Subgroups — C₄×#11718108 #15778173

Order 3328.

498 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₁₀₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718108	C₂	1	2	yes	maximal
3	C₄×#88059	C₂	1	1	yes	maximal, characteristic, complemented
4	#13112945	C₂	1	2	yes	maximal, complemented
5	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₀₄	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₅₂	—	2	1	no	—
8	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88061	—	2	1	no	—
11	C₂×#88059	C₂²	1	2	yes	—
12	C₂×#88058	C₂²	1	1	yes	characteristic
13	C₄×#6099	—	2	1	no	—
14	C₄×D₁₀₄	—	2	1	no	—
15	C₄×D₁₀₄	—	2	1	no	—
16	C₄×D₁₀₄	C₄	1	2	yes	—
17	C₄×D₁₀₄	C₄	1	2	yes	—
18	C₁₃×#346	—	2	1	no	—
19	C₄×C₁₃×C₁₆	—	2	1	no	—
20	#11718144	—	2	1	no	—
21	#11718108	C₄	1	8	yes	direct product factor
22	C₂²×D₅₂	—	2	1	no	—
23	C₂²×C₄×D₁₃	—	4	1	no	—
24	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
27	C₂×#6103	—	4	1	no	—
28	C₂×C₄⋊Dic₁₃	—	2	1	no	—
29	C₂×#6099	—	2	2	no	—
30	C₂×D₁₀₄	—	2	1	no	—
31	C₂×D₁₀₄	—	2	2	no	—
32	C₂×D₁₀₄	C₂×C₄	1	2	yes	—
33	C₂×D₁₀₄	C₂×C₄	1	4	yes	—
34	C₄×D₅₂	—	2	2	no	—
35	C₄×D₅₂	—	2	2	no	—
36	C₄×D₅₂	—	4	1	no	—
37	C₄×D₅₂	—	4	1	no	—
38	C₁₃×M₃₂	—	2	2	no	—
39	C₁₃×M₃₂	C₂×C₄	1	4	yes	complemented
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	#88061	—	2	2	no	—
46	#88061	—	2	2	no	—
47	#88059	—	2	2	no	—
48	#88059	C₂×C₄	1	4	yes	complemented
49	#88058	—	2	1	no	—
50	#88058	C₂×C₄	1	2	yes	—
51	#88043	—	4	1	no	—
52	C₄×#1055	—	4	1	no	—
53	C₄×#361	—	13	1	no	maximal, Sylow
54	C₂³×D₁₃	—	4	1	no	—
55	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×Dic₁₃	—	4	1	no	—
59	C₂×D₅₂	—	2	2	no	—
60	C₂×D₅₂	—	2	4	no	—
61	C₂×D₅₂	—	4	1	no	—
62	C₂×D₅₂	—	4	2	no	—
63	C₂×C₄×D₁₃	—	4	2	no	—
64	C₂×C₄×D₁₃	—	4	2	no	—
65	C₂×C₄×D₁₃	—	8	4	no	—
66	C₂×C₈×C₁₃	—	2	2	no	—
67	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₁₃	#41	1	2	yes	—
73	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₄²×C₁₃	D₈	1	1	yes	characteristic
76	C₄²×C₁₃	D₈	1	1	yes	characteristic
77	#6103	—	4	2	no	—
78	#6103	—	4	2	no	—
79	C₄⋊Dic₁₃	—	2	2	no	—
80	C₄⋊Dic₁₃	—	4	1	no	—
81	#6099	—	2	4	no	—
82	#6099	—	4	2	no	—
83	C₂×#1055	—	4	2	no	—
84	D₁₀₄	—	2	2	no	—
85	D₁₀₄	—	2	4	no	—
86	D₁₀₄	C₄²	1	4	yes	—
87	C₁₃×C₁₆	—	2	4	no	—
88	C₂×C₄×D₈	—	13	1	no	—
89	C₂×#361	—	13	2	no	—
90	C₄×M₃₂	D₁₃	1	1	yes	characteristic, complemented
91	C₄×#108	—	13	1	no	—
92	#1387	—	13	2	no	—
93	C₂³×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
94	C₂²×D₁₃	—	4	2	no	—
95	C₂²×D₁₃	—	4	4	no	—
96	C₂²×D₁₃	—	8	2	no	—
97	C₂²×D₁₃	—	8	2	no	—
98	C₂×C₄×C₁₃	—	2	2	no	—
99	C₂×C₄×C₁₃	—	4	1	no	—
100	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
101	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
102	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
