Subgroups — C₂×#13127983 #16769438

Order 3328.

2074 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×#11719458	C₂	1	1	yes	maximal, characteristic, complemented
2	#13127983	C₂	1	16	yes	maximal, direct product factor
3	C₂×#11718856	C₂	1	3	yes	maximal, complemented
4	C₂×#11718808	C₂	1	3	yes	maximal, complemented
5	C₂×#11718855	C₂	1	3	yes	maximal, complemented
6	C₂²×#88067	C₂	1	3	yes	maximal, complemented
7	C₂×#11718801	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#576	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₃×H(5,𝔽₂)	—	2	1	no	—
10	C₂×#88257	—	2	1	no	—
11	C₁₃×#576	—	2	4	no	—
12	C₁₃×#576	C₂²	1	8	yes	complemented
13	C₂×C₁₃×AGL(1,ℤ/8)	—	2	3	no	—
14	C₂×C₁₃×#135	C₂²	1	3	yes	—
15	C₂²×C₁₃×D₈	C₂²	1	3	yes	—
16	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
17	#11719458	—	2	4	no	—
18	#11719458	C₂²	1	8	yes	complemented
19	C₂×#88163	—	2	3	no	—
20	C₂²×Dic₅₂	C₂²	1	3	yes	complemented
21	C₂×#88158	C₂²	1	3	yes	complemented
22	#11718856	C₂²	1	24	yes	complemented
23	#11718855	C₂²	1	24	yes	complemented
24	C₂×#88069	C₂²	1	3	yes	complemented
25	C₂×#88068	C₂²	1	1	yes	characteristic, complemented
26	C₂×#88067	C₂²	1	3	yes	complemented
27	C₂×#88067	C₂²	1	6	yes	complemented
28	C₂×#88067	C₂²	1	24	yes	complemented
29	#11718808	C₂²	1	24	yes	complemented
30	C₂×#88066	C₂²	1	3	yes	complemented
31	#11718801	C₂²	1	8	yes	complemented
32	C₂×#88052	C₂²	1	3	yes	complemented
33	C₂²×#6090	C₂²	1	3	yes	complemented
34	C₁₃×H(5,𝔽₂)	—	2	8	no	—
35	C₁₃×H(5,𝔽₂)	—	4	4	no	—
36	C₂×C₁₃×#41	—	2	3	no	—
37	C₂×C₁₃×#41	—	4	1	no	—
38	C₂×C₁₃×#41	D₄	1	1	yes	characteristic
39	C₂²×D₄×C₁₃	—	2	3	no	—
40	C₂²×D₄×C₁₃	—	4	3	no	—
41	#88257	—	2	8	no	—
42	#88257	—	4	4	no	—
43	C₂×Q₈×D₁₃	—	4	1	no	—
44	C₂×#6129	—	4	3	no	—
45	C₂×#6127	—	2	3	no	—
46	C₂²×Dic₂₆	—	2	3	no	—
47	C₁₃×AGL(1,ℤ/8)	—	2	24	no	—
48	C₁₃×#135	—	2	6	no	—
49	C₁₃×#135	C₂³	1	12	yes	—
50	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
51	C₂×C₁₃×SD₁₆	—	2	3	no	—
52	C₂×C₁₃×D₈	—	2	3	no	—
53	C₂×C₁₃×D₈	—	2	6	no	—
54	C₂×C₁₃×D₈	C₂³	1	3	yes	—
55	C₂×C₁₃×D₈	C₂³	1	12	yes	—
56	C₁₃×#131	D₄	1	4	yes	—
57	C₁₃×#131	C₂³	1	4	yes	—
58	C₂×C₁₃×M₁₆	D₄	1	3	yes	—
59	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
60	#88163	—	2	24	no	—
61	C₂×Dic₅₂	—	2	3	no	—
62	C₂×Dic₅₂	—	2	6	no	—
63	C₂×Dic₅₂	C₂³	1	3	yes	—
64	C₂×Dic₅₂	C₂³	1	12	yes	complemented
