Subgroups — C₂×#13116980 #16017525

Order 3328.

638 conjugacy classes of subgroups.

The subgroup lattice has 260 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	#13116980	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11718082	C₂	1	2	yes	maximal, complemented
4	C₂×C₄⋊#6090	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718083	C₂	1	2	yes	maximal
6	C₂×C₁₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
8	C₂²×D₁₀₄	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₅₂	—	2	1	no	—
10	C₁₃×#500	—	2	2	no	—
11	C₁₃×#500	C₂²	1	4	yes	complemented
12	C₂×C₁₃×#107	—	2	1	no	—
13	C₂×C₁₃×#107	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×#88061	—	2	1	no	—
16	C₂×#88058	C₂²	1	1	yes	characteristic
17	C₂²×#6090	C₂²	1	2	yes	—
18	C₄×D₁₀₄	—	2	2	no	—
19	C₄×D₁₀₄	C₂²	1	4	yes	—
20	#11718083	C₂²	1	8	yes	—
21	#11718082	C₂²	1	8	yes	complemented
22	C₄⋊#6090	C₂²	1	4	yes	complemented
23	C₂²×Q₈×C₁₃	—	4	1	no	—
24	C₂²×D₅₂	—	2	1	no	—
25	C₂²×C₄×D₁₃	—	4	1	no	—
26	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
28	C₁₃×C₄⋊Q₈	—	2	4	no	—
29	C₁₃×C₄⋊Q₈	—	4	2	no	—
30	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
31	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
32	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
33	C₂×#6103	—	4	1	no	—
34	C₂×C₄⋊Dic₁₃	—	2	1	no	—
35	C₂×D₁₀₄	—	2	1	no	—
36	C₂×D₁₀₄	—	2	2	no	—
37	C₂×D₁₀₄	Q₈	1	4	yes	complemented
38	C₂×D₁₀₄	C₂³	1	2	yes	—
39	C₄×D₅₂	—	2	4	no	—
40	C₄×D₅₂	—	4	2	no	—
41	C₁₃×#107	—	2	4	no	—
42	C₁₃×#107	—	2	2	no	—
43	C₁₃×#107	C₂³	1	4	yes	—
44	C₄×C₈×C₁₃	D₄	1	2	yes	—
45	C₄×C₈×C₁₃	C₂³	1	2	yes	—
46	#88061	—	2	4	no	—
47	#88058	—	2	1	no	—
48	#88058	C₂³	1	2	yes	—
49	C₂×#6090	—	2	4	no	—
50	C₂×#6090	C₂³	1	4	yes	—
51	C₂×#2226	—	13	1	no	maximal, Sylow
52	C₂³×D₁₃	—	4	1	no	—
53	C₂×Q₈×C₁₃	—	4	4	no	—
54	C₂×Q₈×C₁₃	—	8	4	no	—
55	C₂²×C₄×C₁₃	—	4	2	no	—
56	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×Dic₁₃	—	4	1	no	—
60	C₂×D₅₂	—	2	4	no	—
61	C₂×D₅₂	—	2	2	no	—
62	C₂×D₅₂	—	4	1	no	—
63	C₂×D₅₂	—	4	2	no	—
64	C₂×C₄×D₁₃	—	4	4	no	—
65	C₂×C₄×D₁₃	—	8	4	no	—
66	C₂×C₈×C₁₃	—	2	2	no	—
67	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
68	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
70	C₂×C₈×C₁₃	#41	1	2	yes	—
71	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
72	C₁₃×C₄⋊C₄	—	2	4	no	—
73	C₁₃×C₄⋊C₄	—	4	2	no	—
74	C₁₃×C₄⋊C₄	—	4	4	no	—
75	C₄²×C₁₃	D₈	1	2	yes	—
76	C₄²×C₁₃	C₂×D₄	1	2	yes	—
77	#6103	—	4	4	no	—
78	C₄⋊Dic₁₃	—	2	2	no	—
79	C₄⋊Dic₁₃	—	4	1	no	—
80	D₁₀₄	—	2	4	no	—
81	D₁₀₄	—	2	2	no	—
82	D₁₀₄	C₂×Q₈	1	4	yes	complemented
83	#6090	—	2	4	no	—
84	C₂×#500	D₁₃	1	1	yes	characteristic, complemented
85	C₂×C₄×D₈	—	13	1	no	—
86	#2226	—	13	8	no	—
87	C₂×#367	—	13	2	no	—
88	C₂×C₄⋊C₁₆	—	13	1	no	—
89	C₂×#357	—	13	2	no	—
90	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
91	C₂²×D₁₃	—	4	2	no	—
92	C₂²×D₁₃	—	4	4	no	—
93	C₂²×D₁₃	—	8	2	no	—
94	C₂²×D₁₃	—	8	2	no	—
