Subgroups — C₂×#13127975 #16769429

Order 3328.

2126 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719457	C₂	1	1	yes	maximal, characteristic, complemented
2	#13127975	C₂	1	16	yes	maximal, direct product factor
3	C₂×#11718808	C₂	1	3	yes	maximal, complemented
4	C₂×#11718854	C₂	1	3	yes	maximal, complemented
5	C₂×#11718855	C₂	1	3	yes	maximal, complemented
6	C₂²×#88066	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×#88256	—	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	complemented
15	C₂²×C₁₃×D₈	C₂²	1	3	yes	complemented
16	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
17	#11719457	—	2	4	no	—
18	#11719457	C₂²	1	8	yes	complemented
19	C₂×#88162	—	2	3	no	—
20	C₂×#88158	C₂²	1	3	yes	complemented
21	C₂²×D₁₀₄	C₂²	1	3	yes	complemented
22	#11718855	C₂²	1	24	yes	complemented
23	#11718854	C₂²	1	24	yes	complemented
24	C₂×#88069	C₂²	1	1	yes	characteristic, complemented
25	C₂×#88068	C₂²	1	3	yes	complemented
26	C₂×#88067	C₂²	1	3	yes	complemented
27	#11718808	C₂²	1	24	yes	complemented
28	C₂×#88066	C₂²	1	3	yes	complemented
29	C₂×#88066	C₂²	1	6	yes	complemented
30	C₂×#88066	C₂²	1	24	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	#88256	—	2	8	no	—
42	#88256	—	4	4	no	—
43	C₂×#6131	—	4	1	no	—
44	C₂×D₄×D₁₃	—	4	3	no	—
45	C₂×#6127	—	2	3	no	—
46	C₂²×D₅₂	—	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	complemented
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	complemented
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	#88162	—	2	24	no	—
61	C₂×Dic₅₂	C₂³	1	1	yes	characteristic
62	#88158	—	2	6	no	—
63	#88158	C₂³	1	12	yes	complemented
64	C₂×D₁₀₄	—	2	3	no	—
65	C₂×D₁₀₄	—	2	6	no	—
66	C₂×D₁₀₄	C₂³	1	3	yes	—
67	C₂×D₁₀₄	C₂³	1	12	yes	complemented
68	C₂×#6095	—	2	3	no	—
69	#88069	C₂³	1	4	yes	complemented
70	#88068	C₂³	1	12	yes	complemented
71	#88067	C₂³	1	12	yes	complemented
72	#88066	C₂³	1	12	yes	complemented
73	#88066	C₂³	1	24	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×#3415	—	13	1	no	maximal, Sylow
79	C₂⁴×C₁₃	—	4	3	no	—
80	C₂³×D₁₃	—	4	3	no	—
81	C₁₃×#41	—	2	12	no	—
82	C₁₃×#41	—	4	8	no	—
83	C₁₃×#41	—	4	6	no	—
84	C₁₃×#41	D₈	1	4	yes	—
85	C₁₃×#41	C₂×D₄	1	4	yes	—
86	C₂×Q₈×C₁₃	—	2	1	no	—
87	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
88	C₂×D₄×C₁₃	—	2	3	no	—
89	C₂×D₄×C₁₃	—	2	12	no	—
90	C₂×D₄×C₁₃	—	4	3	no	—
91	C₂×D₄×C₁₃	—	4	24	no	—
92	C₂×D₄×C₁₃	—	4	3	no	—
93	C₂×D₄×C₁₃	—	4	6	no	—
94	C₂×D₄×C₁₃	D₈	1	3	yes	—
95	C₂²×C₄×C₁₃	—	4	3	no	—
96	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
97	C₂×C₂²⋊D₁₃	—	4	3	no	—
98	#6131	—	4	8	no	—
99	D₄×D₁₃	—	4	24	no	—
100	#6127	—	2	12	no	—
101	#6127	—	4	6	no	—
102	C₂×D₅₂	—	2	3	no	—
