Subgroups — C₂×C₄×#88100 #16796719

Order 3328.

1790 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88100	C₂	1	4	yes	maximal
2	C₂×C₄×D₁₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄²×#261	C₂	1	1	yes	maximal, characteristic
4	C₄×#88100	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄×#261	C₄	1	4	yes	—
7	C₂²×C₄×#261	C₂²	1	2	yes	—
8	C₂²×D₁₃⋊C₈	C₂²	1	2	yes	—
9	C₂²×C₈×D₁₃	C₂²	1	2	yes	—
10	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×#1057	C₄	1	4	yes	complemented
12	C₂×#88100	C₄	1	32	yes	direct product factor
13	C₂×#88100	C₂²	1	16	yes	—
14	C₄²×#261	C₄	1	4	yes	—
15	C₄²×#261	C₂²	1	4	yes	—
16	C₄×D₁₃⋊C₈	C₄	1	4	yes	complemented
17	C₄×D₁₃⋊C₈	C₂²	1	4	yes	—
18	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic, complemented
19	C₂×C₈×Dic₁₃	C₄	1	2	yes	complemented
20	C₂×C₄×#1055	C₄	1	1	yes	characteristic, complemented
21	C₄×C₈×D₁₃	C₄	1	4	yes	complemented
22	C₄×C₈×D₁₃	C₂²	1	4	yes	—
23	C₂³×#261	C₂×C₄	1	4	yes	—
24	C₂²×C₄×D₁₃	C₂×C₄	1	2	yes	—
25	C₂²×C₄×D₁₃	C₂³	1	1	yes	characteristic
26	C₂²×#1057	C₂×C₄	1	4	yes	—
27	C₂×C₄×#261	—	2	16	no	—
28	C₂×C₄×#261	C₂×C₄	1	16	yes	—
29	C₂×C₄×#261	C₂×C₄	1	4	yes	—
30	C₂×C₄×#261	C₂×C₄	1	16	yes	—
31	C₂×C₄×#261	C₂³	1	4	yes	—
32	C₂×D₁₃⋊C₈	C₂×C₄	1	4	yes	—
33	C₂×D₁₃⋊C₈	C₂×C₄	1	16	yes	—
34	C₂×D₁₃⋊C₈	C₂³	1	4	yes	—
35	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
39	C₂²×#1055	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
41	C₂×C₈×D₁₃	C₂×C₄	1	16	yes	—
42	C₂×C₈×D₁₃	C₂³	1	4	yes	—
43	C₄²×D₁₃	C₂×C₄	1	2	yes	—
44	C₄²×D₁₃	C₂×C₄	1	2	yes	—
45	C₄²×D₁₃	C₂×C₄	1	2	yes	—
46	C₄²×D₁₃	C₂³	1	2	yes	—
47	C₄×#1057	C₂×C₄	1	16	yes	complemented
48	#88100	C₂×C₄	1	64	yes	direct product factor
49	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	complemented
50	C₈×Dic₁₃	C₂×C₄	1	8	yes	complemented
51	C₄×#1055	C₂×C₄	1	4	yes	complemented
52	C₂×C₄×#97	—	13	1	no	maximal, Sylow
53	C₂³×D₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×#261	—	2	16	no	—
55	C₂²×#261	C₄²	1	16	yes	—
56	C₂²×#261	C₂²×C₄	1	8	yes	—
57	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
58	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×Dic₁₃	C₄²	1	2	yes	—
60	C₂²×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×D₁₃	C₄²	1	4	yes	—
62	C₂×C₄×D₁₃	M₁₆	1	16	yes	—
63	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×D₁₃	C₂²×C₄	1	8	yes	—
67	C₂×C₄×D₁₃	C₂²×C₄	1	4	yes	—
68	C₂×C₄×D₁₃	C₂⁴	1	1	yes	characteristic
69	C₂×#1057	C₄²	1	16	yes	complemented
70	C₂×#1057	C₂²×C₄	1	8	yes	—
