Subgroups — C₂×#13131188 #16807029

Order 3328.

718 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719092	C₂	1	1	yes	maximal, characteristic
2	C₂²×#88102	C₂	1	1	yes	maximal, characteristic
3	#13131188	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718222	C₂	1	2	yes	maximal, complemented
5	C₂×#11718221	C₂	1	2	yes	maximal, complemented
6	C₂×D₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
7	C₂×#88237	—	2	1	no	—
8	C₂²×C₄⋊#261	—	2	1	no	—
9	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
10	#11719092	—	2	2	no	—
11	#11719092	C₂²	1	4	yes	—
12	C₂×#88102	—	2	1	no	—
13	C₂×#88102	—	2	2	no	—
14	C₂×#88102	C₂²	1	1	yes	characteristic
15	C₂×#88102	C₂²	1	2	yes	—
16	C₂×#88102	C₂²	1	1	yes	characteristic
17	C₂×#88102	C₂²	1	4	yes	—
18	C₂×#88101	—	2	1	no	—
19	C₂×C₁₃×M₃₂	C₄	1	1	yes	characteristic, complemented
20	C₂×#88052	C₄	1	1	yes	characteristic, complemented
21	D₁₃×M₃₂	C₄	1	4	yes	complemented
22	D₁₃×M₃₂	C₂²	1	4	yes	—
23	C₂×#88038	C₄	1	2	yes	complemented
24	C₂×C₁₆×D₁₃	C₂²	1	2	yes	—
25	#11718222	C₂²	1	8	yes	complemented
26	#11718221	C₂²	1	8	yes	complemented
27	C₂³×#261	—	4	1	no	—
28	C₂²×C₄×D₁₃	D₄	1	1	yes	characteristic
29	C₂×C₂²⋊#261	—	4	1	no	—
30	#88237	—	2	4	no	—
31	#88237	—	4	2	no	—
32	C₂×C₄⋊#261	—	2	1	no	—
33	C₂×C₄⋊#261	—	2	1	no	—
34	C₂×C₄⋊#261	—	2	2	no	—
35	C₂×C₄⋊#261	—	2	4	no	—
36	C₂×C₄⋊#261	—	4	1	no	—
37	C₂×C₄⋊#261	—	4	2	no	—
38	C₂×C₄×#261	—	4	1	no	—
39	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×#1055	Q₈	1	1	yes	characteristic
41	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
43	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
44	C₂×C₈×D₁₃	D₄	1	2	yes	—
45	C₂×C₈×D₁₃	Q₈	1	2	yes	—
46	C₂×C₈×D₁₃	Q₈	1	1	yes	characteristic
47	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
48	C₂×C₈×D₁₃	C₂³	1	2	yes	—
49	#88102	—	2	4	no	—
50	#88102	—	2	2	no	—
51	#88102	—	2	1	no	—
52	#88102	—	2	1	no	—
53	#88102	C₂³	1	2	yes	—
54	#88102	C₂³	1	4	yes	—
55	#88102	C₂³	1	2	yes	—
56	#88101	—	2	4	no	—
57	C₁₃×M₃₂	C₂×C₄	1	4	yes	complemented
58	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	complemented
59	#88052	C₂×C₄	1	4	yes	complemented
60	C₂×#6090	C₂×C₄	1	2	yes	complemented
61	#88038	C₂×C₄	1	8	yes	complemented
62	C₁₆×D₁₃	C₂×C₄	1	4	yes	complemented
63	C₁₆×D₁₃	C₂³	1	4	yes	—
64	C₂×#2159	—	13	1	no	maximal, Sylow
65	C₂³×D₁₃	D₈	1	1	yes	characteristic
