Subgroups — C₂×#11762931 #14537680

Order 2176.

464 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	C₂×#10587634	C₂	1	2	yes	maximal, complemented
2	#11762931	C₂	1	8	yes	maximal, direct product factor
3	C₂×#10587490	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#10587508	C₂	1	2	yes	maximal, complemented
5	C₂×#10587486	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₇×#135	C₂	1	1	yes	maximal, characteristic
7	C₂×#41×C₁₇	C₂²	1	2	yes	—
8	C₁₇×#135	C₄	1	4	yes	complemented
9	C₁₇×#135	C₂²	1	4	yes	—
10	C₂×Q₁₆×C₁₇	C₄	1	1	yes	characteristic, complemented
11	C₂×SD₁₆×C₁₇	C₄	1	2	yes	complemented
12	C₂×D₈×C₁₇	C₄	1	1	yes	characteristic, complemented
13	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
14	#10587634	—	2	4	no	—
15	#10587634	C₂²	1	8	yes	complemented
16	C₂×C₄×Dic₁₇	C₂²	1	1	yes	characteristic
17	C₂×#63223	—	2	2	no	—
18	C₂×#63223	C₂²	1	2	yes	—
19	C₂²×#3622	—	2	2	no	—
20	C₂²×#3622	C₂²	1	1	yes	characteristic
21	#10587508	C₂²	1	8	yes	complemented
22	#10587490	C₂²	1	4	yes	complemented
23	#10587486	C₂²	1	4	yes	complemented
24	#41×C₁₇	—	2	2	no	—
25	#41×C₁₇	—	2	2	no	—
26	#41×C₁₇	C₂×C₄	1	4	yes	—
27	#41×C₁₇	C₂³	1	4	yes	—
28	C₂×Q₈×C₁₇	C₂×C₄	1	2	yes	—
29	C₂×D₄×C₁₇	—	2	2	no	—
30	C₂×D₄×C₁₇	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₁₇	—	2	2	no	—
32	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
33	C₂²×Dic₁₇	—	2	1	no	—
34	Q₁₆×C₁₇	C₂×C₄	1	4	yes	complemented
35	SD₁₆×C₁₇	C₂×C₄	1	8	yes	complemented
36	D₈×C₁₇	C₂×C₄	1	4	yes	complemented
37	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₇	C₂×C₄	1	2	yes	—
40	C₂×C₈×C₁₇	C₂³	1	2	yes	—
41	C₄×Dic₁₇	—	2	1	no	—
42	C₄×Dic₁₇	C₂³	1	2	yes	—
43	#63223	—	2	2	no	—
44	#63223	—	2	8	no	—
45	#63223	C₂³	1	4	yes	—
46	C₂×#3622	—	2	2	no	—
47	C₂×#3622	—	2	8	no	—
48	C₂×#3622	D₄	1	2	yes	—
49	C₂×#3622	D₄	1	2	yes	—
50	C₂×#3622	C₂³	1	2	yes	—
51	C₂³×C₁₇	—	2	2	no	—
52	C₂³×C₁₇	#41	1	1	yes	characteristic
53	Q₈×C₁₇	—	2	2	no	—
54	Q₈×C₁₇	C₂²×C₄	1	4	yes	—
55	D₄×C₁₇	—	2	4	no	—
56	D₄×C₁₇	—	2	4	no	—
57	D₄×C₁₇	—	2	2	no	—
58	D₄×C₁₇	C₂²×C₄	1	4	yes	—
59	C₂×C₄×C₁₇	—	2	2	no	—
60	C₂×C₄×C₁₇	—	2	4	no	—
61	C₂×C₄×C₁₇	—	2	4	no	—
62	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
67	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
68	C₂×Dic₁₇	—	2	2	no	—
69	C₂×Dic₁₇	—	4	2	no	—
70	C₈×C₁₇	C₂²×C₄	1	2	yes	—
71	C₈×C₁₇	C₂²×C₄	1	2	yes	—
72	#3622	—	2	4	no	—
73	#3622	C₂×D₄	1	4	yes	—
74	C₂×#444	—	17	1	no	maximal, Sylow
75	C₂²×C₁₇	—	2	2	no	—
76	C₂²×C₁₇	—	2	4	no	—
77	C₂²×C₁₇	—	4	4	no	—
78	C₂²×C₁₇	C₄×D₄	1	1	yes	characteristic
79	C₂²×C₁₇	C₄×D₄	1	1	yes	characteristic
80	C₂²×C₁₇	C₂×#41	1	1	yes	characteristic
81	C₄×C₁₇	—	2	4	no	—
82	C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
83	C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
84	C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
85	C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
86	Dic₁₇	—	4	2	no	—
87	C₂×#135	D₁₇	1	1	yes	characteristic
88	C₂×#131	—	17	2	no	—
89	#444	—	17	8	no	—
90	C₂×#108	—	17	1	no	—
91	C₂×#104	—	17	2	no	—
92	C₂×#97	—	17	1	no	—
93	C₂×C₁₇	—	2	1	no	—
94	C₂×C₁₇	—	2	1	no	—
95	C₂×C₁₇	—	4	4	no	—
96	C₂×C₁₇	#444	1	2	yes	complemented
97	C₂×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
98	C₂×#41	C₂×D₁₇	1	2	yes	—
99	#135	Dic₁₇	1	4	yes	complemented
100	#135	C₂×D₁₇	1	4	yes	—
101	C₂×Q₁₆	Dic₁₇	1	1	yes	characteristic, complemented
102	C₂×SD₁₆	Dic₁₇	1	2	yes	complemented
103	C₂×D₈	Dic₁₇	1	1	yes	characteristic, complemented
104	#131	—	17	8	no	—
105	#131	—	34	4	no	—
106	C₂×M₁₆	—	17	2	no	—
107	C₂×M₁₆	—	34	2	no	—
108	C₂²×C₈	—	17	1	no	—
