Subgroups — C₂²×#11719205 #16769689

Order 3328.

7630 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719205	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#88142	C₂	1	4	yes	maximal, complemented
3	C₂²×#88144	C₂	1	4	yes	maximal, complemented
4	C₂²×#88178	C₂	1	4	yes	maximal
5	C₂²×#88180	C₂	1	2	yes	maximal
6	C₂³×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×C₄×C₁₃	C₂²	1	2	yes	—
8	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
9	C₂³×Dic₂₆	C₂²	1	4	yes	—
10	C₂²×C₁₃×C₄⋊C₄	C₂²	1	24	yes	—
11	C₂²×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
12	C₂²×C₂²⋊Dic₁₃	C₂²	1	4	yes	—
13	C₂×#88180	C₂²	1	48	yes	—
14	C₂²×C₄⋊Dic₁₃	C₂²	1	2	yes	—
15	C₂²×C₄⋊Dic₁₃	C₂²	1	2	yes	—
16	C₂²×C₄⋊Dic₁₃	C₂²	1	4	yes	—
17	C₂×#88178	C₂²	1	96	yes	—
18	C₂²×#6101	C₂²	1	8	yes	—
19	C₂²×C₄×Dic₁₃	C₂²	1	4	yes	—
20	#11719205	C₂²	1	256	yes	direct product factor
21	C₂×#88144	C₂²	1	96	yes	complemented
22	C₂×#88142	C₂²	1	96	yes	complemented
23	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
24	C₂³×C₄×C₁₃	—	4	8	no	—
25	C₂³×C₄×C₁₃	Q₈	1	16	yes	complemented
26	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
27	C₂³×C₄×C₁₃	C₂³	1	24	yes	—
28	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
29	C₂³×C₄×C₁₃	C₂³	1	12	yes	—
30	C₂³×Dic₁₃	C₂³	1	4	yes	—
31	C₂³×Dic₁₃	C₂³	1	4	yes	—
32	C₂²×Dic₂₆	—	4	16	no	—
33	C₂²×Dic₂₆	C₂³	1	48	yes	—
34	C₂×C₁₃×C₄⋊C₄	C₂³	1	48	yes	—
35	C₂×C₁₃×C₄⋊C₄	C₂³	1	64	yes	—
36	C₂×C₂²⋊Dic₁₃	C₂³	1	48	yes	—
37	#88180	C₂³	1	128	yes	—
38	C₂×C₄⋊Dic₁₃	C₂³	1	24	yes	—
39	C₂×C₄⋊Dic₁₃	C₂³	1	48	yes	—
40	C₂×C₄⋊Dic₁₃	C₂³	1	24	yes	—
41	#88178	C₂³	1	256	yes	—
42	C₂×#6101	C₂³	1	96	yes	—
43	C₂×C₄×Dic₁₃	C₂³	1	48	yes	—
44	#88144	C₂³	1	256	yes	complemented
45	#88142	C₂³	1	256	yes	complemented
46	C₂²×#544	—	13	1	no	maximal, Sylow
47	C₂⁴×C₁₃	—	2	4	no	—
48	C₂⁴×C₁₃	—	2	4	no	—
49	C₂⁴×C₁₃	C₂×Q₈	1	8	yes	—
50	C₂⁴×C₁₃	C₂⁴	1	6	yes	—
51	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	—	2	16	no	—
53	C₂²×C₄×C₁₃	—	4	48	no	—
