Subgroups — C₂×#13115560 #15866191

Order 3328.

1182 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊#1055	C₂	1	2	yes	maximal
3	C₂²×#88043	C₂	1	1	yes	maximal, characteristic
4	#13115560	C₂	1	8	yes	maximal, direct product factor
5	C₂×#11718116	C₂	1	2	yes	maximal
6	C₂×C₄×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×C₄⋊C₄	C₄	1	2	yes	—
8	C₂²×C₄²×C₁₃	C₂²	1	2	yes	—
9	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₂³×#1055	C₂²	1	2	yes	—
11	C₂³×#1055	C₂²	1	1	yes	characteristic
12	C₂³×#1055	C₂²	1	1	yes	characteristic
13	C₄×C₁₃×C₄⋊C₄	C₄	1	4	yes	—
14	C₄×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
15	C₂×C₁₃×#95	C₄	1	1	yes	characteristic
16	C₂×C₁₃×#95	C₄	1	1	yes	characteristic
17	C₂×C₄⋊#1055	—	2	4	no	—
18	C₂×C₄⋊#1055	—	2	8	no	—
19	C₂×C₄⋊#1055	C₂²	1	4	yes	—
20	C₂×C₄⋊#1055	C₂²	1	8	yes	—
21	C₂×#88043	—	2	4	no	—
22	C₂×#88043	C₂²	1	4	yes	—
23	#11718116	C₂²	1	8	yes	—
24	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
28	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
29	C₂×C₁₃×C₄⋊C₄	—	2	8	no	—
30	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
31	C₂×C₄²×C₁₃	—	2	4	no	—
32	C₂×C₄²×C₁₃	—	2	4	no	—
33	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₁₃	D₄	1	4	yes	—
36	C₂×C₄²×C₁₃	Q₈	1	4	yes	—
37	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
38	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
39	C₂²×#1055	—	2	2	no	—
40	C₂²×#1055	—	2	2	no	—
41	C₂²×#1055	—	2	2	no	—
42	C₂²×#1055	—	2	2	no	—
43	C₂²×#1055	—	2	4	no	—
44	C₂²×#1055	—	2	4	no	—
45	C₂²×#1055	—	2	4	no	—
46	C₂²×#1055	D₄	1	4	yes	—
47	C₂²×#1055	Q₈	1	4	yes	—
48	C₂²×#1055	C₂³	1	4	yes	—
49	C₂²×#1055	C₂³	1	2	yes	—
50	C₂²×#1055	C₂³	1	2	yes	—
51	C₁₃×#95	C₂×C₄	1	4	yes	—
52	C₁₃×#95	C₂×C₄	1	4	yes	—
53	C₄⋊#1055	—	2	8	no	—
54	C₄⋊#1055	—	2	16	no	—
55	#88043	—	2	8	no	—
56	C₂×#1940	—	13	1	no	maximal, Sylow
57	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	—	2	4	no	—
59	C₂²×C₄×C₁₃	—	2	4	no	—
60	C₂²×C₄×C₁₃	—	2	4	no	—
61	C₂²×C₄×C₁₃	—	2	4	no	—
62	C₂²×C₄×C₁₃	—	2	4	no	—
63	C₂²×C₄×C₁₃	—	2	4	no	—
64	C₂²×C₄×C₁₃	—	2	4	no	—
65	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
66	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
67	C₂²×C₄×C₁₃	M₁₆	1	8	yes	—
68	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
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	2	yes	—
79	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
80	C₁₃×C₄⋊C₄	—	2	16	no	—
81	C₄²×C₁₃	—	2	8	no	—
82	C₄²×C₁₃	—	2	4	no	—
83	C₄²×C₁₃	—	2	4	no	—
84	C₄²×C₁₃	C₄⋊C₄	1	8	yes	—
85	C₄²×C₁₃	C₂×D₄	1	4	yes	—
86	C₄²×C₁₃	C₂×Q₈	1	4	yes	—
87	C₂×#1055	—	2	8	no	—
88	C₂×#1055	—	2	8	no	—
89	C₂×#1055	—	2	2	no	—
90	C₂×#1055	—	2	2	no	—
91	C₂×#1055	—	2	2	no	—
92	C₂×#1055	—	2	2	no	—
93	C₂×#1055	—	2	4	no	—
94	C₂×#1055	—	2	4	no	—
95	C₂×#1055	—	2	4	no	—
96	C₂×#1055	—	4	8	no	—
97	C₂×#1055	C₂×D₄	1	4	yes	—
98	C₂×#1055	C₂×Q₈	1	4	yes	—
99	C₂²×C₄⋊C₈	—	13	2	no	—
100	C₂²×C₄⋊C₈	—	13	1	no	—
101	C₂²×#97	—	13	1	no	—
102	C₂×C₄×C₄⋊C₄	D₁₃	1	1	yes	characteristic
103	#1940	—	13	8	no	—
104	C₂×#336	—	13	2	no	—
105	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
106	C₂³×C₁₃	C₄×D₄	1	2	yes	—
107	C₂³×C₁₃	C₄×Q₈	1	2	yes	—
108	C₂³×C₁₃	C₂×M₁₆	1	4	yes	—
109	C₂³×C₁₃	#131	1	4	yes	—
110	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃	—	2	4	no	—
112	C₂×C₄×C₁₃	—	2	4	no	—
113	C₂×C₄×C₁₃	—	2	4	no	—
114	C₂×C₄×C₁₃	—	2	4	no	—
115	C₂×C₄×C₁₃	—	2	8	no	—
116	C₂×C₄×C₁₃	—	2	8	no	—
117	C₂×C₄×C₁₃	—	2	16	no	—
118	C₂×C₄×C₁₃	—	2	4	no	—
119	C₂×C₄×C₁₃	—	2	4	no	—
