Subgroups — C₄×#11718150 #15768105

Order 3328.

910 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#6099	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×#1055	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718150	C₂	1	4	yes	maximal
4	C₂×C₄×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
6	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
7	C₂²×#6099	C₂²	1	2	yes	—
8	C₂³×#1055	C₂²	1	2	yes	—
9	C₄×C₁₃×M₁₆	—	2	2	no	—
10	C₄×C₁₃×M₁₆	C₄	1	4	yes	—
11	C₂×C₁₃×#97	—	2	1	no	—
12	C₂×C₄×C₈×C₁₃	—	2	1	no	—
13	C₄×#6099	—	2	2	no	—
14	C₄×#6099	C₄	1	4	yes	—
15	C₂×#88043	—	2	1	no	—
16	C₂×C₄×#1055	—	2	2	no	—
17	C₂×C₄×#1055	—	2	2	no	—
18	C₂×C₄×#1055	—	2	1	no	—
19	C₂×C₄×#1055	C₄	1	4	yes	—
20	#11718150	C₄	1	16	yes	direct product factor
21	C₂³×C₄×C₁₃	Q₈	1	2	yes	—
22	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
23	C₂×C₁₃×M₁₆	—	2	2	no	—
24	C₂×C₁₃×M₁₆	—	2	4	no	—
25	C₂×C₁₃×M₁₆	C₂×C₄	1	8	yes	—
26	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₁₃	—	2	2	no	—
28	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₃	D₄	1	2	yes	—
32	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₃	D₄	1	2	yes	—
34	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
35	C₂×#6099	—	2	2	no	—
36	C₂×#6099	—	2	4	no	—
37	C₂×#6099	C₂×C₄	1	8	yes	—
38	C₂×#6099	C₂×C₄	1	4	yes	—
39	C₂²×#1055	—	2	2	no	—
40	C₂²×#1055	—	2	2	no	—
41	C₂²×#1055	—	2	2	no	—
42	C₂²×#1055	—	2	4	no	—
43	C₂²×#1055	C₂×C₄	1	8	yes	—
44	C₂²×#1055	C₂×C₄	1	4	yes	—
45	C₁₃×#97	—	2	4	no	—
46	C₄×C₈×C₁₃	—	2	4	no	—
47	#88043	—	2	4	no	—
48	C₄×#1055	—	2	8	no	—
49	C₄×#1055	—	2	4	no	—
50	C₄×#341	—	13	1	no	maximal, Sylow
51	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	—	2	8	no	—
53	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
55	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
57	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	#41	1	4	yes	—
65	C₁₃×M₁₆	—	2	4	no	—
66	C₁₃×M₁₆	—	2	8	no	—
67	C₁₃×M₁₆	C₄²	1	8	yes	—
68	C₂×C₈×C₁₃	—	2	4	no	—
69	C₂×C₈×C₁₃	—	2	8	no	—
70	C₄²×C₁₃	C₄²	1	4	yes	—
71	C₄²×C₁₃	C₂²⋊C₄	1	4	yes	—
72	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
73	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
74	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
75	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
76	#6099	—	2	4	no	—
77	#6099	—	2	8	no	—
78	#6099	C₄²	1	8	yes	—
79	C₂×#1055	—	2	4	no	—
80	C₂×#1055	—	2	4	no	—
81	C₂×#1055	—	2	4	no	—
82	C₂×#1055	—	2	8	no	—
83	C₂×#1055	—	2	8	no	—
84	C₂×#1055	—	2	8	no	—
85	C₂×#1055	C₄²	1	8	yes	—
86	C₂×C₄×M₁₆	—	13	1	no	—
87	C₂×C₄×M₁₆	D₁₃	1	1	yes	characteristic
88	C₂²×C₄×C₈	—	13	1	no	—
89	C₂×#341	—	13	4	no	—
90	C₂³×C₁₃	—	2	4	no	—
91	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂³×C₁₃	C₄×Q₈	1	4	yes	—
95	C₂×C₄×C₁₃	—	2	4	no	—
96	C₂×C₄×C₁₃	—	2	8	no	—
97	C₂×C₄×C₁₃	—	2	8	no	—
98	C₂×C₄×C₁₃	#95	1	4	yes	—
99	C₂×C₄×C₁₃	#95	1	4	yes	—
100	C₂×C₄×C₁₃	#108	1	8	yes	—