95	Q₈×C₁₃	—	8	8	no	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	—	4	4	no	—
98	C₂×C₄×C₁₃	—	8	4	no	—
99	C₂×C₄×C₁₃	SD₃₂	1	4	yes	—
100	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
101	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
102	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
104	C₂×C₄×C₁₃	#137	1	2	yes	—
105	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
106	C₂×Dic₁₃	—	4	2	no	—
107	C₂×Dic₁₃	—	8	2	no	—
108	D₅₂	—	2	4	no	—
109	D₅₂	—	4	2	no	—
110	D₅₂	—	4	4	no	—
111	C₄×D₁₃	—	8	8	no	—
112	C₈×C₁₃	—	2	2	no	—
113	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
114	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
115	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
116	C₂²×D₈	—	13	1	no	—
117	C₂×C₄⋊Q₈	—	2	1	no	—
118	C₂×C₄×D₄	—	26	1	no	—
119	C₂²×C₁₆	—	13	2	no	—
120	#500	—	2	2	no	—
121	#500	C₂×D₁₃	1	4	yes	complemented
122	C₄×D₈	—	13	4	no	—
123	C₄×D₈	—	26	2	no	—
124	C₂×#107	—	2	1	no	—
125	C₂×#107	—	13	1	no	—
126	C₂×#107	C₂×D₁₃	1	2	yes	—
127	C₂×#102	—	26	1	no	—
128	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
129	#367	—	13	8	no	—
130	C₄⋊C₁₆	—	13	4	no	—
131	#357	—	13	8	no	—
132	C₂²×C₁₃	#474	1	2	yes	—
133	C₂²×C₁₃	C₂×SD₃₂	1	2	yes	—
134	C₂²×C₁₃	#509	1	2	yes	—
135	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
136	C₂×D₁₃	—	4	4	no	—
137	C₂×D₁₃	—	8	2	no	—
138	C₂×D₁₃	—	8	2	no	—
139	C₂×D₁₃	—	8	8	no	—
140	C₄×C₁₃	—	2	2	no	—
141	C₄×C₁₃	—	8	4	no	—
142	C₄×C₁₃	#474	1	2	yes	—
143	C₄×C₁₃	C₂×SD₃₂	1	4	yes	—
144	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
145	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
146	Dic₁₃	—	8	2	no	—
147	C₂²×Q₈	—	4	1	no	—
148	C₂²×D₄	—	26	1	no	—
149	C₂³×C₄	—	52	1	no	—
150	C₂×D₈	—	13	4	no	—
151	C₂×D₈	—	13	2	no	—
152	C₂×D₈	—	26	2	no	—
153	C₂×D₈	—	26	1	no	—
154	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
155	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
156	C₄⋊Q₈	—	2	4	no	—
157	C₄⋊Q₈	—	4	2	no	—
158	C₄×D₄	—	26	4	no	—
159	C₄×D₄	—	52	2	no	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	—	4	1	no	—
162	C₂×C₄⋊C₄	—	26	1	no	—
163	C₂×C₂²⋊C₄	—	52	1	no	—
164	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
165	C₂×C₁₆	—	13	4	no	—
166	C₂×C₁₆	—	26	4	no	—
167	#107	—	2	4	no	—
168	#107	—	2	2	no	—
169	#107	—	13	2	no	—
170	#107	—	26	1	no	—
171	#107	C₂²×D₁₃	1	4	yes	—
172	#102	—	26	4	no	—
173	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
174	C₄×C₈	C₂²×D₁₃	1	2	yes	—
175	C₂×C₁₃	#2226	1	4	yes	complemented
176	C₂×C₁₃	C₂×#474	1	1	yes	characteristic
177	C₂×C₁₃	C₂²×SD₃₂	1	1	yes	characteristic
178	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
179	D₁₃	—	8	4	no	—
180	C₂⁴	—	52	1	no	—
181	C₂×Q₈	—	4	4	no	—
182	C₂×Q₈	—	8	4	no	—
183	C₂×D₄	—	26	4	no	—
184	C₂×D₄	—	26	2	no	—
185	C₂×D₄	—	52	1	no	—
186	C₂×D₄	—	52	2	no	—
187	C₂²×C₄	—	4	2	no	—
188	C₂²×C₄	—	52	1	no	—
189	C₂²×C₄	—	52	4	no	—
190	C₂²×C₄	—	104	4	no	—