120	C₂×C₄×C₁₃	—	2	4	no	—
121	C₂×C₄×C₁₃	—	4	8	no	—
122	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
123	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	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×Q₈	1	2	yes	—
127	C₂×C₄×C₁₃	#121	1	4	yes	—
128	C₂×C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
129	C₂×C₄×C₁₃	#125	1	4	yes	—
130	C₂×C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
131	C₂×C₄×C₁₃	C₂×M₁₆	1	8	yes	—
132	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
138	#1055	—	2	4	no	—
139	#1055	—	2	4	no	—
140	#1055	—	4	4	no	—
141	C₂³×C₈	—	13	1	no	—
142	C₂³×C₈	—	13	1	no	—
143	C₂³×C₈	—	13	2	no	—
144	C₂²×C₄⋊C₄	Dic₁₃	1	2	yes	—
145	C₂²×C₄²	C₂×D₁₃	1	2	yes	—
146	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
147	C₂×C₄⋊C₈	—	13	8	no	—
148	C₂×C₄⋊C₈	—	13	4	no	—
149	C₂×C₄⋊C₈	—	26	8	no	—
150	C₂×C₄⋊C₈	—	26	4	no	—
151	C₂×#97	—	13	4	no	—
152	C₂×#97	—	26	4	no	—
153	C₄×C₄⋊C₄	Dic₁₃	1	4	yes	—
154	C₄×C₄⋊C₄	C₂×D₁₃	1	4	yes	—
155	C₂×#95	Dic₁₃	1	1	yes	characteristic
156	C₂×#95	Dic₁₃	1	1	yes	characteristic
157	#336	—	13	8	no	—
158	C₂²×C₁₃	#389	1	4	yes	—
159	C₂²×C₁₃	#423	1	4	yes	—
160	C₂²×C₁₃	#424	1	4	yes	—
161	C₂²×C₁₃	#435	1	8	yes	—
162	C₂²×C₁₃	#446	1	8	yes	—
163	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
164	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
165	C₂²×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
166	C₂²×C₁₃	C₂²×M₁₆	1	2	yes	—
167	C₂²×C₁₃	C₂×#131	1	2	yes	—
168	C₄×C₁₃	—	2	4	no	—
169	C₄×C₁₃	—	2	4	no	—
170	C₄×C₁₃	—	2	8	no	—
171	C₄×C₁₃	—	4	4	no	—
172	C₄×C₁₃	#389	1	4	yes	—
173	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
174	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
175	C₄×C₁₃	C₂×#125	1	1	yes	characteristic
176	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
177	C₂³×C₄	C₂×Dic₁₃	1	2	yes	—
178	C₂³×C₄	C₂×Dic₁₃	1	2	yes	—
179	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
180	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
181	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
182	C₂²×C₈	—	13	4	no	—
183	C₂²×C₈	—	13	4	no	—
184	C₂²×C₈	—	13	2	no	—
185	C₂²×C₈	—	13	2	no	—
186	C₂²×C₈	—	13	4	no	—
187	C₂²×C₈	—	26	2	no	—
188	C₂²×C₈	—	26	2	no	—
189	C₂²×C₈	—	26	2	no	—
190	C₂²×C₈	—	26	2	no	—
191	C₂²×C₈	—	26	4	no	—
192	C₂²×C₈	—	26	4	no	—
193	C₂²×C₈	—	26	4	no	—
194	C₂×C₄⋊C₄	—	2	8	no	—
195	C₂×C₄⋊C₄	C₂×Dic₁₃	1	8	yes	—
196	C₂×C₄²	—	2	4	no	—
197	C₂×C₄²	—	2	4	no	—
198	C₂×C₄²	Dic₂₆	1	4	yes	—
199	C₂×C₄²	D₅₂	1	4	yes	—
200	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
201	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
202	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
203	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
204	C₄⋊C₈	—	26	16	no	—
205	C₄⋊C₈	—	26	8	no	—
206	#97	—	26	8	no	—
207	#95	C₂×Dic₁₃	1	4	yes	—
208	#95	C₂×Dic₁₃	1	4	yes	—
209	C₂×C₁₃	#1940	1	8	yes	complemented
210	C₂×C₁₃	C₂×#389	1	1	yes	characteristic
211	C₂×C₁₃	C₂×#423	1	1	yes	characteristic
212	C₂×C₁₃	C₂×#424	1	1	yes	characteristic
213	C₂×C₁₃	C₂×#435	1	2	yes	—
214	C₂×C₁₃	C₂×#446	1	2	yes	—
215	C₂⁴	C₂²×Dic₁₃	1	1	yes	characteristic
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	#6099	1	8	yes	—
224	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
225	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
226	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
227	C₂²×C₄	C₂×D₅₂	1	2	yes	—
228	C₂²×C₄	D₄×D₁₃	1	2	yes	—
229	C₂²×C₄	#6129	1	2	yes	—
230	C₂²×C₄	Q₈×D₁₃	1	2	yes	—