65	#88158	—	2	6	no	—
66	#88158	C₂³	1	12	yes	complemented
67	C₂×D₁₀₄	C₂³	1	1	yes	characteristic
68	C₂×#6095	—	2	3	no	—
69	#88069	C₂³	1	12	yes	complemented
70	#88068	C₂³	1	4	yes	complemented
71	#88067	C₂³	1	24	yes	complemented
72	#88067	C₂³	1	12	yes	complemented
73	#88066	C₂³	1	12	yes	complemented
74	#88052	C₂³	1	12	yes	complemented
75	C₂×#6090	C₂³	1	1	yes	characteristic
76	C₂×#6090	C₂³	1	3	yes	—
77	C₂×#6090	C₂³	1	12	yes	complemented
78	C₂×#3416	—	13	1	no	maximal, Sylow
79	C₂⁴×C₁₃	—	4	3	no	—
80	C₁₃×#41	—	2	12	no	—
81	C₁₃×#41	—	4	8	no	—
82	C₁₃×#41	—	4	6	no	—
83	C₁₃×#41	D₈	1	4	yes	—
84	C₁₃×#41	C₂×D₄	1	4	yes	—
85	C₂×Q₈×C₁₃	—	2	1	no	—
86	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
87	C₂×D₄×C₁₃	—	2	3	no	—
88	C₂×D₄×C₁₃	—	2	12	no	—
89	C₂×D₄×C₁₃	—	4	3	no	—
90	C₂×D₄×C₁₃	—	4	24	no	—
91	C₂×D₄×C₁₃	—	4	3	no	—
92	C₂×D₄×C₁₃	—	4	6	no	—
93	C₂×D₄×C₁₃	D₈	1	3	yes	—
94	C₂²×C₄×C₁₃	—	4	3	no	—
95	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
96	C₂×C₂²⋊D₁₃	—	4	3	no	—
97	C₂²×Dic₁₃	—	4	3	no	—
98	Q₈×D₁₃	—	4	8	no	—
99	#6129	—	4	24	no	—
100	#6127	—	2	12	no	—
101	#6127	—	4	6	no	—
102	C₂×D₅₂	—	2	1	no	—
103	C₂×C₄×D₁₃	—	4	3	no	—
104	C₂×Dic₂₆	—	2	3	no	—
105	C₂×Dic₂₆	—	2	12	no	—
106	C₂×Dic₂₆	—	4	3	no	—
107	C₂×Dic₂₆	—	4	6	no	—
108	C₁₃×Q₁₆	—	2	1	no	—
109	C₁₃×Q₁₆	C₂⁴	1	2	yes	—
110	C₁₃×SD₁₆	—	2	12	no	—
111	C₁₃×D₈	—	2	3	no	—
112	C₁₃×D₈	—	2	12	no	—
113	C₁₃×D₈	C₂⁴	1	6	yes	—
114	C₁₃×M₁₆	C₂×D₄	1	12	yes	—
115	C₂×C₈×C₁₃	C₂×D₄	1	3	yes	—
116	C₂×C₈×C₁₃	C₂×D₄	1	6	yes	—
117	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
118	C₂×C₈×C₁₃	C₂⁴	1	6	yes	—
119	Dic₅₂	—	2	3	no	—
120	Dic₅₂	—	2	12	no	—
121	Dic₅₂	C₂⁴	1	6	yes	—
122	D₁₀₄	—	2	1	no	—
123	D₁₀₄	C₂⁴	1	2	yes	—
124	#6095	—	2	12	no	—
125	#6090	C₂⁴	1	6	yes	—
126	#6090	C₂⁴	1	2	yes	—
127	C₂×#578	—	13	1	no	—
128	C₂×#576	D₁₃	1	1	yes	characteristic, complemented
129	#3416	—	13	16	no	—
130	C₂×#510	—	13	3	no	—
131	C₂×#509	—	13	3	no	—
132	C₂×#508	—	13	3	no	—
133	C₂²×SD₃₂	—	13	3	no	—
134	C₂×#504	—	13	1	no	—
135	C₂³×C₁₃	—	4	3	no	—
136	C₂³×C₁₃	—	4	12	no	—
137	C₂³×C₁₃	—	16	6	no	—
138	C₂³×C₁₃	C₂×D₈	1	3	yes	—
139	C₂²×D₁₃	—	4	1	no	—
140	Q₈×C₁₃	—	2	2	no	—
141	Q₈×C₁₃	—	4	1	no	—
142	Q₈×C₁₃	C₂×D₈	1	4	yes	—
143	D₄×C₁₃	—	2	6	no	—
144	D₄×C₁₃	—	4	12	no	—
145	D₄×C₁₃	—	4	3	no	—
146	D₄×C₁₃	—	4	12	no	—
147	D₄×C₁₃	C₂×D₈	1	12	yes	—
