Subgroups — C₂×#13125264 #16732339

Order 3328.

3366 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88158	C₂	1	4	yes	maximal, complemented
2	C₂×#11719182	C₂	1	2	yes	maximal, complemented
3	#13125264	C₂	1	16	yes	maximal, direct product factor
4	C₂×#11718357	C₂	1	4	yes	maximal, complemented
5	C₂×#11718360	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718356	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11718354	C₂	1	2	yes	maximal, complemented
8	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#6127	C₂²	1	4	yes	—
10	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
11	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×#88184	—	2	4	no	—
13	C₂²×Dic₅₂	C₂²	1	4	yes	—
14	C₂×#88158	—	2	32	no	—
15	C₂×#88158	C₂²	1	32	yes	complemented
16	C₂²×D₁₀₄	C₂²	1	4	yes	—
17	C₂²×#6095	C₂²	1	8	yes	—
18	#11719182	—	2	8	no	—
19	#11719182	C₂²	1	16	yes	complemented
20	C₂×#88129	C₂²	1	4	yes	—
21	C₂×#88128	C₂²	1	2	yes	—
22	C₂×C₄×D₅₂	—	2	4	no	—
23	C₂×#88123	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₁₃	C₂²	1	8	yes	complemented
28	#11718360	C₂²	1	8	yes	complemented
29	#11718357	C₂²	1	32	yes	complemented
30	#11718356	C₂²	1	8	yes	complemented
31	#11718354	C₂²	1	16	yes	complemented
32	C₂³×C₄×C₁₃	D₄	1	2	yes	—
33	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
34	C₂²×C₂²⋊D₁₃	—	2	4	no	—
35	C₂×#6127	—	2	8	no	—
36	C₂×#6127	—	2	32	no	—
37	C₂×#6127	C₂³	1	16	yes	—
38	C₂²×D₅₂	—	2	4	no	—
39	C₂²×D₅₂	C₂³	1	4	yes	—
40	C₂²×C₄×D₁₃	—	2	4	no	—
41	C₂²×Dic₂₆	C₂³	1	4	yes	—
42	C₂²×C₈×C₁₃	D₄	1	16	yes	complemented
43	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
44	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
45	C₂²×C₈×C₁₃	C₂³	1	8	yes	—
46	C₂×C₄²×C₁₃	D₄	1	2	yes	—
47	C₂×C₄²×C₁₃	D₄	1	4	yes	—
48	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
51	#88184	—	2	32	no	—
52	C₂×#6103	—	2	8	no	—
53	C₂×C₄⋊Dic₁₃	—	2	4	no	—
54	C₂×Dic₅₂	—	2	16	no	—
55	C₂×Dic₅₂	C₂³	1	16	yes	—
56	#88158	—	2	128	no	—
57	C₂×D₁₀₄	—	2	16	no	—
58	C₂×D₁₀₄	C₂³	1	16	yes	—
59	C₂×#6095	—	2	32	no	—
60	C₂×#6095	C₂³	1	32	yes	—
61	#88129	—	2	8	no	—
62	#88129	C₂³	1	16	yes	—
63	#88128	—	2	4	no	—
64	#88128	C₂³	1	8	yes	—
65	C₄×D₅₂	—	2	32	no	—
66	#88123	—	2	4	no	—
67	#88123	C₂³	1	8	yes	—
68	C₄×C₈×C₁₃	C₂³	1	4	yes	—
69	C₄×C₈×C₁₃	C₂³	1	4	yes	—
70	C₄×C₈×C₁₃	C₂³	1	8	yes	—
71	C₂×#3147	—	13	1	no	maximal, Sylow
72	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂³×D₁₃	—	2	4	no	—
74	C₂²×C₄×C₁₃	—	2	8	no	—
75	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₁₃	C₂×D₄	1	8	yes	—
77	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₁₃	C₂×D₄	1	8	yes	—
79	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
82	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
83	C₂×C₂²⋊D₁₃	—	2	16	no	—
84	C₂×C₂²⋊D₁₃	—	4	16	no	—
85	C₂²×Dic₁₃	—	2	4	no	—
