Subgroups — C₄×#11718155 #15778172

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×#11718155	C₂	1	2	yes	maximal
2	C₄×#88052	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#88059	C₂	1	1	yes	maximal, characteristic, complemented
4	#13112944	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₄×C₁₃	—	2	1	no	—
8	C₄×C₁₃×D₈	—	2	1	no	—
9	C₄×C₁₃×D₈	—	2	1	no	—
10	C₄×C₁₃×D₈	C₄	1	2	yes	—
11	C₄×C₁₃×D₈	C₄	1	2	yes	—
12	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#102	—	2	1	no	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×#88059	C₂²	1	2	yes	—
16	C₂×#88052	C₂²	1	2	yes	—
17	C₄×#6099	—	2	1	no	—
18	#11718155	C₄	1	8	yes	direct product factor
19	#11718144	—	2	1	no	—
20	#11718053	—	2	1	no	—
21	C₄×#6090	—	2	1	no	—
22	C₂²×D₄×C₁₃	—	2	1	no	—
23	C₂³×C₄×C₁₃	—	4	1	no	—
24	C₂×C₁₃×D₈	—	2	1	no	—
25	C₂×C₁₃×D₈	—	2	2	no	—
26	C₂×C₁₃×D₈	C₂×C₄	1	4	yes	—
27	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
30	C₄×D₄×C₁₃	—	2	2	no	—
31	C₄×D₄×C₁₃	—	2	2	no	—
32	C₄×D₄×C₁₃	—	4	1	no	—
33	C₄×D₄×C₁₃	—	4	1	no	—
34	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
35	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
36	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
37	C₂×#6099	—	2	2	no	—
38	C₁₃×#107	—	2	1	no	—
39	C₁₃×#107	C₂×C₄	1	2	yes	—
40	C₁₃×#102	—	2	2	no	—
41	C₁₃×#102	—	2	2	no	—
42	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
45	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
46	#88059	—	2	2	no	—
47	#88059	C₂×C₄	1	4	yes	complemented
48	#88052	—	2	2	no	—
49	#88052	C₂×C₄	1	4	yes	complemented
50	C₂×#6090	—	2	2	no	—
51	#88043	—	4	1	no	—
52	C₄×#1055	—	4	1	no	—
53	C₄×#361	—	13	1	no	maximal, Sylow
54	C₂⁴×C₁₃	—	4	1	no	—
55	C₂×D₄×C₁₃	—	2	2	no	—
56	C₂×D₄×C₁₃	—	2	4	no	—
57	C₂×D₄×C₁₃	—	4	1	no	—
58	C₂×D₄×C₁₃	—	4	2	no	—
59	C₂²×C₄×C₁₃	—	4	1	no	—
60	C₂²×C₄×C₁₃	—	4	2	no	—
61	C₂²×C₄×C₁₃	—	4	2	no	—
62	C₂²×C₄×C₁₃	—	8	4	no	—
63	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₁₃×D₈	—	2	2	no	—
67	C₁₃×D₈	—	2	4	no	—
68	C₁₃×D₈	C₄²	1	4	yes	—
69	C₂×C₈×C₁₃	—	2	2	no	—
70	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃	#41	1	2	yes	—
76	C₁₃×C₄⋊C₄	—	2	2	no	—
77	C₁₃×C₄⋊C₄	—	4	1	no	—
78	C₁₃×C₂²⋊C₄	—	4	2	no	—
79	C₁₃×C₂²⋊C₄	—	4	2	no	—
80	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₄²×C₁₃	D₈	1	1	yes	characteristic
83	C₄²×C₁₃	D₈	1	1	yes	characteristic
84	#6099	—	2	4	no	—
85	#6099	—	4	2	no	—
86	C₂×#1055	—	4	2	no	—
87	#6090	—	2	4	no	—
88	C₂×C₄×D₈	D₁₃	1	1	yes	characteristic
89	C₂×#361	—	13	2	no	—
90	C₄×M₃₂	—	13	1	no	—
91	C₄×#108	—	13	1	no	—
92	#1387	—	13	2	no	—
93	C₂³×C₁₃	—	4	2	no	—
94	C₂³×C₁₃	—	4	4	no	—
95	C₂³×C₁₃	—	8	2	no	—
96	C₂³×C₁₃	—	8	2	no	—
97	C₂³×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
98	D₄×C₁₃	—	2	4	no	—
99	D₄×C₁₃	—	4	2	no	—
100	D₄×C₁₃	—	4	4	no	—
