Subgroups — C₂×#13115616 #15866459

Order 3328.

918 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88059	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718656	C₂	1	2	yes	maximal
3	C₂²×#88057	C₂	1	1	yes	maximal, characteristic
4	#13115616	C₂	1	8	yes	maximal, direct product factor
5	C₂×#11718146	C₂	1	2	yes	maximal
6	C₂×C₁₃×#405	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊Dic₁₃	C₂²	1	2	yes	—
11	C₂²×#6099	C₂²	1	2	yes	—
12	C₁₃×#405	—	2	2	no	—
13	C₁₃×#405	C₂²	1	4	yes	—
14	C₂×C₁₃×#97	—	2	1	no	—
15	C₂×C₄×C₈×C₁₃	—	2	1	no	—
16	C₂×#88059	C₄	1	8	yes	complemented
17	C₂×#88059	C₂²	1	4	yes	—
18	C₂×#88057	—	2	1	no	—
19	C₂×#88057	—	2	1	no	—
20	C₂×#88057	—	2	2	no	—
21	C₂×#88057	C₂²	1	4	yes	—
22	#11718656	—	2	4	no	—
23	#11718656	C₂²	1	8	yes	—
24	C₂×#88048	—	2	2	no	—
25	C₂×#88047	—	2	2	no	—
26	#11718146	C₂²	1	8	yes	—
27	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
28	C₂³×Dic₁₃	—	2	2	no	—
29	C₂×C₁₃×M₁₆	—	2	1	no	—
30	C₂×C₁₃×M₁₆	—	2	2	no	—
31	C₂×C₁₃×M₁₆	Q₈	1	4	yes	—
32	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
33	C₂²×C₈×C₁₃	—	2	1	no	—
34	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	D₄	1	2	yes	—
38	C₂²×C₈×C₁₃	Q₈	1	4	yes	—
39	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
40	C₁₃×#117	—	2	1	no	—
41	C₁₃×#117	—	2	1	no	—
42	C₁₃×#117	D₄	1	2	yes	—
43	C₁₃×#117	C₂³	1	2	yes	—
44	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
45	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
46	C₂×C₄²×C₁₃	—	2	1	no	—
47	C₂×C₄⋊Dic₁₃	—	2	2	no	—
48	C₂×C₄⋊Dic₁₃	—	2	2	no	—
49	C₂×C₄⋊Dic₁₃	—	2	2	no	—
50	C₂×C₄⋊Dic₁₃	—	2	2	no	—
51	C₂×C₄⋊Dic₁₃	—	2	2	no	—
52	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
53	C₂×#6099	—	2	2	no	—
54	C₂×#6099	—	2	4	no	—
55	C₂×#6099	C₂×C₄	1	8	yes	—
56	C₂×#6099	C₂³	1	4	yes	—
57	C₂²×#1055	—	2	2	no	—
58	C₁₃×#97	—	2	4	no	—
59	C₄×C₈×C₁₃	—	2	4	no	—
60	#88059	C₂×C₄	1	16	yes	complemented
61	#88057	—	2	4	no	—
62	#88057	—	2	4	no	—
63	#88048	—	2	8	no	—
64	#88047	—	2	8	no	—
65	C₂×#1951	—	13	1	no	maximal, Sylow
66	C₂⁴×C₁₃	#41	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	—	2	2	no	—
68	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
74	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
75	C₂²×C₄×C₁₃	#41	1	2	yes	—
76	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
78	C₂²×Dic₁₃	—	2	4	no	—
79	C₂²×Dic₁₃	—	4	2	no	—
80	C₂²×Dic₁₃	—	4	2	no	—
81	C₂²×Dic₁₃	—	8	4	no	—
82	C₁₃×M₁₆	—	2	4	no	—
83	C₁₃×M₁₆	—	2	2	no	—
84	C₁₃×M₁₆	C₂×Q₈	1	4	yes	—
85	C₂×C₈×C₁₃	—	2	4	no	—
86	C₂×C₈×C₁₃	—	2	2	no	—
87	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
88	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
89	C₂×C₈×C₁₃	C₄⋊C₄	1	8	yes	—
90	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
91	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
94	C₁₃×C₄⋊C₄	—	2	2	no	—
95	C₁₃×C₄⋊C₄	C₂×D₄	1	4	yes	—
96	C₁₃×C₂²⋊C₄	—	2	2	no	—
97	C₁₃×C₂²⋊C₄	—	2	2	no	—
98	C₄²×C₁₃	—	2	2	no	—
99	C₄²×C₁₃	—	2	2	no	—
100	C₄⋊Dic₁₃	—	2	4	no	—
101	C₄⋊Dic₁₃	—	2	4	no	—
102	C₄⋊Dic₁₃	—	4	2	no	—
103	C₄⋊Dic₁₃	—	4	2	no	—
104	#6099	—	2	8	no	—
105	#6099	—	2	4	no	—
106	#6099	C₂²×C₄	1	8	yes	—
107	C₂×#1055	—	2	8	no	—
108	C₂×#1055	—	2	4	no	—
109	C₂²×#108	—	13	1	no	—
110	C₂×#428	—	13	2	no	—
111	C₂²×#106	—	13	1	no	—
112	C₂×#405	D₁₃	1	1	yes	characteristic
113	#1951	—	13	8	no	—
114	C₂×#343	—	13	2	no	—
115	C₂³×C₁₃	—	2	4	no	—
116	C₂³×C₁₃	C₄×Q₈	1	2	yes	—
117	C₂³×C₁₃	#121	1	1	yes	characteristic
118	C₂³×C₁₃	#121	1	1	yes	characteristic
119	C₂³×C₁₃	#125	1	2	yes	—
120	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
121	C₂×C₄×C₁₃	—	2	4	no	—
122	C₂×C₄×C₁₃	—	4	4	no	—