109	C₂²×C₈	—	34	2	no	—
110	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
111	C₂×C₄²	—	17	1	no	—
112	#108	—	17	4	no	—
113	#104	—	17	8	no	—
114	#97	—	17	4	no	—
115	C₁₇	C₂×#444	1	1	yes	characteristic, complemented, Sylow
116	#41	—	2	2	no	—
117	#41	—	2	2	no	—
118	#41	C₂×Dic₁₇	1	4	yes	—
119	#41	C₂²×D₁₇	1	4	yes	—
120	C₂×Q₈	C₂×Dic₁₇	1	2	yes	—
121	C₂×D₄	—	2	2	no	—
122	C₂×D₄	C₂×Dic₁₇	1	2	yes	—
123	C₂²×C₄	—	2	2	no	—
124	C₂²×C₄	—	34	1	no	—
125	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
126	Q₁₆	C₂×Dic₁₇	1	4	yes	complemented
127	SD₁₆	C₂×Dic₁₇	1	8	yes	complemented
128	D₈	C₂×Dic₁₇	1	4	yes	complemented
129	M₁₆	—	17	4	no	—
130	M₁₆	—	34	8	no	—
131	M₁₆	—	34	2	no	—
132	C₂×C₈	—	17	2	no	—
133	C₂×C₈	—	17	2	no	—
134	C₂×C₈	—	17	2	no	—
135	C₂×C₈	—	34	8	no	—
136	C₂×C₈	—	34	2	no	—
137	C₂×C₈	C₂×Dic₁₇	1	1	yes	characteristic
138	C₂×C₈	C₂×Dic₁₇	1	1	yes	characteristic
139	C₂×C₈	C₂×Dic₁₇	1	2	yes	—
140	C₂×C₈	C₂²×D₁₇	1	2	yes	—
141	C₄²	—	17	2	no	—
142	C₄²	—	34	1	no	—
143	C₂³	—	2	2	no	—
144	C₂³	#63254	1	1	yes	characteristic
145	Q₈	—	2	2	no	—
146	Q₈	C₂²×Dic₁₇	1	4	yes	—
147	D₄	—	2	4	no	—
148	D₄	—	2	4	no	—
149	D₄	—	2	2	no	—
150	D₄	C₂²×Dic₁₇	1	4	yes	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	34	2	no	—
155	C₂×C₄	—	68	2	no	—
156	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
157	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
158	C₂×C₄	#63254	1	1	yes	characteristic
159	C₂×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
160	C₂×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
161	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
162	C₈	—	17	4	no	—
163	C₈	—	34	4	no	—
164	C₈	C₂²×Dic₁₇	1	2	yes	—
165	C₈	C₂²×Dic₁₇	1	2	yes	—
166	C₂²	—	2	2	no	—
167	C₂²	—	2	4	no	—
168	C₂²	—	4	4	no	—
169	C₂²	D₄×Dic₁₇	1	1	yes	characteristic
170	C₂²	D₄×Dic₁₇	1	1	yes	characteristic
171	C₂²	C₂×#63254	1	1	yes	characteristic
172	C₄	—	2	4	no	—
173	C₄	—	68	2	no	—
174	C₄	D₄×Dic₁₇	1	1	yes	characteristic
175	C₄	D₄×Dic₁₇	1	1	yes	characteristic
176	C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
177	C₄	C₂³×Dic₁₇	1	1	yes	characteristic
178	C₂	—	2	1	no	—
179	C₂	—	2	1	no	—
180	C₂	—	4	4	no	—
181	C₂	#11762931	1	2	yes	direct product factor
182	C₂	C₂×D₄×Dic₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₇
commutator	C₄×C₁₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₇
Fitting, prime radical	C₂×C₁₇×#135	C₂
socle	C₂²×C₁₇	C₄×D₄
cosocle	C₂×C₄×C₁₇	C₂⁴
hypercenter	C₂×#135	D₁₇
hypocenter	C₁₇	C₂×#444
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
1. C₂×#10587634	C₂	semidirect
2. #11762931	C₂	direct (N × H)
3. C₂×#10587490	C₂	semidirect
4. C₂×#10587508	C₂	semidirect
5. C₂×#10587486	C₂	semidirect
8. C₁₇×#135	C₄	semidirect
10. C₂×Q₁₆×C₁₇	C₄	semidirect
11. C₂×SD₁₆×C₁₇	C₄	semidirect
12. C₂×D₈×C₁₇	C₄	semidirect
15. #10587634	C₂²	semidirect
21. #10587508	C₂²	semidirect
22. #10587490	C₂²	semidirect
23. #10587486	C₂²	semidirect
34. Q₁₆×C₁₇	C₂×C₄	semidirect
35. SD₁₆×C₁₇	C₂×C₄	semidirect
36. D₈×C₁₇	C₂×C₄	semidirect
96. C₂×C₁₇	#444	semidirect
99. #135	Dic₁₇	semidirect
101. C₂×Q₁₆	Dic₁₇	semidirect
102. C₂×SD₁₆	Dic₁₇	semidirect
103. C₂×D₈	Dic₁₇	semidirect
115. C₁₇	C₂×#444	semidirect
126. Q₁₆	C₂×Dic₁₇	semidirect
127. SD₁₆	C₂×Dic₁₇	semidirect
128. D₈	C₂×Dic₁₇	semidirect
181. C₂	#11762931	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#10587634	2
#11762931	2
C₂×#10587490	2
C₂×#10587508	2
C₂×#10587486	2
C₂×C₁₇×#135	2
C₂×#444	17

Subgroups — C2×#11762931 #14537680

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11762931 #14537680