Subgroups — C₂×#13114354 #15845547

Order 3328.

682 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718343	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718692	C₂	1	2	yes	maximal
3	#13114354	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718049	C₂	1	1	yes	maximal, characteristic
5	C₂×#11718078	C₂	1	2	yes	maximal
6	C₂×C₁₃×#446	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²×#6107	—	2	2	no	—
9	C₂×#88123	C₂²	1	2	yes	—
10	C₁₃×#446	—	2	2	no	—
11	C₁₃×#446	C₂²	1	4	yes	—
12	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
13	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#97	—	2	1	no	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₂×#88058	—	2	1	no	—
17	C₂×#88058	—	2	1	no	—
18	#11718692	—	2	4	no	—
19	#11718692	C₂²	1	8	yes	—
20	C₂×#88050	—	2	2	no	—
21	C₂×C₄⋊#1055	C₂²	1	2	yes	—
22	#11718343	—	2	2	no	—
23	#11718343	C₂²	1	4	yes	—
24	#11718078	C₂²	1	8	yes	—
25	#11718049	C₂²	1	4	yes	—
26	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
27	C₂²×Dic₂₆	—	2	2	no	—
28	C₂²×C₈×C₁₃	—	2	1	no	—
29	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
31	C₄×Q₈×C₁₃	—	2	1	no	—
32	C₄×Q₈×C₁₃	—	2	1	no	—
33	C₄×Q₈×C₁₃	D₄	1	2	yes	—
34	C₄×Q₈×C₁₃	C₂³	1	2	yes	—
35	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
36	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	—	2	1	no	—
38	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
39	C₂×#6107	—	2	8	no	—
40	C₂×#6107	—	4	8	no	—
41	C₂×C₄⋊Dic₁₃	—	2	2	no	—
42	C₂×C₄⋊Dic₁₃	—	2	2	no	—
43	C₂²×#1055	—	2	2	no	—
44	#88123	—	2	2	no	—
45	#88123	—	2	2	no	—
46	#88123	—	2	2	no	—
47	#88123	C₂³	1	4	yes	—
48	C₁₃×C₄⋊C₈	—	2	4	no	—
49	C₁₃×C₄⋊C₈	—	2	1	no	—
50	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
51	C₁₃×#97	—	2	4	no	—
52	C₄×C₈×C₁₃	D₄	1	2	yes	—
53	C₄×C₈×C₁₃	C₂³	1	2	yes	—
54	#88058	—	2	4	no	—
55	#88058	—	2	4	no	—
56	#88050	—	2	8	no	—
57	C₄⋊#1055	—	2	2	no	—
58	C₄⋊#1055	C₂³	1	4	yes	—
59	C₂×#1694	—	13	1	no	maximal, Sylow
60	C₂×Q₈×C₁₃	—	2	1	no	—
61	C₂×Q₈×C₁₃	—	2	4	no	—
62	C₂×Q₈×C₁₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₁₃	—	2	1	no	—
64	C₂²×C₄×C₁₃	—	2	1	no	—
65	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
68	C₂²×Dic₁₃	—	4	2	no	—
69	C₂×Dic₂₆	—	2	4	no	—
70	C₂×Dic₂₆	—	4	4	no	—
71	C₂×Dic₂₆	—	4	4	no	—
72	C₂×Dic₂₆	—	4	2	no	—
73	C₂×C₈×C₁₃	—	2	2	no	—
74	C₂×C₈×C₁₃	—	2	2	no	—
75	C₂×C₈×C₁₃	—	4	2	no	—
76	C₂×C₈×C₁₃	Q₁₆	1	4	yes	complemented
77	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
79	C₁₃×C₄⋊C₄	—	2	2	no	—
80	C₁₃×C₄⋊C₄	—	2	1	no	—
81	C₁₃×C₄⋊C₄	—	2	2	no	—
82	C₁₃×C₄⋊C₄	C₂×D₄	1	2	yes	—
83	C₄²×C₁₃	—	2	2	no	—
84	C₄²×C₁₃	—	2	2	no	—
85	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₁₃	#41	1	1	yes	characteristic
88	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
89	#6107	—	4	16	no	—
90	C₄⋊Dic₁₃	—	2	4	no	—
91	C₄⋊Dic₁₃	—	2	4	no	—
92	C₄⋊Dic₁₃	—	4	2	no	—
93	C₄⋊Dic₁₃	—	4	2	no	—
94	C₂×#1055	—	2	4	no	—
95	C₂×#1055	—	4	4	no	—
96	C₂×#500	—	13	1	no	—
97	C₂×#462	—	13	2	no	—
98	C₂×#446	D₁₃	1	1	yes	characteristic
99	#1694	—	13	8	no	—
