Subgroups — C₂²×#11718103 #16015914

Order 3328.

782 conjugacy classes of subgroups.

The subgroup lattice has 198 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×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718103	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
6	C₂²×C₄×Dic₁₃	C₄	1	1	yes	characteristic
7	C₂³×#1055	C₄	1	1	yes	characteristic
8	C₂×C₁₃×M₃₂	—	2	3	no	—
9	C₂×C₁₃×M₃₂	C₂²	1	6	yes	—
10	C₂²×C₁₃×C₁₆	—	2	1	no	—
11	C₂×C₈×Dic₁₃	C₄	1	6	yes	—
12	C₂×C₈×Dic₁₃	C₂²	1	6	yes	—
13	C₂×#88052	—	2	3	no	—
14	C₂×#88052	C₂²	1	6	yes	—
15	C₂²×#6090	—	2	1	no	—
16	#11718103	C₂²	1	16	yes	direct product factor
17	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂³×Dic₁₃	—	2	1	no	—
19	C₂²×C₈×C₁₃	C₂×C₄	1	3	yes	—
20	C₂²×C₈×C₁₃	D₄	1	3	yes	—
21	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	Q₈	1	1	yes	characteristic
23	C₂²×C₈×C₁₃	Q₈	1	3	yes	—
24	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
25	C₂×C₄×Dic₁₃	—	2	3	no	—
26	C₂×C₄×Dic₁₃	C₂×C₄	1	6	yes	—
27	C₂²×#1055	C₈	1	6	yes	—
28	C₂²×#1055	C₈	1	2	yes	—
29	C₂²×#1055	C₂×C₄	1	6	yes	—
30	C₁₃×M₃₂	—	2	6	no	—
31	C₁₃×M₃₂	C₂³	1	4	yes	—
32	C₂×C₁₃×C₁₆	—	2	6	no	—
33	C₈×Dic₁₃	C₂×C₄	1	12	yes	—
34	C₈×Dic₁₃	C₂³	1	4	yes	—
35	#88052	—	2	6	no	—
36	#88052	C₂³	1	4	yes	—
37	C₂×#6090	—	2	6	no	—
38	C₂²×#364	—	13	1	no	maximal, Sylow
39	C₂⁴×C₁₃	M₁₆	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₄⋊C₄	1	3	yes	—
41	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂×C₈	1	3	yes	—
43	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	M₁₆	1	3	yes	—
45	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
46	C₂²×Dic₁₃	—	2	8	no	—
47	C₂²×Dic₁₃	—	2	6	no	—
48	C₂×C₈×C₁₃	C₄⋊C₄	1	6	yes	—
49	C₂×C₈×C₁₃	C₄⋊C₄	1	3	yes	—
50	C₂×C₈×C₁₃	C₄⋊C₄	1	3	yes	—
51	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
52	C₂×C₈×C₁₃	C₂×D₄	1	3	yes	—
53	C₂×C₈×C₁₃	C₂×D₄	1	3	yes	—
54	C₂×C₈×C₁₃	C₂×Q₈	1	3	yes	—
55	C₂×C₈×C₁₃	C₂×Q₈	1	3	yes	—
56	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
57	C₄×Dic₁₃	—	2	6	no	—
58	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
59	C₂×#1055	C₂×C₈	1	12	yes	—
60	C₂×#1055	C₂×C₈	1	12	yes	—
61	C₂×#1055	C₂²×C₄	1	4	yes	—
62	C₁₃×C₁₆	—	2	4	no	—
63	#6090	—	2	4	no	—
64	C₂²×M₃₂	—	13	1	no	—
65	C₂²×M₃₂	D₁₃	1	1	yes	characteristic
66	C₂²×C₄×C₈	—	13	1	no	—
67	C₂×#364	—	13	12	no	—
68	C₂³×C₁₃	—	2	4	no	—
69	C₂³×C₁₃	C₄⋊C₈	1	3	yes	—
70	C₂³×C₁₃	C₄⋊C₈	1	1	yes	characteristic
71	C₂³×C₁₃	C₂×M₁₆	1	3	yes	—