231	C₂²×C₄	#6131	1	2	yes	—
232	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
233	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
234	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
235	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
236	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
237	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
238	C₂×C₈	—	13	4	no	—
239	C₂×C₈	—	13	4	no	—
240	C₂×C₈	—	26	8	no	—
241	C₂×C₈	—	26	8	no	—
242	C₂×C₈	—	26	2	no	—
243	C₂×C₈	—	26	2	no	—
244	C₂×C₈	—	26	2	no	—
245	C₂×C₈	—	26	2	no	—
246	C₂×C₈	—	26	4	no	—
247	C₂×C₈	—	26	4	no	—
248	C₂×C₈	—	26	4	no	—
249	C₂×C₈	—	52	8	no	—
250	C₄⋊C₄	—	2	16	no	—
251	C₄²	—	2	8	no	—
252	C₄²	—	2	4	no	—
253	C₄²	—	2	4	no	—
254	C₄²	C₄⋊Dic₁₃	1	8	yes	—
255	C₄²	C₂×Dic₂₆	1	4	yes	—
256	C₄²	C₂×D₅₂	1	4	yes	—
257	C₁₃	C₂×#1940	1	1	yes	characteristic, complemented, Sylow
258	C₂³	C₂×#6099	1	4	yes	—
259	C₂³	C₂×C₄⋊Dic₁₃	1	2	yes	—
260	C₂³	D₄×Dic₁₃	1	2	yes	—
261	C₂³	Q₈×Dic₁₃	1	2	yes	—
262	C₂³	#88202	1	4	yes	—
263	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	2	16	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	8	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	4	8	no	—
275	C₂×C₄	#88142	1	4	yes	—
276	C₂×C₄	#88144	1	4	yes	—
277	C₂×C₄	#88149	1	4	yes	—
278	C₂×C₄	#88151	1	4	yes	—
279	C₂×C₄	C₂×#6099	1	8	yes	—
280	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
281	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
282	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
283	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
284	C₂×C₄	Q₈×Dic₁₃	1	2	yes	—
285	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
286	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
287	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
288	C₂×C₄	C₂×#6129	1	1	yes	characteristic
289	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
290	C₂×C₄	C₂×#6131	1	1	yes	characteristic
291	C₈	—	26	4	no	—
292	C₈	—	26	4	no	—
293	C₈	—	52	4	no	—
294	C₂²	#11718585	1	4	yes	—
295	C₂²	#11718646	1	4	yes	—
296	C₂²	#11718660	1	4	yes	—
297	C₂²	#11718672	1	8	yes	—
298	C₂²	#11718685	1	8	yes	—
299	C₂²	C₂²×#6099	1	2	yes	—
300	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
301	C₂²	C₂×D₄×Dic₁₃	1	1	yes	characteristic
302	C₂²	C₂×Q₈×Dic₁₃	1	1	yes	characteristic
303	C₂²	C₂×#88202	1	2	yes	—
304	C₄	—	2	4	no	—
305	C₄	—	2	4	no	—
306	C₄	—	2	8	no	—
307	C₄	—	4	4	no	—
308	C₄	#11718646	1	4	yes	—
309	C₄	C₂×#88142	1	1	yes	characteristic
310	C₄	C₂×#88144	1	1	yes	characteristic
311	C₄	C₂×#88149	1	1	yes	characteristic
312	C₄	C₂×#88151	1	1	yes	characteristic
313	C₂	#13115560	1	8	yes	direct product factor
314	C₂	C₂×#11718646	1	1	yes	characteristic
315	C₂	C₂×#11718585	1	1	yes	characteristic
316	C₂	C₂×#11718660	1	1	yes	characteristic
317	C₂	C₂×#11718672	1	2	yes	—
318	C₂	C₂×#11718685	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₁₃
commutator	C₂³×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₄×C₁₃×C₄⋊C₄	C₂
socle, prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₄×C₄⋊C₄	D₁₃
hypocenter	C₁₃	C₂×#1940
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
4. #13115560	C₂	direct (N × H)
209. C₂×C₁₃	#1940	semidirect
257. C₁₃	C₂×#1940	semidirect
313. C₂	#13115560	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×C₄⋊#1055	2
C₂²×#88043	2
#13115560	2
C₂×#11718116	2
C₂×C₄×C₁₃×C₄⋊C₄	2
C₂×#1940	13

Subgroups — C2×#13115560 #15866191

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13115560 #15866191