101	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
103	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
104	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
105	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	#117	1	2	yes	—
109	C₂×C₄×C₁₃	#117	1	2	yes	—
110	C₂×C₄×C₁₃	#117	1	2	yes	—
111	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
112	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
113	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
114	C₈×C₁₃	—	2	8	no	—
115	#1055	—	2	8	no	—
116	#1055	—	2	8	no	—
117	C₂²×M₁₆	—	13	2	no	—
118	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
119	C₂³×C₈	—	13	2	no	—
120	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
121	C₄×M₁₆	—	2	2	no	—
122	C₄×M₁₆	—	13	4	no	—
123	C₄×M₁₆	—	26	2	no	—
124	C₄×M₁₆	Dic₁₃	1	4	yes	—
125	C₂×#97	—	2	1	no	—
126	C₂×#97	—	26	1	no	—
127	C₂×C₄×C₈	—	2	1	no	—
128	C₂×C₄×C₈	—	13	4	no	—
129	C₂×C₄×C₈	—	26	2	no	—
130	C₂×C₄×C₈	—	26	2	no	—
131	C₂×C₄×C₈	—	26	1	no	—
132	#341	—	13	16	no	—
133	C₂²×C₁₃	—	2	2	no	—
134	C₂²×C₁₃	—	2	2	no	—
135	C₂²×C₁₃	—	2	8	no	—
136	C₂²×C₁₃	C₂×#95	1	1	yes	characteristic
137	C₂²×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
138	C₂²×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
139	C₂²×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
140	C₂²×C₁₃	C₂×#108	1	4	yes	—
141	C₄×C₁₃	—	2	4	no	—
142	C₄×C₁₃	#341	1	8	yes	complemented
143	C₄×C₁₃	C₄³	1	1	yes	characteristic
144	C₄×C₁₃	#376	1	1	yes	characteristic
145	C₄×C₁₃	C₄×C₂²⋊C₄	1	2	yes	—
146	C₄×C₁₃	C₄×C₂²⋊C₄	1	2	yes	—
147	C₄×C₁₃	C₄×C₂²⋊C₄	1	2	yes	—
148	C₂³×C₄	Dic₂₆	1	2	yes	—
149	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
150	C₂×M₁₆	—	2	4	no	—
151	C₂×M₁₆	—	2	2	no	—
152	C₂×M₁₆	—	13	8	no	—
153	C₂×M₁₆	—	13	4	no	—
154	C₂×M₁₆	—	26	2	no	—
155	C₂×M₁₆	—	26	4	no	—
156	C₂×M₁₆	C₄×D₁₃	1	8	yes	—
157	C₂×M₁₆	C₂×Dic₁₃	1	4	yes	—
158	C₂²×C₈	—	2	2	no	—
159	C₂²×C₈	—	13	8	no	—
160	C₂²×C₈	—	13	4	no	—
161	C₂²×C₈	—	26	2	no	—
162	C₂²×C₈	—	26	4	no	—
163	C₂²×C₈	—	26	2	no	—
164	C₂²×C₈	—	26	2	no	—
165	C₂×C₄²	C₄×D₁₃	1	2	yes	—
166	C₂×C₄²	C₄×D₁₃	1	2	yes	—
167	C₂×C₄²	D₅₂	1	2	yes	—
168	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
169	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
170	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
171	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
172	#97	—	2	4	no	—
173	#97	—	26	4	no	—
174	C₄×C₈	—	2	4	no	—
175	C₄×C₈	—	26	8	no	—
176	C₄×C₈	—	26	4	no	—
177	C₂×C₁₃	—	2	2	no	—
178	C₂×C₁₃	—	2	2	no	—
179	C₂×C₁₃	C₄×#95	1	1	yes	characteristic
180	C₂×C₁₃	C₂×#341	1	2	yes	—
181	C₂×C₁₃	C₄×#108	1	4	yes	—
182	C₂⁴	C₂×Dic₂₆	1	1	yes	characteristic
183	C₂²×C₄	—	2	8	no	—
184	C₂²×C₄	#6101	1	2	yes	—
185	C₂²×C₄	#6101	1	4	yes	—
186	C₂²×C₄	#6101	1	2	yes	—
187	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
188	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₄×D₁₃	1	4	yes	—
