Subgroups — C₄×#11718109 #15778178

Order 3328.

434 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₅₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718109	C₂	1	2	yes	maximal
3	C₄×#88059	C₂	1	1	yes	maximal, characteristic
4	#13112945	C₂	1	2	yes	maximal
5	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₂₆	—	2	1	no	—
8	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88059	C₂²	1	2	yes	—
11	C₂×#88058	C₂²	1	1	yes	characteristic
12	C₂×#88056	—	2	1	no	—
13	C₄×#6099	—	2	1	no	—
14	C₄×Dic₅₂	—	2	1	no	—
15	C₄×Dic₅₂	—	2	1	no	—
16	C₄×Dic₅₂	C₄	1	2	yes	—
17	C₄×Dic₅₂	C₄	1	2	yes	—
18	C₁₃×#346	—	2	1	no	—
19	C₄×C₁₃×C₁₆	—	2	1	no	—
20	#11718144	—	2	1	no	—
21	#11718109	C₄	1	8	yes	direct product factor
22	C₂²×Dic₂₆	—	2	1	no	—
23	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
26	C₂×C₄⋊Dic₁₃	—	2	1	no	—
27	C₂×#6101	—	4	1	no	—
28	C₂×C₄×Dic₁₃	—	4	1	no	—
29	C₂×#6099	—	2	2	no	—
30	C₂×Dic₅₂	—	2	1	no	—
31	C₂×Dic₅₂	—	2	2	no	—
32	C₂×Dic₅₂	C₂×C₄	1	2	yes	—
33	C₂×Dic₅₂	C₂×C₄	1	4	yes	—
34	C₄×Dic₂₆	—	2	2	no	—
35	C₄×Dic₂₆	—	2	2	no	—
36	C₄×Dic₂₆	—	4	1	no	—
37	C₄×Dic₂₆	—	4	1	no	—
38	C₁₃×M₃₂	—	2	2	no	—
39	C₁₃×M₃₂	C₂×C₄	1	4	yes	—
40	C₂×C₁₃×C₁₆	—	2	2	no	—
41	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
45	#88059	—	2	2	no	—
46	#88059	C₂×C₄	1	4	yes	—
47	#88058	—	2	1	no	—
48	#88058	C₂×C₄	1	2	yes	—
49	#88056	—	2	2	no	—
50	#88056	—	2	2	no	—
51	#88043	—	4	1	no	—
52	C₄×#1055	—	4	1	no	—
53	C₄×#362	—	13	1	no	maximal, Sylow
54	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₁₃	—	4	1	no	—
58	C₂²×Dic₁₃	—	4	1	no	—
59	C₂×Dic₂₆	—	2	2	no	—
60	C₂×Dic₂₆	—	2	4	no	—
61	C₂×Dic₂₆	—	4	1	no	—
62	C₂×Dic₂₆	—	4	2	no	—
63	C₂×C₈×C₁₃	—	2	2	no	—
64	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₁₃	#41	1	2	yes	—
70	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₁₃	D₈	1	1	yes	characteristic
73	C₄²×C₁₃	D₈	1	1	yes	characteristic
74	C₄⋊Dic₁₃	—	2	2	no	—
75	C₄⋊Dic₁₃	—	4	1	no	—
76	#6101	—	4	2	no	—
77	#6101	—	4	2	no	—
78	C₄×Dic₁₃	—	4	2	no	—
79	C₄×Dic₁₃	—	4	2	no	—
80	#6099	—	2	4	no	—
81	#6099	—	4	2	no	—
82	C₂×#1055	—	4	2	no	—
83	Dic₅₂	—	2	2	no	—
84	Dic₅₂	—	2	4	no	—
85	Dic₅₂	C₄²	1	4	yes	—
86	C₁₃×C₁₆	—	2	4	no	—
87	C₂×C₄×Q₁₆	—	13	1	no	—
88	C₂×#362	—	13	2	no	—
89	C₄×M₃₂	D₁₃	1	1	yes	characteristic
90	C₄×#108	—	13	1	no	—
91	#1387	—	13	2	no	—
92	C₂³×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	4	1	no	—
95	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
96	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
97	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
98	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
101	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₁₃	#135	1	2	yes	—