148	C₂×C₄×C₁₃	—	4	1	no	—
149	C₂×C₄×C₁₃	—	4	12	no	—
150	C₂×C₄×C₁₃	C₂×D₈	1	3	yes	—
151	C₂×C₄×C₁₃	C₂×D₈	1	6	yes	—
152	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
154	C₂²⋊D₁₃	—	4	12	no	—
155	C₂×Dic₁₃	—	4	3	no	—
156	C₂×Dic₁₃	—	4	12	no	—
157	D₅₂	—	2	2	no	—
158	D₅₂	—	4	1	no	—
159	C₄×D₁₃	—	4	12	no	—
160	Dic₂₆	—	2	6	no	—
161	Dic₂₆	—	4	3	no	—
162	Dic₂₆	—	4	12	no	—
163	C₈×C₁₃	C₂²×D₄	1	6	yes	—
164	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
165	C₈×C₁₃	C₂⁵	1	1	yes	characteristic
166	C₂×2₋¹⁺⁴	—	26	1	no	—
167	C₂×H(5,𝔽₂)	—	2	1	no	—
168	#578	—	13	8	no	—
169	#578	—	26	4	no	—
170	#576	—	2	4	no	—
171	#576	C₂×D₁₃	1	8	yes	complemented
172	C₂×#137	—	26	3	no	—
173	C₂×AGL(1,ℤ/8)	—	2	3	no	—
174	C₂×#135	—	13	3	no	—
175	C₂×#135	C₂×D₁₃	1	3	yes	—
176	C₂²×Q₁₆	—	13	3	no	—
177	C₂²×D₈	C₂×D₁₃	1	3	yes	—
178	C₂×#131	C₂×D₁₃	1	1	yes	characteristic
179	#510	—	13	24	no	—
180	#509	—	13	24	no	—
181	#508	—	13	24	no	—
182	C₂×Q₃₂	—	13	3	no	—
183	C₂×SD₃₂	—	13	1	no	—
184	C₂×SD₃₂	—	13	3	no	—
185	C₂×SD₃₂	—	13	6	no	—
186	C₂×SD₃₂	—	13	24	no	—
187	C₂×D₁₆	—	13	3	no	—
188	#504	—	13	8	no	—
189	C₂×M₃₂	—	13	3	no	—
190	C₂²×C₁₆	—	13	3	no	—
191	C₂²×C₁₃	—	2	6	no	—
192	C₂²×C₁₃	—	4	6	no	—
193	C₂²×C₁₃	—	8	6	no	—
194	C₂²×C₁₃	—	16	12	no	—
195	C₂²×C₁₃	C₂²×D₈	1	6	yes	—
196	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
197	C₂×D₁₃	—	4	2	no	—
198	C₂×D₁₃	—	8	2	no	—
199	C₄×C₁₃	—	4	2	no	—
200	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
201	C₄×C₁₃	C₂²×D₈	1	6	yes	—
202	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
203	Dic₁₃	—	4	6	no	—
204	2₋¹⁺⁴	—	26	8	no	—
205	2₋¹⁺⁴	—	52	4	no	—
206	H(5,𝔽₂)	—	2	8	no	—
207	H(5,𝔽₂)	—	4	4	no	—
208	C₂×#41	—	2	3	no	—
209	C₂×#41	—	4	1	no	—
210	C₂×#41	—	26	3	no	—
211	C₂×#41	—	52	3	no	—
212	C₂×#41	C₂²⋊D₁₃	1	1	yes	characteristic
213	C₂²×Q₈	—	26	3	no	—
214	C₂²×Q₈	—	52	1	no	—
215	C₂²×D₄	—	2	3	no	—
216	C₂²×D₄	—	4	3	no	—
217	#137	—	26	24	no	—
218	AGL(1,ℤ/8)	—	2	24	no	—
219	#135	—	2	6	no	—
220	#135	—	13	12	no	—
221	#135	—	26	6	no	—
222	#135	C₂²×D₁₃	1	12	yes	—
223	C₂×Q₁₆	—	13	3	no	—
224	C₂×Q₁₆	—	13	12	no	—
225	C₂×Q₁₆	—	26	3	no	—
226	C₂×Q₁₆	—	26	6	no	—
227	C₂×Q₁₆	C₂²×D₁₃	1	1	yes	characteristic
228	C₂×SD₁₆	—	2	3	no	—
229	C₂×SD₁₆	—	26	3	no	—
230	C₂×D₈	—	2	3	no	—
231	C₂×D₈	—	2	6	no	—
232	C₂×D₈	—	13	1	no	—
233	C₂×D₈	C₂²×D₁₃	1	3	yes	—
234	C₂×D₈	C₂²×D₁₃	1	12	yes	—
