Subgroups — C₂×#13116415 #15892895

Order 3328.

662 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718780	C₂	1	1	yes	maximal, characteristic
2	#13116415	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11718148	C₂	1	2	yes	maximal
4	C₂×#11718150	C₂	1	2	yes	maximal
5	C₂×#11718152	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#430	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
9	C₂×#88180	C₂²	1	1	yes	characteristic
10	C₂²×#6099	C₂²	1	2	yes	—
11	C₂³×#1055	C₂²	1	1	yes	characteristic
12	C₁₃×#430	—	2	2	no	—
13	C₁₃×#430	C₂²	1	4	yes	—
14	C₂×C₁₃×#108	—	2	2	no	—
15	#11718780	—	2	2	no	—
16	#11718780	C₂²	1	4	yes	—
17	C₂×#88054	—	2	2	no	—
18	#11718152	C₂²	1	4	yes	—
19	#11718150	C₂²	1	8	yes	—
20	#11718148	C₂²	1	8	yes	—
21	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
22	C₂×C₁₃×M₁₆	—	2	2	no	—
23	C₂×C₁₃×M₁₆	—	2	2	no	—
24	C₂×C₁₃×M₁₆	—	2	2	no	—
25	C₂×C₁₃×M₁₆	—	2	2	no	—
26	C₂×C₁₃×M₁₆	—	2	2	no	—
27	C₂×C₁₃×M₁₆	—	2	2	no	—
28	C₂×C₁₃×M₁₆	—	2	2	no	—
29	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
30	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
31	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
32	C₂²×C₈×C₁₃	—	2	2	no	—
33	C₂²×C₈×C₁₃	D₄	1	2	yes	—
34	C₂×C₂²⋊Dic₁₃	Q₈	1	2	yes	—
35	#88180	—	2	1	no	—
36	#88180	—	2	1	no	—
37	#88180	Q₈	1	2	yes	—
38	#88180	C₂³	1	2	yes	—
39	C₂×C₄⋊Dic₁₃	—	2	1	no	—
40	C₂×C₄×Dic₁₃	—	2	1	no	—
41	C₂×#6099	—	2	2	no	—
42	C₂×#6099	—	2	2	no	—
43	C₂×#6099	—	2	2	no	—
44	C₂×#6099	—	2	2	no	—
45	C₂×#6099	—	2	2	no	—
46	C₂×#6099	C₂³	1	4	yes	—
47	C₂²×#1055	—	2	1	no	—
48	C₂²×#1055	—	2	1	no	—
49	C₂²×#1055	—	2	2	no	—
50	C₂²×#1055	—	2	2	no	—
51	C₂²×#1055	—	2	2	no	—
52	C₂²×#1055	C₂³	1	2	yes	—
53	C₁₃×#108	—	2	8	no	—
54	#88054	—	2	8	no	—
55	C₂×#2103	—	13	1	no	maximal, Sylow
56	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	#41	1	2	yes	—
61	C₂²×C₄×C₁₃	#41	1	2	yes	—
62	C₂²×C₄×C₁₃	#41	1	2	yes	—
63	C₂²×C₄×C₁₃	#41	1	2	yes	—
64	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
67	C₂²×Dic₁₃	—	2	2	no	—
68	C₁₃×M₁₆	—	2	4	no	—
69	C₁₃×M₁₆	—	2	4	no	—
70	C₁₃×M₁₆	—	2	4	no	—
71	C₁₃×M₁₆	—	2	4	no	—
72	C₁₃×M₁₆	—	4	2	no	—
73	C₁₃×M₁₆	—	4	2	no	—
74	C₂×C₈×C₁₃	—	2	2	no	—
75	C₂×C₈×C₁₃	—	2	2	no	—
76	C₂×C₈×C₁₃	—	2	4	no	—
77	C₂×C₈×C₁₃	—	4	2	no	—
78	C₂×C₈×C₁₃	—	4	2	no	—
79	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
80	C₂²⋊Dic₁₃	—	2	2	no	—
81	C₂²⋊Dic₁₃	C₂×Q₈	1	4	yes	—
82	C₄⋊Dic₁₃	—	2	2	no	—
83	C₄⋊Dic₁₃	—	2	2	no	—
84	C₄×Dic₁₃	—	2	2	no	—
85	C₄×Dic₁₃	—	2	2	no	—
86	#6099	—	2	4	no	—
87	#6099	—	2	4	no	—
88	#6099	—	4	2	no	—
89	#6099	—	4	2	no	—
90	C₂×#1055	—	2	2	no	—
91	C₂×#1055	—	2	4	no	—
92	C₂×#1055	—	2	4	no	—
93	C₂×#1055	—	4	1	no	—
94	C₂×#1055	—	4	1	no	—
95	C₂×#1055	—	4	2	no	—
96	C₂×#1055	—	4	2	no	—
97	C₂×#1055	—	4	2	no	—
98	C₂×#430	D₁₃	1	1	yes	characteristic
99	C₂×#424	—	13	1	no	—
100	#2103	—	13	8	no	—
101	C₂×#344	—	13	2	no	—
102	C₂×#341	—	13	2	no	—
103	C₂×#338	—	13	1	no	—
104	C₂³×C₁₃	—	4	2	no	—
105	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂³×C₁₃	#125	1	2	yes	—
108	C₂³×C₁₃	#125	1	2	yes	—