103	C₂×Dic₁₃	—	4	2	no	—
104	C₂×Dic₁₃	—	4	4	no	—
105	C₂×Dic₁₃	—	4	2	no	—
106	C₂×Dic₁₃	—	8	2	no	—
107	Dic₂₆	—	2	4	no	—
108	Dic₂₆	—	4	2	no	—
109	Dic₂₆	—	4	4	no	—
110	C₈×C₁₃	—	2	1	no	—
111	C₈×C₁₃	—	2	1	no	—
112	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
113	C₈×C₁₃	#117	1	1	yes	characteristic
114	C₈×C₁₃	C₄×D₄	1	2	yes	—
115	#1055	—	4	4	no	—
116	C₂²×Q₁₆	—	13	1	no	—
117	C₂×C₄×Q₈	—	26	1	no	—
118	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
119	C₄×Q₁₆	—	13	2	no	—
120	C₄×Q₁₆	—	13	2	no	—
121	C₄×Q₁₆	—	26	1	no	—
122	C₄×Q₁₆	—	26	1	no	—
123	C₂×#108	—	13	2	no	—
124	C₂×#107	—	13	1	no	—
125	C₂×#103	—	26	1	no	—
126	C₄×M₁₆	—	26	1	no	—
127	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
128	#362	—	13	8	no	—
129	#346	—	2	1	no	—
130	C₄×C₁₆	—	2	1	no	—
131	#341	—	26	1	no	—
132	C₂²×C₁₃	—	2	1	no	—
133	C₂²×C₁₃	—	2	1	no	—
134	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
135	C₂²×C₁₃	C₄×SD₁₆	1	2	yes	—
136	C₄×C₁₃	—	2	1	no	—
137	C₄×C₁₃	—	2	1	no	—
138	C₄×C₁₃	—	2	1	no	—
139	C₄×C₁₃	#362	1	2	yes	complemented
140	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
141	C₄×C₁₃	#416	1	1	yes	characteristic
142	C₄×C₁₃	C₄×D₈	1	2	yes	—
143	Dic₁₃	—	4	4	no	—
144	Dic₁₃	—	8	2	no	—
145	C₂²×Q₈	—	26	1	no	—
146	C₂×Q₁₆	—	13	4	no	—
147	C₂×Q₁₆	—	13	2	no	—
148	C₂×Q₁₆	—	26	2	no	—
149	C₂×Q₁₆	—	26	1	no	—
150	C₂×M₁₆	—	26	2	no	—
151	C₂²×C₈	D₅₂	1	1	yes	characteristic
152	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
153	C₄×Q₈	—	26	2	no	—
154	C₄×Q₈	—	26	2	no	—
155	C₄×Q₈	—	52	1	no	—
156	C₄×Q₈	—	52	1	no	—
157	C₂×C₄⋊C₄	—	26	1	no	—
158	C₂×C₄⋊C₄	—	52	1	no	—
159	C₂×C₄²	—	52	1	no	—
160	C₂×C₄²	D₅₂	1	1	yes	characteristic
161	M₃₂	—	2	2	no	—
162	M₃₂	C₄×D₁₃	1	4	yes	—
163	C₂×C₁₆	—	2	2	no	—
164	#108	—	13	4	no	—
165	#108	—	26	2	no	—
166	#107	—	13	2	no	—
167	#107	—	26	1	no	—
168	#103	—	26	2	no	—
169	#103	—	26	2	no	—
170	#97	—	52	1	no	—
171	C₄×C₈	—	52	1	no	—
172	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
173	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
174	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
175	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
176	C₂×C₁₃	—	2	2	no	—
177	C₂×C₁₃	C₄×#102	1	1	yes	characteristic
178	C₂×C₁₃	C₂×#362	1	1	yes	characteristic
179	C₂×C₁₃	#2148	1	1	yes	characteristic
180	C₂×Q₈	—	26	4	no	—
181	C₂×Q₈	—	26	2	no	—
182	C₂×Q₈	—	52	1	no	—
183	C₂×Q₈	—	52	2	no	—
184	C₂²×C₄	—	52	1	no	—
185	C₂²×C₄	—	52	1	no	—
186	C₂²×C₄	#6095	1	1	yes	characteristic
187	C₂²×C₄	#6095	1	1	yes	characteristic
188	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
189	Q₁₆	—	13	4	no	—
190	Q₁₆	—	26	2	no	—
191	Q₁₆	—	26	4	no	—