54	C₂²×C₄×C₁₃	C₂×Q₈	1	16	yes	—
55	C₂²×C₄×C₁₃	C₂×Q₈	1	96	yes	complemented
56	C₂²×C₄×C₁₃	C₂⁴	1	24	yes	—
57	C₂²×C₄×C₁₃	C₂⁴	1	32	yes	—
58	C₂²×C₄×C₁₃	C₂⁴	1	12	yes	—
59	C₂²×C₄×C₁₃	C₂⁴	1	16	yes	—
60	C₂²×Dic₁₃	—	2	16	no	—
61	C₂²×Dic₁₃	—	2	16	no	—
62	C₂²×Dic₁₃	C₂⁴	1	24	yes	—
63	C₂²×Dic₁₃	C₂⁴	1	24	yes	—
64	C₂×Dic₂₆	—	4	96	no	—
65	C₂×Dic₂₆	C₂⁴	1	64	yes	—
66	C₁₃×C₄⋊C₄	C₂⁴	1	64	yes	—
67	C₂²⋊Dic₁₃	C₂⁴	1	64	yes	—
68	C₄⋊Dic₁₃	C₂⁴	1	32	yes	—
69	C₄⋊Dic₁₃	C₂⁴	1	64	yes	—
70	C₄⋊Dic₁₃	C₂⁴	1	32	yes	—
71	#6101	C₂⁴	1	128	yes	—
72	C₄×Dic₁₃	C₂⁴	1	64	yes	—
73	C₂×#544	—	13	48	no	—
74	C₂²×C₄⋊Q₈	—	13	4	no	—
75	C₂²×#125	—	13	4	no	—
76	C₂²×C₂²⋊Q₈	—	13	4	no	—
77	C₂²×#117	—	13	2	no	—
78	C₂³×C₄⋊C₄	D₁₃	1	1	yes	characteristic
79	C₂³×C₁₃	—	2	32	no	—
80	C₂³×C₁₃	—	2	12	no	—
81	C₂³×C₁₃	—	2	12	no	—
82	C₂³×C₁₃	C₂²×Q₈	1	24	yes	—
83	C₂³×C₁₃	C₂²×Q₈	1	4	yes	—
84	C₂³×C₁₃	H(5,𝔽₂)	1	4	yes	—
85	C₂³×C₁₃	2₋¹⁺⁴	1	4	yes	—
86	C₂³×C₁₃	C₂⁵	1	4	yes	—
87	C₂³×C₁₃	C₂⁵	1	3	yes	—
88	C₂×C₄×C₁₃	—	2	48	no	—
89	C₂×C₄×C₁₃	—	4	32	no	—
90	C₂×C₄×C₁₃	C₂²×Q₈	1	48	yes	—
91	C₂×C₄×C₁₃	C₂²×Q₈	1	64	yes	complemented
92	C₂×C₄×C₁₃	C₂⁵	1	16	yes	—
93	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
94	C₂×Dic₁₃	—	2	48	no	—
95	C₂×Dic₁₃	—	2	48	no	—
96	C₂×Dic₁₃	C₂⁵	1	16	yes	—
97	C₂×Dic₁₃	C₂⁵	1	16	yes	—
98	Dic₂₆	—	4	64	no	—
99	C₂³×Q₈	—	13	4	no	—
100	C₂⁴×C₄	C₂×D₁₃	1	2	yes	—
101	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
102	#544	—	13	256	no	—
103	C₂×C₄⋊Q₈	—	13	96	no	—
104	C₂×#125	—	13	96	no	—
105	C₂×C₂²⋊Q₈	—	13	96	no	—
106	C₂×#117	—	13	48	no	—
107	C₂²×C₄⋊C₄	—	13	8	no	—
108	C₂²×C₄⋊C₄	—	13	2	no	—
109	C₂²×C₄⋊C₄	—	13	2	no	—
110	C₂²×C₄⋊C₄	—	13	4	no	—
111	C₂²×C₄⋊C₄	C₂×D₁₃	1	4	yes	—
112	C₂²×C₄⋊C₄	C₂×D₁₃	1	24	yes	—
113	C₂²×C₂²⋊C₄	—	13	4	no	—
114	C₂²×C₄²	—	13	4	no	—
115	C₂²×C₁₃	—	2	4	no	—
116	C₂²×C₁₃	—	2	4	no	—
117	C₂²×C₁₃	—	2	48	no	—
118	C₂²×C₁₃	#544	1	16	yes	complemented
119	C₂²×C₁₃	C₂³×Q₈	1	8	yes	—
120	C₂²×C₁₃	C₂³×Q₈	1	6	yes	—
121	C₂²×C₁₃	C₂×H(5,𝔽₂)	1	6	yes	—
122	C₂²×C₁₃	C₂×2₋¹⁺⁴	1	6	yes	—
