Subgroups — C₂²×#11718086 #16015927

Order 3328.

702 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88052	C₂	1	1	yes	maximal, characteristic
2	C₂²×#88057	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718086	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₁₃×#108	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₃×M₁₆	—	2	1	no	—
6	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂²×C₄⋊Dic₁₃	—	2	1	no	—
8	C₂×C₁₃×#108	—	2	3	no	—
9	C₂×C₁₃×#108	C₂²	1	6	yes	—
10	C₂×#88057	—	2	3	no	—
11	C₂×#88057	C₂²	1	6	yes	—
12	C₂×#88052	C₄	1	6	yes	complemented
13	C₂×#88052	C₂²	1	6	yes	—
14	C₂²×#6090	C₄	1	2	yes	complemented
15	#11718086	C₂²	1	16	yes	direct product factor
16	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
17	C₂³×Dic₁₃	—	4	1	no	—
18	C₂×C₁₃×M₁₆	—	2	6	no	—
19	C₂×C₁₃×M₁₆	—	4	3	no	—
20	C₂²×C₈×C₁₃	—	4	1	no	—
21	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	C₂×C₄	1	3	yes	—
23	C₂²×C₈×C₁₃	D₄	1	3	yes	—
24	C₂²×C₈×C₁₃	Q₈	1	3	yes	—
25	C₂²×C₈×C₁₃	Q₈	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
27	C₂×C₄⋊Dic₁₃	—	2	6	no	—
28	C₂×C₄⋊Dic₁₃	—	4	3	no	—
29	C₁₃×#108	—	2	6	no	—
30	C₁₃×#108	C₂³	1	4	yes	—
31	#88057	—	2	6	no	—
32	#88057	C₂³	1	4	yes	—
33	#88052	C₂×C₄	1	12	yes	complemented
34	#88052	C₂³	1	4	yes	—
35	C₂×#6090	C₂×C₄	1	12	yes	complemented
36	C₂²×#365	—	13	1	no	maximal, Sylow
37	C₂⁴×C₁₃	SD₁₆	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	C₄⋊C₄	1	3	yes	—
39	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	SD₁₆	1	3	yes	—
42	C₂²×C₄×C₁₃	SD₁₆	1	3	yes	—
43	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
44	C₂²×Dic₁₃	—	4	6	no	—
45	C₂²×Dic₁₃	—	8	4	no	—
46	C₁₃×M₁₆	—	2	4	no	—
47	C₁₃×M₁₆	—	4	6	no	—
48	C₂×C₈×C₁₃	—	4	6	no	—
49	C₂×C₈×C₁₃	C₄⋊C₄	1	3	yes	—
50	C₂×C₈×C₁₃	C₄⋊C₄	1	6	yes	—
51	C₂×C₈×C₁₃	C₄⋊C₄	1	3	yes	—
52	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
53	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
54	C₂×C₈×C₁₃	C₂×D₄	1	3	yes	—
55	C₂×C₈×C₁₃	C₂×Q₈	1	3	yes	—
56	C₂×C₈×C₁₃	C₂×Q₈	1	3	yes	—
57	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
58	C₄⋊Dic₁₃	—	2	4	no	—
59	C₄⋊Dic₁₃	—	4	6	no	—
60	#6090	C₂²×C₄	1	8	yes	complemented
61	C₂²×M₃₂	—	13	1	no	—
62	C₂²×#108	D₁₃	1	1	yes	characteristic
63	C₂²×#106	—	13	1	no	—
64	C₂×#365	—	13	12	no	—
65	C₂³×C₁₃	—	2	4	no	—
66	C₂³×C₁₃	#106	1	3	yes	—
67	C₂³×C₁₃	#106	1	1	yes	characteristic
68	C₂³×C₁₃	C₂×SD₁₆	1	3	yes	—
69	C₂×C₄×C₁₃	#106	1	6	yes	—
70	C₂×C₄×C₁₃	#106	1	3	yes	—
71	C₂×C₄×C₁₃	#106	1	3	yes	—
72	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
73	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
74	C₂×C₄×C₁₃	C₂×SD₁₆	1	3	yes	—
75	C₂×C₄×C₁₃	C₂×SD₁₆	1	3	yes	—
76	C₂×C₄×C₁₃	C₂×SD₁₆	1	3	yes	—
77	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
78	C₂×Dic₁₃	—	4	4	no	—
79	C₂×Dic₁₃	—	8	12	no	—
80	C₈×C₁₃	—	4	4	no	—
81	C₈×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
82	C₈×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
83	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
84	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
85	C₂²×M₁₆	—	2	1	no	—
86	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
87	C₂²×C₄⋊C₄	—	26	1	no	—
88	C₂×M₃₂	—	13	6	no	—
89	C₂×M₃₂	—	13	6	no	—
90	C₂²×C₁₆	—	13	2	no	—
91	C₂×#108	—	2	3	no	—
92	C₂×#108	C₂×D₁₃	1	6	yes	—
93	C₂×#106	—	13	6	no	—
94	C₂×#106	—	26	3	no	—
95	#365	—	13	16	no	—
96	C₂²×C₁₃	—	2	6	no	—
97	C₂²×C₁₃	—	2	6	no	—
98	C₂²×C₁₃	#365	1	4	yes	complemented
99	C₂²×C₁₃	C₂×#106	1	3	yes	—
100	C₂²×C₁₃	C₂×#106	1	3	yes	—
101	C₂²×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
102	C₄×C₁₃	C₂×#106	1	3	yes	—
103	C₄×C₁₃	C₂×#106	1	3	yes	—
