Subgroups — C₂×#13116411 #15892884

Order 3328.

710 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718786	C₂	1	1	yes	maximal, characteristic
2	#13116411	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11718151	C₂	1	2	yes	maximal
4	C₂×#11718115	C₂	1	2	yes	maximal
5	C₂×#11718149	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#423	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88180	C₂²	1	2	yes	—
11	C₂²×#6099	C₂²	1	2	yes	—
12	C₁₃×#423	—	2	2	no	—
13	C₁₃×#423	C₂²	1	4	yes	—
14	C₂×C₁₃×C₄⋊C₈	—	2	2	no	—
15	#11718786	—	2	2	no	—
16	#11718786	C₂²	1	4	yes	—
17	C₂×#88059	—	2	2	no	—
18	#11718151	C₂²	1	8	yes	—
19	#11718149	C₂²	1	4	yes	—
20	#11718115	C₂²	1	8	yes	—
21	C₂³×C₄×C₁₃	—	2	1	no	—
22	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
23	C₂×C₁₃×M₁₆	—	2	2	no	—
24	C₂×C₁₃×M₁₆	—	2	2	no	—
25	C₂×C₁₃×M₁₆	—	2	1	no	—
26	C₂×C₁₃×M₁₆	—	2	2	no	—
27	C₂×C₁₃×M₁₆	—	2	2	no	—
28	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
29	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
30	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
31	C₂²×C₈×C₁₃	—	2	1	no	—
32	C₂²×C₈×C₁₃	D₄	1	2	yes	—
33	C₂×C₄²×C₁₃	—	2	1	no	—
34	C₂×C₄²×C₁₃	—	2	1	no	—
35	C₂×C₄²×C₁₃	—	2	1	no	—
36	C₂×C₄²×C₁₃	Q₈	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
38	C₂×C₄²×C₁₃	Q₈	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
40	C₂×C₂²⋊Dic₁₃	—	2	2	no	—
41	#88180	—	2	2	no	—
42	#88180	—	2	2	no	—
43	#88180	—	2	2	no	—
44	#88180	C₂³	1	4	yes	—
45	C₂×C₄⋊Dic₁₃	—	2	2	no	—
46	C₂×C₄×Dic₁₃	—	2	2	no	—
47	C₂×#6099	—	2	2	no	—
48	C₂×#6099	—	2	2	no	—
49	C₂×#6099	—	2	2	no	—
50	C₂×#6099	—	2	2	no	—
51	C₂×#6099	—	2	2	no	—
52	C₂×#6099	C₂³	1	4	yes	—
53	C₂²×#1055	—	2	2	no	—
54	C₁₃×C₄⋊C₈	—	2	8	no	—
55	#88059	—	2	8	no	—
56	C₂×#2102	—	13	1	no	maximal, Sylow
57	C₂⁴×C₁₃	#41	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	—	2	1	no	—
59	C₂²×C₄×C₁₃	—	2	2	no	—
60	C₂²×C₄×C₁₃	—	2	1	no	—
61	C₂²×C₄×C₁₃	—	2	2	no	—
62	C₂²×C₄×C₁₃	—	4	4	no	—
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	#41	1	2	yes	—
69	C₂²×C₄×C₁₃	#41	1	2	yes	—
70	C₂²×C₄×C₁₃	#41	1	2	yes	—
71	C₂²×C₄×C₁₃	#41	1	2	yes	—
72	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
73	C₂²×Dic₁₃	—	4	2	no	—
74	C₁₃×M₁₆	—	2	4	no	—
75	C₁₃×M₁₆	—	2	4	no	—
76	C₁₃×M₁₆	—	2	4	no	—
77	C₁₃×M₁₆	—	4	2	no	—
78	C₂×C₈×C₁₃	—	2	2	no	—
79	C₂×C₈×C₁₃	—	2	2	no	—
80	C₂×C₈×C₁₃	—	2	2	no	—
81	C₂×C₈×C₁₃	—	4	1	no	—
82	C₂×C₈×C₁₃	—	4	1	no	—
83	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
84	C₄²×C₁₃	—	2	1	no	—
85	C₄²×C₁₃	—	2	2	no	—
86	C₄²×C₁₃	—	2	1	no	—
87	C₄²×C₁₃	—	2	2	no	—
88	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
89	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
90	C₂²⋊Dic₁₃	—	2	4	no	—
91	C₂²⋊Dic₁₃	—	4	2	no	—
92	C₄⋊Dic₁₃	—	2	4	no	—
93	C₄⋊Dic₁₃	—	4	2	no	—
94	C₄×Dic₁₃	—	2	4	no	—
95	C₄×Dic₁₃	—	4	2	no	—
96	#6099	—	2	4	no	—
97	#6099	—	2	4	no	—
98	#6099	—	4	2	no	—
99	#6099	—	4	2	no	—
100	C₂×#1055	—	2	4	no	—
101	C₂×#1055	—	4	2	no	—
102	C₂×#1055	—	4	2	no	—
103	C₂×#430	—	13	1	no	—
104	C₂×#423	D₁₃	1	1	yes	characteristic
105	#2102	—	13	8	no	—
106	C₂×#344	—	13	2	no	—
107	C₂×#339	—	13	2	no	—
108	C₂×H(3,ℤ/4)	—	13	1	no	—
109	C₂³×C₁₃	—	4	2	no	—
110	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
111	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
112	C₂³×C₁₃	#126	1	2	yes	—
113	C₂³×C₁₃	#126	1	2	yes	—