86	#6127	—	2	64	no	—
87	#6127	—	4	32	no	—
88	C₂×D₅₂	—	2	16	no	—
89	C₂×D₅₂	—	2	16	no	—
90	C₂×D₅₂	—	2	4	no	—
91	C₂×D₅₂	—	4	16	no	—
92	C₂×D₅₂	C₂⁴	1	8	yes	—
93	C₂×C₄×D₁₃	—	2	16	no	—
94	C₂×C₄×D₁₃	—	4	16	no	—
95	C₂×Dic₂₆	—	2	16	no	—
96	C₂×Dic₂₆	—	2	4	no	—
97	C₂×Dic₂₆	C₂⁴	1	8	yes	—
98	C₂×C₈×C₁₃	C₂×D₄	1	8	yes	—
99	C₂×C₈×C₁₃	C₂×D₄	1	8	yes	—
100	C₂×C₈×C₁₃	C₂×D₄	1	32	yes	complemented
101	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
102	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
103	C₄²×C₁₃	C₂×D₄	1	2	yes	—
104	C₄²×C₁₃	C₂×D₄	1	8	yes	—
105	C₄²×C₁₃	C₂×D₄	1	2	yes	—
106	C₄²×C₁₃	C₂⁴	1	2	yes	—
107	C₄²×C₁₃	C₂⁴	1	2	yes	—
108	#6103	—	2	32	no	—
109	C₄⋊Dic₁₃	—	2	16	no	—
110	Dic₅₂	—	2	32	no	—
111	D₁₀₄	—	2	32	no	—
112	#6095	—	2	64	no	—
113	C₂²×#135	—	13	4	no	—
114	C₂×#532	—	13	2	no	—
115	#3147	—	13	16	no	—
116	C₂×#495	—	13	4	no	—
117	C₂×#494	—	13	1	no	—
118	C₂×#493	—	13	1	no	—
119	C₂×#492	—	13	2	no	—
120	C₂²×C₄×C₈	D₁₃	1	1	yes	characteristic, complemented
121	C₂³×C₁₃	—	2	4	no	—
122	C₂³×C₁₃	#127	1	4	yes	—
123	C₂³×C₁₃	C₂²×D₄	1	2	yes	—
124	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
125	C₂²×D₁₃	—	2	8	no	—
126	C₂²×D₁₃	—	4	8	no	—
127	C₂²×D₁₃	—	4	16	no	—
128	C₂×C₄×C₁₃	—	2	4	no	—
129	C₂×C₄×C₁₃	—	2	16	no	—
130	C₂×C₄×C₁₃	#127	1	4	yes	—
131	C₂×C₄×C₁₃	#127	1	8	yes	—
132	C₂×C₄×C₁₃	#135	1	8	yes	complemented
133	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
134	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
136	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
137	C₂×C₄×C₁₃	C₂²×D₄	1	8	yes	—
138	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
139	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
140	C₂²⋊D₁₃	—	4	32	no	—
141	C₂×Dic₁₃	—	2	8	no	—
142	C₂×Dic₁₃	—	4	8	no	—
143	D₅₂	—	2	16	no	—
144	D₅₂	—	4	8	no	—
145	D₅₂	—	4	32	no	—
146	C₄×D₁₃	—	4	32	no	—
147	Dic₂₆	—	2	16	no	—
148	Dic₂₆	—	4	8	no	—
149	C₈×C₁₃	C₂²×D₄	1	8	yes	—
150	C₈×C₁₃	C₂²×D₄	1	8	yes	—
151	C₂²×#41	—	13	4	no	—
152	C₂×#135	—	13	32	no	—
153	C₂×#135	—	26	32	no	—
154	C₂²×Q₁₆	—	13	4	no	—
155	C₂²×SD₁₆	—	13	8	no	—
156	C₂²×D₈	—	13	4	no	—
157	C₂³×C₈	C₂×D₁₃	1	2	yes	—
158	#532	—	13	16	no	—
159	#532	—	26	8	no	—
160	C₂×C₄⋊Q₈	—	13	2	no	—
161	C₂×#127	—	13	2	no	—
162	C₂×#124	—	13	4	no	—
163	C₂×#121	—	26	4	no	—
164	C₂×C₄×D₄	—	26	4	no	—
165	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
166	#495	—	13	32	no	—
167	#494	—	13	8	no	—
168	#493	—	13	8	no	—
169	#492	—	13	16	no	—
170	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
171	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
172	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
173	C₂×C₄×C₈	C₂×D₁₃	1	8	yes	complemented
174	C₂²×C₁₃	—	2	8	no	—
175	C₂²×C₁₃	—	2	4	no	—
176	C₂²×C₁₃	C₂×#127	1	4	yes	—
177	C₂²×C₁₃	C₂×#127	1	2	yes	—
178	C₂²×C₁₃	C₂×#135	1	4	yes	—