71	C₄×#261	—	2	32	no	—
72	C₄×#261	C₄²	1	16	yes	—
73	C₄×#261	C₂²×C₄	1	16	yes	—
74	D₁₃⋊C₈	C₄²	1	16	yes	complemented
75	D₁₃⋊C₈	C₂²×C₄	1	16	yes	—
76	C₂×C₈×C₁₃	C₄²	1	8	yes	complemented
77	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
78	C₄²×C₁₃	C₄²	1	2	yes	—
79	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
80	C₄×Dic₁₃	C₄²	1	4	yes	—
81	C₄×Dic₁₃	C₄²	1	2	yes	—
82	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
83	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
84	C₂×#1055	C₄²	1	8	yes	complemented
85	C₂×#1055	C₂²×C₄	1	4	yes	—
86	C₈×D₁₃	C₄²	1	16	yes	complemented
87	C₈×D₁₃	C₂²×C₄	1	16	yes	—
88	C₂²×#97	—	13	4	no	—
89	C₂²×C₄×C₈	—	13	1	no	—
90	C₂²×C₄×C₈	—	13	1	no	—
91	C₂×C₄³	—	13	1	no	—
92	C₄×#97	—	13	8	no	—
93	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
94	C₂²×D₁₃	C₂×C₄²	1	1	yes	characteristic
95	C₂²×D₁₃	C₂×C₄²	1	4	yes	—
96	C₂²×D₁₃	C₂×M₁₆	1	8	yes	—
97	C₂²×D₁₃	C₂³×C₄	1	1	yes	characteristic
98	C₂×#261	—	2	16	no	—
99	C₂×#261	—	2	32	no	—
100	C₂×#261	C₂×C₄²	1	16	yes	—
101	C₂×C₄×C₁₃	#97	1	8	yes	complemented
102	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
104	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
105	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
106	C₂×Dic₁₃	#97	1	8	yes	complemented
107	C₂×Dic₁₃	C₂×C₄²	1	1	yes	characteristic
108	C₂×Dic₁₃	C₂×C₄²	1	4	yes	—
109	C₂×Dic₁₃	C₂×C₄²	1	4	yes	—
110	C₂×Dic₁₃	C₂³×C₄	1	1	yes	characteristic
111	C₄×D₁₃	#97	1	16	yes	complemented
112	C₄×D₁₃	C₂×C₄²	1	4	yes	—
113	C₄×D₁₃	C₂×C₄²	1	4	yes	—
114	C₄×D₁₃	C₂×C₄²	1	4	yes	—
115	C₄×D₁₃	C₂×M₁₆	1	16	yes	—
116	C₄×D₁₃	C₂³×C₄	1	4	yes	—
117	#1057	C₂×C₄²	1	16	yes	complemented
118	C₈×C₁₃	C₂×C₄²	1	8	yes	complemented
119	#1055	C₂×C₄²	1	8	yes	complemented
120	C₂³×C₈	—	13	2	no	—
121	C₂³×C₈	—	13	2	no	—
122	C₂²×C₄²	—	13	2	no	—
123	C₂²×C₄²	—	13	1	no	—
124	C₂²×C₄²	—	13	4	no	—
125	C₂×#97	—	13	32	no	—
126	C₂×#97	—	13	16	no	—
127	C₂×C₄×C₈	—	13	4	no	—
128	C₂×C₄×C₈	—	13	4	no	—
129	C₂×C₄×C₈	—	13	4	no	—
130	C₂×C₄×C₈	—	13	2	no	—
131	C₂×C₄×C₈	—	13	1	no	—
132	C₂×C₄×C₈	—	13	4	no	—
133	C₂×C₄×C₈	—	13	4	no	—
134	C₂×C₄×C₈	#261	1	1	yes	characteristic, complemented
135	C₄³	—	13	4	no	—
136	C₄³	—	13	4	no	—
137	C₂²×C₁₃	C₄³	1	2	yes	—
138	C₂²×C₁₃	C₂×#97	1	4	yes	—
139	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
140	C₂×D₁₃	C₄³	1	2	yes	—
141	C₂×D₁₃	C₂×#97	1	4	yes	—
142	C₂×D₁₃	C₄×M₁₆	1	16	yes	—
143	C₂×D₁₃	C₂²×C₄²	1	2	yes	—
144	C₂×D₁₃	C₂²×M₁₆	1	4	yes	—
145	#261	—	2	16	no	—