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	4	1	no	—
103	C₂×C₄×C₁₃	—	4	2	no	—
104	C₂×C₄×C₁₃	—	8	4	no	—
105	C₂×C₄×C₁₃	—	8	2	no	—
106	C₂×C₄×C₁₃	—	8	4	no	—
107	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
108	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
109	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
110	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₁₃	#135	1	2	yes	—
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	C₂×D₁₃	1	1	yes	characteristic
122	C₂×C₄×D₄	—	2	1	no	—
123	C₂×M₃₂	—	13	2	no	—
124	C₄×D₈	—	2	1	no	—
125	C₄×D₈	—	2	1	no	—
126	C₄×D₈	Dic₁₃	1	2	yes	—
127	C₄×D₈	Dic₁₃	1	2	yes	—
128	C₂×#108	—	13	2	no	—
129	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
130	C₂×#102	—	2	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	—	26	1	no	—
135	C₄×C₁₆	—	26	1	no	—
136	#341	—	26	1	no	—
137	C₂²×C₁₃	—	2	1	no	—
138	C₂²×C₁₃	—	2	1	no	—
139	C₂²×C₁₃	—	4	4	no	—
140	C₂²×C₁₃	—	8	8	no	—
141	C₂²×C₁₃	—	8	2	no	—
142	C₂²×C₁₃	—	8	2	no	—
143	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
144	C₂²×C₁₃	C₄×SD₁₆	1	2	yes	—
145	C₄×C₁₃	—	2	1	no	—
146	C₄×C₁₃	—	2	1	no	—
147	C₄×C₁₃	—	2	1	no	—
148	C₄×C₁₃	—	8	2	no	—
149	C₄×C₁₃	#361	1	2	yes	complemented
150	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
151	C₄×C₁₃	#416	1	1	yes	characteristic
152	C₄×C₁₃	C₄×D₈	1	2	yes	—
153	C₂²×D₄	—	2	1	no	—
154	C₂³×C₄	—	4	1	no	—
155	C₂×D₈	—	2	2	no	—
156	C₂×D₈	—	2	1	no	—
157	C₂×D₈	C₄×D₁₃	1	4	yes	—
158	C₂×D₈	C₂×Dic₁₃	1	2	yes	—
159	C₂×M₁₆	—	26	2	no	—
160	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
161	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
162	C₄×D₄	—	2	2	no	—
163	C₄×D₄	—	2	2	no	—
164	C₄×D₄	—	4	1	no	—
165	C₄×D₄	—	4	1	no	—
166	C₂×C₄⋊C₄	—	2	1	no	—
167	C₂×C₂²⋊C₄	—	4	1	no	—
168	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
169	M₃₂	—	13	4	no	—
170	M₃₂	—	26	2	no	—
171	C₂×C₁₆	—	26	2	no	—
172	#108	—	13	4	no	—
173	#108	—	26	2	no	—
174	#107	—	2	1	no	—
175	#107	C₂×Dic₁₃	1	2	yes	—
176	#102	—	2	2	no	—
177	#102	—	2	2	no	—
178	#97	—	52	1	no	—
179	C₄×C₈	—	52	1	no	—
180	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
181	C₄×C₈	C₂×Dic₁₃	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	—	8	4	no	—
186	C₂×C₁₃	C₄×#102	1	1	yes	characteristic
187	C₂×C₁₃	C₂×#361	1	1	yes	characteristic
188	C₂×C₁₃	#2148	1	1	yes	characteristic
189	C₂⁴	—	4	1	no	—
190	C₂×D₄	—	2	4	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	4	1	no	—
193	C₂×D₄	—	4	2	no	—
194	C₂²×C₄	—	4	1	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	—	8	4	no	—
198	C₂²×C₄	#6105	1	1	yes	characteristic
199	C₂²×C₄	#6105	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
201	D₈	—	2	2	no	—
202	D₈	—	2	4	no	—
203	D₈	C₄×Dic₁₃	1	4	yes	—
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	C₂²⋊Dic₁₃	1	1	yes	characteristic