235	#131	C₂²⋊D₁₃	1	4	yes	—
236	#131	C₂²×D₁₃	1	4	yes	—
237	C₂×M₁₆	C₂²⋊D₁₃	1	3	yes	—
238	C₂²×C₈	C₂²×D₁₃	1	3	yes	—
239	Q₃₂	—	13	12	no	—
240	SD₃₂	—	13	4	no	—
241	SD₃₂	—	13	12	no	—
242	SD₃₂	—	13	24	no	—
243	D₁₆	—	13	12	no	—
244	M₃₂	—	13	12	no	—
245	C₂×C₁₆	—	13	1	no	—
246	C₂×C₁₆	—	13	3	no	—
247	C₂×C₁₆	—	13	12	no	—
248	C₂×C₁₃	—	2	6	no	—
249	C₂×C₁₃	—	8	6	no	—
250	C₂×C₁₃	#3416	1	2	yes	complemented
251	C₂×C₁₃	C₂³×D₈	1	1	yes	characteristic
252	D₁₃	—	8	2	no	—
253	C₂⁴	—	4	3	no	—
254	#41	—	2	12	no	—
255	#41	—	4	8	no	—
256	#41	—	4	6	no	—
257	#41	—	26	12	no	—
258	#41	—	52	24	no	—
259	#41	—	52	6	no	—
260	#41	#6104	1	4	yes	—
261	#41	C₂×C₂²⋊D₁₃	1	4	yes	—
262	C₂×Q₈	—	2	1	no	—
263	C₂×Q₈	—	26	3	no	—
264	C₂×Q₈	—	26	12	no	—
265	C₂×Q₈	—	52	8	no	—
266	C₂×Q₈	—	52	3	no	—
267	C₂×Q₈	—	52	6	no	—
268	C₂×Q₈	#6104	1	1	yes	characteristic
269	C₂×D₄	—	2	3	no	—
270	C₂×D₄	—	2	12	no	—
271	C₂×D₄	—	4	3	no	—
272	C₂×D₄	—	4	24	no	—
273	C₂×D₄	—	4	3	no	—
274	C₂×D₄	—	4	6	no	—
275	C₂×D₄	—	26	1	no	—
276	C₂×D₄	—	52	3	no	—
277	C₂×D₄	#6104	1	3	yes	—
278	C₂²×C₄	—	4	3	no	—
279	C₂²×C₄	—	52	3	no	—
280	C₂²×C₄	—	52	3	no	—
281	C₂²×C₄	C₂×C₂²⋊D₁₃	1	3	yes	—
282	Q₁₆	—	2	1	no	—
283	Q₁₆	—	13	6	no	—
284	Q₁₆	—	26	3	no	—
285	Q₁₆	—	26	12	no	—
286	Q₁₆	C₂³×D₁₃	1	2	yes	—
287	SD₁₆	—	2	12	no	—
288	SD₁₆	—	26	12	no	—
289	D₈	—	2	3	no	—
290	D₈	—	2	12	no	—
291	D₈	—	13	2	no	—
292	D₈	—	26	1	no	—
293	D₈	C₂³×D₁₃	1	6	yes	—
294	M₁₆	C₂×C₂²⋊D₁₃	1	12	yes	—
295	C₂×C₈	C₂×C₂²⋊D₁₃	1	3	yes	—
296	C₂×C₈	C₂×C₂²⋊D₁₃	1	6	yes	—
297	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
298	C₂×C₈	C₂³×D₁₃	1	6	yes	—
299	C₁₆	—	13	2	no	—
300	C₁₆	—	13	6	no	—
301	C₁₃	C₂×#3416	1	1	yes	characteristic, complemented, Sylow
302	C₂³	—	4	3	no	—
303	C₂³	—	4	12	no	—
304	C₂³	—	16	6	no	—
305	C₂³	—	52	1	no	—
306	C₂³	C₂×#6104	1	3	yes	—
307	Q₈	—	2	2	no	—
308	Q₈	—	4	1	no	—
309	Q₈	—	26	6	no	—
310	Q₈	—	52	12	no	—
311	Q₈	—	52	3	no	—
312	Q₈	C₂×#6104	1	4	yes	—
313	D₄	—	2	6	no	—
314	D₄	—	4	12	no	—
315	D₄	—	4	12	no	—
316	D₄	—	4	3	no	—
317	D₄	—	26	2	no	—
318	D₄	—	52	12	no	—
319	D₄	—	52	1	no	—
320	D₄	C₂×#6104	1	12	yes	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	12	no	—
323	C₂×C₄	—	52	12	no	—
324	C₂×C₄	—	52	3	no	—
325	C₂×C₄	—	52	12	no	—
326	C₂×C₄	C₂×#6104	1	6	yes	—
327	C₂×C₄	C₂×#6104	1	3	yes	—
328	C₂×C₄	C₂²×C₂²⋊D₁₃	1	6	yes	—
329	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