100	C₂×#334	—	13	1	no	—
101	C₂×#332	—	13	2	no	—
102	C₂³×C₁₃	#121	1	1	yes	characteristic
103	Q₈×C₁₃	—	2	4	no	—
104	Q₈×C₁₃	—	4	2	no	—
105	C₂×C₄×C₁₃	—	2	2	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	4	2	no	—
109	C₂×C₄×C₁₃	—	4	2	no	—
110	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
111	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
112	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
114	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
115	C₂×C₄×C₁₃	#137	1	2	yes	—
116	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
119	C₂×Dic₁₃	—	4	4	no	—
120	C₂×Dic₁₃	—	8	4	no	—
121	Dic₂₆	—	4	4	no	—
122	Dic₂₆	—	4	4	no	—
123	Dic₂₆	—	8	4	no	—
124	C₈×C₁₃	—	2	2	no	—
125	C₈×C₁₃	—	4	2	no	—
126	C₈×C₁₃	C₂×Q₁₆	1	4	yes	complemented
127	#1055	—	4	4	no	—
128	C₂²×Q₁₆	—	26	2	no	—
129	C₂×C₄⋊Q₈	—	13	2	no	—
130	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
131	#500	—	13	4	no	—
132	#500	—	26	2	no	—
133	#462	—	13	8	no	—
134	#462	—	26	4	no	—
135	#446	—	2	2	no	—
136	#446	C₂×D₁₃	1	4	yes	—
137	C₂×#107	—	26	1	no	—
138	C₂×#107	—	26	1	no	—
139	C₂×C₄⋊C₈	—	2	1	no	—
140	C₂×C₄⋊C₈	—	13	2	no	—
141	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
142	C₂×#103	—	26	2	no	—
143	C₂×#97	—	2	1	no	—
144	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
145	#334	—	13	4	no	—
146	#332	—	13	8	no	—
147	C₂²×C₁₃	#462	1	2	yes	—
148	C₂²×C₁₃	#469	1	2	yes	—
149	C₂²×C₁₃	#473	1	2	yes	—
150	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
151	C₄×C₁₃	—	2	1	no	—
152	C₄×C₁₃	—	2	1	no	—
153	C₄×C₁₃	—	4	2	no	—
154	C₄×C₁₃	—	4	2	no	—
155	C₄×C₁₃	#462	1	2	yes	—
156	C₄×C₁₃	#469	1	2	yes	—
157	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
158	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
159	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
160	Dic₁₃	—	8	4	no	—
161	C₂²×Q₈	—	26	2	no	—
162	C₂²×Q₈	D₅₂	1	1	yes	characteristic
163	C₂×Q₁₆	—	26	8	no	—
164	C₂×Q₁₆	—	52	8	no	—
165	C₂²×C₈	—	2	1	no	—
166	C₂²×C₈	—	26	2	no	—
167	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
168	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
169	C₄⋊Q₈	—	13	4	no	—
170	C₄⋊Q₈	—	26	2	no	—
171	C₄⋊Q₈	—	26	2	no	—
172	C₄⋊Q₈	—	26	2	no	—
173	C₄×Q₈	—	2	1	no	—
174	C₄×Q₈	—	2	1	no	—
175	C₄×Q₈	D₅₂	1	2	yes	—
176	C₄×Q₈	C₂²×D₁₃	1	2	yes	—
177	C₂×C₄⋊C₄	—	2	1	no	—
178	C₂×C₄⋊C₄	—	26	2	no	—
179	C₂×C₄⋊C₄	—	26	2	no	—
180	C₂×C₄⋊C₄	D₅₂	1	1	yes	characteristic
181	C₂×C₄²	—	2	1	no	—
182	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
183	#107	—	26	4	no	—
184	#107	—	26	4	no	—
185	C₄⋊C₈	—	2	4	no	—
186	C₄⋊C₈	—	2	1	no	—
187	C₄⋊C₈	—	13	4	no	—
188	C₄⋊C₈	—	26	2	no	—
189	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
190	#103	—	26	8	no	—
191	#97	—	2	4	no	—
192	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
193	C₄×C₈	C₂²×D₁₃	1	2	yes	—
194	C₂×C₁₃	#1694	1	4	yes	complemented
195	C₂×C₁₃	C₂×#462	1	1	yes	characteristic
196	C₂×C₁₃	C₂×#469	1	1	yes	characteristic
197	C₂×C₁₃	C₂×#473	1	1	yes	characteristic
198	C₂×Q₈	—	2	1	no	—
199	C₂×Q₈	—	2	4	no	—
