Subgroups — C₂×#13127728 #16743377

Order 3328.

1766 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719272	C₂	1	2	yes	maximal
2	#13127728	C₂	1	16	yes	maximal, direct product factor
3	C₂×#11718343	C₂	1	1	yes	maximal, characteristic
4	C₂×#11718344	C₂	1	2	yes	maximal
5	C₂×#11718490	C₂	1	4	yes	maximal
6	C₂×#11718512	C₂	1	2	yes	maximal
7	C₂×C₄×Dic₅₂	C₂	1	2	yes	maximal
8	C₂×C₄×Dic₅₂	C₂	1	1	yes	maximal, characteristic
9	C₂×C₈×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×#88198	—	2	2	no	—
12	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic
13	#11719272	—	2	8	no	—
14	#11719272	C₂²	1	16	yes	—
15	C₂×#88143	—	2	4	no	—
16	C₂×#88141	—	2	2	no	—
17	C₂×#88124	C₂²	1	4	yes	—
18	C₂×#88123	C₂²	1	2	yes	—
19	C₂×C₄×Dic₂₆	C₂²	1	4	yes	—
20	C₂×C₄×Dic₂₆	C₂²	1	2	yes	—
21	C₈×Q₈×C₁₃	C₂²	1	8	yes	—
22	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
23	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
26	C₂×#88058	C₂²	1	2	yes	—
27	C₂×#88058	C₂²	1	1	yes	characteristic
28	C₂×#88058	C₂²	1	2	yes	—
29	C₂×#88057	C₂²	1	4	yes	—
30	C₂×#88056	C₂²	1	2	yes	—
31	C₂×#88056	C₂²	1	4	yes	—
32	#11718512	C₂²	1	16	yes	—
33	#11718490	C₂²	1	32	yes	—
34	C₄×Dic₅₂	C₂²	1	8	yes	—
35	C₄×Dic₅₂	C₂²	1	16	yes	—
36	#11718344	C₂²	1	16	yes	—
37	#11718343	C₂²	1	8	yes	—
38	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
39	C₂²×Dic₂₆	—	2	2	no	—
40	C₂²×Dic₂₆	C₂³	1	2	yes	—
41	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
42	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
44	C₄×Q₈×C₁₃	D₄	1	4	yes	—
45	C₄×Q₈×C₁₃	C₂³	1	4	yes	—
46	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
47	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
48	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
50	#88198	—	2	16	no	—
51	C₂×C₄⋊Dic₁₃	—	2	2	no	—
52	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
53	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
54	C₂×#6101	—	2	2	no	—
55	C₂×#6101	—	2	4	no	—
56	C₂×#6101	—	2	4	no	—
57	C₂×#6101	—	2	4	no	—
58	C₂×C₄×Dic₁₃	—	2	4	no	—
59	C₂×C₄×Dic₁₃	—	2	2	no	—
60	C₂×Dic₅₂	Q₈	1	8	yes	complemented
61	C₂×Dic₅₂	C₂³	1	4	yes	—
62	#88143	—	2	32	no	—
63	#88141	—	2	16	no	—
64	#88124	—	2	8	no	—
65	#88124	C₂³	1	16	yes	—
66	#88123	—	2	4	no	—
67	#88123	C₂³	1	8	yes	—
68	C₄×Dic₂₆	—	2	8	no	—
69	C₄×Dic₂₆	—	2	4	no	—
70	C₄×Dic₂₆	C₂³	1	16	yes	—
71	C₄×Dic₂₆	C₂³	1	8	yes	—
72	C₁₃×C₄⋊C₈	C₂³	1	8	yes	—
73	C₁₃×C₄⋊C₈	C₂³	1	4	yes	—
74	C₄×C₈×C₁₃	C₂³	1	4	yes	—
75	C₄×C₈×C₁₃	C₂³	1	8	yes	—
76	#88058	C₂³	1	8	yes	—
77	#88058	C₂³	1	8	yes	—
78	#88058	C₂³	1	4	yes	—
79	#88057	C₂³	1	16	yes	—
80	#88056	C₂³	1	8	yes	—
81	#88056	C₂³	1	16	yes	—
82	C₂×#3383	—	13	1	no	maximal, Sylow
83	C₂×Q₈×C₁₃	—	2	4	no	—
