Subgroups — C₂²×#11718779 #16726530

Order 3328.

3438 conjugacy classes of subgroups.

The subgroup lattice has 240 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	1	yes	maximal, characteristic
2	C₂³×C₄×Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718779	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#88054	C₂	1	4	yes	maximal, complemented
5	C₂³×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
7	C₂⁴×Dic₁₃	C₄	1	2	yes	—
8	C₂²×C₁₃×M₁₆	—	2	8	no	—
9	C₂²×C₁₃×M₁₆	C₂²	1	12	yes	—
10	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
11	C₂²×C₄×Dic₁₃	C₄	1	1	yes	characteristic
12	C₂²×C₄×Dic₁₃	C₄	1	12	yes	—
13	C₂²×C₄×Dic₁₃	C₄	1	1	yes	characteristic
14	C₂²×C₄×Dic₁₃	C₂²	1	2	yes	—
15	C₂²×C₄×Dic₁₃	C₂²	1	12	yes	—
16	C₂²×#6099	—	2	8	no	—
17	C₂²×#6099	C₂²	1	12	yes	—
18	C₂³×#1055	C₂²	1	2	yes	—
19	#11718779	C₂²	1	64	yes	direct product factor
20	C₂×#88054	C₂²	1	48	yes	complemented
21	C₂⁵×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₃	C₂×C₄	1	6	yes	—
23	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₃	D₄	1	8	yes	—
25	C₂³×C₄×C₁₃	Q₈	1	8	yes	—
26	C₂³×C₄×C₁₃	C₂³	1	6	yes	—
27	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
28	C₂³×Dic₁₃	—	2	16	no	—
29	C₂³×Dic₁₃	C₂×C₄	1	2	yes	—
30	C₂³×Dic₁₃	C₂×C₄	1	24	yes	—
31	C₂³×Dic₁₃	C₂×C₄	1	2	yes	—
32	C₂×C₁₃×M₁₆	—	2	48	no	—
33	C₂×C₁₃×M₁₆	C₂³	1	16	yes	—
34	C₂²×C₈×C₁₃	—	2	8	no	—
35	C₂²×C₈×C₁₃	C₂³	1	12	yes	—
36	C₂×C₄×Dic₁₃	C₂×C₄	1	12	yes	—
37	C₂×C₄×Dic₁₃	C₂×C₄	1	48	yes	—
38	C₂×C₄×Dic₁₃	C₂×C₄	1	12	yes	—
39	C₂×C₄×Dic₁₃	C₂×C₄	1	12	yes	—
40	C₂×C₄×Dic₁₃	C₂³	1	12	yes	—
41	C₂×C₄×Dic₁₃	C₂³	1	16	yes	—
42	C₂×#6099	—	2	48	no	—
43	C₂×#6099	C₂³	1	16	yes	—
44	C₂²×#1055	—	2	8	no	—
45	C₂²×#1055	C₂³	1	12	yes	—
46	#88054	C₂³	1	64	yes	complemented
47	C₂²×#423	—	13	1	no	maximal, Sylow
48	C₂⁴×C₁₃	—	2	8	no	—
49	C₂⁴×C₁₃	C₄⋊C₄	1	8	yes	—
50	C₂⁴×C₁₃	C₂²×C₄	1	6	yes	—
51	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₄⋊C₄	1	48	yes	—
53	C₂²×C₄×C₁₃	C₄⋊C₄	1	8	yes	—
54	C₂²×C₄×C₁₃	C₂²×C₄	1	12	yes	—
55	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
56	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
57	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
58	C₂²×C₄×C₁₃	C₂×D₄	1	24	yes	—
