Subgroups — C₂×#13131216 #16807122

Order 3328.

754 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	C₂²×#88106	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×#6092	C₂	1	1	yes	maximal, characteristic, complemented
3	#13131216	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊#6092	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂²⋊#6092	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂²⋊#6092	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11718231	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₄×#1055	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×D₄×C₁₃	C₄	1	1	yes	characteristic
10	C₂³×#1055	C₂²	1	1	yes	characteristic
11	C₂³×#1055	C₂²	1	1	yes	characteristic
12	C₂×#88106	—	2	4	no	—
13	C₂×#88106	C₂²	1	4	yes	complemented
14	C₂²×#6092	—	2	4	no	—
15	C₂²×#6092	C₂²	1	1	yes	characteristic
16	C₂²×#6092	C₂²	1	1	yes	characteristic
17	C₂²×#6092	C₂²	1	1	yes	characteristic
18	C₂²×#6092	C₂²	1	1	yes	characteristic
19	C₂²×#6092	C₂²	1	4	yes	complemented
20	C₂×C₂²⋊#1055	C₄	1	1	yes	characteristic
21	C₂×C₂²⋊#1055	C₄	1	1	yes	characteristic
22	D₄×#1055	C₄	1	4	yes	—
23	D₄×#1055	C₂²	1	4	yes	—
24	C₂×C₄⋊#1055	C₄	1	1	yes	characteristic
25	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
26	C₂²⋊#6092	C₂²	1	4	yes	complemented
27	C₂²⋊#6092	C₂²	1	4	yes	complemented
28	#11718231	C₂²	1	4	yes	complemented
29	C₄⋊#6092	C₂²	1	4	yes	complemented
30	C₂²×D₄×C₁₃	C₈	1	1	yes	characteristic
31	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₄×D₄×C₁₃	C₈	1	4	yes	—
34	C₄×D₄×C₁₃	C₂×C₄	1	4	yes	—
35	C₂×C₁₃×C₄⋊C₄	C₈	1	1	yes	characteristic
36	C₂×C₁₃×C₂²⋊C₄	C₈	1	1	yes	characteristic
37	C₂×C₁₃×C₂²⋊C₄	C₈	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×#1055	—	2	2	no	—
40	C₂²×#1055	—	2	2	no	—
41	C₂²×#1055	—	2	2	no	—
42	C₂²×#1055	—	2	2	no	—
43	C₂²×#1055	C₂×C₄	1	2	yes	—
44	C₂²×#1055	C₂×C₄	1	2	yes	—
45	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
46	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
47	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
48	C₂²×#1055	C₂³	1	2	yes	—
49	C₂²×#1055	C₂³	1	2	yes	—
50	C₂²×#1055	C₂³	1	1	yes	characteristic
51	#88106	—	2	8	no	—
52	C₂×#6092	—	2	2	no	—
53	C₂×#6092	—	2	2	no	—
54	C₂×#6092	—	2	2	no	—
55	C₂×#6092	—	2	8	no	—
56	C₂×#6092	D₄	1	4	yes	complemented
57	C₂×#6092	C₂³	1	2	yes	—
58	C₂×#6092	C₂³	1	2	yes	—
59	C₂×#6092	C₂³	1	2	yes	—
60	C₂×#6092	C₂³	1	2	yes	—
61	C₂²⋊#1055	C₂×C₄	1	4	yes	—
62	C₂²⋊#1055	C₂×C₄	1	4	yes	—
63	C₄⋊#1055	C₂×C₄	1	4	yes	—
64	C₄×#1055	C₂×C₄	1	2	yes	—