84	C₂×Q₈×C₁₃	C₂×D₄	1	4	yes	—
85	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
90	C₂²×Dic₁₃	—	2	2	no	—
91	C₂²×Dic₁₃	—	2	2	no	—
92	C₂²×Dic₁₃	—	2	4	no	—
93	C₂×Dic₂₆	—	2	2	no	—
94	C₂×Dic₂₆	—	2	4	no	—
95	C₂×Dic₂₆	—	2	8	no	—
96	C₂×Dic₂₆	—	4	8	no	—
97	C₂×Dic₂₆	C₂×Q₈	1	8	yes	—
98	C₂×Dic₂₆	C₂⁴	1	4	yes	—
99	C₂×C₈×C₁₃	—	2	4	no	—
100	C₂×C₈×C₁₃	—	2	2	no	—
101	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
102	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
103	C₂×C₈×C₁₃	C₂⁴	1	2	yes	—
104	C₂×C₈×C₁₃	C₂⁴	1	2	yes	—
105	C₁₃×C₄⋊C₄	C₂×D₄	1	4	yes	—
106	C₁₃×C₄⋊C₄	C₂×D₄	1	8	yes	—
107	C₄²×C₁₃	C₂×D₄	1	2	yes	—
108	C₄²×C₁₃	C₂×D₄	1	4	yes	—
109	C₄²×C₁₃	C₂⁴	1	2	yes	—
110	C₄²×C₁₃	C₂⁴	1	4	yes	—
111	C₄⋊Dic₁₃	—	2	4	no	—
112	C₄⋊Dic₁₃	—	2	2	no	—
113	C₄⋊Dic₁₃	—	2	8	no	—
114	C₄⋊Dic₁₃	C₂⁴	1	8	yes	—
115	C₄⋊Dic₁₃	C₂⁴	1	4	yes	—
116	#6101	—	2	8	no	—
117	#6101	—	2	16	no	—
118	#6101	—	2	16	no	—
119	#6101	—	2	16	no	—
120	C₄×Dic₁₃	—	2	16	no	—
121	C₄×Dic₁₃	—	2	8	no	—
122	Dic₅₂	C₂×Q₈	1	16	yes	complemented
123	C₂×#557	—	13	2	no	—
124	#3383	—	13	16	no	—
125	C₂×#500	—	13	1	no	—
126	C₂×#499	—	13	2	no	—
127	C₂×#479	—	13	4	no	—
128	C₂×#475	—	13	2	no	—
129	C₂×C₈×Q₈	D₁₃	1	1	yes	characteristic
130	C₂×C₄×Q₁₆	—	13	1	no	—
131	C₂×C₄×Q₁₆	—	13	2	no	—
132	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
133	Q₈×C₁₃	—	2	8	no	—
134	C₂×C₄×C₁₃	—	2	4	no	—
135	C₂×C₄×C₁₃	—	2	4	no	—
136	C₂×C₄×C₁₃	—	2	2	no	—
137	C₂×C₄×C₁₃	#135	1	4	yes	—
138	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
139	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
140	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
141	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
143	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
144	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
145	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
146	C₂×Dic₁₃	—	2	8	no	—
147	C₂×Dic₁₃	—	2	4	no	—
148	C₂×Dic₁₃	—	2	4	no	—
149	C₂×Dic₁₃	—	2	8	no	—
150	C₂×Dic₁₃	—	4	8	no	—
151	C₂×Dic₁₃	—	4	4	no	—
152	Dic₂₆	—	2	8	no	—
153	Dic₂₆	—	2	4	no	—
154	Dic₂₆	—	4	16	no	—
155	Dic₂₆	C₂²×Q₈	1	8	yes	—
156	C₈×C₁₃	—	2	4	no	—
157	C₈×C₁₃	—	2	2	no	—
158	C₈×C₁₃	C₂²×Q₈	1	4	yes	—
159	C₂²×Q₁₆	—	13	1	no	—
160	#557	—	13	16	no	—
161	#557	—	26	8	no	—
162	C₂×C₄⋊Q₈	—	13	2	no	—
163	C₂×C₄⋊Q₈	—	26	2	no	—
164	C₂×#125	—	13	4	no	—
165	C₂×#125	—	26	4	no	—
166	C₂×C₄×Q₈	—	13	2	no	—
167	C₂×C₄×Q₈	—	13	4	no	—
168	C₂×C₄×Q₈	—	26	2	no	—
169	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
170	#500	—	13	8	no	—
171	#499	—	13	16	no	—
172	#479	—	13	32	no	—
173	#475	—	13	16	no	—
174	C₈×Q₈	C₂×D₁₃	1	8	yes	—
175	C₄×Q₁₆	—	13	16	no	—
176	C₄×Q₁₆	—	13	8	no	—
177	C₂×#107	—	13	2	no	—
178	C₂×#107	—	13	1	no	—