209	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
210	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
211	C₂×C₈	#6127	1	2	yes	—
212	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
213	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	—	4	1	no	—
216	C₂²⋊C₄	—	4	2	no	—
217	C₂²⋊C₄	—	4	2	no	—
218	C₄²	#6104	1	1	yes	characteristic
219	C₄²	#6104	1	1	yes	characteristic
220	C₄²	C₂²⋊Dic₁₃	1	1	yes	characteristic
221	C₄²	C₂²⋊Dic₁₃	1	1	yes	characteristic
222	C₁₆	—	26	4	no	—
223	C₁₃	C₄×#361	1	1	yes	characteristic, complemented, Sylow
224	C₂³	—	4	4	no	—
225	C₂³	—	4	2	no	—
226	C₂³	—	8	2	no	—
227	C₂³	—	8	2	no	—
228	C₂³	C₂×#6105	1	1	yes	characteristic
229	D₄	—	2	4	no	—
230	D₄	—	4	4	no	—
231	D₄	—	4	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	8	4	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	—	8	4	no	—
238	C₂×C₄	#88072	1	1	yes	characteristic
239	C₂×C₄	#88072	1	1	yes	characteristic
240	C₂×C₄	#88072	1	1	yes	characteristic
241	C₂×C₄	#88072	1	1	yes	characteristic
242	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
243	C₂×C₄	C₂×#6104	1	1	yes	characteristic
244	C₂×C₄	#88204	1	2	yes	—
245	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
246	C₈	—	2	1	no	—
247	C₈	—	2	1	no	—
248	C₈	—	52	4	no	—
249	C₈	C₂×C₄×Dic₁₃	1	1	yes	characteristic
250	C₈	#88180	1	1	yes	characteristic
251	C₈	C₄×C₂²⋊D₁₃	1	2	yes	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	4	4	no	—
255	C₂²	—	8	8	no	—
256	C₂²	—	8	2	no	—
257	C₂²	—	8	2	no	—
258	C₂²	C₄×#6105	1	2	yes	—
259	C₂²	C₂×#88072	1	1	yes	characteristic
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	8	2	no	—
264	C₄	#11718155	1	2	yes	direct product factor
265	C₄	C₄×#6104	1	2	yes	—
266	C₄	C₄×C₂²⋊Dic₁₃	1	1	yes	characteristic
267	C₄	#11718894	1	1	yes	characteristic
268	C₂	—	2	2	no	—
269	C₂	—	8	4	no	—
270	C₂	C₄×#88072	1	1	yes	characteristic
271	C₂	C₂×#11718155	1	1	yes	characteristic
272	C₂	#13116585	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88072
commutator	C₈×C₁₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₄×C₁₃×D₈	C₂
socle	C₂²×C₁₃	C₂×#102
prime radical	C₂²×C₁₃×D₈	C₂²
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×C₄×D₈	D₁₃
hypocenter	C₁₃	C₄×#361
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
2. C₄×#88052	C₂	semidirect
3. C₄×#88059	C₂	semidirect
4. #13112944	C₂	semidirect
18. #11718155	C₄	direct (N × H)
47. #88059	C₂×C₄	semidirect
49. #88052	C₂×C₄	semidirect
149. C₄×C₁₃	#361	semidirect
223. C₁₃	C₄×#361	semidirect
264. C₄	#11718155	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×#11718155	2
C₄×#88052	2
C₄×#88059	2
#13112944	2
C₂×C₄×C₁₃×D₈	2
C₄×#361	13

Subgroups — C4×#11718155 #15778172

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11718155 #15778172