103	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₁₃	#135	1	2	yes	—
108	C₂×Dic₁₃	—	4	2	no	—
109	C₂×Dic₁₃	—	8	2	no	—
110	D₅₂	—	2	4	no	—
111	D₅₂	—	4	2	no	—
112	D₅₂	—	4	4	no	—
113	C₄×D₁₃	—	8	4	no	—
114	C₄×D₁₃	—	8	4	no	—
115	C₈×C₁₃	—	2	1	no	—
116	C₈×C₁₃	—	2	1	no	—
117	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
118	C₈×C₁₃	#117	1	1	yes	characteristic
119	C₈×C₁₃	C₄×D₄	1	2	yes	—
120	#1055	—	4	4	no	—
121	C₂²×D₈	—	13	1	no	—
122	C₂×C₄×D₄	—	26	1	no	—
123	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
124	C₄×D₈	—	13	2	no	—
125	C₄×D₈	—	13	2	no	—
126	C₄×D₈	—	26	1	no	—
127	C₄×D₈	—	26	1	no	—
128	C₂×#108	—	13	2	no	—
129	C₂×#107	—	13	1	no	—
130	C₂×#102	—	26	1	no	—
131	C₄×M₁₆	—	26	1	no	—
132	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
133	#361	—	13	8	no	—
134	#346	—	2	1	no	—
135	C₄×C₁₆	—	2	1	no	—
136	#341	—	26	1	no	—
137	C₂²×C₁₃	—	2	1	no	—
138	C₂²×C₁₃	—	2	1	no	—
139	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
140	C₂²×C₁₃	C₄×SD₁₆	1	2	yes	—
141	C₂×D₁₃	—	4	4	no	—
142	C₂×D₁₃	—	8	8	no	—
143	C₂×D₁₃	—	8	2	no	—
144	C₂×D₁₃	—	8	2	no	—
145	C₄×C₁₃	—	2	1	no	—
146	C₄×C₁₃	—	2	1	no	—
147	C₄×C₁₃	—	2	1	no	—
148	C₄×C₁₃	#361	1	2	yes	complemented
149	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
150	C₄×C₁₃	#416	1	1	yes	characteristic
151	C₄×C₁₃	C₄×D₈	1	2	yes	—
152	Dic₁₃	—	8	2	no	—
153	C₂²×D₄	—	26	1	no	—
154	C₂³×C₄	—	52	1	no	—
155	C₂×D₈	—	13	4	no	—
156	C₂×D₈	—	13	2	no	—
157	C₂×D₈	—	26	2	no	—
158	C₂×D₈	—	26	1	no	—
159	C₂×M₁₆	—	26	2	no	—
160	C₂²×C₈	D₅₂	1	1	yes	characteristic
161	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
162	C₄×D₄	—	26	2	no	—
163	C₄×D₄	—	26	2	no	—
164	C₄×D₄	—	52	1	no	—
165	C₄×D₄	—	52	1	no	—
166	C₂×C₄⋊C₄	—	26	1	no	—
167	C₂×C₂²⋊C₄	—	52	1	no	—
168	C₂×C₄²	D₅₂	1	1	yes	characteristic
169	M₃₂	—	2	2	no	—
170	M₃₂	C₄×D₁₃	1	4	yes	complemented
171	C₂×C₁₆	—	2	2	no	—
172	#108	—	13	4	no	—
173	#108	—	26	2	no	—
174	#107	—	13	2	no	—
175	#107	—	26	1	no	—
176	#102	—	26	2	no	—
177	#102	—	26	2	no	—
178	#97	—	52	1	no	—
179	C₄×C₈	—	52	1	no	—
180	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
181	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
182	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
183	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
184	C₂×C₁₃	—	2	2	no	—
185	C₂×C₁₃	C₄×#102	1	1	yes	characteristic
186	C₂×C₁₃	C₂×#361	1	1	yes	characteristic
187	C₂×C₁₃	#2148	1	1	yes	characteristic
188	D₁₃	—	8	4	no	—
189	C₂⁴	—	52	1	no	—
190	C₂×D₄	—	26	4	no	—
191	C₂×D₄	—	26	2	no	—
192	C₂×D₄	—	52	1	no	—
193	C₂×D₄	—	52	2	no	—
194	C₂²×C₄	—	52	1	no	—
195	C₂²×C₄	—	52	2	no	—
196	C₂²×C₄	—	52	2	no	—
197	C₂²×C₄	—	104	4	no	—
198	C₂²×C₄	#6095	1	1	yes	characteristic
199	C₂²×C₄	#6095	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
201	D₈	—	13	4	no	—
202	D₈	—	26	2	no	—
203	D₈	—	26	4	no	—