66	C₂²×#261	—	4	1	no	—
67	C₂²×#261	—	4	1	no	—
68	C₂²×#261	—	4	4	no	—
69	C₂²×#261	—	4	2	no	—
70	C₂²×#261	—	8	4	no	—
71	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
72	C₂²×Dic₁₃	Q₁₆	1	1	yes	characteristic
73	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
75	C₂×C₄×D₁₃	D₈	1	1	yes	characteristic
76	C₂×C₄×D₁₃	D₈	1	2	yes	—
77	C₂×C₄×D₁₃	Q₁₆	1	1	yes	characteristic
78	C₂×C₄×D₁₃	Q₁₆	1	2	yes	—
79	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
81	C₂²⋊#261	—	4	4	no	—
82	C₄⋊#261	—	2	2	no	—
83	C₄⋊#261	—	2	2	no	—
84	C₄⋊#261	—	2	4	no	—
85	C₄⋊#261	—	4	4	no	—
86	C₄⋊#261	—	4	2	no	—
87	C₄⋊#261	—	4	1	no	—
88	C₄⋊#261	—	4	1	no	—
89	C₄×#261	—	4	4	no	—
90	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
92	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
94	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
95	C₂×#1055	C₄⋊C₄	1	2	yes	—
96	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
97	C₂×#1055	C₂×Q₈	1	2	yes	—
98	C₈×D₁₃	C₄⋊C₄	1	2	yes	—
99	C₈×D₁₃	C₄⋊C₄	1	2	yes	—
100	C₈×D₁₃	C₄⋊C₄	1	2	yes	—
101	C₈×D₁₃	C₂²×C₄	1	1	yes	characteristic
102	C₈×D₁₃	C₂²×C₄	1	2	yes	—
103	C₈×D₁₃	C₂×D₄	1	2	yes	—
104	C₈×D₁₃	C₂×D₄	1	1	yes	characteristic
105	C₈×D₁₃	C₂×Q₈	1	1	yes	characteristic
106	C₈×D₁₃	C₂×Q₈	1	2	yes	—
107	C₈×D₁₃	C₂⁴	1	1	yes	characteristic
108	C₁₃×C₁₆	C₂²×C₄	1	4	yes	complemented
109	#6090	C₂²×C₄	1	4	yes	complemented
110	C₂²×M₃₂	—	13	1	no	—
111	C₂×#427	—	13	1	no	—
112	C₂²×#107	—	13	1	no	—
113	#2159	—	13	8	no	—
114	C₂×#367	—	13	2	no	—
115	C₂×#366	—	13	2	no	—
116	C₂³×C₁₃	#107	1	1	yes	characteristic
117	C₂²×D₁₃	—	2	2	no	—
118	C₂²×D₁₃	—	2	2	no	—
119	C₂²×D₁₃	#107	1	1	yes	characteristic
120	C₂²×D₁₃	#107	1	2	yes	—
121	C₂²×D₁₃	C₂×D₈	1	2	yes	—
122	C₂²×D₁₃	C₂×D₈	1	1	yes	characteristic
123	C₂×#261	—	4	2	no	—
124	C₂×#261	—	4	2	no	—
125	C₂×#261	—	4	4	no	—
126	C₂×#261	—	8	8	no	—
127	C₂×#261	—	8	2	no	—
128	C₂×#261	—	8	4	no	—
129	C₂×#261	—	8	2	no	—
130	C₂×C₄×C₁₃	#107	1	2	yes	—
131	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
133	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×Dic₁₃	#107	1	2	yes	—
135	C₂×Dic₁₃	#107	1	1	yes	characteristic
136	C₂×Dic₁₃	C₂×Q₁₆	1	2	yes	—
137	C₂×Dic₁₃	C₂×Q₁₆	1	1	yes	characteristic
138	C₄×D₁₃	#107	1	2	yes	—
139	C₄×D₁₃	#107	1	2	yes	—
140	C₄×D₁₃	#107	1	2	yes	—
141	C₄×D₁₃	C₂×C₄⋊C₄	1	2	yes	—
