Subgroups — C₂²×#11718233 #16807009

Order 3328.

782 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88106	C₂	1	2	yes	maximal
2	C₂×#11718233	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₄×#1055	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄²×C₁₃	C₄	1	1	yes	characteristic
5	C₂³×#1055	C₄	1	1	yes	characteristic
6	C₂³×#1055	C₂²	1	1	yes	characteristic
7	C₂×#88106	—	2	6	no	—
8	C₂×#88106	C₂²	1	12	yes	—
9	C₂²×#6092	—	2	2	no	—
10	C₂×C₄×#1055	C₄	1	6	yes	—
11	C₂×C₄×#1055	C₂²	1	6	yes	—
12	#11718233	C₂²	1	16	yes	direct product factor
13	C₂³×C₄×C₁₃	—	2	1	no	—
14	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²×C₁₃	—	2	3	no	—
16	C₂×C₄²×C₁₃	C₂×C₄	1	6	yes	—
17	C₂²×#1055	C₈	1	6	yes	—
18	C₂²×#1055	C₈	1	2	yes	—
19	C₂²×#1055	C₂×C₄	1	3	yes	—
20	C₂²×#1055	C₂×C₄	1	6	yes	—
21	C₂²×#1055	D₄	1	1	yes	characteristic
22	C₂²×#1055	D₄	1	3	yes	—
23	C₂²×#1055	Q₈	1	3	yes	—
24	C₂²×#1055	Q₈	1	1	yes	characteristic
25	C₂²×#1055	C₂³	1	3	yes	—
26	#88106	—	2	12	no	—
27	#88106	C₂³	1	8	yes	—
28	C₂×#6092	—	2	12	no	—
29	C₄×#1055	C₂×C₄	1	12	yes	—
30	C₄×#1055	C₂³	1	4	yes	—
31	C₂²×#364	—	13	1	no	maximal, Sylow
32	C₂⁴×C₁₃	M₁₆	1	1	yes	characteristic
33	C₂²×C₄×C₁₃	—	2	8	no	—
34	C₂²×C₄×C₁₃	—	2	6	no	—
35	C₂²×C₄×C₁₃	C₄⋊C₄	1	3	yes	—
36	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₃	C₂×C₈	1	3	yes	—
38	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	M₁₆	1	3	yes	—
40	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
41	C₄²×C₁₃	—	2	6	no	—
42	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
43	C₂×#1055	C₄⋊C₄	1	3	yes	—
44	C₂×#1055	C₄⋊C₄	1	6	yes	—
45	C₂×#1055	C₄⋊C₄	1	3	yes	—
46	C₂×#1055	C₂×C₈	1	12	yes	—
47	C₂×#1055	C₂×C₈	1	12	yes	—
48	C₂×#1055	C₂²×C₄	1	3	yes	—
49	C₂×#1055	C₂²×C₄	1	4	yes	—
50	C₂×#1055	C₂×D₄	1	3	yes	—
51	C₂×#1055	C₂×D₄	1	3	yes	—
52	C₂×#1055	C₂×Q₈	1	3	yes	—
53	C₂×#1055	C₂×Q₈	1	3	yes	—
54	C₂×#1055	C₂⁴	1	1	yes	characteristic
55	#6092	—	2	8	no	—
56	C₂²×M₃₂	—	13	2	no	—
57	C₂²×C₄×C₈	—	13	1	no	—
58	C₂×#364	—	13	12	no	—
59	C₂³×C₁₃	—	2	4	no	—
60	C₂³×C₁₃	C₄⋊C₈	1	3	yes	—
61	C₂³×C₁₃	C₄⋊C₈	1	1	yes	characteristic
62	C₂³×C₁₃	C₂×M₁₆	1	3	yes	—
63	C₂×C₄×C₁₃	—	2	24	no	—
64	C₂×C₄×C₁₃	—	2	4	no	—
65	C₂×C₄×C₁₃	C₄⋊C₈	1	6	yes	—
66	C₂×C₄×C₁₃	C₄⋊C₈	1	3	yes	—
67	C₂×C₄×C₁₃	C₄⋊C₈	1	3	yes	—
68	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
69	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
70	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
71	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
72	C₂×C₄×C₁₃	C₂×M₁₆	1	3	yes	—
73	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
74	#1055	C₂×C₄⋊C₄	1	3	yes	—
75	#1055	C₂×C₄⋊C₄	1	3	yes	—
76	#1055	C₂²×C₈	1	8	yes	—
77	#1055	C₂²×D₄	1	1	yes	characteristic
78	#1055	C₂²×Q₈	1	1	yes	characteristic
79	C₂³×C₈	—	13	1	no	—
80	C₂³×C₈	—	13	1	no	—
81	C₂²×C₄²	#261	1	1	yes	characteristic
82	C₂×M₃₂	—	13	12	no	—
83	C₂×M₃₂	—	26	6	no	—
84	C₂²×C₁₆	—	26	2	no	—
85	C₂×C₄×C₈	—	13	6	no	—
86	C₂×C₄×C₈	—	13	6	no	—
87	#364	—	13	16	no	—
88	C₂²×C₁₃	—	2	6	no	—
89	C₂²×C₁₃	—	2	6	no	—
90	C₂²×C₁₃	#364	1	4	yes	complemented
91	C₂²×C₁₃	C₂×C₄⋊C₈	1	3	yes	—
92	C₂²×C₁₃	C₂×C₄⋊C₈	1	3	yes	—
93	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
94	C₄×C₁₃	—	2	8	no	—
95	C₄×C₁₃	C₂×C₄⋊C₈	1	3	yes	—
