Subgroups — C₂²×#11718786 #16726612

Order 3328.

2718 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#6099	C₂	1	2	yes	maximal
2	C₂×#11718786	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#88059	C₂	1	4	yes	maximal, complemented
4	C₂³×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
6	C₂²×C₁₃×M₁₆	C₄	1	4	yes	—
7	C₂²×C₁₃×M₁₆	C₄	1	12	yes	—
8	C₂²×C₁₃×M₁₆	C₂²	1	12	yes	—
9	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
10	C₂²×#6099	—	2	2	no	—
11	C₂²×#6099	—	2	12	no	—
12	C₂²×#6099	C₂²	1	4	yes	—
13	C₂²×#6099	C₂²	1	24	yes	—
14	C₂³×#1055	—	2	2	no	—
15	#11718786	C₂²	1	64	yes	direct product factor
16	C₂×#88059	C₂²	1	48	yes	complemented
17	C₂⁵×C₁₃	Q₈	1	1	yes	characteristic
18	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
19	C₂³×C₄×C₁₃	C₂×C₄	1	6	yes	—
20	C₂³×C₄×C₁₃	D₄	1	2	yes	—
21	C₂³×C₄×C₁₃	D₄	1	6	yes	—
22	C₂³×C₄×C₁₃	Q₈	1	1	yes	characteristic
23	C₂³×C₄×C₁₃	Q₈	1	6	yes	—
24	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
25	C₂³×C₄×C₁₃	C₂³	1	6	yes	—
26	C₂×C₁₃×M₁₆	C₂×C₄	1	48	yes	—
27	C₂×C₁₃×M₁₆	C₂×C₄	1	48	yes	—
28	C₂×C₁₃×M₁₆	C₂³	1	16	yes	—
29	C₂²×C₈×C₁₃	C₂×C₄	1	12	yes	—
30	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₁₃	C₂³	1	12	yes	—
32	C₂×#6099	—	2	24	no	—
33	C₂×#6099	—	2	12	no	—
34	C₂×#6099	—	2	48	no	—
35	C₂×#6099	C₂³	1	24	yes	—
36	C₂×#6099	C₂³	1	32	yes	—
37	C₂²×#1055	—	2	24	no	—
38	C₂²×#1055	—	2	4	no	—
39	#88059	C₂³	1	64	yes	complemented
40	C₂²×#430	—	13	1	no	maximal, Sylow
41	C₂⁴×C₁₃	—	4	4	no	—
42	C₂⁴×C₁₃	C₄⋊C₄	1	2	yes	—
43	C₂⁴×C₁₃	C₄⋊C₄	1	6	yes	—
44	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
45	C₂⁴×C₁₃	C₂×Q₈	1	6	yes	—
46	C₂²×C₄×C₁₃	C₄⋊C₄	1	12	yes	—
47	C₂²×C₄×C₁₃	C₄⋊C₄	1	12	yes	—
48	C₂²×C₄×C₁₃	C₄⋊C₄	1	6	yes	—
49	C₂²×C₄×C₁₃	C₄⋊C₄	1	24	yes	—
50	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
52	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂²×C₄	1	12	yes	—
54	C₂²×C₄×C₁₃	C₂²×C₄	1	12	yes	—
55	C₂²×C₄×C₁₃	C₂×D₄	1	12	yes	—
56	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
57	C₂²×C₄×C₁₃	C₂×D₄	1	12	yes	—
58	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂×Q₈	1	3	yes	—
60	C₂²×C₄×C₁₃	C₂×Q₈	1	6	yes	—
61	C₂²×C₄×C₁₃	C₂×Q₈	1	12	yes	—
62	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
63	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