142	C₄×D₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₄×D₁₃	C₂×D₈	1	2	yes	—
144	C₄×D₁₃	C₂×D₈	1	1	yes	characteristic
145	C₄×D₁₃	C₂×Q₁₆	1	2	yes	—
146	C₄×D₁₃	C₂×Q₁₆	1	1	yes	characteristic
147	C₄×D₁₃	C₂²×D₄	1	1	yes	characteristic
148	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
149	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
151	#1055	C₂×C₄⋊C₄	1	1	yes	characteristic
152	#1055	C₂×C₄⋊C₄	1	2	yes	—
153	#1055	C₂²×Q₈	1	1	yes	characteristic
154	C₂³×C₈	—	13	1	no	—
155	C₂×#117	—	26	1	no	—
156	C₂²×C₄⋊C₄	—	26	1	no	—
157	C₂×M₃₂	—	13	1	no	—
158	C₂×M₃₂	—	13	4	no	—
159	C₂×M₃₂	—	13	4	no	—
160	C₂×M₃₂	—	13	2	no	—
161	C₂×M₃₂	#261	1	1	yes	characteristic, complemented
162	C₂²×C₁₆	—	13	2	no	—
163	#427	—	13	4	no	—
164	#427	—	26	2	no	—
165	C₂×#107	—	13	2	no	—
166	C₂×#107	—	13	4	no	—
167	C₂×#107	—	13	1	no	—
168	C₂×#107	—	13	1	no	—
169	C₂×#107	—	26	1	no	—
170	C₂×#107	—	26	2	no	—
171	C₂×#106	—	26	1	no	—
172	#367	—	13	8	no	—
173	#366	—	13	8	no	—
174	C₂²×C₁₃	—	2	2	no	—
175	C₂²×C₁₃	C₂×#107	1	2	yes	—
176	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
177	C₂×D₁₃	—	2	2	no	—
178	C₂×D₁₃	—	2	2	no	—
179	C₂×D₁₃	—	2	2	no	—
180	C₂×D₁₃	—	2	4	no	—
181	C₂×D₁₃	C₂×#107	1	2	yes	—
182	C₂×D₁₃	C₂×#107	1	1	yes	characteristic
183	C₂×D₁₃	#509	1	2	yes	—
184	C₂×D₁₃	#510	1	2	yes	—
185	C₂×D₁₃	C₂²×D₈	1	1	yes	characteristic
186	#261	—	8	2	no	—
187	#261	—	8	4	no	—
188	#261	—	8	2	no	—
189	C₄×C₁₃	C₂×#107	1	2	yes	—
190	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
191	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
192	Dic₁₃	C₂×#107	1	2	yes	—
193	Dic₁₃	C₂×#107	1	1	yes	characteristic
194	Dic₁₃	C₂²×Q₁₆	1	1	yes	characteristic
195	C₂³×C₄	—	13	1	no	—
196	C₂³×C₄	—	52	1	no	—
197	C₂²×C₈	—	13	1	no	—
198	C₂²×C₈	—	13	1	no	—
199	C₂²×C₈	—	13	2	no	—
200	C₂²×C₈	—	13	2	no	—
201	C₂²×C₈	—	13	1	no	—
202	C₂²×C₈	—	13	2	no	—
203	C₂²×C₈	—	13	1	no	—
204	C₂²×C₈	—	13	1	no	—
205	C₂²×C₈	—	13	2	no	—
206	C₂²×C₈	C₂×#261	1	1	yes	characteristic
207	#117	—	26	4	no	—
208	#117	—	52	2	no	—
209	C₂×C₄⋊C₄	—	26	2	no	—
210	C₂×C₄⋊C₄	—	26	4	no	—
211	C₂×C₄⋊C₄	—	26	1	no	—
212	C₂×C₄⋊C₄	—	26	1	no	—
213	C₂×C₄⋊C₄	—	52	1	no	—
214	C₂×C₄⋊C₄	—	52	2	no	—
215	C₂×C₂²⋊C₄	—	52	1	no	—
216	C₂×C₄²	—	52	1	no	—
217	M₃₂	—	13	4	no	—
218	M₃₂	—	13	8	no	—
219	M₃₂	C₂×#261	1	4	yes	complemented
220	C₂×C₁₆	—	13	2	no	—
221	C₂×C₁₆	—	13	4	no	—