59	C₂²×C₄×C₁₃	C₂×Q₈	1	24	yes	—
60	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
61	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
62	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
63	C₂²×Dic₁₃	—	2	8	no	—
64	C₂²×Dic₁₃	—	2	96	no	—
65	C₂²×Dic₁₃	M₁₆	1	16	yes	complemented
66	C₂²×Dic₁₃	C₂²×C₄	1	12	yes	—
67	C₂²×Dic₁₃	C₂²×C₄	1	32	yes	—
68	C₂²×Dic₁₃	C₂²×C₄	1	12	yes	—
69	C₁₃×M₁₆	—	2	32	no	—
70	C₂×C₈×C₁₃	—	2	24	no	—
71	C₂×C₈×C₁₃	C₂⁴	1	8	yes	—
72	C₄×Dic₁₃	C₂²×C₄	1	16	yes	—
73	C₄×Dic₁₃	C₂²×C₄	1	24	yes	—
74	C₄×Dic₁₃	C₂²×C₄	1	16	yes	—
75	C₄×Dic₁₃	C₂⁴	1	8	yes	—
76	#6099	—	2	32	no	—
77	C₂×#1055	—	2	24	no	—
78	C₂×#1055	C₂⁴	1	8	yes	—
79	C₂³×M₁₆	—	13	1	no	—
80	C₂³×M₁₆	D₁₃	1	1	yes	characteristic
81	C₂³×C₄²	—	13	1	no	—
82	C₂×#423	—	13	24	no	—
83	C₂²×C₄⋊C₈	—	13	4	no	—
84	C₂³×C₁₃	—	2	24	no	—
85	C₂³×C₁₃	—	2	32	no	—
86	C₂³×C₁₃	C₂×C₄⋊C₄	1	24	yes	—
87	C₂³×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
88	C₂³×C₁₃	C₂×M₁₆	1	8	yes	—
89	C₂³×C₁₃	C₂³×C₄	1	4	yes	—
90	C₂³×C₁₃	C₂³×C₄	1	3	yes	—
91	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	48	yes	—
92	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	12	yes	—
93	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	24	yes	—
94	C₂×C₄×C₁₃	C₂³×C₄	1	3	yes	—
95	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
96	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
97	C₂×C₄×C₁₃	C₂²×D₄	1	8	yes	—
98	C₂×C₄×C₁₃	C₂²×Q₈	1	8	yes	—
99	C₂×C₄×C₁₃	C₂²×Q₈	1	6	yes	—
100	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
101	C₂×Dic₁₃	—	2	24	no	—
102	C₂×Dic₁₃	—	2	64	no	—
103	C₂×Dic₁₃	C₂×M₁₆	1	48	yes	complemented
104	C₂×Dic₁₃	C₂³×C₄	1	8	yes	—
105	C₂×Dic₁₃	C₂³×C₄	1	8	yes	—
106	C₈×C₁₃	—	2	8	no	—
107	#1055	—	2	8	no	—
108	C₂⁴×C₄	—	13	2	no	—
109	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
110	C₂²×M₁₆	—	2	8	no	—
111	C₂²×M₁₆	—	13	12	no	—
112	C₂²×M₁₆	—	26	8	no	—
113	C₂²×M₁₆	C₂×D₁₃	1	12	yes	—
114	C₂³×C₈	—	13	2	no	—
115	C₂³×C₈	C₂×D₁₃	1	2	yes	—
116	C₂²×C₄²	—	13	1	no	—
117	C₂²×C₄²	—	13	12	no	—
118	C₂²×C₄²	—	13	1	no	—
119	C₂²×C₄²	—	13	12	no	—
120	C₂²×C₄²	—	13	2	no	—
121	#423	—	13	64	no	—
122	C₂×C₄⋊C₈	—	13	48	no	—
123	C₂²×C₁₃	—	2	48	no	—
124	C₂²×C₁₃	—	2	8	no	—
