Subgroups — C₂×#13115621 #15866465

Order 3328.

918 conjugacy classes of subgroups.

The subgroup lattice has 358 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×#88058	C₂	1	1	yes	maximal, characteristic
4	#13115621	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×#88058	—	2	1	no	—
19	C₂×#88058	—	2	1	no	—
20	C₂×#88058	—	2	2	no	—
21	C₂×#88058	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	1	yes	characteristic
45	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
46	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
47	C₂×C₄²×C₁₃	—	2	1	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	—	2	2	no	—
53	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
54	C₂×#6099	—	2	2	no	—
55	C₂×#6099	—	2	4	no	—
56	C₂×#6099	C₂×C₄	1	8	yes	—
57	C₂×#6099	C₂³	1	4	yes	—
58	C₂²×#1055	—	2	2	no	—
59	C₁₃×#97	—	2	4	no	—
60	C₄×C₈×C₁₃	—	2	4	no	—
61	#88059	C₂×C₄	1	16	yes	complemented
62	#88058	—	2	4	no	—
63	#88058	—	2	4	no	—
64	#88048	—	2	8	no	—
65	#88047	—	2	8	no	—
66	C₂×#1952	—	13	1	no	maximal, Sylow
67	C₂⁴×C₁₃	#41	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	—	2	1	no	—
69	C₂²×C₄×C₁₃	—	2	1	no	—
70	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
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×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₁₃	#41	1	2	yes	—
78	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
80	C₂²×Dic₁₃	—	2	4	no	—
81	C₂²×Dic₁₃	—	4	2	no	—
82	C₂²×Dic₁₃	—	4	2	no	—
83	C₂²×Dic₁₃	—	8	4	no	—
84	C₁₃×M₁₆	—	2	4	no	—
85	C₁₃×M₁₆	—	2	2	no	—
86	C₁₃×M₁₆	C₂×Q₈	1	4	yes	—
87	C₂×C₈×C₁₃	—	2	4	no	—
88	C₂×C₈×C₁₃	—	2	2	no	—
89	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
90	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
91	C₂×C₈×C₁₃	C₄⋊C₄	1	8	yes	—
92	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
93	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
96	C₁₃×C₄⋊C₄	—	2	1	no	—
97	C₁₃×C₄⋊C₄	—	2	1	no	—
98	C₁₃×C₄⋊C₄	C₂×D₄	1	2	yes	—
99	C₁₃×C₄⋊C₄	C₂×D₄	1	2	yes	—
100	C₁₃×C₂²⋊C₄	—	2	2	no	—
101	C₁₃×C₂²⋊C₄	—	2	2	no	—
102	C₄²×C₁₃	—	2	2	no	—
103	C₄²×C₁₃	—	2	2	no	—
104	C₄⋊Dic₁₃	—	2	4	no	—
105	C₄⋊Dic₁₃	—	2	4	no	—
106	C₄⋊Dic₁₃	—	4	2	no	—
107	C₄⋊Dic₁₃	—	4	2	no	—
108	#6099	—	2	8	no	—
109	#6099	—	2	4	no	—
110	#6099	C₂²×C₄	1	8	yes	—
111	C₂×#1055	—	2	8	no	—
112	C₂×#1055	—	2	4	no	—
113	C₂²×#108	—	13	1	no	—
114	C₂×#428	—	13	2	no	—
115	C₂²×#107	—	13	1	no	—
116	C₂×#405	D₁₃	1	1	yes	characteristic
117	#1952	—	13	8	no	—
118	C₂×#343	—	13	2	no	—
119	C₂³×C₁₃	—	2	2	no	—
120	C₂³×C₁₃	—	2	2	no	—
121	C₂³×C₁₃	C₄×Q₈	1	2	yes	—
122	C₂³×C₁₃	#121	1	1	yes	characteristic
123	C₂³×C₁₃	#121	1	1	yes	characteristic
124	C₂³×C₁₃	#125	1	2	yes	—
125	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
126	C₂×C₄×C₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	—	2	2	no	—
128	C₂×C₄×C₁₃	—	4	2	no	—
129	C₂×C₄×C₁₃	—	4	2	no	—
130	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
131	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
132	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
133	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
135	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
136	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
138	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
139	C₂×C₄×C₁₃	#125	1	2	yes	—
140	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
141	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
142	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
146	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