109	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
112	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
113	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
114	C₂×C₄×C₁₃	#123	1	2	yes	—
115	C₂×C₄×C₁₃	#124	1	2	yes	—
116	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
117	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
118	C₂×C₄×C₁₃	#125	1	2	yes	—
119	C₂×C₄×C₁₃	#125	1	2	yes	—
120	C₂×C₄×C₁₃	#126	1	2	yes	—
121	C₂×C₄×C₁₃	#126	1	2	yes	—
122	C₂×C₄×C₁₃	#126	1	2	yes	—
123	C₂×C₄×C₁₃	#126	1	2	yes	—
124	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
126	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
127	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
128	C₂×Dic₁₃	—	2	4	no	—
129	C₂×Dic₁₃	—	4	4	no	—
130	C₈×C₁₃	—	2	4	no	—
131	C₈×C₁₃	—	4	4	no	—
132	#1055	—	4	2	no	—
133	#1055	—	4	4	no	—
134	C₂²×M₁₆	—	13	2	no	—
135	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
136	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
137	C₂³×C₈	—	13	1	no	—
138	C₂×#117	—	13	1	no	—
139	#430	—	2	2	no	—
140	#430	C₂×D₁₃	1	4	yes	—
141	C₂×#108	—	2	2	no	—
142	#424	—	13	4	no	—
143	#424	—	26	2	no	—
144	C₂×C₄⋊C₈	—	26	2	no	—
145	#344	—	13	8	no	—
146	#341	—	13	8	no	—
147	#338	—	13	4	no	—
148	C₂²×C₁₃	—	2	2	no	—
149	C₂²×C₁₃	—	2	4	no	—
150	C₂²×C₁₃	—	4	1	no	—
151	C₂²×C₁₃	—	4	2	no	—
152	C₂²×C₁₃	—	4	1	no	—
153	C₂²×C₁₃	#400	1	1	yes	characteristic
154	C₂²×C₁₃	#400	1	2	yes	—
155	C₂²×C₁₃	#400	1	1	yes	characteristic
156	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
157	C₂²×C₁₃	C₂×#125	1	2	yes	—
158	C₄×C₁₃	#400	1	1	yes	characteristic
159	C₄×C₁₃	#400	1	2	yes	—
160	C₄×C₁₃	#400	1	1	yes	characteristic
161	C₄×C₁₃	C₂×#123	1	1	yes	characteristic
162	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
163	C₄×C₁₃	C₂×#126	1	2	yes	—
164	Dic₁₃	—	4	4	no	—
165	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	13	4	no	—
174	C₂×M₁₆	—	26	2	no	—
175	C₂×M₁₆	—	26	2	no	—
176	C₂×M₁₆	—	26	2	no	—
177	C₂×M₁₆	—	26	2	no	—
178	C₂×M₁₆	—	26	2	no	—
179	C₂×M₁₆	C₂²⋊D₁₃	1	2	yes	—
180	C₂×M₁₆	C₂²×D₁₃	1	4	yes	—
181	C₂×M₁₆	C₂²×D₁₃	1	2	yes	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	13	2	no	—
184	C₂²×C₈	—	26	2	no	—
185	C₂²×C₈	—	26	2	no	—
186	C₂²×C₈	—	26	2	no	—
187	C₂²×C₈	—	26	1	no	—
188	C₂²×C₈	—	26	1	no	—
189	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
190	#117	—	13	2	no	—
191	#117	—	13	2	no	—
192	#117	—	26	1	no	—
193	#117	—	26	1	no	—
194	C₂×C₄⋊C₄	—	26	1	no	—
195	C₂×C₂²⋊C₄	—	13	2	no	—
196	C₂×C₄²	—	26	1	no	—
197	#108	—	2	8	no	—
198	C₄⋊C₈	—	26	8	no	—
199	C₂×C₁₃	—	2	1	no	—
200	C₂×C₁₃	—	2	2	no	—
201	C₂×C₁₃	—	2	1	no	—
202	C₂×C₁₃	—	2	2	no	—
203	C₂×C₁₃	#2103	1	2	yes	complemented
204	C₂×C₁₃	C₂×#400	1	1	yes	characteristic
205	C₂⁴	Q₈×D₁₃	1	1	yes	characteristic
206	C₂²×C₄	—	26	2	no	—
207	C₂²×C₄	#6127	1	2	yes	—
208	C₂²×C₄	#6127	1	2	yes	—
209	C₂²×C₄	#6129	1	2	yes	—
210	C₂²×C₄	#6129	1	2	yes	—
211	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
212	C₂²×C₄	#6131	1	1	yes	characteristic
213	C₂²×C₄	#6131	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