64	C₁₃×M₁₆	C₂²×C₄	1	64	yes	—
65	C₂×C₈×C₁₃	C₂²×C₄	1	24	yes	—
66	C₂×C₈×C₁₃	C₂²×C₄	1	24	yes	—
67	C₂×C₈×C₁₃	C₂⁴	1	8	yes	—
68	#6099	—	2	32	no	—
69	#6099	—	2	24	no	—
70	#6099	C₂⁴	1	16	yes	—
71	C₂×#1055	—	2	24	no	—
72	C₂×#1055	—	2	32	no	—
73	C₂³×M₁₆	—	13	2	no	—
74	C₂³×M₁₆	D₁₃	1	1	yes	characteristic
75	C₂×#430	—	13	24	no	—
76	C₂²×#108	—	13	4	no	—
77	C₂³×C₁₃	—	2	8	no	—
78	C₂³×C₁₃	—	2	4	no	—
79	C₂³×C₁₃	—	4	12	no	—
80	C₂³×C₁₃	—	4	12	no	—
81	C₂³×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
82	C₂³×C₁₃	C₂×C₄⋊C₄	1	12	yes	—
83	C₂³×C₁₃	C₂×C₄⋊C₄	1	12	yes	—
84	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂³×C₁₃	C₂²×Q₈	1	3	yes	—
86	C₂³×C₁₃	C₂²×Q₈	1	4	yes	—
87	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
88	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	24	yes	—
89	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	24	yes	—
90	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	12	yes	—
91	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
92	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	12	yes	—
93	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
94	C₂×C₄×C₁₃	C₂³×C₄	1	3	yes	—
95	C₂×C₄×C₁₃	C₂³×C₄	1	3	yes	—
96	C₂×C₄×C₁₃	C₂³×C₄	1	8	yes	—
97	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
98	C₂×C₄×C₁₃	C₂²×D₄	1	8	yes	—
99	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
100	C₂×C₄×C₁₃	C₂²×Q₈	1	3	yes	—
101	C₂×C₄×C₁₃	C₂²×Q₈	1	4	yes	—
102	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
103	C₈×C₁₃	C₂³×C₄	1	16	yes	—
104	#1055	—	2	16	no	—
105	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
106	C₂²×M₁₆	—	13	4	no	—
107	C₂²×M₁₆	—	13	24	no	—
108	C₂²×M₁₆	—	26	12	no	—
109	C₂²×M₁₆	—	26	2	no	—
110	C₂²×M₁₆	Dic₁₃	1	4	yes	—
111	C₂²×M₁₆	Dic₁₃	1	12	yes	—
112	C₂²×M₁₆	C₂×D₁₃	1	12	yes	—
113	C₂³×C₈	—	26	2	no	—
114	C₂³×C₈	C₂×D₁₃	1	2	yes	—
115	#430	—	13	64	no	—
116	C₂×#108	—	13	48	no	—
117	C₂²×C₁₃	—	2	24	no	—
118	C₂²×C₁₃	—	2	6	no	—
119	C₂²×C₁₃	—	2	6	no	—
120	C₂²×C₁₃	—	4	12	no	—
121	C₂²×C₁₃	—	4	4	no	—
122	C₂²×C₁₃	#430	1	4	yes	complemented
123	C₂²×C₁₃	C₂²×C₄⋊C₄	1	3	yes	—
124	C₂²×C₁₃	C₂²×C₄⋊C₄	1	8	yes	—
125	C₂²×C₁₃	C₂²×C₄⋊C₄	1	3	yes	—
126	C₂²×C₁₃	C₂³×Q₈	1	1	yes	characteristic
127	C₄×C₁₃	C₂²×C₄⋊C₄	1	3	yes	—
128	C₄×C₁₃	C₂²×C₄⋊C₄	1	3	yes	—
