Subgroups — C₂×#5197 #83909

Order 384.

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×C₃×#135	C₂	1	1	yes	maximal, characteristic
2	C₂×#948	C₂	1	2	yes	maximal, complemented
3	C₂×#837	C₂	1	2	yes	maximal, complemented
4	#5197	C₂	1	8	yes	maximal, direct product factor
5	C₂×#819	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#815	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#444	—	3	1	no	maximal, Sylow
8	C₂×C₃×#41	C₂²	1	2	yes	—
9	C₃×#135	C₄	1	4	yes	complemented
10	C₃×#135	C₂²	1	4	yes	—
11	C₂×C₃×Q₁₆	C₄	1	1	yes	characteristic, complemented
12	C₂×C₃×SD₁₆	C₄	1	2	yes	complemented
13	C₂×C₃×D₈	C₄	1	1	yes	characteristic, complemented
14	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
15	#948	—	2	4	no	—
16	#948	C₂²	1	8	yes	complemented
17	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
18	C₂×#208	—	2	2	no	—
19	C₂×#208	C₂²	1	2	yes	—
20	C₂²×C₃⋊C₈	—	2	2	no	—
21	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
22	#837	C₂²	1	8	yes	complemented
23	#819	C₂²	1	4	yes	complemented
24	#815	C₂²	1	4	yes	complemented
25	C₂×#135	D₃	1	1	yes	characteristic
26	C₂×#131	—	3	2	no	—
27	#444	—	3	8	no	—
28	C₂×#108	—	3	1	no	—
29	C₂×#104	—	3	2	no	—
30	C₂×#97	—	3	1	no	—
31	C₃×#41	—	2	2	no	—
32	C₃×#41	—	2	2	no	—
33	C₃×#41	C₂×C₄	1	4	yes	—
34	C₃×#41	C₂³	1	4	yes	—
35	C₂×C₃×Q₈	C₂×C₄	1	2	yes	—
36	C₂×C₃×D₄	—	2	2	no	—
37	C₂×C₃×D₄	C₂×C₄	1	2	yes	—
38	C₂²×C₃×C₄	—	2	2	no	—
39	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
40	C₂²×Dic₃	—	2	1	no	—
41	C₃×Q₁₆	C₂×C₄	1	4	yes	complemented
42	C₃×SD₁₆	C₂×C₄	1	8	yes	complemented
43	C₃×D₈	C₂×C₄	1	4	yes	complemented
44	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
47	C₂×C₃×C₈	C₂³	1	2	yes	—
48	C₄×Dic₃	—	2	1	no	—
49	C₄×Dic₃	C₂³	1	2	yes	—
50	#208	—	2	2	no	—
51	#208	—	2	8	no	—
52	#208	C₂³	1	4	yes	—
53	C₂×C₃⋊C₈	—	2	2	no	—
54	C₂×C₃⋊C₈	—	2	8	no	—
55	C₂×C₃⋊C₈	D₄	1	2	yes	—
56	C₂×C₃⋊C₈	D₄	1	2	yes	—
57	C₂×C₃⋊C₈	C₂³	1	2	yes	—
58	C₂×#41	C₂×D₃	1	2	yes	—
59	#135	Dic₃	1	4	yes	complemented
60	#135	C₂×D₃	1	4	yes	—
61	C₂×Q₁₆	Dic₃	1	1	yes	characteristic, complemented
62	C₂×SD₁₆	Dic₃	1	2	yes	complemented
63	C₂×D₈	Dic₃	1	1	yes	characteristic, complemented
64	#131	—	3	8	no	—
65	#131	—	6	4	no	—
66	C₂×M₁₆	—	3	2	no	—
67	C₂×M₁₆	—	6	2	no	—
68	C₂²×C₈	—	3	1	no	—
69	C₂²×C₈	—	6	2	no	—
70	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
71	C₂×C₄²	—	3	1	no	—
72	#108	—	3	4	no	—
73	#104	—	3	8	no	—
74	#97	—	3	4	no	—
75	C₂³×C₃	—	2	2	no	—
76	C₂³×C₃	#41	1	1	yes	characteristic
77	C₃×Q₈	—	2	2	no	—
78	C₃×Q₈	C₂²×C₄	1	4	yes	—
79	C₃×D₄	—	2	4	no	—
80	C₃×D₄	—	2	4	no	—
81	C₃×D₄	—	2	2	no	—
82	C₃×D₄	C₂²×C₄	1	4	yes	—
83	C₂×C₃×C₄	—	2	2	no	—
84	C₂×C₃×C₄	—	2	4	no	—
85	C₂×C₃×C₄	—	2	4	no	—
86	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
89	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
90	C₂×C₃×C₄	#41	1	1	yes	characteristic
91	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
92	C₂×Dic₃	—	2	2	no	—
93	C₂×Dic₃	—	4	2	no	—
94	C₃×C₈	C₂²×C₄	1	2	yes	—
95	C₃×C₈	C₂²×C₄	1	2	yes	—
96	C₃⋊C₈	—	2	4	no	—
97	C₃⋊C₈	C₂×D₄	1	4	yes	—
98	#41	—	2	2	no	—
99	#41	—	2	2	no	—
100	#41	C₂×Dic₃	1	4	yes	—
101	#41	C₂²×D₃	1	4	yes	—
102	C₂×Q₈	C₂×Dic₃	1	2	yes	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	C₂×Dic₃	1	2	yes	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	—	6	1	no	—
107	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
108	Q₁₆	C₂×Dic₃	1	4	yes	complemented
109	SD₁₆	C₂×Dic₃	1	8	yes	complemented