72	C₂×C₄×C₁₃	C₄⋊C₈	1	6	yes	—
73	C₂×C₄×C₁₃	C₄⋊C₈	1	3	yes	—
74	C₂×C₄×C₁₃	C₄⋊C₈	1	3	yes	—
75	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
76	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
77	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
78	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
79	C₂×C₄×C₁₃	C₂×M₁₆	1	3	yes	—
80	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
81	C₂×Dic₁₃	—	2	24	no	—
82	C₂×Dic₁₃	—	2	4	no	—
83	C₈×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
84	C₈×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
85	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
86	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
87	#1055	C₂²×C₈	1	8	yes	—
88	C₂³×C₈	—	13	1	no	—
89	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
90	C₂²×C₄²	—	13	1	no	—
91	C₂×M₃₂	—	2	3	no	—
92	C₂×M₃₂	—	13	6	no	—
93	C₂×M₃₂	—	26	3	no	—
94	C₂×M₃₂	C₂×D₁₃	1	6	yes	—
95	C₂²×C₁₆	—	2	1	no	—
96	C₂²×C₁₆	—	26	1	no	—
97	C₂×C₄×C₈	—	13	6	no	—
98	C₂×C₄×C₈	—	13	6	no	—
99	#364	—	13	16	no	—
100	C₂²×C₁₃	—	2	6	no	—
101	C₂²×C₁₃	—	2	6	no	—
102	C₂²×C₁₃	#364	1	4	yes	complemented
103	C₂²×C₁₃	C₂×C₄⋊C₈	1	3	yes	—
104	C₂²×C₁₃	C₂×C₄⋊C₈	1	3	yes	—
105	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
106	C₄×C₁₃	C₂×C₄⋊C₈	1	3	yes	—
107	C₄×C₁₃	C₂×C₄⋊C₈	1	3	yes	—
108	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
109	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
110	Dic₁₃	—	2	8	no	—
111	C₂³×C₄	—	26	1	no	—
112	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
113	C₂²×C₈	—	13	6	no	—
114	C₂²×C₈	—	13	2	no	—
115	C₂²×C₈	—	13	6	no	—
116	C₂²×C₈	Dic₂₆	1	1	yes	characteristic
117	C₂²×C₈	Dic₂₆	1	3	yes	—
118	C₂²×C₈	C₄×D₁₃	1	3	yes	—
119	C₂²×C₈	C₂²⋊D₁₃	1	3	yes	—
120	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
121	C₂²×C₈	C₂²×D₁₃	1	3	yes	—
122	C₂×C₄²	—	13	6	no	—
123	C₂×C₄²	—	26	3	no	—
124	M₃₂	—	2	6	no	—
125	M₃₂	—	13	4	no	—
126	M₃₂	—	26	6	no	—
127	M₃₂	C₂²×D₁₃	1	4	yes	—
128	C₂×C₁₆	—	2	6	no	—
129	C₂×C₁₆	—	26	6	no	—
130	C₄×C₈	—	13	12	no	—
131	C₄×C₈	—	13	4	no	—
132	C₂×C₁₃	—	2	1	no	—
133	C₂×C₁₃	—	2	3	no	—
134	C₂×C₁₃	C₂×#364	1	6	yes	complemented
135	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
136	C₂⁴	#6094	1	1	yes	characteristic
137	C₂²×C₄	—	26	8	no	—
138	C₂²×C₄	—	26	6	no	—
139	C₂²×C₄	C₈×D₁₃	1	3	yes	—
140	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
141	C₂²×C₄	#6094	1	3	yes	—
142	C₂²×C₄	#6101	1	3	yes	—
