Subgroups — C₂×#13116912 #16017299

Order 3328.

790 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718854	C₂	1	2	yes	maximal, complemented
2	C₂×#11718356	C₂	1	1	yes	maximal, characteristic, complemented
3	#13116912	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718055	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718087	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×#443	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×#131	—	2	1	no	—
9	C₂²×D₁₀₄	—	2	1	no	—
10	C₂²×D₁₀₄	C₂²	1	2	yes	—
11	C₂×#88128	—	2	1	no	—
12	C₁₃×#443	—	2	2	no	—
13	C₁₃×#443	C₂²	1	4	yes	complemented
14	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×#104	—	2	1	no	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	#11718854	—	2	4	no	—
18	#11718854	C₂²	1	8	yes	complemented
19	C₂×#88068	—	2	2	no	—
20	C₂×#88066	—	2	2	no	—
21	C₂×#88052	C₂²	1	2	yes	—
22	#11718356	—	2	2	no	—
23	#11718356	C₂²	1	4	yes	complemented
24	#11718087	C₂²	1	8	yes	complemented
25	#11718055	C₂²	1	4	yes	complemented
26	C₂×C₁₃×#41	—	2	1	no	—
27	C₂²×D₅₂	—	2	2	no	—
28	C₂²×D₅₂	—	4	2	no	—
29	C₁₃×#135	—	2	1	no	—
30	C₁₃×#135	—	2	1	no	—
31	C₁₃×#135	D₄	1	2	yes	complemented
32	C₁₃×#135	C₂³	1	2	yes	—
33	C₂×C₁₃×Q₁₆	D₄	1	1	yes	characteristic, complemented
34	C₂×C₁₃×SD₁₆	—	2	1	no	—
35	C₂×C₁₃×D₈	D₄	1	1	yes	characteristic, complemented
36	C₁₃×#131	—	2	4	no	—
37	C₁₃×#131	—	4	2	no	—
38	C₂×C₁₃×M₁₆	—	2	1	no	—
39	C₂×C₁₃×M₁₆	—	4	1	no	—
40	C₂²×C₈×C₁₃	—	4	1	no	—
41	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
44	C₂×D₁₀₄	—	2	2	no	—
45	C₂×D₁₀₄	—	2	2	no	—
46	C₂×D₁₀₄	—	2	2	no	—
47	C₂×D₁₀₄	—	2	2	no	—
48	C₂×D₁₀₄	—	2	4	no	—
49	C₂×D₁₀₄	—	2	2	no	—
50	C₂×D₁₀₄	—	4	1	no	—
51	C₂×D₁₀₄	—	4	1	no	—
52	C₂×D₁₀₄	—	4	2	no	—
53	C₂×D₁₀₄	C₂³	1	4	yes	—
54	#88128	—	2	4	no	—
55	#88128	—	4	2	no	—
56	C₁₃×#108	—	2	1	no	—
57	C₁₃×#108	C₂³	1	2	yes	—
58	C₁₃×#104	—	2	4	no	—
59	C₄×C₈×C₁₃	D₄	1	2	yes	—
60	C₄×C₈×C₁₃	C₂³	1	2	yes	—
61	#88068	—	2	8	no	—
62	#88066	—	2	8	no	—
63	#88052	—	2	2	no	—
64	#88052	C₂³	1	4	yes	—
65	C₂×#6090	—	2	2	no	—
66	C₂×#2213	—	13	1	no	maximal, Sylow
67	C₂³×D₁₃	—	4	2	no	—
68	C₁₃×#41	—	2	2	no	—
69	C₁₃×#41	—	2	2	no	—
70	C₁₃×#41	—	4	1	no	—
71	C₁₃×#41	—	4	1	no	—
72	C₂×Q₈×C₁₃	—	2	1	no	—
73	C₂×D₄×C₁₃	—	2	1	no	—
74	C₂×D₄×C₁₃	—	4	1	no	—
75	C₂²×C₄×C₁₃	—	2	1	no	—
76	C₂²×C₄×C₁₃	—	4	1	no	—
77	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂×D₅₂	—	2	4	no	—
79	C₂×D₅₂	—	4	2	no	—
80	C₂×D₅₂	—	4	2	no	—
81	C₂×D₅₂	—	4	2	no	—
82	C₂×D₅₂	—	4	2	no	—
83	C₂×D₅₂	—	4	8	no	—
84	C₂×D₅₂	—	4	2	no	—
85	C₂×D₅₂	—	8	8	no	—
86	C₁₃×Q₁₆	—	2	1	no	—
87	C₁₃×Q₁₆	C₂×D₄	1	2	yes	complemented
88	C₁₃×SD₁₆	—	2	2	no	—
89	C₁₃×SD₁₆	—	2	2	no	—
90	C₁₃×D₈	—	2	1	no	—
91	C₁₃×D₈	C₂×D₄	1	2	yes	complemented
92	C₁₃×M₁₆	—	2	2	no	—
93	C₁₃×M₁₆	—	4	1	no	—
94	C₁₃×M₁₆	—	4	4	no	—