129	C₄×C₁₃	C₂²×C₄⋊C₄	1	8	yes	—
130	C₄×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
131	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
132	C₂⁵	Dic₂₆	1	1	yes	characteristic
133	C₂³×C₄	Dic₂₆	1	1	yes	characteristic
134	C₂³×C₄	Dic₂₆	1	6	yes	—
135	C₂³×C₄	D₅₂	1	6	yes	—
136	C₂³×C₄	D₅₂	1	2	yes	—
137	C₂³×C₄	C₂×Dic₁₃	1	2	yes	—
138	C₂³×C₄	C₂×Dic₁₃	1	6	yes	—
139	C₂³×C₄	C₂²×D₁₃	1	6	yes	—
140	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
141	C₂×M₁₆	—	13	24	no	—
142	C₂×M₁₆	—	13	32	no	—
143	C₂×M₁₆	—	26	24	no	—
144	C₂×M₁₆	—	26	12	no	—
145	C₂×M₁₆	—	26	48	no	—
146	C₂×M₁₆	C₂×Dic₁₃	1	48	yes	—
147	C₂×M₁₆	C₂×Dic₁₃	1	48	yes	—
148	C₂×M₁₆	C₂²×D₁₃	1	16	yes	—
149	C₂²×C₈	—	26	24	no	—
150	C₂²×C₈	—	26	4	no	—
151	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
152	C₂²×C₈	C₂×Dic₁₃	1	12	yes	—
153	C₂²×C₈	C₂²×D₁₃	1	12	yes	—
154	#108	—	13	64	no	—
155	C₂×C₁₃	—	2	8	no	—
156	C₂×C₁₃	—	2	3	no	—
157	C₂×C₁₃	—	2	1	no	—
158	C₂×C₁₃	C₂×#430	1	6	yes	complemented
159	C₂×C₁₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
160	C₂⁴	—	4	4	no	—
161	C₂⁴	C₄⋊Dic₁₃	1	2	yes	—
162	C₂⁴	C₄⋊Dic₁₃	1	6	yes	—
163	C₂⁴	C₂×Dic₂₆	1	1	yes	characteristic
164	C₂⁴	C₂×Dic₂₆	1	6	yes	—
165	C₂²×C₄	C₄⋊Dic₁₃	1	6	yes	—
166	C₂²×C₄	C₄⋊Dic₁₃	1	12	yes	—
167	C₂²×C₄	C₄⋊Dic₁₃	1	24	yes	—
168	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
169	C₂²×C₄	C₄⋊Dic₁₃	1	12	yes	—
170	C₂²×C₄	C₂×Dic₂₆	1	6	yes	—
171	C₂²×C₄	C₂×Dic₂₆	1	12	yes	—
172	C₂²×C₄	C₂×Dic₂₆	1	3	yes	—
173	C₂²×C₄	C₂×D₅₂	1	12	yes	—
174	C₂²×C₄	C₂×D₅₂	1	3	yes	—
175	C₂²×C₄	C₂×D₅₂	1	12	yes	—
176	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
177	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
178	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
179	C₂²×C₄	C₂²×Dic₁₃	1	12	yes	—
180	C₂²×C₄	C₂²×Dic₁₃	1	12	yes	—
181	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
182	C₂²×C₄	C₂³×D₁₃	1	4	yes	—
183	M₁₆	—	13	16	no	—
184	M₁₆	—	26	32	no	—
185	M₁₆	—	26	24	no	—
186	M₁₆	C₂²×Dic₁₃	1	64	yes	—
187	C₂×C₈	—	26	24	no	—
188	C₂×C₈	—	26	32	no	—
189	C₂×C₈	C₂²×Dic₁₃	1	24	yes	—
190	C₂×C₈	C₂²×Dic₁₃	1	24	yes	—
191	C₂×C₈	C₂³×D₁₃	1	8	yes	—
192	C₁₃	C₂²×#430	1	1	yes	characteristic, complemented, Sylow
193	C₂³	—	2	8	no	—
194	C₂³	—	2	4	no	—
195	C₂³	—	4	12	no	—
196	C₂³	—	4	12	no	—
197	C₂³	C₂×C₄⋊Dic₁₃	1	12	yes	—
