄	1	3	yes	—
48	C₄×Dic₁₇	C₂×Q₈	1	3	yes	—
49	C₄×Dic₁₇	C₂⁴	1	1	yes	characteristic
50	C₂×#3622	—	2	12	no	—
51	C₂×#3622	C₂²×C₄	1	4	yes	—
52	C₂²×#334	—	17	1	no	maximal, Sylow
53	C₂³×C₁₇	C₄⋊C₈	1	4	yes	—
54	C₂³×C₁₇	#106	1	4	yes	—
55	C₂³×C₁₇	#108	1	4	yes	—
56	C₂³×C₁₇	C₂×C₄⋊C₄	1	3	yes	—
57	C₂×C₄×C₁₇	C₄⋊C₈	1	12	yes	—
58	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	3	yes	—
59	C₂×C₄×C₁₇	C₂²×C₈	1	6	yes	—
60	C₂×C₄×C₁₇	C₂×M₁₆	1	6	yes	—
61	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
62	C₂×Dic₁₇	—	2	6	no	—
63	C₂×Dic₁₇	—	2	6	no	—
64	C₂×Dic₁₇	#106	1	12	yes	—
65	C₂×Dic₁₇	C₂×C₄⋊C₄	1	3	yes	—
66	C₂×Dic₁₇	C₂×C₄⋊C₄	1	3	yes	—
67	C₂×Dic₁₇	C₂×SD₁₆	1	12	yes	—
68	C₂×Dic₁₇	C₂²×D₄	1	1	yes	characteristic
69	C₂×Dic₁₇	C₂²×Q₈	1	1	yes	characteristic
70	#3624	—	4	8	no	—
71	C₈×C₁₇	C₂²×C₈	1	8	yes	complemented
72	#3622	—	2	4	no	—
73	C₂²×C₄⋊C₈	—	17	2	no	—
74	C₂²×C₄×C₈	—	17	1	no	—
75	C₂×#334	—	17	12	no	—
76	C₂²×C₁₇	#334	1	16	yes	complemented
77	C₂²×C₁₇	C₂×C₄⋊C₈	1	6	yes	—
78	C₂²×C₁₇	C₂×#106	1	6	yes	—
79	C₂²×C₁₇	C₂×#108	1	6	yes	—
80	C₂²×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
81	C₄×C₁₇	C₂×C₄⋊C₈	1	6	yes	—
82	C₄×C₁₇	C₂³×C₈	1	1	yes	characteristic
83	C₄×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
84	Dic₁₇	—	2	3	no	—
85	Dic₁₇	—	2	1	no	—
86	Dic₁₇	C₂×#106	1	6	yes	—
87	Dic₁₇	C₂²×SD₁₆	1	2	yes	—
88	C₂³×C₈	—	17	1	no	—
89	C₂³×C₈	—	34	2	no	—
90	C₂³×C₈	#595	1	1	yes	characteristic
91	C₂²×C₄²	—	17	1	no	—
92	C₂×C₄⋊C₈	—	17	12	no	—
93	C₂×C₄⋊C₈	—	34	6	no	—
94	C₂×C₄×C₈	—	17	6	no	—
95	C₂×C₄×C₈	—	17	6	no	—
96	#334	—	17	16	no	—
97	C₂×C₁₇	C₂×#334	1	12	yes	complemented
98	C₂×C₁₇	C₂²×C₄⋊C₈	1	1	yes	characteristic
99	C₂×C₁₇	C₂²×#106	1	1	yes	characteristic
100	C₂×C₁₇	C₂²×#108	1	1	yes	characteristic
101	C₂³×C₄	—	17	1	no	—
102	C₂³×C₄	—	17	1	no	—
103	C₂³×C₄	C₂×#595	1	1	yes	characteristic
104	C₂²×C₈	—	17	6	no	—
105	C₂²×C₈	—	34	4	no	—
106	C₂²×C₈	—	34	12	no	—
107	C₂²×C₈	—	68	8	no	—
108	C₂²×C₈	#3624	1	8	yes	complemented
109	C₂²×C₈	C₂×#595	1	6	yes	—
110	C₂×C₄²	—	17	3	no	—
111	C₂×C₄²	—	17	3	no	—
112	C₂×C₄²	—	17	3	no	—
113	C₂×C₄²	—	17	3	no	—
114	C₄⋊C₈	—	17	8	no	—
115	C₄⋊C₈	—	34	12	no	—
116	C₄×C₈	—	17	4	no	—
117	C₄×C₈	—	17	12	no	—
118	C₁₇	C₂²×#334	1	1	yes	characteristic, complemented, Sylow
119	C₂⁴	C₄⋊#595	1	1	yes	characteristic
120	C₂²×C₄	—	17	3	no	—
121	C₂²×C₄	—	17	3	no	—
122	C₂²×C₄	—	17	8	no	—
123	C₂²×C₄	—	17	3	no	—
124	C₂²×C₄	—	17	3	no	—