191	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
192	C₂²×C₄	#6127	1	4	yes	—
193	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
196	M₁₆	—	2	4	no	—
197	M₁₆	—	2	8	no	—
198	M₁₆	—	13	8	no	—
199	M₁₆	—	26	4	no	—
200	M₁₆	—	26	8	no	—
201	M₁₆	C₄×Dic₁₃	1	8	yes	—
202	C₂×C₈	—	2	8	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	13	8	no	—
205	C₂×C₈	—	26	4	no	—
206	C₂×C₈	—	26	8	no	—
207	C₂×C₈	—	26	4	no	—
208	C₂×C₈	—	26	4	no	—
209	C₂×C₈	—	26	8	no	—
210	C₂×C₈	—	26	8	no	—
211	C₄²	C₄×Dic₁₃	1	4	yes	—
212	C₄²	#6103	1	4	yes	—
213	C₄²	#6103	1	2	yes	—
214	C₄²	#6103	1	2	yes	—
215	C₄²	C₂²⋊Dic₁₃	1	2	yes	—
216	C₄²	C₂²⋊Dic₁₃	1	2	yes	—
217	C₁₃	C₄×#341	1	1	yes	characteristic, complemented, Sylow
218	C₂³	—	2	4	no	—
219	C₂³	C₄×Dic₂₆	1	4	yes	—
220	C₂³	C₂×#6101	1	1	yes	characteristic
221	C₂³	C₂×#6101	1	1	yes	characteristic
222	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	8	no	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	#88062	1	8	yes	—
227	C₂×C₄	#88071	1	4	yes	—
228	C₂×C₄	#88071	1	4	yes	—
229	C₂×C₄	C₄²×D₁₃	1	2	yes	—
230	C₂×C₄	C₄²×D₁₃	1	2	yes	—
231	C₂×C₄	#88126	1	2	yes	—
232	C₂×C₄	#88126	1	2	yes	—
233	C₂×C₄	C₄×D₅₂	1	4	yes	—
234	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
235	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
236	C₂×C₄	#88180	1	2	yes	—
237	C₂×C₄	C₂×#6103	1	1	yes	characteristic
238	C₂×C₄	C₂×#6103	1	1	yes	characteristic
239	C₂×C₄	C₄×C₂²⋊D₁₃	1	4	yes	—
240	C₂×C₄	C₄×C₂²⋊D₁₃	1	4	yes	—
241	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
242	C₈	—	2	8	no	—
243	C₈	—	26	8	no	—
244	C₈	—	26	8	no	—
245	C₂²	—	2	8	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	2	no	—
248	C₂²	C₄×#6101	1	2	yes	—
249	C₂²	C₄×#6101	1	2	yes	—
250	C₂²	C₄×C₄⋊Dic₁₃	1	2	yes	—
251	C₂²	C₂×#88062	1	4	yes	—
252	C₂²	C₂×#88071	1	1	yes	characteristic
253	C₄	—	2	4	no	—
254	C₄	#11718150	1	8	yes	direct product factor
255	C₄	C₄²×Dic₁₃	1	1	yes	characteristic
256	C₄	#11718593	1	1	yes	characteristic
257	C₄	C₄×#6103	1	2	yes	—
258	C₄	C₄×#6103	1	2	yes	—
259	C₄	C₄×C₂²⋊Dic₁₃	1	2	yes	—
260	C₂	—	2	2	no	—
261	C₂	—	2	2	no	—
262	C₂	C₄×#88071	1	1	yes	characteristic
263	C₂	C₂×#11718150	1	2	yes	—
264	C₂	C₄×#88062	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²⋊D₁₃
commutator	C₄×C₁₃	C₄³
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₂×C₄×C₁₃×M₁₆	C₂
socle	C₂³×C₁₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₂×Q₈
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	C₂×C₄×M₁₆	D₁₃
hypocenter	C₁₃	C₄×#341
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
20. #11718150	C₄	direct (N × H)
142. C₄×C₁₃	#341	semidirect
217. C₁₃	C₄×#341	semidirect
254. C₄	#11718150	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×C₄×#6099	2
C₂²×C₄×#1055	2
C₂×#11718150	2
C₂×C₄×C₁₃×M₁₆	2
C₄×#341	13

Subgroups — C4×#11718150 #15768105

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11718150 #15768105