148	C₂×Dic₁₃	—	4	4	no	—
149	C₂×Dic₁₃	—	8	4	no	—
150	C₂×Dic₁₃	—	8	4	no	—
151	C₂×Dic₁₃	—	8	4	no	—
152	C₈×C₁₃	—	2	4	no	—
153	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
154	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
155	#1055	—	2	8	no	—
156	C₂²×M₁₆	—	13	2	no	—
157	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
158	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
159	C₂×#117	C₂×D₁₃	1	1	yes	characteristic
160	C₂²×C₄⋊C₄	—	13	2	no	—
161	C₂×#108	—	13	8	no	—
162	C₂×#108	—	13	4	no	—
163	#428	—	13	8	no	—
164	#428	—	26	4	no	—
165	C₂×#107	—	13	4	no	—
166	C₂×#107	—	26	1	no	—
167	C₂×#107	—	26	1	no	—
168	C₂×#107	—	26	2	no	—
169	C₂×#107	—	26	2	no	—
170	C₂×#106	—	26	2	no	—
171	#405	—	2	2	no	—
172	#405	C₂×D₁₃	1	4	yes	—
173	C₂×#97	—	2	1	no	—
174	C₂×C₄×C₈	—	2	1	no	—
175	#343	—	13	8	no	—
176	C₂²×C₁₃	—	2	4	no	—
177	C₂²×C₁₃	—	2	4	no	—
178	C₂²×C₁₃	—	2	1	no	—
179	C₂²×C₁₃	—	2	1	no	—
180	C₂²×C₁₃	—	2	1	no	—
181	C₂²×C₁₃	—	2	1	no	—
182	C₂²×C₁₃	#389	1	2	yes	—
183	C₂²×C₁₃	#389	1	2	yes	—
184	C₂²×C₁₃	#472	1	2	yes	—
185	C₂²×C₁₃	#473	1	2	yes	—
186	C₂²×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
187	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
188	C₂²×C₁₃	C₂×#125	1	1	yes	characteristic
189	C₄×C₁₃	—	4	2	no	—
190	C₄×C₁₃	—	4	2	no	—
191	C₄×C₁₃	#389	1	2	yes	—
192	C₄×C₁₃	#389	1	2	yes	—
193	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
194	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
195	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
196	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
197	Dic₁₃	—	8	4	no	—
198	C₂³×C₄	—	26	2	no	—
199	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
200	C₂×M₁₆	—	2	2	no	—
201	C₂×M₁₆	—	2	1	no	—
202	C₂×M₁₆	—	13	8	no	—
203	C₂×M₁₆	—	13	4	no	—
204	C₂×M₁₆	—	26	2	no	—
205	C₂×M₁₆	—	26	4	no	—
206	C₂×M₁₆	Dic₂₆	1	4	yes	—
207	C₂×M₁₆	C₂²×D₁₃	1	2	yes	—
208	C₂²×C₈	—	2	1	no	—
209	C₂²×C₈	—	26	2	no	—
210	C₂²×C₈	Dic₂₆	1	4	yes	—
211	C₂²×C₈	C₄×D₁₃	1	4	yes	—
212	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
213	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
214	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
215	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
216	#117	—	2	1	no	—
217	#117	—	2	1	no	—
218	#117	D₅₂	1	2	yes	—
219	#117	C₂²×D₁₃	1	2	yes	—
220	C₂×C₄⋊C₄	—	13	4	no	—
221	C₂×C₄⋊C₄	—	26	2	no	—
222	C₂×C₄⋊C₄	—	26	2	no	—
223	C₂×C₄⋊C₄	—	26	2	no	—
224	C₂×C₄⋊C₄	—	26	2	no	—
225	C₂×C₄⋊C₄	—	26	2	no	—
226	C₂×C₄⋊C₄	D₅₂	1	1	yes	characteristic
227	C₂×C₄⋊C₄	D₅₂	1	1	yes	characteristic
228	C₂×C₂²⋊C₄	—	2	1	no	—
229	C₂×C₄²	—	2	1	no	—
230	#108	—	13	16	no	—
231	#107	—	26	4	no	—
232	#107	—	26	4	no	—
233	#107	—	26	8	no	—
234	#106	—	26	8	no	—
235	#97	—	2	4	no	—
236	C₄×C₈	—	2	4	no	—
237	C₂×C₁₃	—	2	1	no	—
238	C₂×C₁₃	—	2	1	no	—
239	C₂×C₁₃	—	2	2	no	—
240	C₂×C₁₃	#1952	1	4	yes	complemented
241	C₂×C₁₃	C₂×#389	1	1	yes	characteristic
242	C₂×C₁₃	C₂×#472	1	1	yes	characteristic
243	C₂×C₁₃	C₂×#473	1	1	yes	characteristic
244	C₂⁴	#6127	1	1	yes	characteristic
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	26	4	no	—
248	C₂²×C₄	—	52	2	no	—
249	C₂²×C₄	—	52	2	no	—
250	C₂²×C₄	—	104	4	no	—
251	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
252	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
253	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
254	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
256	C₂²×C₄	#6127	1	1	yes	characteristic
257	C₂²×C₄	#6127	1	2	yes	—
