Subgroups — C₂×#13130557 #16796861

Order 3328.

930 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719089	C₂	1	2	yes	maximal
2	#13130557	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11718269	C₂	1	1	yes	maximal, characteristic
4	C₂×#11718267	C₂	1	1	yes	maximal, characteristic
5	C₂×#11718235	C₂	1	2	yes	maximal
6	C₂×#11718350	C₂	1	1	yes	maximal, characteristic
7	C₂×#88237	C₂²	1	2	yes	—
8	C₂×D₁₃⋊M₁₆	C₂²	1	2	yes	—
9	C₂²×#6094	C₂²	1	1	yes	characteristic
10	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
11	C₂×#88126	C₂²	1	1	yes	characteristic
12	#11719089	C₄	1	8	yes	complemented
13	#11719089	C₂²	1	8	yes	—
14	C₂×#88100	C₄	1	4	yes	complemented
15	C₂×C₈×#261	C₄	1	4	yes	complemented
16	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic, complemented
17	C₂×#88055	C₄	1	1	yes	characteristic, complemented
18	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic, complemented
19	C₂×#88043	C₄	1	1	yes	characteristic, complemented
20	#11718350	C₄	1	4	yes	complemented
21	#11718350	C₂²	1	4	yes	—
22	#11718269	C₂²	1	4	yes	—
23	#11718267	C₂²	1	4	yes	—
24	#11718235	C₂²	1	8	yes	—
25	C₂²×C₄×D₁₃	C₂³	1	1	yes	characteristic
26	C₂×C₂²⋊#261	—	2	2	no	—
27	C₂×#6122	—	2	2	no	—
28	#88237	—	2	2	no	—
29	#88237	—	2	2	no	—
30	#88237	C₂×C₄	1	4	yes	—
31	#88237	C₂³	1	4	yes	—
32	C₂×C₄⋊#261	—	2	2	no	—
33	C₂×C₄×#261	C₂×C₄	1	4	yes	—
34	D₁₃⋊M₁₆	—	2	2	no	—
35	D₁₃⋊M₁₆	—	2	2	no	—
36	D₁₃⋊M₁₆	C₂×C₄	1	4	yes	—
37	D₁₃⋊M₁₆	C₂³	1	4	yes	—
38	C₂×#6118	—	2	2	no	—
39	C₂×D₁₃⋊C₈	C₂×C₄	1	4	yes	—
40	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂×#6103	—	2	1	no	—
44	C₂×#6101	—	2	1	no	—
45	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
46	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
47	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
48	C₂×#6094	—	2	2	no	—
49	C₂×#6094	—	2	2	no	—
50	C₂×#6094	C₂×C₄	1	2	yes	—
51	C₂×#6094	C₂³	1	2	yes	—
52	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
53	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
54	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
55	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
56	C₂×C₈×D₁₃	C₂³	1	2	yes	—
57	#88126	—	2	1	no	—
58	#88126	—	2	1	no	—
59	#88126	C₂×C₄	1	2	yes	—
60	#88126	C₂³	1	2	yes	—
61	#88100	C₂×C₄	1	16	yes	complemented
62	C₈×#261	C₂×C₄	1	16	yes	complemented
63	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	complemented