114	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
115	C₂×C₄×C₁₃	—	2	4	no	—
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	4	4	no	—
119	C₂×C₄×C₁₃	—	4	4	no	—
120	C₂×C₄×C₁₃	—	4	2	no	—
121	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
123	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
124	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
125	C₂×C₄×C₁₃	#123	1	2	yes	—
126	C₂×C₄×C₁₃	#124	1	2	yes	—
127	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
128	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
129	C₂×C₄×C₁₃	#125	1	2	yes	—
130	C₂×C₄×C₁₃	#125	1	2	yes	—
131	C₂×C₄×C₁₃	#125	1	2	yes	—
132	C₂×C₄×C₁₃	#125	1	2	yes	—
133	C₂×C₄×C₁₃	#126	1	2	yes	—
134	C₂×C₄×C₁₃	#126	1	2	yes	—
135	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
138	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
139	C₂×Dic₁₃	—	4	4	no	—
140	C₂×Dic₁₃	—	8	4	no	—
141	C₈×C₁₃	—	2	4	no	—
142	C₈×C₁₃	—	4	2	no	—
143	#1055	—	4	4	no	—
144	C₂²×M₁₆	—	13	2	no	—
145	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
146	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
147	C₂×#117	—	13	2	no	—
148	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
149	#430	—	13	4	no	—
150	#430	—	26	2	no	—
151	C₂×#108	—	26	2	no	—
152	#423	—	2	2	no	—
153	#423	C₂×D₁₃	1	4	yes	—
154	C₂×C₄⋊C₈	—	2	2	no	—
155	#344	—	13	8	no	—
156	#339	—	13	8	no	—
157	H(3,ℤ/4)	—	13	4	no	—
158	C₂²×C₁₃	—	2	4	no	—
159	C₂²×C₁₃	—	2	2	no	—
160	C₂²×C₁₃	—	4	1	no	—
161	C₂²×C₁₃	—	4	1	no	—
162	C₂²×C₁₃	—	4	2	no	—
163	C₂²×C₁₃	#400	1	1	yes	characteristic
164	C₂²×C₁₃	#400	1	2	yes	—
165	C₂²×C₁₃	#400	1	1	yes	characteristic
166	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
167	C₂²×C₁₃	C₂×#126	1	2	yes	—
168	C₄×C₁₃	—	2	4	no	—
169	C₄×C₁₃	—	4	2	no	—
170	C₄×C₁₃	#400	1	1	yes	characteristic
171	C₄×C₁₃	#400	1	2	yes	—
172	C₄×C₁₃	#400	1	1	yes	characteristic
173	C₄×C₁₃	C₂×#123	1	1	yes	characteristic
174	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
175	C₄×C₁₃	C₂×#125	1	2	yes	—
176	Dic₁₃	—	8	4	no	—
177	C₂³×C₄	—	2	1	no	—
178	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	1	no	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	13	4	no	—
185	C₂×M₁₆	—	26	2	no	—
186	C₂×M₁₆	—	26	2	no	—
187	C₂×M₁₆	—	26	2	no	—
188	C₂×M₁₆	—	26	2	no	—
189	C₂×M₁₆	—	26	2	no	—
190	C₂×M₁₆	C₂²⋊D₁₃	1	2	yes	—
191	C₂×M₁₆	C₂²×D₁₃	1	2	yes	—
192	C₂×M₁₆	C₂²×D₁₃	1	2	yes	—
193	C₂²×C₈	—	2	1	no	—
194	C₂²×C₈	—	26	2	no	—
195	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
196	#117	—	13	4	no	—
197	#117	—	26	2	no	—
198	#117	—	26	2	no	—
199	#117	—	26	2	no	—
200	C₂×C₄⋊C₄	—	26	2	no	—
201	C₂×C₂²⋊C₄	—	26	2	no	—
202	C₂×C₄²	—	2	1	no	—
203	C₂×C₄²	—	2	1	no	—
204	C₂×C₄²	—	2	1	no	—
205	C₂×C₄²	—	26	2	no	—
206	C₂×C₄²	Dic₂₆	1	1	yes	characteristic
207	C₂×C₄²	Dic₂₆	1	2	yes	—
208	C₂×C₄²	Dic₂₆	1	1	yes	characteristic
209	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
210	#108	—	26	8	no	—
211	C₄⋊C₈	—	2	8	no	—
212	C₂×C₁₃	—	2	2	no	—
213	C₂×C₁₃	—	2	1	no	—
214	C₂×C₁₃	—	2	2	no	—
215	C₂×C₁₃	—	2	1	no	—
216	C₂×C₁₃	#2102	1	2	yes	complemented
217	C₂×C₁₃	C₂×#400	1	1	yes	characteristic
218	C₂⁴	#6127	1	1	yes	characteristic
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	4	4	no	—
224	C₂²×C₄	—	52	2	no	—
225	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
226	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