123	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
125	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
126	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
128	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
129	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
131	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
132	C₂×C₄×C₁₃	#125	1	2	yes	—
133	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
134	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
135	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
141	C₂×Dic₁₃	—	4	4	no	—
142	C₂×Dic₁₃	—	8	4	no	—
143	C₂×Dic₁₃	—	8	4	no	—
144	C₂×Dic₁₃	—	8	4	no	—
145	C₈×C₁₃	—	2	4	no	—
146	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
147	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
148	#1055	—	2	8	no	—
149	C₂²×M₁₆	—	13	2	no	—
150	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
151	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
152	C₂×#117	C₂×D₁₃	1	1	yes	characteristic
153	C₂²×C₄⋊C₄	—	13	2	no	—
154	C₂×#108	—	13	8	no	—
155	C₂×#108	—	13	4	no	—
156	#428	—	13	8	no	—
157	#428	—	26	4	no	—
158	C₂×#107	—	26	2	no	—
159	C₂×#106	—	13	4	no	—
160	C₂×#106	—	26	1	no	—
161	C₂×#106	—	26	1	no	—
162	C₂×#106	—	26	2	no	—
163	C₂×#106	—	26	2	no	—
164	#405	—	2	2	no	—
165	#405	C₂×D₁₃	1	4	yes	—
166	C₂×#97	—	2	1	no	—
167	C₂×C₄×C₈	—	2	1	no	—
168	#343	—	13	8	no	—
169	C₂²×C₁₃	—	2	4	no	—
170	C₂²×C₁₃	—	2	4	no	—
171	C₂²×C₁₃	—	2	2	no	—
172	C₂²×C₁₃	—	2	2	no	—
173	C₂²×C₁₃	#389	1	2	yes	—
174	C₂²×C₁₃	#389	1	2	yes	—
175	C₂²×C₁₃	#471	1	4	yes	—
176	C₂²×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
177	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
178	C₂²×C₁₃	C₂×#125	1	1	yes	characteristic
179	C₄×C₁₃	—	4	4	no	—
180	C₄×C₁₃	#389	1	2	yes	—
181	C₄×C₁₃	#389	1	2	yes	—
182	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
183	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
184	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
185	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
186	Dic₁₃	—	8	4	no	—
187	C₂³×C₄	—	26	2	no	—
188	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	2	1	no	—
191	C₂×M₁₆	—	13	8	no	—
192	C₂×M₁₆	—	13	4	no	—
193	C₂×M₁₆	—	26	2	no	—
194	C₂×M₁₆	—	26	4	no	—
195	C₂×M₁₆	Dic₂₆	1	4	yes	—
196	C₂×M₁₆	C₂²×D₁₃	1	2	yes	—
197	C₂²×C₈	—	2	1	no	—
198	C₂²×C₈	—	26	2	no	—
199	C₂²×C₈	Dic₂₆	1	4	yes	—
200	C₂²×C₈	C₄×D₁₃	1	4	yes	—
201	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
202	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
203	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
204	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
205	#117	—	2	1	no	—
206	#117	—	2	1	no	—
207	#117	D₅₂	1	2	yes	—
208	#117	C₂²×D₁₃	1	2	yes	—
209	C₂×C₄⋊C₄	—	13	4	no	—
210	C₂×C₄⋊C₄	—	26	2	no	—
211	C₂×C₄⋊C₄	—	26	2	no	—
212	C₂×C₄⋊C₄	—	26	2	no	—
213	C₂×C₄⋊C₄	—	26	2	no	—
214	C₂×C₄⋊C₄	—	26	2	no	—
215	C₂×C₄⋊C₄	D₅₂	1	2	yes	—
216	C₂×C₂²⋊C₄	—	2	1	no	—
217	C₂×C₄²	—	2	1	no	—
218	#108	—	13	16	no	—
219	#107	—	26	8	no	—
220	#106	—	26	4	no	—
221	#106	—	26	4	no	—
222	#106	—	26	8	no	—
223	#97	—	2	4	no	—
224	C₄×C₈	—	2	4	no	—
225	C₂×C₁₃	—	2	1	no	—
226	C₂×C₁₃	—	2	1	no	—
227	C₂×C₁₃	—	2	2	no	—
228	C₂×C₁₃	#1951	1	4	yes	complemented
229	C₂×C₁₃	C₂×#389	1	1	yes	characteristic
230	C₂×C₁₃	C₂×#471	1	2	yes	—
231	C₂⁴	#6127	1	1	yes	characteristic
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	26	4	no	—
234	C₂²×C₄	—	52	2	no	—
235	C₂²×C₄	—	52	2	no	—
236	C₂²×C₄	—	104	4	no	—
237	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
238	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
239	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
