Subgroups — C₂²×#11718726 #16732530

Order 3328.

2142 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88123	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718726	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#88047	C₂	1	2	yes	maximal
4	C₂²×#88048	C₂	1	2	yes	maximal
5	C₂²×#88043	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×#125	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₂₆	—	2	1	no	—
8	C₂×C₁₃×#125	—	2	6	no	—
9	C₂×C₁₃×#125	C₂²	1	12	yes	—
10	C₂²×C₁₃×C₄⋊C₄	—	2	1	no	—
11	C₂²×C₁₃×C₄⋊C₄	—	2	1	no	—
12	C₂²×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂²×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄⋊Dic₁₃	—	2	1	no	—
16	C₂²×C₄⋊Dic₁₃	C₂²	1	2	yes	—
17	C₂³×#1055	C₂²	1	2	yes	—
18	C₂×#88123	—	2	6	no	—
19	C₂×#88123	C₂²	1	12	yes	—
20	#11718726	C₂²	1	64	yes	direct product factor
21	C₂×#88048	C₂²	1	24	yes	—
22	C₂×#88047	C₂²	1	24	yes	—
23	C₂×#88043	C₂²	1	12	yes	—
24	C₂³×C₄×C₁₃	—	2	1	no	—
25	C₂³×C₄×C₁₃	—	2	1	no	—
26	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
29	C₂³×Dic₁₃	—	2	2	no	—
30	C₂²×Dic₂₆	—	2	12	no	—
31	C₂²×Dic₂₆	—	4	8	no	—
32	C₁₃×#125	—	2	24	no	—
33	C₁₃×#125	C₂³	1	16	yes	—
34	C₂×C₁₃×C₄⋊C₄	—	2	12	no	—
35	C₂×C₁₃×C₄⋊C₄	—	2	12	no	—
36	C₂×C₁₃×C₄⋊C₄	—	2	3	no	—
37	C₂×C₁₃×C₄⋊C₄	—	2	3	no	—
38	C₂×C₁₃×C₄⋊C₄	C₂³	1	6	yes	—
39	C₂×C₁₃×C₄⋊C₄	C₂³	1	6	yes	—
40	C₂×C₄²×C₁₃	D₄	1	6	yes	—
41	C₂×C₄²×C₁₃	C₂³	1	6	yes	—
42	C₂×C₄⋊Dic₁₃	—	2	6	no	—
43	C₂×C₄⋊Dic₁₃	—	2	12	no	—
44	C₂×C₄⋊Dic₁₃	C₂³	1	12	yes	—
45	C₂²×#1055	Q₈	1	16	yes	complemented
46	C₂²×#1055	C₂³	1	12	yes	—
47	#88123	—	2	24	no	—
48	#88123	C₂³	1	16	yes	—
49	#88048	C₂³	1	32	yes	—
50	#88047	C₂³	1	32	yes	—
51	#88043	C₂³	1	16	yes	—
52	C₂²×#501	—	13	1	no	maximal, Sylow
53	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	—	2	4	no	—
55	C₂²×C₄×C₁₃	—	2	4	no	—
56	C₂²×C₄×C₁₃	—	2	6	no	—
57	C₂²×C₄×C₁₃	—	2	6	no	—
58	C₂²×C₄×C₁₃	—	4	4	no	—
59	C₂²×C₄×C₁₃	—	4	4	no	—
60	C₂²×C₄×C₁₃	C₂×D₄	1	6	yes	—
61	C₂²×C₄×C₁₃	C₂×D₄	1	6	yes	—
62	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
63	C₂²×C₄×C₁₃	C₂×Q₈	1	8	yes	—
64	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
65	C₂²×Dic₁₃	—	2	12	no	—
66	C₂²×Dic₁₃	—	4	8	no	—
67	C₂×Dic₂₆	—	2	16	no	—
68	C₂×Dic₂₆	—	4	48	no	—
69	C₁₃×C₄⋊C₄	—	2	16	no	—
70	C₁₃×C₄⋊C₄	—	2	16	no	—
71	C₁₃×C₄⋊C₄	—	2	6	no	—
72	C₁₃×C₄⋊C₄	—	2	6	no	—
73	C₁₃×C₄⋊C₄	C₂⁴	1	4	yes	—
74	C₁₃×C₄⋊C₄	C₂⁴	1	4	yes	—
75	C₄²×C₁₃	C₂×D₄	1	12	yes	—
76	C₄²×C₁₃	C₂⁴	1	4	yes	—
77	C₄⋊Dic₁₃	—	2	12	no	—
78	C₄⋊Dic₁₃	—	2	16	no	—
79	C₄⋊Dic₁₃	C₂⁴	1	8	yes	—
80	C₂×#1055	C₂×Q₈	1	48	yes	complemented
81	C₂×#1055	C₂⁴	1	8	yes	—
82	C₂²×C₄⋊Q₈	—	13	1	no	—
83	C₂²×#125	D₁₃	1	1	yes	characteristic
84	C₂×#501	—	13	24	no	—
85	C₂²×#107	—	13	2	no	—
86	C₂²×#106	—	13	2	no	—
87	C₂²×#97	—	13	1	no	—