65	C₄×#1055	C₂³	1	2	yes	—
66	C₂×#2168	—	13	1	no	maximal, Sylow
67	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
68	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
69	C₂×D₄×C₁₃	—	2	4	no	—
70	C₂×D₄×C₁₃	C₂×C₈	1	4	yes	—
71	C₂²×C₄×C₁₃	—	2	2	no	—
72	C₂²×C₄×C₁₃	—	2	2	no	—
73	C₂²×C₄×C₁₃	—	2	2	no	—
74	C₂²×C₄×C₁₃	—	2	2	no	—
75	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
80	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
81	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₁₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
85	C₁₃×C₂²⋊C₄	C₂×C₈	1	4	yes	—
86	C₁₃×C₂²⋊C₄	C₂×C₈	1	4	yes	—
87	C₄²×C₁₃	C₂×C₈	1	2	yes	—
88	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
89	C₂×#1055	—	2	2	no	—
90	C₂×#1055	—	2	2	no	—
91	C₂×#1055	—	2	2	no	—
92	C₂×#1055	—	2	4	no	—
93	C₂×#1055	—	2	4	no	—
94	C₂×#1055	—	2	2	no	—
95	C₂×#1055	—	2	2	no	—
96	C₂×#1055	—	2	2	no	—
97	C₂×#1055	—	2	2	no	—
98	C₂×#1055	C₂²×C₄	1	2	yes	—
99	C₂×#1055	C₂²×C₄	1	2	yes	—
100	C₂×#1055	C₂²×C₄	1	2	yes	—
101	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
102	C₂×#1055	C₂×D₄	1	2	yes	—
103	C₂×#1055	#41	1	2	yes	—
104	C₂×#1055	C₂⁴	1	1	yes	characteristic
105	#6092	—	2	2	no	—
106	#6092	—	2	2	no	—
107	#6092	—	2	2	no	—
108	#6092	C₂×D₄	1	4	yes	complemented
109	C₂²×M₃₂	—	13	1	no	—
110	C₂³×C₁₆	—	13	1	no	—
111	C₂×C₈×D₄	—	13	1	no	—
112	#2168	—	13	8	no	—
113	C₂×C₄⋊C₁₆	—	13	1	no	—
114	C₂×C₂²⋊C₁₆	—	13	1	no	—
115	C₂×C₂²⋊C₁₆	—	13	1	no	—
116	C₂×#346	—	13	1	no	—
117	C₂³×C₁₃	—	2	2	no	—
118	C₂³×C₁₃	—	2	2	no	—
119	C₂³×C₁₃	—	2	2	no	—
120	C₂³×C₁₃	—	2	2	no	—
121	C₂³×C₁₃	—	2	2	no	—
122	C₂³×C₁₃	M₃₂	1	4	yes	complemented
123	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
124	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
125	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
126	D₄×C₁₃	—	2	8	no	—
127	C₂×C₄×C₁₃	—	2	2	no	—
128	C₂×C₄×C₁₃	—	2	2	no	—
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	—	2	2	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	2	2	no	—
133	C₂×C₄×C₁₃	—	2	2	no	—
134	C₂×C₄×C₁₃	—	2	2	no	—
135	C₂×C₄×C₁₃	—	2	4	no	—
136	C₂×C₄×C₁₃	—	2	4	no	—
137	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
138	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
139	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
140	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
141	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
142	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
143	C₂×C₄×C₁₃	#131	1	2	yes	—