95	C₂×C₈×C₁₃	—	4	1	no	—
96	C₂×C₈×C₁₃	—	4	4	no	—
97	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
98	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
99	C₂×C₈×C₁₃	D₈	1	2	yes	—
100	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
104	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
105	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
106	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
107	C₄²×C₁₃	—	2	1	no	—
108	C₄²×C₁₃	C₂×D₄	1	2	yes	—
109	D₁₀₄	—	2	4	no	—
110	D₁₀₄	—	2	4	no	—
111	D₁₀₄	—	4	2	no	—
112	D₁₀₄	—	4	2	no	—
113	D₁₀₄	—	4	4	no	—
114	D₁₀₄	—	4	4	no	—
115	#6090	—	2	4	no	—
116	C₂×#509	—	13	2	no	—
117	C₂×#493	—	13	1	no	—
118	C₂×#443	D₁₃	1	1	yes	characteristic, complemented
119	#2213	—	13	8	no	—
120	C₂×#364	—	13	1	no	—
121	C₂×#361	—	13	2	no	—
122	C₂³×C₁₃	—	4	1	no	—
123	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₂²×D₁₃	—	4	4	no	—
125	C₂²×D₁₃	—	8	8	no	—
126	C₂²×D₁₃	—	8	2	no	—
127	C₂²×D₁₃	—	8	2	no	—
128	Q₈×C₁₃	—	2	2	no	—
129	Q₈×C₁₃	—	4	1	no	—
130	D₄×C₁₃	—	2	2	no	—
131	D₄×C₁₃	—	4	2	no	—
132	D₄×C₁₃	—	4	2	no	—
133	D₄×C₁₃	—	4	1	no	—
134	C₂×C₄×C₁₃	—	2	4	no	—
135	C₂×C₄×C₁₃	—	2	2	no	—
136	C₂×C₄×C₁₃	—	4	1	no	—
137	C₂×C₄×C₁₃	—	4	2	no	—
138	C₂×C₄×C₁₃	—	4	2	no	—
139	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
140	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
145	D₅₂	—	4	4	no	—
146	D₅₂	—	4	4	no	—
147	D₅₂	—	8	16	no	—
148	D₅₂	—	8	2	no	—
149	D₅₂	—	8	2	no	—
150	C₈×C₁₃	—	4	2	no	—
151	C₈×C₁₃	#121	1	1	yes	characteristic
152	C₈×C₁₃	#121	1	1	yes	characteristic
153	C₈×C₁₃	C₂×D₈	1	2	yes	—
154	C₈×C₁₃	C₂×D₈	1	2	yes	—
155	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
156	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
157	C₂×#135	C₂×D₁₃	1	1	yes	characteristic
158	C₂²×D₈	—	13	2	no	—
159	C₂²×D₈	—	26	1	no	—
160	C₂×#131	—	2	1	no	—
161	C₂×#127	—	26	1	no	—
162	#509	—	13	8	no	—
163	#509	—	26	4	no	—
164	C₂×SD₃₂	—	26	2	no	—
165	C₂×D₁₆	—	26	2	no	—
166	C₂×M₃₂	—	13	2	no	—
167	#493	—	13	4	no	—
168	#493	—	26	2	no	—
169	#443	—	2	2	no	—
170	#443	C₂×D₁₃	1	4	yes	complemented
171	C₂×#108	C₂×D₁₃	1	1	yes	characteristic
172	C₂×#104	—	2	1	no	—
173	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
174	#364	—	13	4	no	—
175	#361	—	13	8	no	—
176	C₂²×C₁₃	—	2	2	no	—
177	C₂²×C₁₃	—	4	2	no	—
178	C₂²×C₁₃	—	8	2	no	—
179	C₂²×C₁₃	#459	1	1	yes	characteristic
180	C₂²×C₁₃	#459	1	1	yes	characteristic
181	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
182	C₂×D₁₃	—	8	8	no	—
183	C₂×D₁₃	—	8	4	no	—
184	C₂×D₁₃	—	16	4	no	—
185	C₂×D₁₃	—	16	4	no	—
186	C₄×C₁₃	—	2	4	no	—
187	C₄×C₁₃	—	4	2	no	—
188	C₄×C₁₃	#459	1	1	yes	characteristic
189	C₄×C₁₃	#459	1	1	yes	characteristic
190	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
191	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