179	C₂×#107	—	13	2	no	—
180	C₂×#106	—	13	4	no	—
181	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
182	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
183	C₂×#103	—	13	4	no	—
184	C₂×#103	—	13	2	no	—
185	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
186	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
187	C₂²×C₁₃	D₄×Q₈	1	2	yes	—
188	C₂²×C₁₃	C₂×#135	1	2	yes	—
189	C₂²×C₁₃	#578	1	2	yes	—
190	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
191	C₄×C₁₃	—	2	4	no	—
192	C₄×C₁₃	—	2	4	no	—
193	C₄×C₁₃	—	2	2	no	—
194	C₄×C₁₃	D₄×Q₈	1	2	yes	—
195	C₄×C₁₃	C₂×#135	1	4	yes	—
196	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
197	C₄×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
198	Dic₁₃	—	2	8	no	—
199	Dic₁₃	—	4	8	no	—
200	Dic₁₃	—	4	4	no	—
201	C₂²×Q₈	—	13	2	no	—
202	C₂²×Q₈	—	26	2	no	—
203	C₂²×Q₈	D₅₂	1	1	yes	characteristic
204	C₂×Q₁₆	—	13	8	no	—
205	C₂×Q₁₆	—	13	4	no	—
206	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
207	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
208	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
209	C₄⋊Q₈	—	13	8	no	—
210	C₄⋊Q₈	—	26	16	no	—
211	C₄⋊Q₈	—	26	4	no	—
212	#125	—	13	16	no	—
213	#125	—	26	32	no	—
214	#125	—	26	8	no	—
215	C₄×Q₈	—	13	8	no	—
216	C₄×Q₈	—	13	16	no	—
217	C₄×Q₈	—	26	16	no	—
218	C₄×Q₈	—	26	8	no	—
219	C₄×Q₈	—	26	4	no	—
220	C₄×Q₈	D₅₂	1	4	yes	—
221	C₄×Q₈	C₂²×D₁₃	1	4	yes	—
222	C₂×C₄⋊C₄	—	13	2	no	—
223	C₂×C₄⋊C₄	—	13	4	no	—
224	C₂×C₄⋊C₄	—	26	2	no	—
225	C₂×C₄⋊C₄	—	26	4	no	—
226	C₂×C₄⋊C₄	—	26	4	no	—
227	C₂×C₄⋊C₄	—	26	4	no	—
228	C₂×C₄⋊C₄	—	26	2	no	—
229	C₂×C₄⋊C₄	D₅₂	1	1	yes	characteristic
230	C₂×C₄⋊C₄	D₅₂	1	2	yes	—
231	C₂×C₄²	—	26	4	no	—
232	C₂×C₄²	—	26	2	no	—
233	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
234	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
235	#107	—	13	4	no	—
236	#107	—	13	8	no	—
237	#107	—	13	8	no	—
238	#106	—	13	16	no	—
239	C₄⋊C₈	C₂²×D₁₃	1	4	yes	—
240	C₄⋊C₈	C₂²×D₁₃	1	8	yes	—
241	#103	—	13	16	no	—
242	#103	—	13	8	no	—
243	C₄×C₈	C₂²×D₁₃	1	8	yes	—
244	C₄×C₈	C₂²×D₁₃	1	4	yes	—
245	C₂×C₁₃	#3383	1	4	yes	complemented
246	C₂×C₁₃	C₂×D₄×Q₈	1	1	yes	characteristic
247	C₂×C₁₃	C₂²×#135	1	1	yes	characteristic
248	C₂×C₁₃	C₂×#578	1	1	yes	characteristic
249	C₂×Q₈	—	2	4	no	—
250	C₂×Q₈	—	13	8	no	—
251	C₂×Q₈	—	13	4	no	—
252	C₂×Q₈	—	26	2	no	—
253	C₂×Q₈	—	26	8	no	—
254	C₂×Q₈	—	26	4	no	—
255	C₂×Q₈	—	52	8	no	—
256	C₂×Q₈	C₂×D₅₂	1	4	yes	—
257	C₂²×C₄	—	26	2	no	—
258	C₂²×C₄	—	26	2	no	—
259	C₂²×C₄	—	26	4	no	—
260	C₂²×C₄	C₂×D₅₂	1	2	yes	—
261	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
262	C₂²×C₄	C₂×D₅₂	1	2	yes	—
263	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