146	C₄×C₁₃	C₄³	1	2	yes	—
147	C₄×C₁₃	C₂×#97	1	8	yes	complemented
148	C₄×C₁₃	C₂²×C₄²	1	2	yes	—
149	Dic₁₃	C₄³	1	2	yes	—
150	Dic₁₃	C₂×#97	1	8	yes	complemented
151	Dic₁₃	C₂²×C₄²	1	2	yes	—
152	C₂³×C₄	—	13	2	no	—
153	C₂³×C₄	—	13	1	no	—
154	C₂³×C₄	—	13	4	no	—
155	C₂²×C₈	—	13	4	no	—
156	C₂²×C₈	—	13	4	no	—
157	C₂²×C₈	—	13	4	no	—
158	C₂²×C₈	—	13	16	no	—
159	C₂²×C₈	—	13	2	no	—
160	C₂²×C₈	—	13	16	no	—
161	C₂²×C₈	—	13	4	no	—
162	C₂²×C₈	—	13	4	no	—
163	C₂²×C₈	C₂×#261	1	2	yes	—
164	C₂×C₄²	—	13	4	no	—
165	C₂×C₄²	—	13	4	no	—
166	C₂×C₄²	—	13	16	no	—
167	C₂×C₄²	—	13	2	no	—
168	C₂×C₄²	—	13	1	no	—
169	C₂×C₄²	—	13	2	no	—
170	C₂×C₄²	—	13	2	no	—
171	C₂×C₄²	—	13	2	no	—
172	C₂×C₄²	—	13	2	no	—
173	C₂×C₄²	—	13	16	no	—
174	C₂×C₄²	—	26	16	no	—
175	C₂×C₄²	C₂×#261	1	1	yes	characteristic
176	#97	—	13	64	no	—
177	C₄×C₈	—	13	16	no	—
178	C₄×C₈	—	13	8	no	—
179	C₄×C₈	—	13	4	no	—
180	C₄×C₈	C₂×#261	1	4	yes	complemented
181	C₂×C₁₃	C₄×#97	1	4	yes	complemented
182	C₂×C₁₃	C₂×C₄³	1	1	yes	characteristic
183	C₂×C₁₃	C₂²×#97	1	2	yes	—
184	D₁₃	C₄×#97	1	4	yes	complemented
185	D₁₃	C₂×C₄×M₁₆	1	4	yes	—
186	C₂⁴	—	13	1	no	—
187	C₂²×C₄	—	13	1	no	—
188	C₂²×C₄	—	13	2	no	—
189	C₂²×C₄	—	13	16	no	—
190	C₂²×C₄	—	13	1	no	—
191	C₂²×C₄	—	13	4	no	—
192	C₂²×C₄	—	13	8	no	—
193	C₂²×C₄	—	13	1	no	—
194	C₂²×C₄	—	13	4	no	—
195	C₂²×C₄	—	13	1	no	—
196	C₂²×C₄	—	13	1	no	—
197	C₂²×C₄	—	13	8	no	—
198	C₂²×C₄	—	13	16	no	—
199	C₂²×C₄	—	26	16	no	—
200	C₂²×C₄	C₄×#261	1	2	yes	—
201	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
202	C₂×C₈	—	13	8	no	—
203	C₂×C₈	—	13	16	no	—
204	C₂×C₈	—	13	16	no	—
205	C₂×C₈	—	13	16	no	—
206	C₂×C₈	—	13	8	no	—
207	C₂×C₈	—	13	4	no	—
208	C₂×C₈	—	13	16	no	—
209	C₂×C₈	—	13	16	no	—
210	C₂×C₈	C₄×#261	1	8	yes	complemented
211	C₂×C₈	C₂²×#261	1	4	yes	—
212	C₄²	—	13	16	no	—
213	C₄²	—	13	16	no	—
214	C₄²	—	13	4	no	—
215	C₄²	—	13	2	no	—
216	C₄²	—	13	4	no	—
217	C₄²	—	13	2	no	—
218	C₄²	—	26	32	no	—
219	C₄²	C₄×#261	1	2	yes	—
220	C₄²	C₂²×#261	1	2	yes	—
221	C₁₃	C₂×C₄×#97	1	1	yes	characteristic, complemented, Sylow
222	C₂³	—	13	4	no	—
223	C₂³	—	13	8	no	—
224	C₂³	—	13	1	no	—
225	C₂³	—	13	1	no	—
226	C₂³	C₂×C₄×#261	1	1	yes	characteristic
227	C₂×C₄	—	13	4	no	—
228	C₂×C₄	—	13	8	no	—
229	C₂×C₄	—	13	1	no	—
230	C₂×C₄	—	13	4	no	—
231	C₂×C₄	—	13	1	no	—
232	C₂×C₄	—	13	4	no	—
233	C₂×C₄	—	13	4	no	—
234	C₂×C₄	—	13	16	no	—
235	C₂×C₄	—	13	16	no	—
236	C₂×C₄	—	13	4	no	—
237	C₂×C₄	—	13	4	no	—
238	C₂×C₄	—	13	16	no	—
239	C₂×C₄	—	26	16	no	—
240	C₂×C₄	—	26	32	no	—