123	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
124	C₄×C₁₃	—	2	16	no	—
125	C₄×C₁₃	C₂³×Q₈	1	16	yes	—
126	Dic₁₃	—	2	16	no	—
127	Dic₁₃	—	2	16	no	—
128	C₂⁵	C₂²×D₁₃	1	1	yes	characteristic
129	C₂²×Q₈	—	13	48	no	—
130	C₂²×Q₈	—	52	16	no	—
131	C₂³×C₄	—	4	8	no	—
132	C₂³×C₄	—	13	4	no	—
133	C₂³×C₄	—	13	4	no	—
134	C₂³×C₄	Dic₂₆	1	16	yes	complemented
135	C₂³×C₄	C₂²×D₁₃	1	4	yes	—
136	C₂³×C₄	C₂²×D₁₃	1	24	yes	—
137	C₂³×C₄	C₂²×D₁₃	1	2	yes	—
138	C₂³×C₄	C₂²×D₁₃	1	12	yes	—
139	C₄⋊Q₈	—	13	256	no	—
140	#125	—	13	256	no	—
141	C₂²⋊Q₈	—	13	256	no	—
142	#117	—	13	128	no	—
143	C₂×C₄⋊C₄	—	13	24	no	—
144	C₂×C₄⋊C₄	—	13	24	no	—
145	C₂×C₄⋊C₄	—	13	48	no	—
146	C₂×C₄⋊C₄	—	13	96	no	—
147	C₂×C₄⋊C₄	C₂²×D₁₃	1	48	yes	—
148	C₂×C₄⋊C₄	C₂²×D₁₃	1	64	yes	—
149	C₂×C₂²⋊C₄	—	13	48	no	—
150	C₂×C₄²	—	13	48	no	—
151	C₂×C₁₃	—	2	8	no	—
152	C₂×C₁₃	C₂×#544	1	12	yes	complemented
153	C₂×C₁₃	C₂⁴×Q₈	1	1	yes	characteristic
154	C₂×C₁₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
155	C₂×C₁₃	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
156	C₂⁴	—	2	4	no	—
157	C₂⁴	—	2	4	no	—
158	C₂⁴	C₂×Dic₂₆	1	8	yes	—
159	C₂⁴	C₂³×D₁₃	1	6	yes	—
160	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
161	C₂×Q₈	—	13	64	no	—
162	C₂×Q₈	—	52	96	no	—
163	C₂²×C₄	—	2	16	no	—
164	C₂²×C₄	—	4	48	no	—
165	C₂²×C₄	—	13	24	no	—
166	C₂²×C₄	—	13	24	no	—
167	C₂²×C₄	—	26	16	no	—
168	C₂²×C₄	—	26	16	no	—
169	C₂²×C₄	C₂×Dic₂₆	1	16	yes	—
170	C₂²×C₄	C₂×Dic₂₆	1	96	yes	complemented
171	C₂²×C₄	C₂³×D₁₃	1	24	yes	—
172	C₂²×C₄	C₂³×D₁₃	1	32	yes	—
173	C₂²×C₄	C₂³×D₁₃	1	12	yes	—
174	C₂²×C₄	C₂³×D₁₃	1	16	yes	—
175	C₄⋊C₄	—	13	32	no	—
176	C₄⋊C₄	—	13	32	no	—
177	C₄⋊C₄	—	13	64	no	—
178	C₄⋊C₄	—	13	128	no	—
179	C₄⋊C₄	C₂³×D₁₃	1	64	yes	—
180	C₂²⋊C₄	—	13	64	no	—
181	C₄²	—	13	64	no	—
182	C₁₃	C₂²×#544	1	1	yes	characteristic, complemented, Sylow
183	C₂³	—	2	32	no	—
184	C₂³	—	2	12	no	—
185	C₂³	—	2	12	no	—
186	C₂³	C₂²×Dic₂₆	1	24	yes	—
187	C₂³	C₂²×Dic₂₆	1	4	yes	—
188	C₂³	#88251	1	4	yes	—
189	C₂³	#88254	1	4	yes	—