192	C₂×#41	—	2	1	no	—
193	C₂²×D₄	—	26	2	no	—
194	C₂²×D₄	—	52	2	no	—
195	#135	—	2	1	no	—
196	#135	—	2	1	no	—
197	#135	D₅₂	1	2	yes	complemented
198	#135	C₂²×D₁₃	1	2	yes	—
199	C₂×Q₁₆	D₅₂	1	1	yes	characteristic, complemented
200	C₂×SD₁₆	—	2	1	no	—
201	C₂×D₈	—	13	4	no	—
202	C₂×D₈	—	26	2	no	—
203	C₂×D₈	—	26	2	no	—
204	C₂×D₈	—	26	2	no	—
205	C₂×D₈	—	26	2	no	—
206	C₂×D₈	—	26	2	no	—
207	C₂×D₈	—	26	4	no	—
208	C₂×D₈	—	52	1	no	—
209	C₂×D₈	—	52	1	no	—
210	C₂×D₈	—	52	2	no	—
211	C₂×D₈	D₅₂	1	1	yes	characteristic, complemented
212	#131	—	2	4	no	—
213	#131	—	4	2	no	—
214	C₂×M₁₆	—	2	1	no	—
215	C₂×M₁₆	—	4	1	no	—
216	C₂²×C₈	—	4	1	no	—
217	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
218	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
219	#127	—	26	4	no	—
220	#127	—	52	2	no	—
221	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
222	SD₃₂	—	26	8	no	—
223	D₁₆	—	26	8	no	—
224	M₃₂	—	13	4	no	—
225	M₃₂	—	26	2	no	—
226	C₂×C₁₆	—	26	2	no	—
227	#108	—	2	1	no	—
228	#108	C₂²×D₁₃	1	2	yes	—
229	#104	—	2	4	no	—
230	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
231	C₄×C₈	C₂²×D₁₃	1	2	yes	—
232	C₂×C₁₃	—	2	1	no	—
233	C₂×C₁₃	—	2	1	no	—
234	C₂×C₁₃	—	8	2	no	—
235	C₂×C₁₃	#2213	1	2	yes	complemented
236	C₂×C₁₃	C₂×#459	1	1	yes	characteristic
237	D₁₃	—	16	4	no	—
238	C₂⁴	—	52	2	no	—
239	#41	—	2	2	no	—
240	#41	—	2	2	no	—
241	#41	—	4	1	no	—
242	#41	—	4	1	no	—
243	C₂×Q₈	—	2	1	no	—
244	C₂×D₄	—	2	1	no	—
245	C₂×D₄	—	4	1	no	—
246	C₂×D₄	—	26	4	no	—
247	C₂×D₄	—	52	8	no	—
248	C₂×D₄	—	52	2	no	—
249	C₂×D₄	—	52	2	no	—
250	C₂×D₄	—	52	2	no	—
251	C₂×D₄	—	52	2	no	—
252	C₂×D₄	—	52	2	no	—
253	C₂×D₄	—	104	8	no	—
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	4	1	no	—
256	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
257	Q₁₆	—	2	1	no	—
258	Q₁₆	C₂×D₅₂	1	2	yes	complemented
259	SD₁₆	—	2	2	no	—
260	SD₁₆	—	2	2	no	—
261	D₈	—	2	1	no	—
262	D₈	—	26	4	no	—
263	D₈	—	26	4	no	—
264	D₈	—	52	2	no	—
265	D₈	—	52	2	no	—
266	D₈	—	52	4	no	—
267	D₈	—	52	4	no	—
268	D₈	C₂×D₅₂	1	2	yes	complemented
269	M₁₆	—	2	2	no	—
270	M₁₆	—	4	1	no	—
271	M₁₆	—	4	4	no	—
272	C₂×C₈	—	4	1	no	—
273	C₂×C₈	—	4	4	no	—
274	C₂×C₈	#6104	1	1	yes	characteristic
275	C₂×C₈	#6104	1	1	yes	characteristic
276	C₂×C₈	#6104	1	2	yes	—
277	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
278	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
279	C₂×C₈	#6127	1	1	yes	characteristic
280	C₂×C₈	#6127	1	1	yes	characteristic
281	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
282	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
283	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
284	C₄²	—	2	1	no	—
285	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
286	C₁₆	—	26	4	no	—
287	C₁₃	C₂×#2213	1	1	yes	characteristic, complemented, Sylow