179	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
180	C₂×D₁₃	—	4	8	no	—
181	C₂×D₁₃	—	4	16	no	—
182	C₂×D₁₃	—	8	16	no	—
183	C₄×C₁₃	—	2	4	no	—
184	C₄×C₁₃	C₂×#127	1	2	yes	—
185	C₄×C₁₃	C₂×#127	1	4	yes	—
186	C₄×C₁₃	C₂×#135	1	8	yes	complemented
187	C₄×C₁₃	C₂³×D₄	1	2	yes	—
188	Dic₁₃	—	4	8	no	—
189	C₂×#41	—	13	16	no	—
190	C₂×#41	—	26	8	no	—
191	C₂×#41	—	26	32	no	—
192	C₂²×Q₈	—	13	4	no	—
193	C₂²×D₄	—	13	4	no	—
194	C₂²×D₄	—	26	4	no	—
195	C₂²×D₄	—	26	4	no	—
196	C₂³×C₄	—	26	4	no	—
197	C₂³×C₄	D₅₂	1	2	yes	—
198	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
199	#135	—	26	128	no	—
200	C₂×Q₁₆	—	13	16	no	—
201	C₂×Q₁₆	—	26	16	no	—
202	C₂×SD₁₆	—	13	32	no	—
203	C₂×SD₁₆	—	26	32	no	—
204	C₂×D₈	—	13	16	no	—
205	C₂×D₈	—	26	16	no	—
206	C₂²×C₈	D₅₂	1	16	yes	complemented
207	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
208	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
209	C₂²×C₈	C₂²×D₁₃	1	8	yes	—
210	C₄⋊Q₈	—	13	8	no	—
211	C₄⋊Q₈	—	26	4	no	—
212	#127	—	13	8	no	—
213	#127	—	26	4	no	—
214	#124	—	13	16	no	—
215	#124	—	26	8	no	—
216	#121	—	26	32	no	—
217	C₄×D₄	—	26	32	no	—
218	C₂×C₄⋊C₄	—	26	4	no	—
219	C₂×C₂²⋊C₄	—	26	8	no	—
220	C₂×C₄²	D₅₂	1	4	yes	—
221	C₂×C₄²	D₅₂	1	2	yes	—
222	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
223	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
224	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
225	C₄×C₈	C₂²×D₁₃	1	4	yes	—
226	C₄×C₈	C₂²×D₁₃	1	4	yes	—
227	C₄×C₈	C₂²×D₁₃	1	8	yes	—
228	C₂×C₁₃	—	2	4	no	—
229	C₂×C₁₃	#3147	1	4	yes	complemented
230	C₂×C₁₃	C₂²×#127	1	1	yes	characteristic
231	C₂×C₁₃	C₂²×#135	1	2	yes	—
232	D₁₃	—	8	8	no	—
233	C₂⁴	—	26	4	no	—
234	C₂⁴	C₂×D₅₂	1	1	yes	characteristic
235	#41	—	26	64	no	—
236	#41	—	52	32	no	—
237	C₂×Q₈	—	13	8	no	—
238	C₂×Q₈	—	26	4	no	—
239	C₂×Q₈	—	26	16	no	—
240	C₂×D₄	—	13	8	no	—
241	C₂×D₄	—	26	16	no	—
242	C₂×D₄	—	26	16	no	—
243	C₂×D₄	—	26	16	no	—
244	C₂×D₄	—	26	4	no	—
245	C₂×D₄	—	52	16	no	—
246	C₂×D₄	—	52	16	no	—
247	C₂²×C₄	—	2	8	no	—
248	C₂²×C₄	—	26	4	no	—
249	C₂²×C₄	—	26	16	no	—
250	C₂²×C₄	—	52	16	no	—
251	C₂²×C₄	C₂×D₅₂	1	2	yes	—
252	C₂²×C₄	C₂×D₅₂	1	8	yes	—
253	C₂²×C₄	C₂×D₅₂	1	2	yes	—
254	C₂²×C₄	C₂×D₅₂	1	8	yes	—
255	C₂²×C₄	C₂×D₅₂	1	2	yes	—
256	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
257	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
258	C₂²×C₄	C₂³×D₁₃	1	2	yes	—
259	Q₁₆	—	26	32	no	—
260	SD₁₆	—	26	64	no	—
261	D₈	—	26	32	no	—
262	C₂×C₈	C₂×D₅₂	1	8	yes	—
263	C₂×C₈	C₂×D₅₂	1	8	yes	—
264	C₂×C₈	C₂×D₅₂	1	32	yes	complemented
265	C₂×C₈	C₂³×D₁₃	1	4	yes	—
266	C₂×C₈	C₂³×D₁₃	1	4	yes	—
267	C₄⋊C₄	—	26	16	no	—
268	C₂²⋊C₄	—	26	32	no	—
269	C₄²	C₂×D₅₂	1	8	yes	—
270	C₄²	C₂×D₅₂	1	2	yes	—
271	C₄²	C₂×D₅₂	1	2	yes	—
272	C₄²	C₂³×D₁₃	1	2	yes	—
273	C₄²	C₂³×D₁₃	1	2	yes	—
274	C₁₃	C₂×#3147	1	1	yes	characteristic, complemented, Sylow