222	C₂×C₁₆	—	13	4	no	—
223	C₂×C₁₆	C₂×#261	1	2	yes	complemented
224	#107	—	13	2	no	—
225	#107	—	13	4	no	—
226	#107	—	13	2	no	—
227	#107	—	26	4	no	—
228	#107	—	26	2	no	—
229	#107	—	26	1	no	—
230	#107	—	26	1	no	—
231	#106	—	26	4	no	—
232	C₂×C₁₃	—	2	2	no	—
233	C₂×C₁₃	#2159	1	2	yes	complemented
234	C₂×C₁₃	C₂²×#107	1	1	yes	characteristic
235	D₁₃	—	2	2	no	—
236	D₁₃	#2159	1	2	yes	complemented
237	D₁₃	C₂×#509	1	1	yes	characteristic
238	D₁₃	C₂×#510	1	1	yes	characteristic
239	C₂⁴	—	13	1	no	—
240	C₂²×C₄	—	13	1	no	—
241	C₂²×C₄	—	13	1	no	—
242	C₂²×C₄	—	13	2	no	—
243	C₂²×C₄	—	13	2	no	—
244	C₂²×C₄	—	13	1	no	—
245	C₂²×C₄	—	13	1	no	—
246	C₂²×C₄	—	13	2	no	—
247	C₂²×C₄	—	13	2	no	—
248	C₂²×C₄	—	13	1	no	—
249	C₂²×C₄	—	52	1	no	—
250	C₂²×C₄	—	52	1	no	—
251	C₂²×C₄	—	52	2	no	—
252	C₂²×C₄	—	52	4	no	—
253	C₂²×C₄	—	104	4	no	—
254	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
255	C₂×C₈	—	13	2	no	—
256	C₂×C₈	—	13	1	no	—
257	C₂×C₈	—	13	2	no	—
258	C₂×C₈	—	13	1	no	—
259	C₂×C₈	—	13	2	no	—
260	C₂×C₈	—	13	1	no	—
261	C₂×C₈	—	13	2	no	—
262	C₂×C₈	—	13	2	no	—
263	C₂×C₈	—	13	2	no	—
264	C₂×C₈	—	13	1	no	—
265	C₂×C₈	—	13	2	no	—
266	C₂×C₈	—	13	1	no	—
267	C₂×C₈	—	13	1	no	—
268	C₂×C₈	—	13	2	no	—
269	C₂×C₈	C₄⋊#261	1	2	yes	—
270	C₂×C₈	C₄⋊#261	1	1	yes	characteristic
271	C₂×C₈	C₂²×#261	1	1	yes	characteristic
272	C₂×C₈	C₂²×#261	1	2	yes	—
273	C₄⋊C₄	—	26	2	no	—
274	C₄⋊C₄	—	26	2	no	—
275	C₄⋊C₄	—	26	4	no	—
276	C₄⋊C₄	—	52	4	no	—
277	C₄⋊C₄	—	52	2	no	—
278	C₄⋊C₄	—	52	1	no	—
279	C₄⋊C₄	—	52	1	no	—
280	C₂²⋊C₄	—	52	4	no	—
281	C₄²	—	52	4	no	—
282	C₁₆	—	13	4	no	—
283	C₁₆	C₂²×#261	1	4	yes	complemented
284	C₁₃	C₂×#2159	1	1	yes	characteristic, complemented, Sylow
285	C₂³	—	13	1	no	—
286	C₂³	—	13	2	no	—
287	C₂³	—	13	2	no	—
288	C₂³	—	13	1	no	—
289	C₂³	—	26	2	no	—
290	C₂³	—	26	2	no	—
291	C₂³	#88102	1	1	yes	characteristic
292	C₂×C₄	—	13	2	no	—
293	C₂×C₄	—	13	2	no	—
294	C₂×C₄	—	13	1	no	—
295	C₂×C₄	—	13	1	no	—
296	C₂×C₄	—	13	2	no	—
297	C₂×C₄	—	13	1	no	—
298	C₂×C₄	—	13	2	no	—
299	C₂×C₄	—	13	2	no	—
300	C₂×C₄	—	13	2	no	—
301	C₂×C₄	—	13	1	no	—
302	C₂×C₄	—	13	2	no	—
303	C₂×C₄	—	13	1	no	—
304	C₂×C₄	—	13	1	no	—
305	C₂×C₄	—	13	2	no	—
306	C₂×C₄	—	52	2	no	—
307	C₂×C₄	—	52	2	no	—
308	C₂×C₄	—	52	4	no	—
309	C₂×C₄	—	104	4	no	—
310	C₂×C₄	—	104	2	no	—
311	C₂×C₄	—	104	2	no	—
312	C₂×C₄	—	104	8	no	—
313	C₂×C₄	#88102	1	2	yes	—