198	C₂³	C₂×C₄⋊Dic₁₃	1	3	yes	—
199	C₂³	C₂×C₄⋊Dic₁₃	1	12	yes	—
200	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
201	C₂³	C₂²×Dic₂₆	1	3	yes	—
202	C₂³	C₂²×Dic₂₆	1	4	yes	—
203	C₂×C₄	C₂×C₄⋊Dic₁₃	1	24	yes	—
204	C₂×C₄	C₂×C₄⋊Dic₁₃	1	6	yes	—
205	C₂×C₄	C₂×C₄⋊Dic₁₃	1	12	yes	—
206	C₂×C₄	C₂×C₄⋊Dic₁₃	1	24	yes	—
207	C₂×C₄	C₂×C₄⋊Dic₁₃	1	12	yes	—
208	C₂×C₄	C₂×C₄⋊Dic₁₃	1	3	yes	—
209	C₂×C₄	C₂×C₄⋊Dic₁₃	1	3	yes	—
210	C₂×C₄	C₂²×Dic₂₆	1	3	yes	—
211	C₂×C₄	C₂²×Dic₂₆	1	4	yes	—
212	C₂×C₄	C₂²×D₅₂	1	3	yes	—
213	C₂×C₄	C₂²×D₅₂	1	8	yes	—
214	C₂×C₄	C₂²×D₅₂	1	3	yes	—
215	C₂×C₄	C₂³×Dic₁₃	1	3	yes	—
216	C₂×C₄	C₂³×Dic₁₃	1	3	yes	—
217	C₂×C₄	C₂³×Dic₁₃	1	8	yes	—
218	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
219	C₈	—	26	16	no	—
220	C₈	C₂³×Dic₁₃	1	16	yes	—
221	C₂²	—	2	24	no	—
222	C₂²	—	2	6	no	—
223	C₂²	—	2	6	no	—
224	C₂²	—	4	12	no	—
225	C₂²	—	4	4	no	—
226	C₂²	#11718786	1	4	yes	direct product factor
227	C₂²	C₂²×C₄⋊Dic₁₃	1	8	yes	—
228	C₂²	C₂²×C₄⋊Dic₁₃	1	3	yes	—
229	C₂²	C₂²×C₄⋊Dic₁₃	1	3	yes	—
230	C₂²	C₂³×Dic₂₆	1	1	yes	characteristic
231	C₄	C₂²×C₄⋊Dic₁₃	1	8	yes	—
232	C₄	C₂²×C₄⋊Dic₁₃	1	3	yes	—
233	C₄	C₂²×C₄⋊Dic₁₃	1	3	yes	—
234	C₄	C₂³×D₅₂	1	1	yes	characteristic
235	C₄	C₂⁴×Dic₁₃	1	1	yes	characteristic
236	C₂	—	2	8	no	—
237	C₂	—	2	3	no	—
238	C₂	—	2	1	no	—
239	C₂	C₂×#11718786	1	6	yes	direct product factor
240	C₂	C₂³×C₄⋊Dic₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₅₂
commutator	C₄×C₁₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂³×C₁₃×M₁₆	C₂
socle	C₂³×C₁₃	C₂×C₄⋊C₄
prime radical	C₂⁵×C₁₃	Q₈
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂³×M₁₆	D₁₃
hypocenter	C₁₃	C₂²×#430
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ C₂³×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#11718786	C₂	direct (N × H)
3. C₂²×#88059	C₂	semidirect
15. #11718786	C₂²	direct (N × H)
16. C₂×#88059	C₂²	semidirect
39. #88059	C₂³	semidirect
122. C₂²×C₁₃	#430	semidirect
158. C₂×C₁₃	C₂×#430	semidirect
192. C₁₃	C₂²×#430	semidirect
226. C₂²	#11718786	direct (N × H)
239. C₂	C₂×#11718786	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×#6099	2
C₂×#11718786	2
C₂²×#88059	2
C₂³×C₁₃×M₁₆	2
C₂²×#430	13

Subgroups — C22×#11718786 #16726612

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718786 #16726612