191	C₂²×C₄	#6104	1	1	yes	characteristic
192	C₂²×C₄	#6104	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
194	D₈	—	13	4	no	—
195	D₈	—	26	4	no	—
196	D₈	—	26	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	Q₈×D₁₃	1	2	yes	—
199	C₂×C₈	#6131	1	2	yes	—
200	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
201	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
202	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
203	C₄⋊C₄	—	2	4	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₄⋊C₄	—	4	4	no	—
206	C₄⋊C₄	—	26	2	no	—
207	C₄⋊C₄	—	52	1	no	—
208	C₂²⋊C₄	—	52	4	no	—
209	C₄²	#6104	1	2	yes	—
210	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
211	C₁₆	—	26	4	no	—
212	C₁₃	C₂×#2226	1	1	yes	characteristic, complemented, Sylow
213	C₂³	—	52	2	no	—
214	C₂³	—	52	4	no	—
215	C₂³	—	104	2	no	—
216	C₂³	—	104	2	no	—
217	C₂³	C₂×#6104	1	1	yes	characteristic
218	Q₈	—	8	8	no	—
219	D₄	—	26	4	no	—
220	D₄	—	52	2	no	—
221	D₄	—	52	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	8	4	no	—
225	C₂×C₄	—	52	2	no	—
226	C₂×C₄	—	104	2	no	—
227	C₂×C₄	—	104	8	no	—
228	C₂×C₄	#88068	1	4	yes	—
229	C₂×C₄	C₂×#6104	1	2	yes	—
230	C₂×C₄	C₂×#6104	1	1	yes	characteristic
231	C₂×C₄	C₂×#6104	1	2	yes	—
232	C₂×C₄	#88196	1	2	yes	—
233	C₂×C₄	#88200	1	2	yes	—
234	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
235	C₈	—	2	2	no	—
236	C₈	#88200	1	2	yes	—
237	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
238	C₈	C₂×#6131	1	1	yes	characteristic
239	C₂²	—	52	4	no	—
240	C₂²	—	104	2	no	—
241	C₂²	—	104	2	no	—
242	C₂²	—	104	8	no	—
243	C₂²	#11718748	1	2	yes	—
244	C₂²	#11718808	1	2	yes	—
245	C₂²	C₂×#88068	1	2	yes	—
246	C₂²	C₂²×#6104	1	1	yes	characteristic
247	C₄	—	2	2	no	—
248	C₄	—	8	4	no	—
249	C₄	—	104	2	no	—
250	C₄	#11718748	1	2	yes	—
251	C₄	C₂×#88068	1	4	yes	—
252	C₄	C₂×#88196	1	1	yes	characteristic
253	C₄	C₂×#88200	1	1	yes	characteristic
254	C₂	—	104	4	no	—
255	C₂	#13116980	1	4	yes	direct product factor
256	C₂	C₂×#11718748	1	1	yes	characteristic
257	C₂	C₂²×#88068	1	1	yes	characteristic
258	C₂	C₂×#11718808	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#500	C₂
socle	C₂³×C₁₃	C₂×D₈
prime radical	C₂²×D₁₀₄	C₂²
hypercenter	C₂×#500	D₁₃
hypocenter	C₁₃	C₂×#2226
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
2. #13116980	C₂	direct (N × H)
3. C₂×#11718082	C₂	semidirect
4. C₂×C₄⋊#6090	C₂	semidirect
6. C₂×C₁₃×#500	C₂	semidirect
11. C₁₃×#500	C₂²	semidirect
21. #11718082	C₂²	semidirect
22. C₄⋊#6090	C₂²	semidirect
37. C₂×D₁₀₄	Q₈	semidirect
82. D₁₀₄	C₂×Q₈	semidirect
84. C₂×#500	D₁₃	semidirect
121. #500	C₂×D₁₃	semidirect
175. C₂×C₁₃	#2226	semidirect
212. C₁₃	C₂×#2226	semidirect
255. C₂	#13116980	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
#13116980	2
C₂×#11718082	2
C₂×C₄⋊#6090	2
C₂×#11718083	2
C₂×C₁₃×#500	2
C₂×#2226	13

Subgroups — C2×#13116980 #16017525

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116980 #16017525