64	#88055	C₂×C₄	1	4	yes	complemented
65	C₈×Dic₁₃	C₂×C₄	1	4	yes	complemented
66	#88043	C₂×C₄	1	4	yes	complemented
67	C₂×#1747	—	13	1	no	maximal, Sylow
68	C₂³×D₁₃	#41	1	1	yes	characteristic
69	C₂²×#261	—	2	4	no	—
70	C₂²×C₄×C₁₃	—	2	1	no	—
71	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×Dic₁₃	—	2	1	no	—
73	C₂²×Dic₁₃	#41	1	1	yes	characteristic
74	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
76	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
77	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
78	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
79	C₂×C₄×D₁₃	#41	1	2	yes	—
80	C₂×C₄×D₁₃	#41	1	2	yes	—
81	C₂×C₄×D₁₃	C₂⁴	1	1	yes	characteristic
82	C₂²⋊#261	—	2	4	no	—
83	C₂²⋊#261	—	2	4	no	—
84	#6122	—	2	4	no	—
85	#6122	—	2	4	no	—
86	C₂×#1057	—	2	4	no	—
87	C₄⋊#261	—	2	4	no	—
88	C₄⋊#261	—	2	4	no	—
89	C₄×#261	—	2	4	no	—
90	C₄×#261	C₂²×C₄	1	8	yes	—
91	#6118	—	2	4	no	—
92	#6118	—	2	4	no	—
93	D₁₃⋊C₈	—	2	4	no	—
94	D₁₃⋊C₈	C₂²×C₄	1	8	yes	—
95	C₂×C₈×C₁₃	—	2	1	no	—
96	C₂×C₈×C₁₃	—	2	1	no	—
97	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
98	C₂×C₈×C₁₃	C₄²	1	2	yes	—
99	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
100	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
102	C₄²×C₁₃	—	2	1	no	—
103	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
104	#6103	—	2	2	no	—
105	#6103	—	2	2	no	—
106	#6101	—	2	2	no	—
107	#6101	—	2	2	no	—
108	C₄×Dic₁₃	—	2	1	no	—
109	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
110	C₂×#1055	—	2	1	no	—
111	C₂×#1055	—	2	1	no	—
112	C₂×#1055	C₄²	1	2	yes	—
113	C₂×#1055	C₄²	1	2	yes	—
114	C₂×#1055	C₂²×C₄	1	2	yes	—
115	C₂×#1055	C₂²×C₄	1	2	yes	—
116	#6094	—	2	4	no	—
117	#6094	—	2	4	no	—
118	C₈×D₁₃	C₄²	1	4	yes	—
119	C₈×D₁₃	C₄²	1	4	yes	—
120	C₈×D₁₃	C₂²×C₄	1	4	yes	—
121	C₈×D₁₃	C₂²×C₄	1	4	yes	—
122	C₂×#405	—	13	2	no	—
123	C₂×#405	—	13	1	no	—
124	#1747	—	13	8	no	—
125	C₂×#338	—	13	1	no	—
126	C₂×H(3,ℤ/4)	—	13	2	no	—
127	C₂×H(3,ℤ/4)	—	13	1	no	—
128	C₂³×C₁₃	#117	1	1	yes	characteristic
129	C₂²×D₁₃	—	4	2	no	—
130	C₂²×D₁₃	#117	1	2	yes	—
131	C₂²×D₁₃	#117	1	2	yes	—
132	C₂²×D₁₃	#117	1	1	yes	characteristic
133	C₂²×D₁₃	C₂×#41	1	1	yes	characteristic
134	C₂×#261	—	2	8	no	—
135	C₂×#261	—	4	8	no	—
136	C₂×C₄×C₁₃	—	2	2	no	—
137	C₂×C₄×C₁₃	—	4	2	no	—
138	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
140	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