125	C₂²×C₄	—	34	4	no	—
126	C₂²×C₄	#63243	1	4	yes	—
127	C₂²×C₄	C₄⋊#595	1	3	yes	—
128	C₂²×C₄	C₂×#3624	1	4	yes	—
129	C₂²×C₄	C₂²×#595	1	3	yes	—
130	C₂×C₈	—	17	4	no	—
131	C₂×C₈	—	34	12	no	—
132	C₂×C₈	—	34	8	no	—
133	C₂×C₈	—	68	24	no	—
134	C₂×C₈	C₂×#3624	1	24	yes	complemented
135	C₂×C₈	C₂²×#595	1	4	yes	—
136	C₄²	—	17	3	no	—
137	C₄²	—	17	6	no	—
138	C₄²	—	17	3	no	—
139	C₄²	—	17	1	no	—
140	C₄²	—	17	3	no	—
141	C₂³	#10587533	1	4	yes	—
142	C₂³	#10587535	1	4	yes	—
143	C₂³	C₄⋊#3624	1	4	yes	—
144	C₂³	C₂×C₄⋊#595	1	3	yes	—
145	C₂×C₄	—	17	12	no	—
146	C₂×C₄	—	17	12	no	—
147	C₂×C₄	—	17	1	no	—
148	C₂×C₄	—	17	3	no	—
149	C₂×C₄	—	17	3	no	—
150	C₂×C₄	—	17	1	no	—
151	C₂×C₄	—	34	6	no	—
152	C₂×C₄	—	34	6	no	—
153	C₂×C₄	C₄⋊#3624	1	12	yes	—
154	C₂×C₄	C₂×#63243	1	6	yes	—
155	C₂×C₄	C₂×C₄⋊#595	1	3	yes	—
156	C₂×C₄	C₂²×#3624	1	6	yes	—
157	C₂×C₄	C₂³×#595	1	1	yes	characteristic
158	C₈	—	34	4	no	—
159	C₈	—	68	8	no	—
160	C₈	C₂²×#3624	1	8	yes	complemented
161	C₂²	#11763670	1	16	yes	direct product factor
162	C₂²	C₂×C₄⋊#3624	1	6	yes	—
163	C₂²	C₂×#10587533	1	6	yes	—
164	C₂²	C₂×#10587535	1	6	yes	—
165	C₂²	C₂²×C₄⋊#595	1	1	yes	characteristic
166	C₄	—	17	2	no	—
167	C₄	—	17	6	no	—
168	C₄	—	34	3	no	—
169	C₄	—	34	1	no	—
170	C₄	C₂×C₄⋊#3624	1	6	yes	—
171	C₄	C₂³×#3624	1	1	yes	characteristic
172	C₄	C₂²×#63243	1	1	yes	characteristic
173	C₂	C₂×#11763670	1	12	yes	direct product factor
174	C₂	C₂²×C₄⋊#3624	1	1	yes	characteristic
175	C₂	C₂²×#10587533	1	1	yes	characteristic
176	C₂	C₂²×#10587535	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₄⋊#595
commutator	C₄×C₁₇	C₂³×C₈
Frattini	C₂×C₄	C₂³×#595
Fitting	C₂³×C₈×C₁₇	C₄
socle, prime radical	C₂⁴×C₁₇	C₄⋊C₄
cosocle	C₄×Dic₁₇	C₂⁴
hypercenter	C₂³×C₈	#595
hypocenter	C₁₇	C₂²×#334
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#11763670	C₂	direct (N × H)
7. #11763670	C₂²	direct (N × H)
18. C₂²×C₈×C₁₇	C₈	semidirect
43. C₂×C₈×C₁₇	C₂×C₈	semidirect
71. C₈×C₁₇	C₂²×C₈	semidirect
76. C₂²×C₁₇	#334	semidirect
97. C₂×C₁₇	C₂×#334	semidirect
108. C₂²×C₈	#3624	semidirect
118. C₁₇	C₂²×#334	semidirect
134. C₂×C₈	C₂×#3624	semidirect
160. C₈	C₂²×#3624	semidirect
161. C₂²	#11763670	direct (N × H)
173. C₂	C₂×#11763670	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⋊#3624	2
C₂×#11763670	2
C₂²×C₈×Dic₁₇	2
C₂²×#334	17

Subgroups — C22×#11763670 #17966304

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11763670 #17966304