104	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
105	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
106	Dic₁₃	—	8	4	no	—
107	C₂³×C₄	—	52	1	no	—
108	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
109	C₂×M₁₆	—	2	6	no	—
110	C₂×M₁₆	—	4	3	no	—
111	C₂²×C₈	—	4	1	no	—
112	C₂²×C₈	Dic₂₆	1	3	yes	—
113	C₂²×C₈	Dic₂₆	1	1	yes	characteristic
114	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
115	C₂²×C₈	C₄×D₁₃	1	3	yes	—
116	C₂²×C₈	C₂²⋊D₁₃	1	3	yes	—
117	C₂²×C₈	C₂²×D₁₃	1	3	yes	—
118	C₂×C₄⋊C₄	—	26	6	no	—
119	C₂×C₄⋊C₄	—	52	3	no	—
120	M₃₂	—	13	12	no	—
121	M₃₂	—	13	4	no	—
122	C₂×C₁₆	—	13	12	no	—
123	#108	—	2	6	no	—
124	#108	C₂²×D₁₃	1	4	yes	—
125	#106	—	13	4	no	—
126	#106	—	26	6	no	—
127	C₂×C₁₃	—	2	1	no	—
128	C₂×C₁₃	—	2	3	no	—
129	C₂×C₁₃	C₂×#365	1	6	yes	complemented
130	C₂×C₁₃	C₂²×#106	1	1	yes	characteristic
131	C₂⁴	#6106	1	1	yes	characteristic
132	C₂²×C₄	—	52	6	no	—
133	C₂²×C₄	—	104	4	no	—
134	C₂²×C₄	#6101	1	3	yes	—
135	C₂²×C₄	#6101	1	1	yes	characteristic
136	C₂²×C₄	#6105	1	3	yes	—
137	C₂²×C₄	#6105	1	1	yes	characteristic
138	C₂²×C₄	#6106	1	3	yes	—
139	C₂²×C₄	C₂×C₂²⋊D₁₃	1	3	yes	—
140	M₁₆	—	2	4	no	—
141	M₁₆	—	4	6	no	—
142	C₂×C₈	—	4	6	no	—
143	C₂×C₈	#6101	1	3	yes	—
144	C₂×C₈	#6101	1	6	yes	—
145	C₂×C₈	#6101	1	3	yes	—
146	C₂×C₈	C₂×Dic₂₆	1	3	yes	—
147	C₂×C₈	C₂×Dic₂₆	1	3	yes	—
148	C₂×C₈	C₂×C₄×D₁₃	1	3	yes	—
149	C₂×C₈	C₂×C₄×D₁₃	1	3	yes	—
150	C₂×C₈	C₂×C₂²⋊D₁₃	1	3	yes	—
151	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
152	C₄⋊C₄	—	26	4	no	—
153	C₄⋊C₄	—	52	6	no	—
154	C₁₆	—	13	8	no	—
155	C₁₃	C₂²×#365	1	1	yes	characteristic, complemented, Sylow
156	C₂³	—	2	4	no	—
157	C₂³	#88048	1	3	yes	—
158	C₂³	#88048	1	1	yes	characteristic
159	C₂³	C₂×#6106	1	3	yes	—
160	C₂×C₄	—	52	4	no	—
161	C₂×C₄	—	104	12	no	—
162	C₂×C₄	#88048	1	6	yes	—
163	C₂×C₄	#88048	1	3	yes	—
164	C₂×C₄	#88048	1	3	yes	—
165	C₂×C₄	C₂×#6101	1	3	yes	—
166	C₂×C₄	C₂×#6101	1	3	yes	—
167	C₂×C₄	C₂×#6105	1	3	yes	—
168	C₂×C₄	C₂×#6105	1	3	yes	—
169	C₂×C₄	C₂×#6106	1	3	yes	—
170	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
171	C₈	—	4	4	no	—
172	C₈	C₂×#6101	1	3	yes	—
173	C₈	C₂×#6101	1	3	yes	—
174	C₈	C₂²×Dic₂₆	1	1	yes	characteristic
175	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
176	C₂²	—	2	6	no	—
177	C₂²	—	2	6	no	—
178	C₂²	#11718086	1	4	yes	direct product factor
179	C₂²	C₂×#88048	1	3	yes	—
180	C₂²	C₂×#88048	1	3	yes	—
181	C₂²	C₂²×#6106	1	1	yes	characteristic
182	C₄	—	104	4	no	—
183	C₄	C₂×#88048	1	3	yes	—
184	C₄	C₂×#88048	1	3	yes	—
185	C₄	C₂²×#6101	1	1	yes	characteristic
186	C₄	C₂²×#6105	1	1	yes	characteristic
187	C₂	—	2	1	no	—
188	C₂	—	2	3	no	—
189	C₂	C₂×#11718086	1	6	yes	direct product factor
190	C₂	C₂²×#88048	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88048
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂²×C₁₃×#108	C₂
socle	C₂³×C₁₃	#106
prime radical	C₂⁴×C₁₃	SD₁₆
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂²×#108	D₁₃
hypocenter	C₁₃	C₂²×#365
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
3. C₂×#11718086	C₂	direct (N × H)
12. C₂×#88052	C₄	semidirect
14. C₂²×#6090	C₄	semidirect
15. #11718086	C₂²	direct (N × H)
33. #88052	C₂×C₄	semidirect
35. C₂×#6090	C₂×C₄	semidirect
60. #6090	C₂²×C₄	semidirect
98. C₂²×C₁₃	#365	semidirect
129. C₂×C₁₃	C₂×#365	semidirect
155. C₁₃	C₂²×#365	semidirect
178. C₂²	#11718086	direct (N × H)
189. C₂	C₂×#11718086	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×#88052	2
C₂²×#88057	2
C₂×#11718086	2
C₂²×C₁₃×#108	2
C₂²×#365	13

Subgroups — C22×#11718086 #16015927

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718086 #16015927