125	C₂²×C₁₃	#423	1	16	yes	complemented
126	C₂²×C₁₃	C₂²×C₄⋊C₄	1	8	yes	—
127	C₂²×C₁₃	C₂²×C₄⋊C₄	1	6	yes	—
128	C₂²×C₁₃	C₂²×M₁₆	1	12	yes	—
129	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
130	C₄×C₁₃	C₂²×C₄⋊C₄	1	6	yes	—
131	C₄×C₁₃	C₂²×C₄⋊C₄	1	8	yes	—
132	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
133	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
134	Dic₁₃	—	2	8	no	—
135	Dic₁₃	C₂²×M₁₆	1	16	yes	complemented
136	C₂⁵	C₄×D₁₃	1	1	yes	characteristic
137	C₂³×C₄	—	13	2	no	—
138	C₂³×C₄	—	13	24	no	—
139	C₂³×C₄	—	13	2	no	—
140	C₂³×C₄	—	26	16	no	—
141	C₂³×C₄	Dic₂₆	1	8	yes	—
142	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
143	C₂³×C₄	C₄×D₁₃	1	6	yes	—
144	C₂³×C₄	C₂²⋊D₁₃	1	8	yes	—
145	C₂³×C₄	C₂²×D₁₃	1	6	yes	—
146	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
147	C₂×M₁₆	—	2	48	no	—
148	C₂×M₁₆	—	13	16	no	—
149	C₂×M₁₆	—	26	48	no	—
150	C₂×M₁₆	C₂²×D₁₃	1	16	yes	—
151	C₂²×C₈	—	2	8	no	—
152	C₂²×C₈	—	13	12	no	—
153	C₂²×C₈	—	26	8	no	—
154	C₂²×C₈	C₂²×D₁₃	1	12	yes	—
155	C₂×C₄²	—	13	12	no	—
156	C₂×C₄²	—	13	12	no	—
157	C₂×C₄²	—	13	48	no	—
158	C₂×C₄²	—	13	12	no	—
159	C₂×C₄²	—	13	12	no	—
160	C₂×C₄²	—	13	16	no	—
161	C₄⋊C₈	—	13	64	no	—
162	C₂×C₁₃	—	2	8	no	—
163	C₂×C₁₃	C₂×#423	1	12	yes	complemented
164	C₂×C₁₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
165	C₂×C₁₃	C₂³×M₁₆	1	2	yes	—
166	C₂⁴	—	2	8	no	—
167	C₂⁴	#6101	1	8	yes	—
168	C₂⁴	C₂×C₄×D₁₃	1	6	yes	—
169	C₂⁴	C₂×C₄×D₁₃	1	1	yes	characteristic
170	C₂²×C₄	—	13	16	no	—
171	C₂²×C₄	—	13	12	no	—
172	C₂²×C₄	—	13	32	no	—
173	C₂²×C₄	—	13	12	no	—
174	C₂²×C₄	—	26	96	no	—
175	C₂²×C₄	—	26	8	no	—
176	C₂²×C₄	#6101	1	48	yes	—
177	C₂²×C₄	#6101	1	8	yes	—
178	C₂²×C₄	C₂×Dic₂₆	1	24	yes	—
179	C₂²×C₄	C₂×Dic₂₆	1	4	yes	—
180	C₂²×C₄	C₂×C₄×D₁₃	1	12	yes	—
181	C₂²×C₄	C₂×C₄×D₁₃	1	6	yes	—
182	C₂²×C₄	C₂×C₄×D₁₃	1	3	yes	—
183	C₂²×C₄	C₂×C₂²⋊D₁₃	1	4	yes	—
184	C₂²×C₄	C₂×C₂²⋊D₁₃	1	24	yes	—
185	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
186	C₂²×C₄	C₂³×D₁₃	1	4	yes	—
187	M₁₆	—	2	32	no	—
188	M₁₆	—	26	32	no	—
189	C₂×C₈	—	2	24	no	—
190	C₂×C₈	—	13	8	no	—
191	C₂×C₈	—	26	24	no	—
192	C₂×C₈	C₂³×D₁₃	1	8	yes	—
193	C₄²	—	13	16	no	—
194	C₄²	—	13	24	no	—
195	C₄²	—	13	16	no	—
196	C₄²	—	13	8	no	—