330	C₈	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
331	C₈	C₂²×C₂²⋊D₁₃	1	6	yes	—
332	C₈	C₂⁴×D₁₃	1	1	yes	characteristic
333	C₂²	—	2	6	no	—
334	C₂²	—	4	6	no	—
335	C₂²	—	8	6	no	—
336	C₂²	—	16	12	no	—
337	C₂²	—	52	2	no	—
338	C₂²	—	104	2	no	—
339	C₂²	C₂²×#6104	1	6	yes	—
340	C₂²	C₂²×#6104	1	1	yes	characteristic
341	C₄	—	4	2	no	—
342	C₄	—	52	6	no	—
343	C₄	C₂²×#6104	1	6	yes	—
344	C₄	C₂²×#6104	1	1	yes	characteristic
345	C₄	C₂³×C₂²⋊D₁₃	1	1	yes	characteristic
346	C₂	—	2	6	no	—
347	C₂	—	8	6	no	—
348	C₂	—	104	2	no	—
349	C₂	#13127983	1	2	yes	direct product factor
350	C₂	C₂³×#6104	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₈×C₁₃	C₂⁵
Frattini	C₈	C₂⁴×D₁₃
Fitting	C₂×C₁₃×#576	C₂
socle	C₂²×C₁₃	C₂²×D₈
hypercenter	C₂×#576	D₁₃
hypocenter	C₁₃	C₂×#3416
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#576

Complements

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

N	H ≅ G/N	type
1. C₂×#11719458	C₂	semidirect
2. #13127983	C₂	direct (N × H)
3. C₂×#11718856	C₂	semidirect
4. C₂×#11718808	C₂	semidirect
5. C₂×#11718855	C₂	semidirect
6. C₂²×#88067	C₂	semidirect
7. C₂×#11718801	C₂	semidirect
8. C₂×C₁₃×#576	C₂	semidirect
12. C₁₃×#576	C₂²	semidirect
18. #11719458	C₂²	semidirect
20. C₂²×Dic₅₂	C₂²	semidirect
21. C₂×#88158	C₂²	semidirect
22. #11718856	C₂²	semidirect
23. #11718855	C₂²	semidirect
24. C₂×#88069	C₂²	semidirect
25. C₂×#88068	C₂²	semidirect
26. C₂×#88067	C₂²	semidirect
27. C₂×#88067	C₂²	semidirect
28. C₂×#88067	C₂²	semidirect
29. #11718808	C₂²	semidirect
30. C₂×#88066	C₂²	semidirect
31. #11718801	C₂²	semidirect
32. C₂×#88052	C₂²	semidirect
33. C₂²×#6090	C₂²	semidirect
64. C₂×Dic₅₂	C₂³	semidirect
66. #88158	C₂³	semidirect
69. #88069	C₂³	semidirect
70. #88068	C₂³	semidirect
71. #88067	C₂³	semidirect
72. #88067	C₂³	semidirect
73. #88066	C₂³	semidirect
74. #88052	C₂³	semidirect
77. C₂×#6090	C₂³	semidirect
128. C₂×#576	D₁₃	semidirect
171. #576	C₂×D₁₃	semidirect
250. C₂×C₁₃	#3416	semidirect
301. C₁₃	C₂×#3416	semidirect
349. C₂	#13127983	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×#11719458	2
#13127983	2
C₂×#11718856	2
C₂×#11718808	2
C₂×#11718855	2
C₂²×#88067	2
C₂×#11718801	2
C₂×C₁₃×#576	2
C₂×#3416	13

Subgroups — C2×#13127983 #16769438

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13127983 #16769438