275	C₂³	—	2	4	no	—
276	C₂³	—	26	8	no	—
277	C₂³	—	52	8	no	—
278	C₂³	—	52	16	no	—
279	C₂³	#88128	1	4	yes	—
280	C₂³	C₂²×D₅₂	1	2	yes	—
281	C₂³	C₂²×D₅₂	1	1	yes	characteristic
282	Q₈	—	26	16	no	—
283	Q₈	—	52	8	no	—
284	D₄	—	26	16	no	—
285	D₄	—	52	32	no	—
286	D₄	—	52	8	no	—
287	D₄	—	52	32	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	16	no	—
290	C₂×C₄	—	26	8	no	—
291	C₂×C₄	—	52	8	no	—
292	C₂×C₄	—	52	32	no	—
293	C₂×C₄	#88128	1	4	yes	—
294	C₂×C₄	#88128	1	8	yes	—
295	C₂×C₄	#88158	1	8	yes	complemented
296	C₂×C₄	C₂²×D₅₂	1	4	yes	—
297	C₂×C₄	C₂²×D₅₂	1	2	yes	—
298	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
299	C₂×C₄	C₂²×D₅₂	1	4	yes	—
300	C₂×C₄	C₂²×D₅₂	1	8	yes	—
301	C₂×C₄	C₂²×D₅₂	1	4	yes	—
302	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
303	C₈	C₂²×D₅₂	1	8	yes	—
304	C₈	C₂²×D₅₂	1	8	yes	—
305	C₂²	—	2	4	no	—
306	C₂²	—	2	8	no	—
307	C₂²	—	52	16	no	—
308	C₂²	—	52	8	no	—
309	C₂²	—	104	16	no	—
310	C₂²	C₂×#88128	1	4	yes	—
311	C₂²	C₂×#88128	1	2	yes	—
312	C₂²	C₂×#88158	1	4	yes	—
313	C₂²	C₂³×D₅₂	1	1	yes	characteristic
314	C₄	—	2	4	no	—
315	C₄	—	52	8	no	—
316	C₄	C₂×#88128	1	4	yes	—
317	C₄	C₂×#88128	1	2	yes	—
318	C₄	C₂×#88158	1	8	yes	complemented
319	C₄	C₂³×D₅₂	1	2	yes	—
320	C₂	—	2	4	no	—
321	C₂	—	104	8	no	—
322	C₂	#13125264	1	4	yes	direct product factor
323	C₂	C₂²×#88128	1	1	yes	characteristic
324	C₂	C₂²×#88158	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₅₂
commutator, cosocle	C₂×C₄×C₁₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂²×C₄×C₈×C₁₃	C₂
socle	C₂³×C₁₃	C₂²×D₄
hypercenter	C₂²×C₄×C₈	D₁₃
hypocenter	C₁₃	C₂×#3147
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#88158	C₂	semidirect
2. C₂×#11719182	C₂	semidirect
3. #13125264	C₂	direct (N × H)
4. C₂×#11718357	C₂	semidirect
5. C₂×#11718360	C₂	semidirect
6. C₂×#11718356	C₂	semidirect
7. C₂×#11718354	C₂	semidirect
8. C₂²×C₄×C₈×C₁₃	C₂	semidirect
15. C₂×#88158	C₂²	semidirect
19. #11719182	C₂²	semidirect
27. C₂×C₄×C₈×C₁₃	C₂²	semidirect
28. #11718360	C₂²	semidirect
29. #11718357	C₂²	semidirect
30. #11718356	C₂²	semidirect
31. #11718354	C₂²	semidirect
42. C₂²×C₈×C₁₃	D₄	semidirect
100. C₂×C₈×C₁₃	C₂×D₄	semidirect
120. C₂²×C₄×C₈	D₁₃	semidirect
132. C₂×C₄×C₁₃	#135	semidirect
173. C₂×C₄×C₈	C₂×D₁₃	semidirect
186. C₄×C₁₃	C₂×#135	semidirect
206. C₂²×C₈	D₅₂	semidirect
229. C₂×C₁₃	#3147	semidirect
264. C₂×C₈	C₂×D₅₂	semidirect
274. C₁₃	C₂×#3147	semidirect
295. C₂×C₄	#88158	semidirect
318. C₄	C₂×#88158	semidirect
322. C₂	#13125264	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×#88158	2
C₂×#11719182	2
#13125264	2
C₂×#11718357	2
C₂×#11718360	2
C₂×#11718356	2
C₂×#11718354	2
C₂²×C₄×C₈×C₁₃	2
C₂×#3147	13

Subgroups — C2×#13125264 #16732339

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13125264 #16732339