96	C₄×C₁₃	C₂×C₄⋊C₈	1	3	yes	—
97	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
98	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
99	C₂³×C₄	—	2	1	no	—
100	C₂³×C₄	C₂×#261	1	1	yes	characteristic
101	C₂²×C₈	—	13	2	no	—
102	C₂²×C₈	—	13	6	no	—
103	C₂²×C₈	—	13	3	no	—
104	C₂²×C₈	—	13	3	no	—
105	C₂²×C₈	—	13	1	no	—
106	C₂²×C₈	—	13	1	no	—
107	C₂²×C₈	—	13	6	no	—
108	C₂²×C₈	—	13	3	no	—
109	C₂²×C₈	—	13	3	no	—
110	C₂×C₄²	—	2	3	no	—
111	C₂×C₄²	C₂×#261	1	6	yes	—
112	M₃₂	—	13	8	no	—
113	M₃₂	—	26	12	no	—
114	C₂×C₁₆	—	26	12	no	—
115	C₄×C₈	—	13	4	no	—
116	C₄×C₈	—	13	12	no	—
117	C₂×C₁₃	—	2	1	no	—
118	C₂×C₁₃	—	2	3	no	—
119	C₂×C₁₃	C₂×#364	1	6	yes	complemented
120	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
121	C₂⁴	#6122	1	1	yes	characteristic
122	C₂²×C₄	—	2	8	no	—
123	C₂²×C₄	—	2	6	no	—
124	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
125	C₂²×C₄	D₁₃⋊C₈	1	3	yes	—
126	C₂²×C₄	C₄⋊#261	1	3	yes	—
127	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
128	C₂²×C₄	#6122	1	3	yes	—
129	C₂²×C₄	C₂²×#261	1	3	yes	—
130	C₂×C₈	—	13	12	no	—
131	C₂×C₈	—	13	12	no	—
132	C₂×C₈	—	13	3	no	—
133	C₂×C₈	—	13	3	no	—
134	C₂×C₈	—	13	6	no	—
135	C₂×C₈	—	13	3	no	—
136	C₂×C₈	—	13	3	no	—
137	C₂×C₈	—	13	3	no	—
138	C₂×C₈	—	13	3	no	—
139	C₂×C₈	—	13	1	no	—
140	C₂×C₈	—	13	4	no	—
141	C₂×C₈	—	13	3	no	—
142	C₄²	—	2	6	no	—
143	C₄²	C₂²×#261	1	4	yes	—
144	C₁₆	—	26	8	no	—
145	C₁₃	C₂²×#364	1	1	yes	characteristic, complemented, Sylow
146	C₂³	—	2	4	no	—
147	C₂³	#88112	1	3	yes	—
148	C₂³	#88112	1	1	yes	characteristic
149	C₂³	C₂×#6122	1	3	yes	—
150	C₂×C₄	—	2	24	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	#88112	1	6	yes	—
153	C₂×C₄	#88112	1	3	yes	—
154	C₂×C₄	#88112	1	3	yes	—
155	C₂×C₄	C₂×D₁₃⋊C₈	1	3	yes	—
156	C₂×C₄	C₂×D₁₃⋊C₈	1	3	yes	—
157	C₂×C₄	C₂×C₄⋊#261	1	3	yes	—
158	C₂×C₄	C₂×C₄⋊#261	1	3	yes	—
159	C₂×C₄	C₂×#6122	1	3	yes	—
160	C₂×C₄	C₂³×#261	1	1	yes	characteristic
161	C₈	—	13	8	no	—
162	C₈	—	13	1	no	—
163	C₈	—	13	3	no	—
164	C₈	—	13	3	no	—
165	C₈	—	13	1	no	—
166	C₂²	—	2	6	no	—
167	C₂²	—	2	6	no	—
168	C₂²	#11718233	1	4	yes	direct product factor
169	C₂²	C₂×#88112	1	3	yes	—
170	C₂²	C₂×#88112	1	3	yes	—
171	C₂²	C₂²×#6122	1	1	yes	characteristic
172	C₄	—	2	8	no	—
173	C₄	C₂×#88112	1	3	yes	—
174	C₄	C₂×#88112	1	3	yes	—
175	C₄	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
176	C₄	C₂²×C₄⋊#261	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	3	no	—
179	C₂	C₂×#11718233	1	6	yes	direct product factor
180	C₂	C₂²×#88112	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₄⋊#261
commutator	C₄×C₁₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂²×C₄²×C₁₃	C₄
socle	C₂³×C₁₃	C₄⋊C₈
prime radical	C₂⁴×C₁₃	M₁₆
cosocle	C₂×#1055	C₂⁴
hypercenter	C₂²×C₄²	#261
hypocenter	C₁₃	C₂²×#364
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×#11718233	C₂	direct (N × H)
12. #11718233	C₂²	direct (N × H)
90. C₂²×C₁₃	#364	semidirect
119. C₂×C₁₃	C₂×#364	semidirect
145. C₁₃	C₂²×#364	semidirect
168. C₂²	#11718233	direct (N × H)
179. C₂	C₂×#11718233	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×#88106	2
C₂×#11718233	2
C₂²×C₄×#1055	2
C₂²×#364	13

Subgroups — C22×#11718233 #16807009

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718233 #16807009