240	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
242	C₂²×C₄	#6127	1	1	yes	characteristic
243	C₂²×C₄	#6127	1	2	yes	—
244	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
246	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
247	M₁₆	—	2	4	no	—
248	M₁₆	—	2	2	no	—
249	M₁₆	—	13	8	no	—
250	M₁₆	—	26	8	no	—
251	M₁₆	—	26	4	no	—
252	M₁₆	C₂×Dic₂₆	1	4	yes	—
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	26	8	no	—
256	C₂×C₈	—	26	4	no	—
257	C₂×C₈	#6101	1	4	yes	—
258	C₂×C₈	#6101	1	4	yes	—
259	C₂×C₈	#6101	1	8	yes	—
260	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
261	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
262	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
263	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	26	4	no	—
266	C₄⋊C₄	—	26	4	no	—
267	C₄⋊C₄	—	52	2	no	—
268	C₄⋊C₄	—	52	2	no	—
269	C₄⋊C₄	C₂×D₅₂	1	4	yes	—
270	C₂²⋊C₄	—	2	2	no	—
271	C₂²⋊C₄	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	2	2	no	—
274	C₁₃	C₂×#1951	1	1	yes	characteristic, complemented, Sylow
275	C₂³	—	2	4	no	—
276	C₂³	C₄×Dic₂₆	1	2	yes	—
277	C₂³	#88124	1	2	yes	—
278	C₂³	#88184	1	1	yes	characteristic
279	C₂³	#88184	1	1	yes	characteristic
280	C₂³	C₂×#6127	1	1	yes	characteristic
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	52	4	no	—
284	C₂×C₄	—	104	4	no	—
285	C₂×C₄	—	104	4	no	—
286	C₂×C₄	—	104	4	no	—
287	C₂×C₄	C₄×Dic₂₆	1	2	yes	—
288	C₂×C₄	#88123	1	2	yes	—
289	C₂×C₄	#88123	1	2	yes	—
290	C₂×C₄	#88124	1	2	yes	—
291	C₂×C₄	C₄×D₅₂	1	2	yes	—
292	C₂×C₄	C₄×D₅₂	1	2	yes	—
293	C₂×C₄	C₂×#6101	1	2	yes	—
294	C₂×C₄	C₂×#6101	1	2	yes	—
295	C₂×C₄	#88178	1	2	yes	—
296	C₂×C₄	#88178	1	2	yes	—
297	C₂×C₄	#88184	1	1	yes	characteristic
298	C₂×C₄	#88184	1	1	yes	characteristic
299	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
300	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
301	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
302	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
303	C₂×C₄	C₂×#6127	1	1	yes	characteristic
304	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
305	C₈	—	2	4	no	—
306	C₈	—	26	8	no	—
307	C₈	C₂×#6101	1	4	yes	—
308	C₈	C₂×#6101	1	4	yes	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	4	no	—
312	C₂²	—	2	4	no	—
313	C₂²	#11718589	1	2	yes	—
314	C₂²	#11718589	1	2	yes	—
315	C₂²	#11718802	1	4	yes	—
316	C₂²	C₂×C₄×Dic₂₆	1	1	yes	characteristic
317	C₂²	C₂×#88124	1	1	yes	characteristic
318	C₂²	C₂×#88184	1	1	yes	characteristic
319	C₄	—	4	4	no	—
320	C₄	—	104	4	no	—
321	C₄	#11718589	1	2	yes	—
322	C₄	#11718589	1	2	yes	—
323	C₄	C₂×#88123	1	1	yes	characteristic
324	C₄	C₂×C₄×D₅₂	1	1	yes	characteristic
325	C₄	C₂²×#6101	1	1	yes	characteristic
326	C₄	C₂×#88178	1	1	yes	characteristic
327	C₂	—	2	1	no	—
328	C₂	—	2	1	no	—
329	C₂	—	2	2	no	—
330	C₂	#13115616	1	4	yes	direct product factor
331	C₂	C₂×#11718589	1	1	yes	characteristic
332	C₂	C₂×#11718802	1	2	yes	—

	subgroup	G/H
center	C₂³	#88184
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×#405	C₂
socle	C₂³×C₁₃	#121
prime radical	C₂⁴×C₁₃	#41
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×#405	D₁₃
hypocenter	C₁₃	C₂×#1951
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
4. #13115616	C₂	direct (N × H)
16. C₂×#88059	C₄	semidirect
60. #88059	C₂×C₄	semidirect
228. C₂×C₁₃	#1951	semidirect
274. C₁₃	C₂×#1951	semidirect
330. C₂	#13115616	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×#88059	2
C₂×#11718656	2
C₂²×#88057	2
#13115616	2
C₂×#11718146	2
C₂×C₁₃×#405	2
C₂×#1951	13

Subgroups — C2×#13115616 #15866459

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13115616 #15866459