141	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
142	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
144	C₂×Dic₁₃	—	2	2	no	—
145	C₂×Dic₁₃	—	4	2	no	—
146	C₂×Dic₁₃	#117	1	2	yes	—
147	C₂×Dic₁₃	#117	1	2	yes	—
148	C₂×Dic₁₃	#117	1	1	yes	characteristic
149	C₂×Dic₁₃	C₂×#41	1	1	yes	characteristic
150	C₄×D₁₃	C₂×C₄²	1	2	yes	—
151	C₄×D₁₃	#117	1	2	yes	—
152	C₄×D₁₃	#117	1	2	yes	—
153	C₄×D₁₃	#117	1	2	yes	—
154	C₄×D₁₃	#117	1	2	yes	—
155	C₄×D₁₃	#117	1	2	yes	—
156	C₄×D₁₃	C₂³×C₄	1	2	yes	—
157	C₄×D₁₃	C₂×#41	1	2	yes	—
158	#1057	—	2	8	no	—
159	C₈×C₁₃	—	2	2	no	—
160	C₈×C₁₃	C₂×C₄²	1	2	yes	—
161	C₈×C₁₃	C₂×C₄²	1	2	yes	—
162	#1055	—	2	2	no	—
163	#1055	C₂×C₄²	1	4	yes	—
164	C₂²×M₁₆	—	13	2	no	—
165	C₂²×M₁₆	—	13	1	no	—
166	C₂³×C₈	—	13	1	no	—
167	C₂×#117	—	13	2	no	—
168	C₂×#117	—	13	1	no	—
169	#405	—	13	8	no	—
170	#405	—	13	8	no	—
171	#405	—	13	4	no	—
172	#405	—	13	4	no	—
173	C₂×#97	—	13	4	no	—
174	C₂×#97	—	13	1	no	—
175	C₂×#97	—	13	1	no	—
176	C₂×C₄×C₈	—	13	4	no	—
177	C₂×C₄×C₈	—	13	1	no	—
178	C₂×C₄×C₈	#261	1	1	yes	characteristic, complemented
179	#338	—	13	4	no	—
180	H(3,ℤ/4)	—	13	8	no	—
181	H(3,ℤ/4)	—	13	4	no	—
182	C₂²×C₁₃	—	2	2	no	—
183	C₂²×C₁₃	#376	1	1	yes	characteristic
184	C₂²×C₁₃	#376	1	1	yes	characteristic
185	C₂²×C₁₃	C₂×#117	1	1	yes	characteristic
186	C₂×D₁₃	—	2	4	no	—
187	C₂×D₁₃	—	4	2	no	—
188	C₂×D₁₃	—	4	1	no	—
189	C₂×D₁₃	—	4	1	no	—
190	C₂×D₁₃	#376	1	2	yes	—
191	C₂×D₁₃	C₂×#117	1	2	yes	—
192	#261	—	4	8	no	—
193	C₄×C₁₃	—	4	2	no	—
194	C₄×C₁₃	#376	1	1	yes	characteristic
195	C₄×C₁₃	#376	1	1	yes	characteristic
196	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
197	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
198	Dic₁₃	—	4	2	no	—
199	Dic₁₃	#376	1	2	yes	—
200	Dic₁₃	C₂×#117	1	2	yes	—
201	C₂³×C₄	—	13	1	no	—
202	C₂×M₁₆	—	13	4	no	—
203	C₂×M₁₆	—	13	4	no	—
204	C₂×M₁₆	—	13	2	no	—
205	C₂×M₁₆	—	13	2	no	—
206	C₂×M₁₆	—	26	2	no	—
207	C₂×M₁₆	—	26	2	no	—
208	C₂×M₁₆	—	26	2	no	—
209	C₂×M₁₆	—	26	2	no	—
210	C₂×M₁₆	—	26	2	no	—
211	C₂×M₁₆	—	26	2	no	—
212	C₂²×C₈	—	13	1	no	—
213	C₂²×C₈	—	13	2	no	—
214	C₂²×C₈	—	13	4	no	—
215	C₂²×C₈	—	13	2	no	—
216	C₂²×C₈	—	13	2	no	—
217	C₂²×C₈	—	13	2	no	—
218	C₂²×C₈	—	13	4	no	—
219	C₂²×C₈	—	13	1	no	—
220	C₂²×C₈	C₂×#261	1	1	yes	characteristic
221	C₂²×C₈	C₂×#261	1	1	yes	characteristic
222	#117	—	13	4	no	—
223	#117	—	13	4	no	—
224	#117	—	13	2	no	—
225	#117	—	13	2	no	—
226	#117	—	26	1	no	—
