Subgroups — C₂×#68717 #10792691

Order 768.

930 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×#5122	C₂	1	3	yes	maximal
2	C₂×C₃×#338	C₂	1	1	yes	maximal, characteristic
3	C₂×#4557	C₂	1	3	yes	maximal
4	#68717	C₂	1	8	yes	maximal, direct product factor
5	C₂×#1747	—	3	1	no	maximal, Sylow
6	C₂²×C₃×M₁₆	C₂²	1	3	yes	—
7	C₂×#926	C₂²	1	3	yes	—
8	C₂³×C₃⋊C₈	C₂²	1	1	yes	characteristic
9	#5122	C₄	1	12	yes	complemented
10	#5122	C₂²	1	12	yes	—
11	C₂×#815	C₄	1	6	yes	complemented
12	C₂×C₈×Dic₃	C₄	1	6	yes	complemented
13	C₃×#338	C₂²	1	4	yes	—
14	#4557	C₂²	1	12	yes	—
15	C₂×#405	—	3	3	no	—
16	#1747	—	3	8	no	—
17	C₂×#338	D₃	1	1	yes	characteristic
18	C₂×H(3,ℤ/4)	—	3	3	no	—
19	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
20	C₂×C₃×M₁₆	—	2	6	no	—
21	C₂×C₃×M₁₆	—	2	6	no	—
22	C₂×C₃×M₁₆	C₂×C₄	1	6	yes	—
23	C₂×C₃×M₁₆	C₂³	1	6	yes	—
24	C₂²×C₃×C₈	C₂×C₄	1	6	yes	—
25	C₂×#217	—	2	3	no	—
26	#926	—	2	3	no	—
27	#926	—	2	3	no	—
28	#926	C₂×C₄	1	6	yes	—
29	#926	C₂³	1	6	yes	—
30	C₂×C₄⋊Dic₃	—	2	3	no	—
31	C₂×C₄×Dic₃	C₂×C₄	1	6	yes	—
32	C₂²×C₃⋊C₈	C₂×C₄	1	6	yes	—
33	C₂²×C₃⋊C₈	C₂×C₄	1	3	yes	—
34	C₂²×C₃⋊C₈	C₂×C₄	1	3	yes	—
35	C₂²×C₃⋊C₈	C₂³	1	2	yes	—
36	#815	C₂×C₄	1	24	yes	complemented
37	C₈×Dic₃	C₂×C₄	1	24	yes	complemented
38	C₂²×M₁₆	C₂×D₃	1	3	yes	—
39	C₂³×C₈	—	3	1	no	—
40	C₂×#117	—	3	3	no	—
41	#405	—	3	12	no	—
42	#405	—	3	12	no	—
43	C₂×#97	—	3	6	no	—
44	C₂×C₄×C₈	—	3	6	no	—
45	#338	C₂×D₃	1	4	yes	—
46	H(3,ℤ/4)	—	3	12	no	—
47	C₂⁴×C₃	#41	1	1	yes	characteristic
48	C₂²×C₃×C₄	C₂²×C₄	1	3	yes	—
49	C₂²×C₃×C₄	C₂²×C₄	1	3	yes	—
50	C₂²×C₃×C₄	#41	1	3	yes	—
51	C₂²×C₃×C₄	#41	1	3	yes	—
52	C₂²×C₃×C₄	#41	1	1	yes	characteristic
53	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
54	C₂²×Dic₃	—	2	6	no	—
55	C₃×M₁₆	—	2	12	no	—
56	C₃×M₁₆	—	2	12	no	—
57	C₂×C₃×C₈	—	2	6	no	—
58	C₂×C₃×C₈	—	2	6	no	—
59	C₂×C₃×C₈	C₂²×C₄	1	12	yes	—
60	#217	—	2	6	no	—
61	#217	—	2	6	no	—
62	C₄⋊Dic₃	—	2	6	no	—
63	C₄⋊Dic₃	—	2	6	no	—
64	C₄×Dic₃	—	2	6	no	—
65	C₄×Dic₃	C₂²×C₄	1	12	yes	—
66	C₂×C₃⋊C₈	C₄²	1	6	yes	—