264	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
265	Q₁₆	—	13	16	no	—
266	C₂×C₈	—	2	4	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	Q₈×D₁₃	1	4	yes	—
269	C₂×C₈	C₂³×D₁₃	1	4	yes	—
270	C₂×C₈	C₂³×D₁₃	1	2	yes	—
271	C₂×C₈	C₂³×D₁₃	1	2	yes	—
272	C₄⋊C₄	—	13	4	no	—
273	C₄⋊C₄	—	13	8	no	—
274	C₄⋊C₄	—	26	2	no	—
275	C₄⋊C₄	—	26	4	no	—
276	C₄⋊C₄	—	26	8	no	—
277	C₄⋊C₄	—	26	16	no	—
278	C₄⋊C₄	—	26	16	no	—
279	C₄⋊C₄	—	26	16	no	—
280	C₄⋊C₄	—	26	8	no	—
281	C₄⋊C₄	C₂×D₅₂	1	4	yes	—
282	C₄⋊C₄	C₂×D₅₂	1	8	yes	—
283	C₄²	—	26	16	no	—
284	C₄²	—	26	8	no	—
285	C₄²	C₂×D₅₂	1	4	yes	—
286	C₄²	C₂×D₅₂	1	2	yes	—
287	C₄²	C₂³×D₁₃	1	4	yes	—
288	C₄²	C₂³×D₁₃	1	2	yes	—
289	C₁₃	C₂×#3383	1	1	yes	characteristic, complemented, Sylow
290	C₂³	C₂²×D₅₂	1	1	yes	characteristic
291	Q₈	—	2	8	no	—
292	Q₈	—	13	8	no	—
293	Q₈	—	26	8	no	—
294	Q₈	—	26	4	no	—
295	Q₈	—	52	16	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	26	8	no	—
300	C₂×C₄	—	26	4	no	—
301	C₂×C₄	—	26	4	no	—
302	C₂×C₄	—	26	8	no	—
303	C₂×C₄	—	52	8	no	—
304	C₂×C₄	—	52	4	no	—
305	C₂×C₄	#88158	1	4	yes	—
306	C₂×C₄	C₂²×D₅₂	1	4	yes	—
307	C₂×C₄	C₂²×D₅₂	1	2	yes	—
308	C₂×C₄	C₂²×D₅₂	1	4	yes	—
309	C₂×C₄	C₂²×D₅₂	1	2	yes	—
310	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
311	C₂×C₄	C₂×Q₈×D₁₃	1	2	yes	—
312	C₂×C₄	#88257	1	2	yes	—
313	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
314	C₈	—	2	4	no	—
315	C₈	—	2	2	no	—
316	C₈	C₂×Q₈×D₁₃	1	4	yes	—
317	C₂²	Q₈×D₅₂	1	2	yes	—
318	C₂²	C₂×#88158	1	2	yes	—
319	C₂²	#11719458	1	2	yes	—
320	C₂²	C₂³×D₅₂	1	1	yes	characteristic
321	C₄	—	2	4	no	—
322	C₄	—	2	4	no	—
323	C₄	—	2	2	no	—
324	C₄	—	26	8	no	—
325	C₄	—	52	8	no	—
326	C₄	—	52	4	no	—
327	C₄	Q₈×D₅₂	1	2	yes	—
328	C₄	C₂×#88158	1	4	yes	—
329	C₄	C₂²×Q₈×D₁₃	1	1	yes	characteristic
330	C₄	C₂×#88257	1	1	yes	characteristic
331	C₂	#13127728	1	4	yes	direct product factor
332	C₂	C₂×Q₈×D₅₂	1	1	yes	characteristic
333	C₂	C₂²×#88158	1	1	yes	characteristic
334	C₂	C₂×#11719458	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₅₂
commutator, cosocle	C₂×C₄×C₁₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂×C₈×Q₈×C₁₃	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
hypercenter	C₂×C₈×Q₈	D₁₃
hypocenter	C₁₃	C₂×#3383
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. #13127728	C₂	direct (N × H)
60. C₂×Dic₅₂	Q₈	semidirect
122. Dic₅₂	C₂×Q₈	semidirect
245. C₂×C₁₃	#3383	semidirect
289. C₁₃	C₂×#3383	semidirect
331. C₂	#13127728	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×#11719272	2
#13127728	2
C₂×#11718343	2
C₂×#11718344	2
C₂×#11718490	2
C₂×#11718512	2
C₂×C₄×Dic₅₂	2
C₂×C₈×Q₈×C₁₃	2
C₂×#3383	13

Subgroups — C2×#13127728 #16743377

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13127728 #16743377