197	C₁₃	C₂²×#423	1	1	yes	characteristic, complemented, Sylow
198	C₂³	—	2	24	no	—
199	C₂³	—	2	32	no	—
200	C₂³	M₁₆×D₁₃	1	8	yes	—
201	C₂³	C₂×#6101	1	24	yes	—
202	C₂³	C₂×#6101	1	4	yes	—
203	C₂³	C₂²×C₄×D₁₃	1	4	yes	—
204	C₂³	C₂²×C₄×D₁₃	1	3	yes	—
205	C₂×C₄	—	13	48	no	—
206	C₂×C₄	—	13	8	no	—
207	C₂×C₄	—	13	8	no	—
208	C₂×C₄	—	26	64	no	—
209	C₂×C₄	—	26	24	no	—
210	C₂×C₄	C₂×#6101	1	48	yes	—
211	C₂×C₄	C₂×#6101	1	24	yes	—
212	C₂×C₄	C₂×#6101	1	12	yes	—
213	C₂×C₄	C₂²×Dic₂₆	1	8	yes	—
214	C₂×C₄	C₂²×Dic₂₆	1	6	yes	—
215	C₂×C₄	C₂²×C₄×D₁₃	1	4	yes	—
216	C₂×C₄	C₂²×C₄×D₁₃	1	3	yes	—
217	C₂×C₄	C₂²×C₂²⋊D₁₃	1	6	yes	—
218	C₂×C₄	C₂²×C₂²⋊D₁₃	1	8	yes	—
219	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
220	C₈	—	2	8	no	—
221	C₈	—	26	8	no	—
222	C₂²	—	2	48	no	—
223	C₂²	—	2	8	no	—
224	C₂²	#11718779	1	16	yes	direct product factor
225	C₂²	C₂×M₁₆×D₁₃	1	12	yes	—
226	C₂²	C₂²×#6101	1	8	yes	—
227	C₂²	C₂²×#6101	1	6	yes	—
228	C₂²	C₂³×C₄×D₁₃	1	1	yes	characteristic
229	C₄	—	13	16	no	—
230	C₄	—	26	8	no	—
231	C₄	C₂²×#6101	1	6	yes	—
232	C₄	C₂²×#6101	1	8	yes	—
233	C₄	C₂³×Dic₂₆	1	1	yes	characteristic
234	C₄	C₂³×C₂²⋊D₁₃	1	1	yes	characteristic
235	C₂	—	2	8	no	—
236	C₂	C₂×#11718779	1	12	yes	direct product factor
237	C₂	C₂³×#6101	1	1	yes	characteristic
238	C₂	C₂²×M₁₆×D₁₃	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₁₃
commutator	C₂²×C₁₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂³×C₁₃×M₁₆	C₂
socle	C₂⁴×C₁₃	C₂²×C₄
prime radical	C₂⁵×C₁₃	C₂×C₄
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂³×M₁₆	D₁₃
hypocenter	C₁₃	C₂²×#423
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×#11718779	C₂	direct (N × H)
4. C₂²×#88054	C₂	semidirect
19. #11718779	C₂²	direct (N × H)
20. C₂×#88054	C₂²	semidirect
46. #88054	C₂³	semidirect
65. C₂²×Dic₁₃	M₁₆	semidirect
103. C₂×Dic₁₃	C₂×M₁₆	semidirect
125. C₂²×C₁₃	#423	semidirect
135. Dic₁₃	C₂²×M₁₆	semidirect
163. C₂×C₁₃	C₂×#423	semidirect
197. C₁₃	C₂²×#423	semidirect
224. C₂²	#11718779	direct (N × H)
236. C₂	C₂×#11718779	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
C₂³×C₄×Dic₁₃	2
C₂×#11718779	2
C₂²×#88054	2
C₂³×C₁₃×M₁₆	2
C₂²×#423	13

Subgroups — C22×#11718779 #16726530

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718779 #16726530