288	C₂³	—	4	1	no	—
289	C₂³	—	52	4	no	—
290	C₂³	—	104	8	no	—
291	C₂³	—	104	2	no	—
292	C₂³	—	104	2	no	—
293	C₂³	#88203	1	1	yes	characteristic
294	Q₈	—	2	2	no	—
295	Q₈	—	4	1	no	—
296	D₄	—	2	2	no	—
297	D₄	—	4	2	no	—
298	D₄	—	4	2	no	—
299	D₄	—	4	1	no	—
300	D₄	—	52	4	no	—
301	D₄	—	52	4	no	—
302	D₄	—	104	16	no	—
303	D₄	—	104	2	no	—
304	D₄	—	104	2	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	#88184	1	1	yes	characteristic
311	C₂×C₄	#88184	1	1	yes	characteristic
312	C₂×C₄	C₂×#6104	1	1	yes	characteristic
313	C₂×C₄	C₂×#6104	1	1	yes	characteristic
314	C₂×C₄	#88203	1	1	yes	characteristic
315	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
316	C₈	—	4	2	no	—
317	C₈	#88184	1	1	yes	characteristic
318	C₈	#88184	1	1	yes	characteristic
319	C₈	C₂×#6104	1	2	yes	—
320	C₈	C₂×#6104	1	2	yes	—
321	C₈	C₂²×D₅₂	1	1	yes	characteristic
322	C₈	C₂×#6127	1	1	yes	characteristic
323	C₂²	—	2	2	no	—
324	C₂²	—	4	2	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	104	8	no	—
327	C₂²	—	104	4	no	—
328	C₂²	—	208	4	no	—
329	C₂²	—	208	4	no	—
330	C₂²	#11718676	1	1	yes	characteristic
331	C₂²	#11718676	1	1	yes	characteristic
332	C₂²	C₂×#88203	1	1	yes	characteristic
333	C₄	—	2	4	no	—
334	C₄	—	4	2	no	—
335	C₄	#11718676	1	1	yes	characteristic
336	C₄	#11718676	1	1	yes	characteristic
337	C₄	C₂×#88184	1	1	yes	characteristic
338	C₄	C₂²×#6104	1	1	yes	characteristic
339	C₂	—	2	1	no	—
340	C₂	—	2	1	no	—
341	C₂	—	8	2	no	—
342	C₂	—	208	4	no	—
343	C₂	#13116912	1	2	yes	direct product factor
344	C₂	C₂×#11718676	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11718676
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#443	C₂
socle	C₂²×C₁₃	#459
hypercenter	C₂×#443	D₁₃
hypocenter	C₁₃	C₂×#2213
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
1. C₂×#11718854	C₂	semidirect
2. C₂×#11718356	C₂	semidirect
3. #13116912	C₂	direct (N × H)
4. C₂×#11718055	C₂	semidirect
5. C₂×#11718087	C₂	semidirect
6. C₂×C₁₃×#443	C₂	semidirect
13. C₁₃×#443	C₂²	semidirect
18. #11718854	C₂²	semidirect
23. #11718356	C₂²	semidirect
24. #11718087	C₂²	semidirect
25. #11718055	C₂²	semidirect
31. C₁₃×#135	D₄	semidirect
33. C₂×C₁₃×Q₁₆	D₄	semidirect
35. C₂×C₁₃×D₈	D₄	semidirect
87. C₁₃×Q₁₆	C₂×D₄	semidirect
91. C₁₃×D₈	C₂×D₄	semidirect
118. C₂×#443	D₁₃	semidirect
170. #443	C₂×D₁₃	semidirect
197. #135	D₅₂	semidirect
199. C₂×Q₁₆	D₅₂	semidirect
211. C₂×D₈	D₅₂	semidirect
235. C₂×C₁₃	#2213	semidirect
258. Q₁₆	C₂×D₅₂	semidirect
268. D₈	C₂×D₅₂	semidirect
287. C₁₃	C₂×#2213	semidirect
343. C₂	#13116912	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×#11718854	2
C₂×#11718356	2
#13116912	2
C₂×#11718055	2
C₂×#11718087	2
C₂×C₁₃×#443	2
C₂×#2213	13

Subgroups — C2×#13116912 #16017299

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116912 #16017299