88	C₂³×C₁₃	C₄⋊Q₈	1	4	yes	—
89	C₂³×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
90	C₂³×C₁₃	#137	1	4	yes	—
91	C₂³×C₁₃	C₂²×D₄	1	3	yes	—
92	C₂×C₄×C₁₃	—	2	12	no	—
93	C₂×C₄×C₁₃	—	2	12	no	—
94	C₂×C₄×C₁₃	—	2	4	no	—
95	C₂×C₄×C₁₃	—	2	4	no	—
96	C₂×C₄×C₁₃	—	4	12	no	—
97	C₂×C₄×C₁₃	—	4	12	no	—
98	C₂×C₄×C₁₃	C₄⋊Q₈	1	12	yes	—
99	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
100	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
101	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
102	C₂×C₄×C₁₃	C₂²×Q₈	1	12	yes	—
103	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
104	C₂×Dic₁₃	—	2	8	no	—
105	C₂×Dic₁₃	—	4	24	no	—
106	Dic₂₆	—	4	32	no	—
107	#1055	C₂²×Q₈	1	16	yes	complemented
108	C₂³×Q₈	—	26	1	no	—
109	C₂³×C₈	—	13	2	no	—
110	C₂×C₄⋊Q₈	—	13	12	no	—
111	C₂×C₄⋊Q₈	—	26	6	no	—
112	C₂×#125	—	2	6	no	—
113	C₂×#125	C₂×D₁₃	1	12	yes	—
114	C₂²×C₄⋊C₄	—	2	1	no	—
115	C₂²×C₄⋊C₄	—	2	1	no	—
116	C₂²×C₄⋊C₄	—	13	2	no	—
117	C₂²×C₄⋊C₄	—	26	1	no	—
118	C₂²×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
119	C₂²×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
120	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
121	#501	—	13	64	no	—
122	C₂×#107	—	13	24	no	—
123	C₂×#106	—	13	24	no	—
124	C₂×#97	—	13	12	no	—
125	C₂²×C₁₃	#501	1	16	yes	complemented
126	C₂²×C₁₃	C₂×C₄⋊Q₈	1	6	yes	—
127	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	6	yes	—
128	C₂²×C₁₃	C₂×#137	1	6	yes	—
129	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
130	C₄×C₁₃	—	2	4	no	—
131	C₄×C₁₃	—	2	4	no	—
132	C₄×C₁₃	—	4	4	no	—
133	C₄×C₁₃	—	4	4	no	—
134	C₄×C₁₃	C₂×C₄⋊Q₈	1	6	yes	—
135	C₄×C₁₃	C₂³×Q₈	1	2	yes	—
136	Dic₁₃	—	4	8	no	—
137	C₂²×Q₈	—	26	12	no	—
138	C₂²×Q₈	—	52	8	no	—
139	C₂³×C₄	—	2	1	no	—
140	C₂³×C₄	—	2	1	no	—
141	C₂³×C₄	—	26	2	no	—
142	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
143	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
144	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
145	C₂²×C₈	—	13	16	no	—
146	C₂²×C₈	—	13	12	no	—
147	C₄⋊Q₈	—	13	16	no	—
148	C₄⋊Q₈	—	26	24	no	—
149	#125	—	2	24	no	—
150	#125	C₂²×D₁₃	1	16	yes	—
151	C₂×C₄⋊C₄	—	2	12	no	—
152	C₂×C₄⋊C₄	—	2	12	no	—
153	C₂×C₄⋊C₄	—	2	3	no	—
154	C₂×C₄⋊C₄	—	2	3	no	—
155	C₂×C₄⋊C₄	—	13	12	no	—
156	C₂×C₄⋊C₄	—	26	6	no	—
157	C₂×C₄⋊C₄	—	26	12	no	—
158	C₂×C₄⋊C₄	C₂²×D₁₃	1	6	yes	—
159	C₂×C₄⋊C₄	C₂²×D₁₃	1	6	yes	—
160	C₂×C₄²	C₂²⋊D₁₃	1	6	yes	—
161	C₂×C₄²	C₂²×D₁₃	1	6	yes	—
162	#107	—	13	32	no	—
163	#106	—	13	32	no	—
164	#97	—	13	16	no	—
165	C₂×C₁₃	C₂×#501	1	12	yes	complemented
166	C₂×C₁₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
167	C₂×C₁₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
168	C₂×C₁₃	C₂²×#137	1	1	yes	characteristic
169	C₂⁴	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