227	#117	—	26	1	no	—
228	#117	—	26	2	no	—
229	#117	—	26	2	no	—
230	C₂×C₄⋊C₄	—	26	1	no	—
231	C₂×C₄⋊C₄	—	26	2	no	—
232	C₂×C₂²⋊C₄	—	26	1	no	—
233	C₂×C₂²⋊C₄	—	26	2	no	—
234	C₂×C₄²	—	13	4	no	—
235	C₂×C₄²	—	13	1	no	—
236	C₂×C₄²	C₂×#261	1	1	yes	characteristic
237	#97	—	13	16	no	—
238	#97	—	13	4	no	—
239	#97	—	13	4	no	—
240	C₄×C₈	—	13	16	no	—
241	C₄×C₈	—	13	4	no	—
242	C₄×C₈	C₂×#261	1	4	yes	complemented
243	C₂×C₁₃	—	2	1	no	—
244	C₂×C₁₃	—	2	1	no	—
245	C₂×C₁₃	#1747	1	2	yes	complemented
246	C₂×C₁₃	C₂×#376	1	1	yes	characteristic
247	D₁₃	—	2	2	no	—
248	D₁₃	—	2	2	no	—
249	C₂⁴	—	13	1	no	—
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	—	13	1	no	—
252	C₂²×C₄	—	13	2	no	—
253	C₂²×C₄	—	13	2	no	—
254	C₂²×C₄	—	13	2	no	—
255	C₂²×C₄	—	13	1	no	—
256	C₂²×C₄	—	13	2	no	—
257	C₂²×C₄	—	13	1	no	—
258	C₂²×C₄	—	13	1	no	—
259	C₂²×C₄	—	13	1	no	—
260	C₂²×C₄	—	26	4	no	—
261	C₂²×C₄	—	26	1	no	—
262	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
263	M₁₆	—	26	4	no	—
264	M₁₆	—	26	4	no	—
265	M₁₆	—	26	4	no	—
266	M₁₆	—	26	4	no	—
267	M₁₆	—	26	4	no	—
268	M₁₆	—	26	4	no	—
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	13	2	no	—
272	C₂×C₈	—	13	2	no	—
273	C₂×C₈	—	13	2	no	—
274	C₂×C₈	—	13	4	no	—
275	C₂×C₈	—	13	4	no	—
276	C₂×C₈	—	13	4	no	—
277	C₂×C₈	—	13	4	no	—
278	C₂×C₈	—	13	8	no	—
279	C₂×C₈	—	13	2	no	—
280	C₂×C₈	—	26	4	no	—
281	C₂×C₈	—	26	4	no	—
282	C₂×C₈	—	26	1	no	—
283	C₂×C₈	—	26	1	no	—
284	C₂×C₈	C₄×#261	1	1	yes	characteristic
285	C₂×C₈	C₄×#261	1	2	yes	—
286	C₂×C₈	C₄×#261	1	1	yes	characteristic
287	C₂×C₈	C₂²×#261	1	2	yes	—
288	C₂×C₈	C₂²×#261	1	2	yes	—
289	C₄⋊C₄	—	26	2	no	—
290	C₄⋊C₄	—	26	2	no	—
291	C₄⋊C₄	—	26	4	no	—
292	C₄⋊C₄	—	26	4	no	—
293	C₂²⋊C₄	—	26	2	no	—
294	C₂²⋊C₄	—	26	2	no	—
295	C₂²⋊C₄	—	26	4	no	—
296	C₂²⋊C₄	—	26	4	no	—
297	C₄²	—	2	1	no	—
298	C₄²	—	13	8	no	—
299	C₄²	—	13	2	no	—
300	C₄²	—	26	4	no	—
301	C₄²	—	26	1	no	—
302	C₄²	C₂²×#261	1	2	yes	—
303	C₁₃	C₂×#1747	1	1	yes	characteristic, complemented, Sylow
304	C₂³	—	13	2	no	—
305	C₂³	—	13	2	no	—
306	C₂³	—	13	1	no	—
307	C₂³	—	13	1	no	—
308	C₂³	—	52	2	no	—
309	C₂³	#88237	1	1	yes	characteristic
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	13	2	no	—
313	C₂×C₄	—	13	2	no	—
314	C₂×C₄	—	13	1	no	—
315	C₂×C₄	—	13	1	no	—
316	C₂×C₄	—	13	2	no	—
317	C₂×C₄	—	13	2	no	—
318	C₂×C₄	—	13	2	no	—
319	C₂×C₄	—	13	2	no	—
320	C₂×C₄	—	13	2	no	—
321	C₂×C₄	—	13	2	no	—
322	C₂×C₄	—	13	2	no	—