216	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
217	M₁₆	—	2	4	no	—
218	M₁₆	—	2	4	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	2	4	no	—
221	M₁₆	—	4	2	no	—
222	M₁₆	—	4	2	no	—
223	M₁₆	—	26	4	no	—
224	M₁₆	—	26	4	no	—
225	M₁₆	—	52	2	no	—
226	M₁₆	—	52	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	26	4	no	—
233	C₂×C₈	—	26	4	no	—
234	C₂×C₈	—	26	2	no	—
235	C₂×C₈	—	52	2	no	—
236	C₂×C₈	—	52	2	no	—
237	C₂×C₈	—	52	1	no	—
238	C₂×C₈	—	52	1	no	—
239	C₂×C₈	—	52	2	no	—
240	C₂×C₈	C₂×C₂²⋊D₁₃	1	4	yes	—
241	C₄⋊C₄	—	26	2	no	—
242	C₄⋊C₄	—	26	2	no	—
243	C₂²⋊C₄	—	13	4	no	—
244	C₂²⋊C₄	—	26	2	no	—
245	C₄²	—	26	2	no	—
246	C₄²	—	26	2	no	—
247	C₁₃	C₂×#2103	1	1	yes	characteristic, complemented, Sylow
248	C₂³	—	4	2	no	—
249	C₂³	#88143	1	2	yes	—
250	C₂³	#88143	1	2	yes	—
251	C₂³	#88200	1	1	yes	characteristic
252	C₂³	#88200	1	1	yes	characteristic
253	C₂³	C₂×Q₈×D₁₃	1	1	yes	characteristic
254	C₂×C₄	—	26	4	no	—
255	C₂×C₄	—	52	4	no	—
256	C₂×C₄	#88143	1	2	yes	—
257	C₂×C₄	#88143	1	2	yes	—
258	C₂×C₄	#88152	1	2	yes	—
259	C₂×C₄	#88152	1	2	yes	—
260	C₂×C₄	#88152	1	2	yes	—
261	C₂×C₄	#88152	1	2	yes	—
262	C₂×C₄	#88183	1	1	yes	characteristic
263	C₂×C₄	#88183	1	1	yes	characteristic
264	C₂×C₄	#88183	1	2	yes	—
265	C₂×C₄	#88190	1	2	yes	—
266	C₂×C₄	#88190	1	1	yes	characteristic
267	C₂×C₄	#88190	1	1	yes	characteristic
268	C₂×C₄	#88200	1	1	yes	characteristic
269	C₂×C₄	#88200	1	1	yes	characteristic
270	C₂×C₄	C₂×#6127	1	2	yes	—
271	C₂×C₄	C₂×#6129	1	2	yes	—
272	C₂×C₄	C₂×#6131	1	1	yes	characteristic
273	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
274	C₈	—	2	4	no	—
275	C₈	—	4	4	no	—
276	C₈	—	52	4	no	—
277	C₈	—	52	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	1	no	—
282	C₂²	—	4	1	no	—
283	C₂²	#11718643	1	2	yes	—
284	C₂²	#11718643	1	1	yes	characteristic
285	C₂²	#11718643	1	1	yes	characteristic
286	C₂²	C₂×#88143	1	2	yes	—
287	C₂²	C₂×#88200	1	1	yes	characteristic
288	C₄	—	52	4	no	—
289	C₄	#11718643	1	2	yes	—
290	C₄	#11718643	1	1	yes	characteristic
291	C₄	#11718643	1	1	yes	characteristic
292	C₄	C₂×#88152	1	2	yes	—
293	C₄	C₂×#88183	1	1	yes	characteristic
294	C₄	C₂×#88190	1	1	yes	characteristic
295	C₂	—	2	1	no	—
296	C₂	—	2	2	no	—
297	C₂	—	2	1	no	—
298	C₂	—	2	2	no	—
299	C₂	#13116415	1	2	yes	direct product factor
300	C₂	C₂×#11718643	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88190
commutator, cosocle	C₂²×C₄×C₁₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×#430	C₂
socle	C₂²×C₁₃	#400
prime radical	C₂⁴×C₁₃	C₂×Q₈
hypercenter	C₂×#430	D₁₃
hypocenter	C₁₃	C₂×#2103
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#430

Complements

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

N	H ≅ G/N	type
2. #13116415	C₂	direct (N × H)
203. C₂×C₁₃	#2103	semidirect
247. C₁₃	C₂×#2103	semidirect
299. C₂	#13116415	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×#11718780	2
#13116415	2
C₂×#11718148	2
C₂×#11718150	2
C₂×#11718152	2
C₂×C₁₃×#430	2
C₂×#2103	13

Subgroups — C2×#13116415 #15892895

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116415 #15892895