170	C₂×Q₈	—	26	16	no	—
171	C₂×Q₈	—	52	48	no	—
172	C₂²×C₄	—	2	4	no	—
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	—	2	6	no	—
175	C₂²×C₄	—	2	6	no	—
176	C₂²×C₄	—	4	4	no	—
177	C₂²×C₄	—	4	4	no	—
178	C₂²×C₄	—	26	12	no	—
179	C₂²×C₄	—	52	8	no	—
180	C₂²×C₄	Q₈×D₁₃	1	8	yes	—
181	C₂²×C₄	C₂×C₂²⋊D₁₃	1	6	yes	—
182	C₂²×C₄	C₂×C₂²⋊D₁₃	1	6	yes	—
183	C₂²×C₄	C₂×C₂²⋊D₁₃	1	3	yes	—
184	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
185	C₂×C₈	—	13	48	no	—
186	C₂×C₈	—	13	8	no	—
187	C₄⋊C₄	—	2	16	no	—
188	C₄⋊C₄	—	2	16	no	—
189	C₄⋊C₄	—	2	6	no	—
190	C₄⋊C₄	—	2	6	no	—
191	C₄⋊C₄	—	13	8	no	—
192	C₄⋊C₄	—	26	12	no	—
193	C₄⋊C₄	—	26	16	no	—
194	C₄⋊C₄	C₂³×D₁₃	1	4	yes	—
195	C₄⋊C₄	C₂³×D₁₃	1	4	yes	—
196	C₄²	C₂×C₂²⋊D₁₃	1	12	yes	—
197	C₄²	C₂³×D₁₃	1	4	yes	—
198	C₁₃	C₂²×#501	1	1	yes	characteristic, complemented, Sylow
199	C₂³	#88198	1	4	yes	—
200	C₂³	#88203	1	4	yes	—
201	C₂³	#88205	1	4	yes	—
202	C₂³	C₂²×C₂²⋊D₁₃	1	3	yes	—
203	Q₈	—	52	32	no	—
204	C₂×C₄	—	2	12	no	—
205	C₂×C₄	—	2	12	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	4	12	no	—
209	C₂×C₄	—	4	12	no	—
210	C₂×C₄	—	26	8	no	—
211	C₂×C₄	—	52	24	no	—
212	C₂×C₄	#88198	1	12	yes	—
213	C₂×C₄	C₂×Q₈×D₁₃	1	12	yes	—
214	C₂×C₄	C₂²×C₂²⋊D₁₃	1	4	yes	—
215	C₂×C₄	C₂²×C₂²⋊D₁₃	1	4	yes	—
216	C₂×C₄	C₂²×C₂²⋊D₁₃	1	3	yes	—
217	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
218	C₈	—	13	16	no	—
219	C₂²	#11718726	1	16	yes	direct product factor
220	C₂²	C₂×#88198	1	6	yes	—
221	C₂²	C₂×#88203	1	6	yes	—
222	C₂²	C₂×#88205	1	6	yes	—
223	C₂²	C₂³×C₂²⋊D₁₃	1	1	yes	characteristic
224	C₄	—	2	4	no	—
225	C₄	—	2	4	no	—
226	C₄	—	4	4	no	—
227	C₄	—	4	4	no	—
228	C₄	—	52	8	no	—
229	C₄	C₂×#88198	1	6	yes	—
230	C₄	C₂²×Q₈×D₁₃	1	2	yes	—
231	C₂	C₂×#11718726	1	12	yes	direct product factor
232	C₂	C₂²×#88198	1	1	yes	characteristic
233	C₂	C₂²×#88203	1	1	yes	characteristic
234	C₂	C₂²×#88205	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂²×C₁₃×#125	C₂
socle, prime radical	C₂⁴×C₁₃	C₂×D₄
hypercenter	C₂²×#125	D₁₃
hypocenter	C₁₃	C₂²×#501
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×#125

Complements

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

N	H ≅ G/N	type
2. C₂×#11718726	C₂	direct (N × H)
20. #11718726	C₂²	direct (N × H)
45. C₂²×#1055	Q₈	semidirect
80. C₂×#1055	C₂×Q₈	semidirect
107. #1055	C₂²×Q₈	semidirect
125. C₂²×C₁₃	#501	semidirect
165. C₂×C₁₃	C₂×#501	semidirect
198. C₁₃	C₂²×#501	semidirect
219. C₂²	#11718726	direct (N × H)
231. C₂	C₂×#11718726	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×#88123	2
C₂×#11718726	2
C₂²×#88047	2
C₂²×#88048	2
C₂²×#88043	2
C₂²×C₁₃×#125	2
C₂²×#501	13

Subgroups — C22×#11718726 #16732530

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718726 #16732530