103	C₂×D₅₂	—	2	12	no	—
104	C₂×D₅₂	—	4	3	no	—
105	C₂×D₅₂	—	4	6	no	—
106	C₂×C₄×D₁₃	—	4	3	no	—
107	C₂×Dic₂₆	—	2	1	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	1	no	—
120	Dic₅₂	C₂⁴	1	2	yes	—
121	D₁₀₄	—	2	3	no	—
122	D₁₀₄	—	2	12	no	—
123	D₁₀₄	C₂⁴	1	6	yes	—
124	#6095	—	2	12	no	—
125	#6090	C₂⁴	1	2	yes	—
126	#6090	C₂⁴	1	6	yes	—
127	C₂×#576	—	13	1	no	—
128	C₂×#576	D₁₃	1	1	yes	characteristic, complemented
129	#3415	—	13	16	no	—
130	C₂×#509	—	13	3	no	—
131	C₂×#509	—	13	3	no	—
132	C₂×#508	—	13	3	no	—
133	C₂²×D₁₆	—	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	3	no	—
140	C₂²×D₁₃	—	4	12	no	—
141	C₂²×D₁₃	—	16	6	no	—
142	Q₈×C₁₃	—	2	2	no	—
143	Q₈×C₁₃	—	4	1	no	—
144	Q₈×C₁₃	C₂×D₈	1	4	yes	—
145	D₄×C₁₃	—	2	6	no	—
146	D₄×C₁₃	—	4	12	no	—
147	D₄×C₁₃	—	4	3	no	—
148	D₄×C₁₃	—	4	12	no	—
149	D₄×C₁₃	C₂×D₈	1	12	yes	—
150	C₂×C₄×C₁₃	—	4	1	no	—
151	C₂×C₄×C₁₃	—	4	12	no	—
152	C₂×C₄×C₁₃	C₂×D₈	1	3	yes	—
153	C₂×C₄×C₁₃	C₂×D₈	1	6	yes	—
154	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
156	C₂²⋊D₁₃	—	4	12	no	—
157	C₂×Dic₁₃	—	4	1	no	—
158	D₅₂	—	2	6	no	—
159	D₅₂	—	4	3	no	—
160	D₅₂	—	4	12	no	—
161	C₄×D₁₃	—	4	12	no	—
162	Dic₂₆	—	2	2	no	—
163	Dic₂₆	—	4	1	no	—
164	C₈×C₁₃	C₂²×D₄	1	6	yes	—
165	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
166	C₈×C₁₃	C₂⁵	1	1	yes	characteristic
167	C₂×H(5,𝔽₂)	—	2	1	no	—
168	C₂×H(5,𝔽₂)	—	26	1	no	—
169	#576	—	2	4	no	—
170	#576	—	13	8	no	—
171	#576	—	26	4	no	—
172	#576	C₂×D₁₃	1	8	yes	complemented
173	C₂×AGL(1,ℤ/8)	—	2	3	no	—
174	C₂×AGL(1,ℤ/8)	—	26	3	no	—
175	C₂×#135	—	13	3	no	—
176	C₂×#135	C₂×D₁₃	1	3	yes	complemented
177	C₂²×D₈	—	13	3	no	—
178	C₂²×D₈	C₂×D₁₃	1	3	yes	complemented
179	C₂×#131	C₂×D₁₃	1	1	yes	characteristic
180	#509	—	13	24	no	—
181	#509	—	13	24	no	—
182	#508	—	13	24	no	—
183	C₂×Q₃₂	—	13	1	no	—
184	C₂×SD₃₂	—	13	3	no	—
185	C₂×SD₃₂	—	13	3	no	—
186	C₂×D₁₆	—	13	3	no	—
187	C₂×D₁₆	—	13	6	no	—
188	C₂×D₁₆	—	13	24	no	—
189	#504	—	13	8	no	—
190	C₂×M₃₂	—	13	3	no	—
191	C₂²×C₁₆	—	13	3	no	—
192	C₂²×C₁₃	—	2	6	no	—
193	C₂²×C₁₃	—	4	6	no	—
194	C₂²×C₁₃	—	8	6	no	—
195	C₂²×C₁₃	—	16	12	no	—
196	C₂²×C₁₃	C₂²×D₈	1	6	yes	—
197	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
198	C₂×D₁₃	—	4	6	no	—
199	C₂×D₁₃	—	8	6	no	—
200	C₂×D₁₃	—	16	12	no	—
201	C₄×C₁₃	—	4	2	no	—
202	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
203	C₄×C₁₃	C₂²×D₈	1	6	yes	—
204	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
205	Dic₁₃	—	4	2	no	—
206	H(5,𝔽₂)	—	2	8	no	—
207	H(5,𝔽₂)	—	4	4	no	—
208	H(5,𝔽₂)	—	26	8	no	—
209	H(5,𝔽₂)	—	52	4	no	—
210	C₂×#41	—	2	3	no	—
211	C₂×#41	—	4	1	no	—
212	C₂×#41	—	26	3	no	—
213	C₂×#41	—	52	1	no	—
214	C₂×#41	C₂²⋊D₁₃	1	1	yes	characteristic