241	C₂×C₄	#88100	1	8	yes	direct product factor
242	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
243	C₂×C₄	C₂×C₄×#261	1	4	yes	—
244	C₂×C₄	C₂×C₄×#261	1	4	yes	—
245	C₂×C₄	C₂³×#261	1	1	yes	characteristic
246	C₈	—	13	16	no	—
247	C₈	—	13	8	no	—
248	C₈	C₂×C₄×#261	1	8	yes	complemented
249	C₂²	—	13	4	no	—
250	C₂²	—	13	4	no	—
251	C₂²	—	13	16	no	—
252	C₂²	—	13	2	no	—
253	C₂²	—	13	2	no	—
254	C₂²	C₄²×#261	1	2	yes	—
255	C₂²	C₂×#88100	1	4	yes	—
256	C₂²	C₂²×C₄×#261	1	1	yes	characteristic
257	C₄	—	13	8	no	—
258	C₄	—	13	2	no	—
259	C₄	—	13	2	no	—
260	C₄	—	26	16	no	—
261	C₄	C₄²×#261	1	2	yes	—
262	C₄	C₂×#88100	1	8	yes	direct product factor
263	C₄	C₂²×C₄×#261	1	2	yes	—
264	C₂	—	13	4	no	—
265	C₂	—	13	4	no	—
266	C₂	C₄×#88100	1	4	yes	direct product factor
267	C₂	C₂×C₄²×#261	1	1	yes	characteristic
268	C₂	C₂²×#88100	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂×#261
commutator	C₂×C₁₃	C₂×C₄³
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂³×C₁₃	C₂×C₄²
prime radical	C₂³×D₁₃	C₂²×C₄
cosocle	C₂×C₄×D₁₃	C₂⁴
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	C₂×C₄×#97
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
4. C₄×#88100	C₂	direct (N × H)
11. C₂×C₄×#1057	C₄	semidirect
12. C₂×#88100	C₄	direct (N × H)
16. C₄×D₁₃⋊C₈	C₄	semidirect
18. C₂×C₄×C₈×C₁₃	C₄	semidirect
19. C₂×C₈×Dic₁₃	C₄	semidirect
20. C₂×C₄×#1055	C₄	semidirect
21. C₄×C₈×D₁₃	C₄	semidirect
47. C₄×#1057	C₂×C₄	semidirect
48. #88100	C₂×C₄	direct (N × H)
49. C₄×C₈×C₁₃	C₂×C₄	semidirect
50. C₈×Dic₁₃	C₂×C₄	semidirect
51. C₄×#1055	C₂×C₄	semidirect
69. C₂×#1057	C₄²	semidirect
74. D₁₃⋊C₈	C₄²	semidirect
76. C₂×C₈×C₁₃	C₄²	semidirect
84. C₂×#1055	C₄²	semidirect
86. C₈×D₁₃	C₄²	semidirect
101. C₂×C₄×C₁₃	#97	semidirect
106. C₂×Dic₁₃	#97	semidirect
111. C₄×D₁₃	#97	semidirect
117. #1057	C₂×C₄²	semidirect
118. C₈×C₁₃	C₂×C₄²	semidirect
119. #1055	C₂×C₄²	semidirect
134. C₂×C₄×C₈	#261	semidirect
147. C₄×C₁₃	C₂×#97	semidirect
150. Dic₁₃	C₂×#97	semidirect
180. C₄×C₈	C₂×#261	semidirect
181. C₂×C₁₃	C₄×#97	semidirect
184. D₁₃	C₄×#97	semidirect
210. C₂×C₈	C₄×#261	semidirect
221. C₁₃	C₂×C₄×#97	semidirect
241. C₂×C₄	#88100	direct (N × H)
248. C₈	C₂×C₄×#261	semidirect
262. C₄	C₂×#88100	direct (N × H)
266. C₂	C₄×#88100	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#88100	2
C₂×C₄×D₁₃⋊C₈	2
C₂×C₄²×#261	2
C₄×#88100	2
C₂×C₄×C₈×D₁₃	2
C₂×C₄×#97	13

Subgroups — C2×C4×#88100 #16796719

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#88100 #16796719