258	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
259	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
260	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
261	M₁₆	—	2	4	no	—
262	M₁₆	—	2	2	no	—
263	M₁₆	—	13	8	no	—
264	M₁₆	—	26	8	no	—
265	M₁₆	—	26	4	no	—
266	M₁₆	C₂×Dic₂₆	1	4	yes	—
267	C₂×C₈	—	2	4	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	26	8	no	—
270	C₂×C₈	—	26	4	no	—
271	C₂×C₈	#6101	1	4	yes	—
272	C₂×C₈	#6101	1	4	yes	—
273	C₂×C₈	#6101	1	8	yes	—
274	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
275	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
276	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
277	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	26	4	no	—
281	C₄⋊C₄	—	26	4	no	—
282	C₄⋊C₄	—	52	2	no	—
283	C₄⋊C₄	—	52	2	no	—
284	C₄⋊C₄	C₂×D₅₂	1	2	yes	—
285	C₄⋊C₄	C₂×D₅₂	1	2	yes	—
286	C₂²⋊C₄	—	2	2	no	—
287	C₂²⋊C₄	—	2	2	no	—
288	C₄²	—	2	2	no	—
289	C₄²	—	2	2	no	—
290	C₁₃	C₂×#1952	1	1	yes	characteristic, complemented, Sylow
291	C₂³	—	2	2	no	—
292	C₂³	—	2	2	no	—
293	C₂³	C₄×Dic₂₆	1	2	yes	—
294	C₂³	#88124	1	2	yes	—
295	C₂³	#88184	1	1	yes	characteristic
296	C₂³	#88184	1	1	yes	characteristic
297	C₂³	C₂×#6127	1	1	yes	characteristic
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	52	4	no	—
303	C₂×C₄	—	104	4	no	—
304	C₂×C₄	—	104	4	no	—
305	C₂×C₄	—	104	4	no	—
306	C₂×C₄	C₄×Dic₂₆	1	2	yes	—
307	C₂×C₄	#88123	1	2	yes	—
308	C₂×C₄	#88123	1	2	yes	—
309	C₂×C₄	#88124	1	2	yes	—
310	C₂×C₄	C₄×D₅₂	1	2	yes	—
311	C₂×C₄	C₄×D₅₂	1	2	yes	—
312	C₂×C₄	C₂×#6101	1	2	yes	—
313	C₂×C₄	C₂×#6101	1	2	yes	—
314	C₂×C₄	#88178	1	2	yes	—
315	C₂×C₄	#88178	1	2	yes	—
316	C₂×C₄	#88184	1	1	yes	characteristic
317	C₂×C₄	#88184	1	1	yes	characteristic
318	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
319	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
320	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
321	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
322	C₂×C₄	C₂×#6127	1	1	yes	characteristic
323	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
324	C₈	—	2	4	no	—
325	C₈	—	26	8	no	—
326	C₈	C₂×#6101	1	4	yes	—
327	C₈	C₂×#6101	1	4	yes	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	4	no	—
333	C₂²	—	2	4	no	—
334	C₂²	#11718589	1	2	yes	—
335	C₂²	#11718589	1	2	yes	—
336	C₂²	#11718803	1	2	yes	—
337	C₂²	#11718804	1	2	yes	—
338	C₂²	C₂×C₄×Dic₂₆	1	1	yes	characteristic
339	C₂²	C₂×#88124	1	1	yes	characteristic
340	C₂²	C₂×#88184	1	1	yes	characteristic
341	C₄	—	4	2	no	—
342	C₄	—	4	2	no	—
343	C₄	—	104	4	no	—
344	C₄	#11718589	1	2	yes	—
345	C₄	#11718589	1	2	yes	—
346	C₄	C₂×#88123	1	1	yes	characteristic
347	C₄	C₂×C₄×D₅₂	1	1	yes	characteristic
348	C₄	C₂²×#6101	1	1	yes	characteristic
349	C₄	C₂×#88178	1	1	yes	characteristic
350	C₂	—	2	1	no	—
351	C₂	—	2	1	no	—
352	C₂	—	2	2	no	—
353	C₂	#13115621	1	4	yes	direct product factor
354	C₂	C₂×#11718589	1	1	yes	characteristic
355	C₂	C₂×#11718803	1	1	yes	characteristic
356	C₂	C₂×#11718804	1	1	yes	characteristic

	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×#1952
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. #13115621	C₂	direct (N × H)
16. C₂×#88059	C₄	semidirect
61. #88059	C₂×C₄	semidirect
240. C₂×C₁₃	#1952	semidirect
290. C₁₃	C₂×#1952	semidirect
353. C₂	#13115621	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×#88058	2
#13115621	2
C₂×#11718146	2
C₂×C₁₃×#405	2
C₂×#1952	13

Subgroups — C2×#13115621 #15866465

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13115621 #15866465