200	C₂×Q₈	—	26	4	no	—
201	C₂×Q₈	—	52	2	no	—
202	C₂×Q₈	—	52	4	no	—
203	C₂×Q₈	—	52	4	no	—
204	C₂×Q₈	C₂×D₅₂	1	2	yes	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	52	2	no	—
208	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
209	C₂²×C₄	#6127	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
211	Q₁₆	—	52	16	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	—	26	4	no	—
216	C₂×C₈	—	52	4	no	—
217	C₂×C₈	#6107	1	4	yes	complemented
218	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
219	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	1	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	26	4	no	—
224	C₄⋊C₄	—	26	4	no	—
225	C₄⋊C₄	—	52	2	no	—
226	C₄⋊C₄	—	52	2	no	—
227	C₄⋊C₄	C₂×D₅₂	1	2	yes	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	C₂×D₅₂	1	1	yes	characteristic
231	C₄²	#6127	1	1	yes	characteristic
232	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
233	C₄²	C₂³×D₁₃	1	1	yes	characteristic
234	C₁₃	C₂×#1694	1	1	yes	characteristic, complemented, Sylow
235	C₂³	#88184	1	1	yes	characteristic
236	Q₈	—	2	4	no	—
237	Q₈	—	4	2	no	—
238	Q₈	—	52	4	no	—
239	Q₈	—	52	4	no	—
240	Q₈	—	104	4	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	52	4	no	—
247	C₂×C₄	—	104	4	no	—
248	C₂×C₄	#88162	1	4	yes	—
249	C₂×C₄	#88184	1	1	yes	characteristic
250	C₂×C₄	#88184	1	1	yes	characteristic
251	C₂×C₄	#88184	1	1	yes	characteristic
252	C₂×C₄	C₂×#6107	1	2	yes	—
253	C₂×C₄	#88205	1	2	yes	—
254	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
255	C₂×C₄	C₂×#6127	1	1	yes	characteristic
256	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
257	C₈	—	2	2	no	—
258	C₈	—	4	2	no	—
259	C₈	—	52	4	no	—
260	C₈	C₂×#6107	1	4	yes	complemented
261	C₂²	#11718692	1	2	yes	—
262	C₂²	#11718796	1	2	yes	—
263	C₂²	#11718804	1	2	yes	—
264	C₂²	C₂×#88184	1	1	yes	characteristic
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	4	2	no	—
268	C₄	—	4	2	no	—
269	C₄	—	104	4	no	—
270	C₄	#11718692	1	2	yes	—
271	C₄	#11718796	1	2	yes	—
272	C₄	C₂×#88162	1	2	yes	—
273	C₄	C₂²×#6107	1	1	yes	characteristic
274	C₄	C₂×#88205	1	1	yes	characteristic
275	C₂	#13114354	1	4	yes	direct product factor
276	C₂	C₂×#11718692	1	1	yes	characteristic
277	C₂	C₂×#11718796	1	1	yes	characteristic
278	C₂	C₂×#11718804	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88184
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₁₃×#446	C₂
socle, prime radical	C₂³×C₁₃	#121
hypercenter	C₂×#446	D₁₃
hypocenter	C₁₃	C₂×#1694
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
3. #13114354	C₂	direct (N × H)
76. C₂×C₈×C₁₃	Q₁₆	semidirect
126. C₈×C₁₃	C₂×Q₁₆	semidirect
194. C₂×C₁₃	#1694	semidirect
217. C₂×C₈	#6107	semidirect
234. C₁₃	C₂×#1694	semidirect
260. C₈	C₂×#6107	semidirect
275. C₂	#13114354	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×#11718343	2
C₂×#11718692	2
#13114354	2
C₂×#11718049	2
C₂×#11718078	2
C₂×C₁₃×#446	2
C₂×#1694	13

Subgroups — C2×#13114354 #15845547

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13114354 #15845547