314	C₂×C₄	#88102	1	1	yes	characteristic
315	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
316	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
317	C₈	—	13	2	no	—
318	C₈	—	13	1	no	—
319	C₈	—	13	1	no	—
320	C₈	C₂×C₄⋊#261	1	2	yes	—
321	C₈	C₂×C₄⋊#261	1	1	yes	characteristic
322	C₈	C₂³×#261	1	1	yes	characteristic
323	C₂²	—	2	2	no	—
324	C₂²	—	13	2	no	—
325	C₂²	—	13	2	no	—
326	C₂²	—	13	1	no	—
327	C₂²	—	13	1	no	—
328	C₂²	—	13	2	no	—
329	C₂²	—	26	2	no	—
330	C₂²	—	26	2	no	—
331	C₂²	—	26	4	no	—
332	C₂²	—	26	2	no	—
333	C₂²	C₂×#88102	1	2	yes	—
334	C₂²	C₂×#88102	1	1	yes	characteristic
335	C₄	—	13	1	no	—
336	C₄	—	13	1	no	—
337	C₄	—	13	2	no	—
338	C₄	—	104	2	no	—
339	C₄	—	104	2	no	—
340	C₄	—	104	4	no	—
341	C₄	C₂×#88102	1	2	yes	—
342	C₄	C₂×#88102	1	1	yes	characteristic
343	C₄	C₂²×C₄⋊#261	1	1	yes	characteristic
344	C₂	—	2	2	no	—
345	C₂	—	13	1	no	—
346	C₂	—	13	1	no	—
347	C₂	—	13	2	no	—
348	C₂	—	26	2	no	—
349	C₂	#13131188	1	2	yes	direct product factor
350	C₂	C₂²×#88102	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#88102
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₈	C₂³×#261
Fitting	C₂×C₁₃×M₃₂	C₄
socle	C₂²×C₁₃	C₂×#107
prime radical	C₂³×D₁₃	D₈
cosocle	C₈×D₁₃	C₂⁴
hypercenter	C₂×M₃₂	#261
hypocenter	C₁₃	C₂×#2159
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
3. #13131188	C₂	direct (N × H)
4. C₂×#11718222	C₂	semidirect
5. C₂×#11718221	C₂	semidirect
19. C₂×C₁₃×M₃₂	C₄	semidirect
20. C₂×#88052	C₄	semidirect
21. D₁₃×M₃₂	C₄	semidirect
23. C₂×#88038	C₄	semidirect
25. #11718222	C₂²	semidirect
26. #11718221	C₂²	semidirect
57. C₁₃×M₃₂	C₂×C₄	semidirect
58. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
59. #88052	C₂×C₄	semidirect
60. C₂×#6090	C₂×C₄	semidirect
61. #88038	C₂×C₄	semidirect
62. C₁₆×D₁₃	C₂×C₄	semidirect
108. C₁₃×C₁₆	C₂²×C₄	semidirect
109. #6090	C₂²×C₄	semidirect
161. C₂×M₃₂	#261	semidirect
219. M₃₂	C₂×#261	semidirect
223. C₂×C₁₆	C₂×#261	semidirect
233. C₂×C₁₃	#2159	semidirect
236. D₁₃	#2159	semidirect
283. C₁₆	C₂²×#261	semidirect
284. C₁₃	C₂×#2159	semidirect
349. C₂	#13131188	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×#11719092	2
C₂²×#88102	2
#13131188	2
C₂×#11718222	2
C₂×#11718221	2
C₂×D₁₃×M₃₂	2
C₂×#2159	13

Subgroups — C2×#13131188 #16807029

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13131188 #16807029