215	C₂²×D₄	—	2	3	no	—
216	C₂²×D₄	—	4	3	no	—
217	C₂²×D₄	—	26	3	no	—
218	C₂²×D₄	—	52	3	no	—
219	AGL(1,ℤ/8)	—	2	24	no	—
220	AGL(1,ℤ/8)	—	26	24	no	—
221	#135	—	2	6	no	—
222	#135	—	13	12	no	—
223	#135	—	26	6	no	—
224	#135	C₂²×D₁₃	1	12	yes	complemented
225	C₂×Q₁₆	—	13	1	no	—
226	C₂×Q₁₆	C₂²×D₁₃	1	1	yes	characteristic
227	C₂×SD₁₆	—	2	3	no	—
228	C₂×SD₁₆	—	26	3	no	—
229	C₂×D₈	—	2	3	no	—
230	C₂×D₈	—	2	6	no	—
231	C₂×D₈	—	13	3	no	—
232	C₂×D₈	—	13	12	no	—
233	C₂×D₈	—	26	3	no	—
234	C₂×D₈	—	26	6	no	—
235	C₂×D₈	C₂²×D₁₃	1	3	yes	—
236	C₂×D₈	C₂²×D₁₃	1	12	yes	complemented
237	#131	C₂²⋊D₁₃	1	4	yes	—
238	#131	C₂²×D₁₃	1	4	yes	—
239	C₂×M₁₆	C₂²⋊D₁₃	1	3	yes	—
240	C₂²×C₈	C₂²×D₁₃	1	3	yes	—
241	Q₃₂	—	13	4	no	—
242	SD₃₂	—	13	12	no	—
243	SD₃₂	—	13	12	no	—
244	D₁₆	—	13	12	no	—
245	D₁₆	—	13	24	no	—
246	M₃₂	—	13	12	no	—
247	C₂×C₁₆	—	13	1	no	—
248	C₂×C₁₆	—	13	3	no	—
249	C₂×C₁₆	—	13	12	no	—
250	C₂×C₁₃	—	2	6	no	—
251	C₂×C₁₃	—	8	6	no	—
252	C₂×C₁₃	#3415	1	2	yes	complemented
253	C₂×C₁₃	C₂³×D₈	1	1	yes	characteristic
254	D₁₃	—	8	6	no	—
255	C₂⁴	—	4	3	no	—
256	C₂⁴	—	52	3	no	—
257	#41	—	2	12	no	—
258	#41	—	4	8	no	—
259	#41	—	4	6	no	—
260	#41	—	26	12	no	—
261	#41	—	52	8	no	—
262	#41	—	52	6	no	—
263	#41	#6104	1	4	yes	—
264	#41	C₂×C₂²⋊D₁₃	1	4	yes	—
265	C₂×Q₈	—	2	1	no	—
266	C₂×Q₈	—	26	1	no	—
267	C₂×Q₈	#6104	1	1	yes	characteristic
268	C₂×D₄	—	2	3	no	—
269	C₂×D₄	—	2	12	no	—
270	C₂×D₄	—	4	3	no	—
271	C₂×D₄	—	4	24	no	—
272	C₂×D₄	—	4	3	no	—
273	C₂×D₄	—	4	6	no	—
274	C₂×D₄	—	26	3	no	—
275	C₂×D₄	—	26	12	no	—
276	C₂×D₄	—	52	3	no	—
277	C₂×D₄	—	52	24	no	—
278	C₂×D₄	—	52	3	no	—
279	C₂×D₄	—	52	6	no	—
280	C₂×D₄	#6104	1	3	yes	—
281	C₂²×C₄	—	4	3	no	—
282	C₂²×C₄	—	52	3	no	—
283	C₂²×C₄	C₂×C₂²⋊D₁₃	1	3	yes	—
284	Q₁₆	—	2	1	no	—
285	Q₁₆	—	13	2	no	—
286	Q₁₆	—	26	1	no	—
287	Q₁₆	C₂³×D₁₃	1	2	yes	—
288	SD₁₆	—	2	12	no	—
289	SD₁₆	—	26	12	no	—
290	D₈	—	2	3	no	—
291	D₈	—	2	12	no	—
292	D₈	—	13	6	no	—
293	D₈	—	26	3	no	—
294	D₈	—	26	12	no	—
295	D₈	C₂³×D₁₃	1	6	yes	—
296	M₁₆	C₂×C₂²⋊D₁₃	1	12	yes	—
297	C₂×C₈	C₂×C₂²⋊D₁₃	1	3	yes	—
298	C₂×C₈	C₂×C₂²⋊D₁₃	1	6	yes	—
299	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
300	C₂×C₈	C₂³×D₁₃	1	6	yes	—
301	C₁₆	—	13	2	no	—
302	C₁₆	—	13	6	no	—
303	C₁₃	C₂×#3415	1	1	yes	characteristic, complemented, Sylow
304	C₂³	—	4	3	no	—
305	C₂³	—	4	12	no	—
306	C₂³	—	16	6	no	—
307	C₂³	—	52	3	no	—
308	C₂³	—	52	12	no	—
309	C₂³	—	208	6	no	—
310	C₂³	C₂×#6104	1	3	yes	—
311	Q₈	—	2	2	no	—
312	Q₈	—	4	1	no	—
313	Q₈	—	26	2	no	—
314	Q₈	—	52	1	no	—
315	Q₈	C₂×#6104	1	4	yes	—
316	D₄	—	2	6	no	—
317	D₄	—	4	12	no	—
318	D₄	—	4	12	no	—
319	D₄	—	4	3	no	—
320	D₄	—	26	6	no	—
321	D₄	—	52	12	no	—
322	D₄	—	52	12	no	—