67	C₂×C₃⋊C₈	C₄²	1	6	yes	—
68	C₂×C₃⋊C₈	C₄²	1	1	yes	characteristic
69	C₂×C₃⋊C₈	C₄²	1	3	yes	—
70	C₂×C₃⋊C₈	C₂²×C₄	1	6	yes	—
71	C₂×C₃⋊C₈	C₂²×C₄	1	6	yes	—
72	C₂³×C₄	C₂²×D₃	1	1	yes	characteristic
73	C₂×M₁₆	—	2	6	no	—
74	C₂×M₁₆	—	2	6	no	—
75	C₂×M₁₆	C₄×D₃	1	6	yes	—
76	C₂×M₁₆	C₂²×D₃	1	6	yes	—
77	C₂²×C₈	—	3	3	no	—
78	C₂²×C₈	—	3	3	no	—
79	C₂²×C₈	—	3	6	no	—
80	C₂²×C₈	—	3	2	no	—
81	C₂²×C₈	C₄×D₃	1	6	yes	—
82	#117	—	3	6	no	—
83	#117	—	3	6	no	—
84	#117	—	6	3	no	—
85	#117	—	6	3	no	—
86	C₂×C₄⋊C₄	—	6	3	no	—
87	C₂×C₂²⋊C₄	—	6	3	no	—
88	C₂×C₄²	—	3	6	no	—
89	#97	—	3	24	no	—
90	C₄×C₈	—	3	24	no	—
91	C₂³×C₃	—	4	2	no	—
92	C₂³×C₃	#117	1	3	yes	—
93	C₂³×C₃	#117	1	3	yes	—
94	C₂³×C₃	C₂×#41	1	1	yes	characteristic
95	C₂×C₃×C₄	C₂×C₄²	1	3	yes	—
96	C₂×C₃×C₄	C₂×C₄²	1	1	yes	characteristic
97	C₂×C₃×C₄	#117	1	3	yes	—
98	C₂×C₃×C₄	#117	1	3	yes	—
99	C₂×C₃×C₄	#117	1	6	yes	—
100	C₂×C₃×C₄	#117	1	3	yes	—
101	C₂×C₃×C₄	#117	1	3	yes	—
102	C₂×C₃×C₄	C₂³×C₄	1	3	yes	—
103	C₂×C₃×C₄	C₂×#41	1	3	yes	—
104	C₂×Dic₃	—	2	12	no	—
105	C₂×Dic₃	—	4	12	no	—
106	C₃×C₈	—	2	12	no	—
107	C₃⋊C₈	C₂×C₄²	1	2	yes	—
108	C₃⋊C₈	C₂×C₄²	1	6	yes	—
109	C₂⁴	#239	1	1	yes	characteristic
110	C₂²×C₄	—	6	6	no	—
111	C₂²×C₄	C₂×C₄×D₃	1	3	yes	—
112	C₂²×C₄	C₂×C₄×D₃	1	3	yes	—
113	C₂²×C₄	#237	1	3	yes	—
114	C₂²×C₄	#237	1	3	yes	—
115	C₂²×C₄	#239	1	1	yes	characteristic
116	C₂²×C₄	C₂³×D₃	1	1	yes	characteristic
117	M₁₆	—	2	12	no	—
118	M₁₆	—	2	12	no	—
119	C₂×C₈	—	2	6	no	—
120	C₂×C₈	—	2	6	no	—
121	C₂×C₈	—	3	6	no	—
122	C₂×C₈	—	3	6	no	—
123	C₂×C₈	—	3	1	no	—
124	C₂×C₈	—	3	3	no	—
125	C₂×C₈	—	3	6	no	—
126	C₂×C₈	—	3	6	no	—
127	C₂×C₈	C₂×C₄×D₃	1	12	yes	—
128	C₄⋊C₄	—	6	6	no	—
129	C₄⋊C₄	—	6	6	no	—
130	C₂²⋊C₄	—	6	6	no	—
131	C₂²⋊C₄	—	6	6	no	—
132	C₄²	—	3	12	no	—
133	C₄²	—	6	6	no	—
134	C₂²×C₃	—	2	6	no	—
135	C₂²×C₃	—	4	3	no	—
136	C₂²×C₃	—	4	1	no	—
137	C₂²×C₃	#376	1	3	yes	—
138	C₂²×C₃	#376	1	1	yes	characteristic
139	C₂²×C₃	C₂×#117	1	3	yes	—
140	C₃×C₄	#376	1	3	yes	—
141	C₃×C₄	#376	1	1	yes	characteristic