143	C₂²×C₄	#6101	1	1	yes	characteristic
144	C₂²×C₄	C₂×C₄×D₁₃	1	3	yes	—
145	C₂×C₈	—	13	12	no	—
146	C₂×C₈	—	13	12	no	—
147	C₂×C₈	—	13	4	no	—
148	C₂×C₈	#6101	1	3	yes	—
149	C₂×C₈	#6101	1	6	yes	—
150	C₂×C₈	#6101	1	3	yes	—
151	C₂×C₈	C₂×Dic₂₆	1	3	yes	—
152	C₂×C₈	C₂×Dic₂₆	1	3	yes	—
153	C₂×C₈	C₂×C₄×D₁₃	1	3	yes	—
154	C₂×C₈	C₂×C₂²⋊D₁₃	1	3	yes	—
155	C₂×C₈	C₂×C₂²⋊D₁₃	1	3	yes	—
156	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
157	C₄²	—	13	4	no	—
158	C₄²	—	26	6	no	—
159	C₁₆	—	2	4	no	—
160	C₁₆	—	26	4	no	—
161	C₁₃	C₂²×#364	1	1	yes	characteristic, complemented, Sylow
162	C₂³	—	2	4	no	—
163	C₂³	#88054	1	3	yes	—
164	C₂³	#88054	1	1	yes	characteristic
165	C₂³	C₂×#6094	1	3	yes	—
166	C₂×C₄	—	26	24	no	—
167	C₂×C₄	—	26	4	no	—
168	C₂×C₄	#88054	1	6	yes	—
169	C₂×C₄	#88054	1	3	yes	—
170	C₂×C₄	#88054	1	3	yes	—
171	C₂×C₄	C₂×C₈×D₁₃	1	3	yes	—
172	C₂×C₄	C₂×C₈×D₁₃	1	3	yes	—
173	C₂×C₄	C₂×#6094	1	3	yes	—
174	C₂×C₄	C₂×#6101	1	3	yes	—
175	C₂×C₄	C₂×#6101	1	3	yes	—
176	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
177	C₈	—	13	8	no	—
178	C₈	C₂×#6101	1	3	yes	—
179	C₈	C₂×#6101	1	3	yes	—
180	C₈	C₂²×Dic₂₆	1	1	yes	characteristic
181	C₈	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
182	C₂²	—	2	6	no	—
183	C₂²	—	2	6	no	—
184	C₂²	#11718103	1	4	yes	direct product factor
185	C₂²	C₂×#88054	1	3	yes	—
186	C₂²	C₂×#88054	1	3	yes	—
187	C₂²	C₂²×#6094	1	1	yes	characteristic
188	C₄	—	26	8	no	—
189	C₄	C₂×#88054	1	3	yes	—
190	C₄	C₂×#88054	1	3	yes	—
191	C₄	C₂²×C₈×D₁₃	1	1	yes	characteristic
192	C₄	C₂²×#6101	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	3	no	—
195	C₂	C₂×#11718103	1	6	yes	direct product factor
196	C₂	C₂²×#88054	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²⋊D₁₃
commutator	C₄×C₁₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂²×C₁₃×M₃₂	C₂
socle	C₂³×C₁₃	C₄⋊C₈
prime radical	C₂⁴×C₁₃	M₁₆
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂²×M₃₂	D₁₃
hypocenter	C₁₃	C₂²×#364
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. C₂×#11718103	C₂	direct (N × H)
16. #11718103	C₂²	direct (N × H)
102. C₂²×C₁₃	#364	semidirect
134. C₂×C₁₃	C₂×#364	semidirect
161. C₁₃	C₂²×#364	semidirect
184. C₂²	#11718103	direct (N × H)
195. C₂	C₂×#11718103	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×C₈×Dic₁₃	2
C₂×#11718103	2
C₂²×C₁₃×M₃₂	2
C₂²×#364	13

Subgroups — C22×#11718103 #16015914

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718103 #16015914