227	C₂²×C₄	#6127	1	1	yes	characteristic
228	C₂²×C₄	#6129	1	2	yes	—
229	C₂²×C₄	#6129	1	2	yes	—
230	C₂²×C₄	#6131	1	2	yes	—
231	C₂²×C₄	#6131	1	2	yes	—
232	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
234	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
235	M₁₆	—	2	4	no	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	4	2	no	—
239	M₁₆	—	26	4	no	—
240	M₁₆	—	26	4	no	—
241	M₁₆	—	52	2	no	—
242	M₁₆	—	52	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	4	1	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	—	26	4	no	—
249	C₂×C₈	—	52	2	no	—
250	C₂×C₈	—	52	2	no	—
251	C₂×C₈	C₂×C₂²⋊D₁₃	1	4	yes	—
252	C₄⋊C₄	—	26	4	no	—
253	C₄⋊C₄	—	52	2	no	—
254	C₂²⋊C₄	—	26	4	no	—
255	C₂²⋊C₄	—	52	2	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	1	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	26	4	no	—
261	C₄²	—	52	2	no	—
262	C₄²	C₂×Dic₂₆	1	2	yes	—
263	C₄²	C₂×Dic₂₆	1	2	yes	—
264	C₁₃	C₂×#2102	1	1	yes	characteristic, complemented, Sylow
265	C₂³	—	4	2	no	—
266	C₂³	#88152	1	2	yes	—
267	C₂³	#88152	1	2	yes	—
268	C₂³	#88178	1	1	yes	characteristic
269	C₂³	#88178	1	1	yes	characteristic
270	C₂³	C₂×#6127	1	1	yes	characteristic
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	52	4	no	—
278	C₂×C₄	—	104	4	no	—
279	C₂×C₄	#88144	1	2	yes	—
280	C₂×C₄	#88144	1	2	yes	—
281	C₂×C₄	#88144	1	2	yes	—
282	C₂×C₄	#88144	1	2	yes	—
283	C₂×C₄	#88152	1	2	yes	—
284	C₂×C₄	#88152	1	2	yes	—
285	C₂×C₄	#88178	1	1	yes	characteristic
286	C₂×C₄	#88178	1	1	yes	characteristic
287	C₂×C₄	#88189	1	1	yes	characteristic
288	C₂×C₄	#88189	1	1	yes	characteristic
289	C₂×C₄	#88189	1	2	yes	—
290	C₂×C₄	#88201	1	2	yes	—
291	C₂×C₄	#88201	1	1	yes	characteristic
292	C₂×C₄	#88201	1	1	yes	characteristic
293	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
294	C₂×C₄	C₂×#6129	1	2	yes	—
295	C₂×C₄	C₂×#6131	1	2	yes	—
296	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
297	C₈	—	2	4	no	—
298	C₈	—	4	2	no	—
299	C₈	—	52	4	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	4	no	—
302	C₂²	—	4	1	no	—
303	C₂²	—	4	1	no	—
304	C₂²	—	4	2	no	—
305	C₂²	#11718647	1	1	yes	characteristic
306	C₂²	#11718647	1	2	yes	—
307	C₂²	#11718647	1	1	yes	characteristic
308	C₂²	C₂×#88152	1	2	yes	—
309	C₂²	C₂×#88178	1	1	yes	characteristic
310	C₄	—	2	4	no	—
311	C₄	—	4	2	no	—
312	C₄	—	104	4	no	—
313	C₄	#11718647	1	2	yes	—
314	C₄	#11718647	1	1	yes	characteristic
315	C₄	#11718647	1	1	yes	characteristic
316	C₄	C₂×#88144	1	2	yes	—
317	C₄	C₂×#88189	1	1	yes	characteristic
318	C₄	C₂×#88201	1	1	yes	characteristic
319	C₂	—	2	1	no	—
320	C₂	—	2	2	no	—
321	C₂	—	2	1	no	—
322	C₂	—	2	2	no	—
323	C₂	#13116411	1	2	yes	direct product factor
324	C₂	C₂×#11718647	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88201
commutator, cosocle	C₂²×C₄×C₁₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×#423	C₂
socle	C₂²×C₁₃	#400
prime radical	C₂⁴×C₁₃	#41
hypercenter	C₂×#423	D₁₃
hypocenter	C₁₃	C₂×#2102
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
2. #13116411	C₂	direct (N × H)
216. C₂×C₁₃	#2102	semidirect
264. C₁₃	C₂×#2102	semidirect
323. C₂	#13116411	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×#11718786	2
#13116411	2
C₂×#11718151	2
C₂×#11718115	2
C₂×#11718149	2
C₂×C₁₃×#423	2
C₂×#2102	13

Subgroups — C2×#13116411 #15892884

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116411 #15892884