190	C₂³	C₂⁴×D₁₃	1	4	yes	—
191	C₂³	C₂⁴×D₁₃	1	3	yes	—
192	Q₈	—	52	64	no	—
193	C₂×C₄	—	2	48	no	—
194	C₂×C₄	—	4	32	no	—
195	C₂×C₄	—	13	16	no	—
196	C₂×C₄	—	13	16	no	—
197	C₂×C₄	—	26	48	no	—
198	C₂×C₄	—	26	48	no	—
199	C₂×C₄	C₂²×Dic₂₆	1	48	yes	—
200	C₂×C₄	C₂²×Dic₂₆	1	64	yes	complemented
201	C₂×C₄	C₂⁴×D₁₃	1	16	yes	—
202	C₂×C₄	C₂⁴×D₁₃	1	8	yes	—
203	C₂²	—	2	48	no	—
204	C₂²	—	2	4	no	—
205	C₂²	—	2	4	no	—
206	C₂²	#11719205	1	16	yes	direct product factor
207	C₂²	C₂³×Dic₂₆	1	8	yes	—
208	C₂²	C₂³×Dic₂₆	1	6	yes	—
209	C₂²	C₂×#88251	1	6	yes	—
210	C₂²	C₂×#88254	1	6	yes	—
211	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
212	C₄	—	2	16	no	—
213	C₄	—	26	16	no	—
214	C₄	—	26	16	no	—
215	C₄	C₂³×Dic₂₆	1	16	yes	—
216	C₂	—	2	8	no	—
217	C₂	C₂×#11719205	1	12	yes	direct product factor
218	C₂	C₂⁴×Dic₂₆	1	1	yes	characteristic
219	C₂	C₂²×#88251	1	1	yes	characteristic
220	C₂	C₂²×#88254	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁶
Frattini	C₂²	C₂⁵×D₁₃
Fitting	C₂³×C₁₃×C₄⋊C₄	C₂
socle	C₂⁴×C₁₃	C₂⁴
prime radical	C₂⁵×C₁₃	C₂³
hypercenter	C₂³×C₄⋊C₄	D₁₃
hypocenter	C₁₃	C₂²×#544
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#11719205	C₂	direct (N × H)
2. C₂²×#88142	C₂	semidirect
3. C₂²×#88144	C₂	semidirect
20. #11719205	C₂²	direct (N × H)
21. C₂×#88144	C₂²	semidirect
22. C₂×#88142	C₂²	semidirect
25. C₂³×C₄×C₁₃	Q₈	semidirect
44. #88144	C₂³	semidirect
45. #88142	C₂³	semidirect
55. C₂²×C₄×C₁₃	C₂×Q₈	semidirect
91. C₂×C₄×C₁₃	C₂²×Q₈	semidirect
118. C₂²×C₁₃	#544	semidirect
134. C₂³×C₄	Dic₂₆	semidirect
152. C₂×C₁₃	C₂×#544	semidirect
170. C₂²×C₄	C₂×Dic₂₆	semidirect
182. C₁₃	C₂²×#544	semidirect
200. C₂×C₄	C₂²×Dic₂₆	semidirect
206. C₂²	#11719205	direct (N × H)
217. C₂	C₂×#11719205	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×#11719205	2
C₂²×#88142	2
C₂²×#88144	2
C₂²×#88178	2
C₂²×#88180	2
C₂³×C₁₃×C₄⋊C₄	2
C₂²×#544	13

Subgroups — C22×#11719205 #16769689

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11719205 #16769689