192	M₁₆	—	26	4	no	—
193	M₁₆	—	52	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	52	2	no	—
196	C₂×C₈	#6103	1	1	yes	characteristic
197	C₂×C₈	#6103	1	1	yes	characteristic
198	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
199	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
200	C₂×C₈	#6127	1	2	yes	—
201	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
202	C₄⋊C₄	—	26	2	no	—
203	C₄⋊C₄	—	52	1	no	—
204	C₄⋊C₄	—	52	2	no	—
205	C₄⋊C₄	—	52	2	no	—
206	C₄²	—	52	2	no	—
207	C₄²	—	52	2	no	—
208	C₄²	D₁₀₄	1	1	yes	characteristic
209	C₄²	D₁₀₄	1	1	yes	characteristic
210	C₄²	#6103	1	1	yes	characteristic
211	C₄²	#6103	1	1	yes	characteristic
212	C₁₆	—	2	4	no	—
213	C₁₃	C₄×#362	1	1	yes	characteristic, complemented, Sylow
214	C₂³	C₂×#6095	1	1	yes	characteristic
215	Q₈	—	26	4	no	—
216	Q₈	—	52	2	no	—
217	Q₈	—	52	4	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	4	1	no	—
220	C₂×C₄	—	52	2	no	—
221	C₂×C₄	—	52	4	no	—
222	C₂×C₄	—	52	2	no	—
223	C₂×C₄	—	104	2	no	—
224	C₂×C₄	#88061	1	1	yes	characteristic
225	C₂×C₄	#88061	1	1	yes	characteristic
226	C₂×C₄	#88061	1	1	yes	characteristic
227	C₂×C₄	#88061	1	1	yes	characteristic
228	C₂×C₄	C₄×D₅₂	1	2	yes	—
229	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
230	C₂×C₄	#88158	1	2	yes	—
231	C₂×C₄	C₂×#6103	1	1	yes	characteristic
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₈	—	52	4	no	—
235	C₈	C₄²×D₁₃	1	1	yes	characteristic
236	C₈	#88126	1	1	yes	characteristic
237	C₈	C₄×C₂²⋊D₁₃	1	2	yes	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	C₄×#6095	1	2	yes	—
241	C₂²	C₂×#88061	1	1	yes	characteristic
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	52	4	no	—
246	C₄	—	104	2	no	—
247	C₄	#11718109	1	2	yes	direct product factor
248	C₄	C₄×D₁₀₄	1	2	yes	—
249	C₄	C₄×#6103	1	1	yes	characteristic
250	C₄	#11718774	1	1	yes	characteristic
251	C₂	—	2	2	no	—
252	C₂	C₄×#88061	1	1	yes	characteristic
253	C₂	C₂×#11718109	1	1	yes	characteristic
254	C₂	#13116586	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88061
commutator	C₈×C₁₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₄×C₁₃×M₃₂	C₂
socle	C₂²×C₁₃	C₂×#102
prime radical	C₂³×C₁₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₄×M₃₂	D₁₃
hypocenter	C₁₃	C₄×#362
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
21. #11718109	C₄	direct (N × H)
139. C₄×C₁₃	#362	semidirect
213. C₁₃	C₄×#362	semidirect
247. C₄	#11718109	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×Dic₅₂	2
C₂×#11718109	2
C₄×#88059	2
#13112945	2
C₄×C₁₃×M₃₂	2
C₄×#362	13

Subgroups — C4×#11718109 #15778178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11718109 #15778178