323	C₂×C₄	—	13	2	no	—
324	C₂×C₄	—	26	8	no	—
325	C₂×C₄	—	26	2	no	—
326	C₂×C₄	—	52	2	no	—
327	C₂×C₄	—	52	8	no	—
328	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
329	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
330	C₂×C₄	#88237	1	1	yes	characteristic
331	C₂×C₄	#88237	1	1	yes	characteristic
332	C₂×C₄	#88237	1	1	yes	characteristic
333	C₂×C₄	C₂³×#261	1	1	yes	characteristic
334	C₈	—	2	2	no	—
335	C₈	—	13	4	no	—
336	C₈	—	26	8	no	—
337	C₈	—	26	2	no	—
338	C₈	C₂×C₄×#261	1	2	yes	—
339	C₈	C₂×C₄×#261	1	2	yes	—
340	C₂²	—	2	2	no	—
341	C₂²	—	13	2	no	—
342	C₂²	—	13	2	no	—
343	C₂²	—	26	4	no	—
344	C₂²	—	52	1	no	—
345	C₂²	—	52	1	no	—
346	C₂²	—	52	2	no	—
347	C₂²	#11719057	1	1	yes	characteristic
348	C₂²	#11719057	1	1	yes	characteristic
349	C₂²	C₂×#88237	1	1	yes	characteristic
350	C₄	—	4	2	no	—
351	C₄	—	13	2	no	—
352	C₄	—	13	2	no	—
353	C₄	—	52	2	no	—
354	C₄	—	52	8	no	—
355	C₄	#11719057	1	1	yes	characteristic
356	C₄	#11719057	1	1	yes	characteristic
357	C₄	C₂²×C₄×#261	1	1	yes	characteristic
358	C₄	C₂×#88237	1	1	yes	characteristic
359	C₂	—	2	1	no	—
360	C₂	—	2	1	no	—
361	C₂	—	26	2	no	—
362	C₂	—	26	2	no	—
363	C₂	#13130557	1	2	yes	direct product factor
364	C₂	C₂×#11719057	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄×#261
commutator	C₄×C₁₃	C₂²×C₄²
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂²×C₁₃	#376
prime radical	C₂³×D₁₃	#41
cosocle	C₂×C₄×D₁₃	C₂⁴
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	C₂×#1747
LCS	G ▷ C₄×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #13130557	C₂	direct (N × H)
12. #11719089	C₄	semidirect
14. C₂×#88100	C₄	semidirect
15. C₂×C₈×#261	C₄	semidirect
16. C₂×C₄×C₈×C₁₃	C₄	semidirect
17. C₂×#88055	C₄	semidirect
18. C₂×C₈×Dic₁₃	C₄	semidirect
19. C₂×#88043	C₄	semidirect
20. #11718350	C₄	semidirect
61. #88100	C₂×C₄	semidirect
62. C₈×#261	C₂×C₄	semidirect
63. C₄×C₈×C₁₃	C₂×C₄	semidirect
64. #88055	C₂×C₄	semidirect
65. C₈×Dic₁₃	C₂×C₄	semidirect
66. #88043	C₂×C₄	semidirect
178. C₂×C₄×C₈	#261	semidirect
242. C₄×C₈	C₂×#261	semidirect
245. C₂×C₁₃	#1747	semidirect
303. C₁₃	C₂×#1747	semidirect
363. C₂	#13130557	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×#11719089	2
#13130557	2
C₂×#11718269	2
C₂×#11718267	2
C₂×#11718235	2
C₂×#11718350	2
C₂×#1747	13

Subgroups — C2×#13130557 #16796861

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13130557 #16796861