204	M₁₆	—	26	4	no	—
205	M₁₆	—	52	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	52	2	no	—
208	C₂×C₈	#6103	1	1	yes	characteristic
209	C₂×C₈	#6103	1	1	yes	characteristic
210	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
211	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
212	C₂×C₈	#6127	1	2	yes	—
213	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
214	C₄⋊C₄	—	26	2	no	—
215	C₄⋊C₄	—	52	1	no	—
216	C₂²⋊C₄	—	52	2	no	—
217	C₂²⋊C₄	—	52	2	no	—
218	C₄²	D₁₀₄	1	1	yes	characteristic
219	C₄²	D₁₀₄	1	1	yes	characteristic
220	C₄²	#6103	1	1	yes	characteristic
221	C₄²	#6103	1	1	yes	characteristic
222	C₁₆	—	2	4	no	—
223	C₁₃	C₄×#361	1	1	yes	characteristic, complemented, Sylow
224	C₂³	—	52	2	no	—
225	C₂³	—	52	4	no	—
226	C₂³	—	104	2	no	—
227	C₂³	—	104	2	no	—
228	C₂³	C₂×#6095	1	1	yes	characteristic
229	D₄	—	26	4	no	—
230	D₄	—	52	2	no	—
231	D₄	—	52	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	52	2	no	—
235	C₂×C₄	—	104	4	no	—
236	C₂×C₄	—	104	2	no	—
237	C₂×C₄	—	104	4	no	—
238	C₂×C₄	#88061	1	1	yes	characteristic
239	C₂×C₄	#88061	1	1	yes	characteristic
240	C₂×C₄	#88061	1	1	yes	characteristic
241	C₂×C₄	#88061	1	1	yes	characteristic
242	C₂×C₄	C₄×D₅₂	1	2	yes	—
243	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
244	C₂×C₄	#88158	1	2	yes	—
245	C₂×C₄	C₂×#6103	1	1	yes	characteristic
246	C₈	—	2	1	no	—
247	C₈	—	2	1	no	—
248	C₈	—	52	4	no	—
249	C₈	C₄²×D₁₃	1	1	yes	characteristic
250	C₈	#88126	1	1	yes	characteristic
251	C₈	C₄×C₂²⋊D₁₃	1	2	yes	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	52	4	no	—
255	C₂²	—	104	8	no	—
256	C₂²	—	104	2	no	—
257	C₂²	—	104	2	no	—
258	C₂²	C₄×#6095	1	2	yes	—
259	C₂²	C₂×#88061	1	1	yes	characteristic
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	104	2	no	—
264	C₄	#11718108	1	2	yes	direct product factor
265	C₄	C₄×D₁₀₄	1	2	yes	—
266	C₄	C₄×#6103	1	1	yes	characteristic
267	C₄	#11718774	1	1	yes	characteristic
268	C₂	—	2	2	no	—
269	C₂	—	104	4	no	—
270	C₂	C₄×#88061	1	1	yes	characteristic
271	C₂	C₂×#11718108	1	1	yes	characteristic
272	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	C₂²×D₁₀₄	C₂²
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₄×M₃₂	D₁₃
hypocenter	C₁₃	C₄×#361
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
3. C₄×#88059	C₂	semidirect
4. #13112945	C₂	semidirect
5. C₄×C₁₃×M₃₂	C₂	semidirect
21. #11718108	C₄	direct (N × H)
39. C₁₃×M₃₂	C₂×C₄	semidirect
48. #88059	C₂×C₄	semidirect
90. C₄×M₃₂	D₁₃	semidirect
148. C₄×C₁₃	#361	semidirect
170. M₃₂	C₄×D₁₃	semidirect
223. C₁₃	C₄×#361	semidirect
264. C₄	#11718108	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×D₁₀₄	2
C₂×#11718108	2
C₄×#88059	2
#13112945	2
C₄×C₁₃×M₃₂	2
C₄×#361	13

Subgroups — C4×#11718108 #15778173

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11718108 #15778173