142	C₃×C₄	C₂²×C₄²	1	1	yes	characteristic
143	C₃×C₄	C₂×#117	1	3	yes	—
144	Dic₃	—	4	12	no	—
145	C₂³	—	4	2	no	—
146	C₂³	#892	1	3	yes	—
147	C₂³	#892	1	3	yes	—
148	C₂³	C₂×#239	1	1	yes	characteristic
149	C₂×C₄	—	6	12	no	—
150	C₂×C₄	—	12	12	no	—
151	C₂×C₄	C₄²×D₃	1	3	yes	—
152	C₂×C₄	C₄²×D₃	1	1	yes	characteristic
153	C₂×C₄	#877	1	3	yes	—
154	C₂×C₄	#877	1	3	yes	—
155	C₂×C₄	#877	1	6	yes	—
156	C₂×C₄	#892	1	3	yes	—
157	C₂×C₄	#892	1	3	yes	—
158	C₂×C₄	C₂²×C₄×D₃	1	3	yes	—
159	C₂×C₄	C₂×#237	1	3	yes	—
160	C₈	—	2	12	no	—
161	C₈	—	3	2	no	—
162	C₈	—	3	6	no	—
163	C₂×C₃	—	2	3	no	—
164	C₂×C₃	—	2	3	no	—
165	C₂×C₃	#1747	1	2	yes	complemented
166	C₂×C₃	C₂×#376	1	1	yes	characteristic
167	C₂²	—	2	6	no	—
168	C₂²	—	4	3	no	—
169	C₂²	—	4	1	no	—
170	C₂²	#4648	1	3	yes	—
171	C₂²	#4648	1	1	yes	characteristic
172	C₂²	C₂×#892	1	3	yes	—
173	C₄	—	12	12	no	—
174	C₄	#4648	1	3	yes	—
175	C₄	#4648	1	1	yes	characteristic
176	C₄	C₂×C₄²×D₃	1	1	yes	characteristic
177	C₄	C₂×#877	1	3	yes	—
178	C₃	C₂×#1747	1	1	yes	characteristic, complemented, Sylow
179	C₂	—	2	3	no	—
180	C₂	—	2	3	no	—
181	C₂	#68717	1	2	yes	direct product factor
182	C₂	C₂×#4648	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄²×D₃
commutator	C₃×C₄	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₃
Fitting	C₂×C₃×#338	C₂
socle	C₂²×C₃	#376
prime radical	C₂⁴×C₃	#41
cosocle	C₂²×C₃×C₄	C₂⁴
hypercenter	C₂×#338	D₃
hypocenter	C₃	C₂×#1747
LCS	G ▷ C₃×C₄ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#338

Complements

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

N	H ≅ G/N	type
4. #68717	C₂	direct (N × H)
9. #5122	C₄	semidirect
11. C₂×#815	C₄	semidirect
12. C₂×C₈×Dic₃	C₄	semidirect
36. #815	C₂×C₄	semidirect
37. C₈×Dic₃	C₂×C₄	semidirect
165. C₂×C₃	#1747	semidirect
178. C₃	C₂×#1747	semidirect
181. C₂	#68717	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×#5122	2
C₂×C₃×#338	2
C₂×#4557	2
#68717	2
C₂×#1747	3

Subgroups — C2×#68717 #10792691

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#68717 #10792691