144	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
145	#1055	—	2	2	no	—
146	#1055	—	2	2	no	—
147	#1055	—	2	2	no	—
148	#1055	C₄×D₄	1	2	yes	—
149	#1055	C₂²×D₄	1	1	yes	characteristic
150	#1055	C₂×#41	1	1	yes	characteristic
151	C₂³×C₈	—	13	1	no	—
152	C₂³×C₈	—	13	1	no	—
153	C₂×C₄×D₄	#261	1	1	yes	characteristic
154	C₂×M₃₂	—	13	4	no	—
155	C₂×M₃₂	—	26	4	no	—
156	C₂²×C₁₆	—	13	1	no	—
157	C₂²×C₁₆	—	13	1	no	—
158	C₂²×C₁₆	—	13	1	no	—
159	C₂²×C₁₆	—	13	1	no	—
160	C₂²×C₁₆	—	13	4	no	—
161	C₂²×C₁₆	—	26	4	no	—
162	C₈×D₄	—	13	4	no	—
163	C₈×D₄	—	13	4	no	—
164	C₂×C₄⋊C₈	—	13	1	no	—
165	C₂×C₂²⋊C₈	—	13	1	no	—
166	C₂×C₂²⋊C₈	—	13	1	no	—
167	C₂×C₄×C₈	—	13	1	no	—
168	C₄⋊C₁₆	—	13	4	no	—
169	C₂²⋊C₁₆	—	13	4	no	—
170	C₂²⋊C₁₆	—	13	4	no	—
171	#346	—	13	4	no	—
172	C₂²×C₁₃	—	2	2	no	—
173	C₂²×C₁₃	—	2	2	no	—
174	C₂²×C₁₃	—	2	8	no	—
175	C₂²×C₁₃	—	2	2	no	—
176	C₂²×C₁₃	—	2	2	no	—
177	C₂²×C₁₃	—	2	2	no	—
178	C₂²×C₁₃	—	4	4	no	—
179	C₂²×C₁₃	C₈×D₄	1	2	yes	—
180	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
181	C₂²×C₁₃	C₂×M₃₂	1	4	yes	complemented
182	C₂²×C₁₃	#504	1	2	yes	—
183	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
184	C₄×C₁₃	—	2	2	no	—
185	C₄×C₁₃	—	2	2	no	—
186	C₄×C₁₃	—	2	2	no	—
187	C₄×C₁₃	—	2	2	no	—
188	C₄×C₁₃	C₈×D₄	1	2	yes	—
189	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
190	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
191	C₂²×D₄	#1057	1	1	yes	characteristic
192	C₂³×C₄	C₂×#261	1	1	yes	characteristic
193	C₂³×C₄	C₂×#261	1	1	yes	characteristic
194	C₂²×C₈	—	13	1	no	—
195	C₂²×C₈	—	13	2	no	—
196	C₂²×C₈	—	13	2	no	—
197	C₂²×C₈	—	13	2	no	—
198	C₂²×C₈	—	13	2	no	—
199	C₂²×C₈	—	13	1	no	—
200	C₂²×C₈	—	13	1	no	—
201	C₂²×C₈	—	13	1	no	—
202	C₂²×C₈	—	26	2	no	—
203	C₂²×C₈	—	26	2	no	—
204	C₂²×C₈	—	26	2	no	—
205	C₂²×C₈	—	26	2	no	—
206	C₄×D₄	#1057	1	4	yes	—
207	C₄×D₄	C₂×#261	1	4	yes	—
208	C₂×C₄⋊C₄	#1057	1	1	yes	characteristic
209	C₂×C₂²⋊C₄	#1057	1	1	yes	characteristic
210	C₂×C₂²⋊C₄	#1057	1	1	yes	characteristic
211	C₂×C₄²	C₂×#261	1	1	yes	characteristic
212	M₃₂	—	26	8	no	—
213	C₂×C₁₆	—	13	4	no	—
214	C₂×C₁₆	—	13	2	no	—
215	C₂×C₁₆	—	13	2	no	—
216	C₂×C₁₆	—	13	2	no	—
217	C₂×C₁₆	—	13	2	no	—
218	C₂×C₁₆	—	26	8	no	—
219	C₂×C₁₆	—	26	2	no	—
220	C₂×C₁₆	—	26	2	no	—
221	C₂×C₁₆	—	26	2	no	—
222	C₄⋊C₈	—	13	4	no	—
223	C₂²⋊C₈	—	13	4	no	—
224	C₂²⋊C₈	—	13	4	no	—
225	C₄×C₈	—	13	2	no	—
226	C₄×C₈	—	13	2	no	—
227	C₂×C₁₃	—	2	4	no	—
228	C₂×C₁₃	—	4	2	no	—
229	C₂×C₁₃	#2168	1	4	yes	complemented
230	C₂×C₁₃	C₂×C₈×D₄	1	1	yes	characteristic
231	C₂×C₁₃	C₂²×M₃₂	1	1	yes	characteristic