323	D₄	—	52	3	no	—
324	D₄	C₂×#6104	1	12	yes	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	4	12	no	—
327	C₂×C₄	—	52	1	no	—
328	C₂×C₄	—	52	12	no	—
329	C₂×C₄	C₂×#6104	1	6	yes	—
330	C₂×C₄	C₂×#6104	1	3	yes	—
331	C₂×C₄	C₂²×C₂²⋊D₁₃	1	6	yes	—
332	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
333	C₈	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
334	C₈	C₂²×C₂²⋊D₁₃	1	6	yes	—
335	C₈	C₂⁴×D₁₃	1	1	yes	characteristic
336	C₂²	—	2	6	no	—
337	C₂²	—	4	6	no	—
338	C₂²	—	8	6	no	—
339	C₂²	—	16	12	no	—
340	C₂²	—	52	6	no	—
341	C₂²	—	104	6	no	—
342	C₂²	—	208	12	no	—
343	C₂²	C₂²×#6104	1	6	yes	—
344	C₂²	C₂²×#6104	1	1	yes	characteristic
345	C₄	—	4	2	no	—
346	C₄	—	52	2	no	—
347	C₄	C₂²×#6104	1	6	yes	—
348	C₄	C₂²×#6104	1	1	yes	characteristic
349	C₄	C₂³×C₂²⋊D₁₃	1	1	yes	characteristic
350	C₂	—	2	6	no	—
351	C₂	—	8	6	no	—
352	C₂	—	104	6	no	—
353	C₂	#13127975	1	2	yes	direct product factor
354	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×#3415
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×#11719457	C₂	semidirect
2. #13127975	C₂	direct (N × H)
3. C₂×#11718808	C₂	semidirect
4. C₂×#11718854	C₂	semidirect
5. C₂×#11718855	C₂	semidirect
6. C₂²×#88066	C₂	semidirect
7. C₂×#11718801	C₂	semidirect
8. C₂×C₁₃×#576	C₂	semidirect
12. C₁₃×#576	C₂²	semidirect
14. C₂×C₁₃×#135	C₂²	semidirect
15. C₂²×C₁₃×D₈	C₂²	semidirect
18. #11719457	C₂²	semidirect
20. C₂×#88158	C₂²	semidirect
21. C₂²×D₁₀₄	C₂²	semidirect
22. #11718855	C₂²	semidirect
23. #11718854	C₂²	semidirect
24. C₂×#88069	C₂²	semidirect
25. C₂×#88068	C₂²	semidirect
26. C₂×#88067	C₂²	semidirect
27. #11718808	C₂²	semidirect
28. C₂×#88066	C₂²	semidirect
29. C₂×#88066	C₂²	semidirect
30. C₂×#88066	C₂²	semidirect
31. #11718801	C₂²	semidirect
32. C₂×#88052	C₂²	semidirect
33. C₂²×#6090	C₂²	semidirect
49. C₁₃×#135	C₂³	semidirect
55. C₂×C₁₃×D₈	C₂³	semidirect
63. #88158	C₂³	semidirect
67. C₂×D₁₀₄	C₂³	semidirect
69. #88069	C₂³	semidirect
70. #88068	C₂³	semidirect
71. #88067	C₂³	semidirect
72. #88066	C₂³	semidirect
73. #88066	C₂³	semidirect
74. #88052	C₂³	semidirect
77. C₂×#6090	C₂³	semidirect
128. C₂×#576	D₁₃	semidirect
172. #576	C₂×D₁₃	semidirect
176. C₂×#135	C₂×D₁₃	semidirect
178. C₂²×D₈	C₂×D₁₃	semidirect
224. #135	C₂²×D₁₃	semidirect
236. C₂×D₈	C₂²×D₁₃	semidirect
252. C₂×C₁₃	#3415	semidirect
303. C₁₃	C₂×#3415	semidirect
353. C₂	#13127975	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×#11719457	2
#13127975	2
C₂×#11718808	2
C₂×#11718854	2
C₂×#11718855	2
C₂²×#88066	2
C₂×#11718801	2
C₂×C₁₃×#576	2
C₂×#3415	13

Subgroups — C2×#13127975 #16769429

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13127975 #16769429