110	D₈	C₂×Dic₃	1	4	yes	complemented
111	M₁₆	—	3	4	no	—
112	M₁₆	—	6	8	no	—
113	M₁₆	—	6	2	no	—
114	C₂×C₈	—	3	2	no	—
115	C₂×C₈	—	3	2	no	—
116	C₂×C₈	—	3	2	no	—
117	C₂×C₈	—	6	8	no	—
118	C₂×C₈	—	6	2	no	—
119	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
120	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
121	C₂×C₈	C₂×Dic₃	1	2	yes	—
122	C₂×C₈	C₂²×D₃	1	2	yes	—
123	C₄²	—	3	2	no	—
124	C₄²	—	6	1	no	—
125	C₂²×C₃	—	2	2	no	—
126	C₂²×C₃	—	2	4	no	—
127	C₂²×C₃	—	4	4	no	—
128	C₂²×C₃	C₄×D₄	1	1	yes	characteristic
129	C₂²×C₃	C₄×D₄	1	1	yes	characteristic
130	C₂²×C₃	C₂×#41	1	1	yes	characteristic
131	C₃×C₄	—	2	4	no	—
132	C₃×C₄	C₄×D₄	1	1	yes	characteristic
133	C₃×C₄	C₄×D₄	1	1	yes	characteristic
134	C₃×C₄	C₂³×C₄	1	1	yes	characteristic
135	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
136	Dic₃	—	4	2	no	—
137	C₂³	—	2	2	no	—
138	C₂³	#237	1	1	yes	characteristic
139	Q₈	—	2	2	no	—
140	Q₈	C₂²×Dic₃	1	4	yes	—
141	D₄	—	2	4	no	—
142	D₄	—	2	4	no	—
143	D₄	—	2	2	no	—
144	D₄	C₂²×Dic₃	1	4	yes	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	6	2	no	—
149	C₂×C₄	—	12	2	no	—
150	C₂×C₄	D₃×D₄	1	1	yes	characteristic
151	C₂×C₄	D₃×D₄	1	1	yes	characteristic
152	C₂×C₄	#237	1	1	yes	characteristic
153	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
154	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
155	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
156	C₈	—	3	4	no	—
157	C₈	—	6	4	no	—
158	C₈	C₂²×Dic₃	1	2	yes	—
159	C₈	C₂²×Dic₃	1	2	yes	—
160	C₂×C₃	—	2	1	no	—
161	C₂×C₃	—	2	1	no	—
162	C₂×C₃	—	4	4	no	—
163	C₂×C₃	#444	1	2	yes	complemented
164	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
165	C₂²	—	2	2	no	—
166	C₂²	—	2	4	no	—
167	C₂²	—	4	4	no	—
168	C₂²	D₄×Dic₃	1	1	yes	characteristic
169	C₂²	D₄×Dic₃	1	1	yes	characteristic
170	C₂²	C₂×#237	1	1	yes	characteristic
171	C₄	—	2	4	no	—
172	C₄	—	12	2	no	—
173	C₄	D₄×Dic₃	1	1	yes	characteristic
174	C₄	D₄×Dic₃	1	1	yes	characteristic
175	C₄	C₂×D₃×D₄	1	1	yes	characteristic
176	C₄	C₂³×Dic₃	1	1	yes	characteristic
177	C₃	C₂×#444	1	1	yes	characteristic, complemented, Sylow
178	C₂	—	2	1	no	—
179	C₂	—	2	1	no	—
180	C₂	—	4	4	no	—
181	C₂	#5197	1	2	yes	direct product factor
182	C₂	C₂×D₄×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₃×D₄
commutator	C₃×C₄	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃
Fitting, prime radical	C₂×C₃×#135	C₂
socle	C₂²×C₃	C₄×D₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₂×#135	D₃
hypocenter	C₃	C₂×#444
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
2. C₂×#948	C₂	semidirect
3. C₂×#837	C₂	semidirect
4. #5197	C₂	direct (N × H)
5. C₂×#819	C₂	semidirect
6. C₂×#815	C₂	semidirect
9. C₃×#135	C₄	semidirect
11. C₂×C₃×Q₁₆	C₄	semidirect
12. C₂×C₃×SD₁₆	C₄	semidirect
13. C₂×C₃×D₈	C₄	semidirect
16. #948	C₂²	semidirect
22. #837	C₂²	semidirect
23. #819	C₂²	semidirect
24. #815	C₂²	semidirect
41. C₃×Q₁₆	C₂×C₄	semidirect
42. C₃×SD₁₆	C₂×C₄	semidirect
43. C₃×D₈	C₂×C₄	semidirect
59. #135	Dic₃	semidirect
61. C₂×Q₁₆	Dic₃	semidirect
62. C₂×SD₁₆	Dic₃	semidirect
63. C₂×D₈	Dic₃	semidirect
108. Q₁₆	C₂×Dic₃	semidirect
109. SD₁₆	C₂×Dic₃	semidirect
110. D₈	C₂×Dic₃	semidirect
163. C₂×C₃	#444	semidirect
177. C₃	C₂×#444	semidirect
181. C₂	#5197	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×C₃×#135	2
C₂×#948	2
C₂×#837	2
#5197	2
C₂×#819	2
C₂×#815	2
C₂×#444	3

Subgroups — C2×#5197 #83909

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#5197 #83909