232	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
233	C₂⁴	C₂×#1057	1	1	yes	characteristic
234	C₂⁴	C₂×#1057	1	1	yes	characteristic
235	C₂×D₄	—	2	4	no	—
236	C₂×D₄	C₂×#1057	1	4	yes	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
242	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
243	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
244	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
245	C₂²×C₄	C₂×#1057	1	2	yes	—
246	C₂²×C₄	C₂×#1057	1	2	yes	—
247	C₂²×C₄	C₂²×#261	1	2	yes	—
248	C₂²×C₄	C₂²×#261	1	2	yes	—
249	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
250	C₂×C₈	—	13	2	no	—
251	C₂×C₈	—	13	2	no	—
252	C₂×C₈	—	13	1	no	—
253	C₂×C₈	—	13	1	no	—
254	C₂×C₈	—	13	2	no	—
255	C₂×C₈	—	13	2	no	—
256	C₂×C₈	—	13	2	no	—
257	C₂×C₈	—	26	2	no	—
258	C₂×C₈	—	26	2	no	—
259	C₂×C₈	—	26	2	no	—
260	C₂×C₈	—	26	2	no	—
261	C₂×C₈	—	26	4	no	—
262	C₂×C₈	—	26	4	no	—
263	C₂×C₈	—	26	2	no	—
264	C₂×C₈	—	26	2	no	—
265	C₂×C₈	—	26	2	no	—
266	C₄⋊C₄	C₂×#1057	1	4	yes	—
267	C₂²⋊C₄	C₂×#1057	1	4	yes	—
268	C₂²⋊C₄	C₂×#1057	1	4	yes	—
269	C₄²	C₂×#1057	1	2	yes	—
270	C₄²	C₂²×#261	1	2	yes	—
271	C₁₆	—	13	4	no	—
272	C₁₆	—	26	2	no	—
273	C₁₆	—	26	2	no	—
274	C₁₆	—	26	2	no	—
275	C₁₃	C₂×#2168	1	1	yes	characteristic, complemented, Sylow
276	C₂³	—	2	2	no	—
277	C₂³	—	2	2	no	—
278	C₂³	—	2	2	no	—
279	C₂³	—	2	2	no	—
280	C₂³	—	2	2	no	—
281	C₂³	#88106	1	4	yes	complemented
282	C₂³	C₂²×#1057	1	2	yes	—
283	C₂³	C₂²×#1057	1	2	yes	—
284	C₂³	C₂²×#1057	1	1	yes	characteristic
285	D₄	—	2	8	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	#88238	1	2	yes	—
297	C₂×C₄	D₄×#261	1	2	yes	—
298	C₂×C₄	C₂²×#1057	1	2	yes	—
299	C₂×C₄	C₂²×#1057	1	2	yes	—
300	C₂×C₄	C₂²×#1057	1	2	yes	—
301	C₂×C₄	C₂²×#1057	1	2	yes	—
302	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
303	C₂×C₄	C₂³×#261	1	1	yes	characteristic
304	C₈	—	13	1	no	—
305	C₈	—	13	1	no	—
306	C₈	—	13	2	no	—
307	C₈	—	26	2	no	—
308	C₈	—	26	2	no	—
309	C₈	—	26	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	8	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	4	4	no	—
317	C₂²	C₂×#88106	1	2	yes	—
318	C₂²	C₂×#88106	1	4	yes	complemented
319	C₂²	D₄×#1057	1	2	yes	—
320	C₂²	#11719162	1	2	yes	—
321	C₂²	C₂³×#1057	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	2	2	no	—
325	C₄	—	2	2	no	—
326	C₄	D₄×#1057	1	2	yes	—
327	C₄	C₂×#88238	1	1	yes	characteristic
328	C₄	C₂×D₄×#261	1	1	yes	characteristic
329	C₂	—	2	4	no	—
330	C₂	—	4	2	no	—
331	C₂	#13131216	1	4	yes	direct product factor
332	C₂	C₂×D₄×#1057	1	1	yes	characteristic
333	C₂	C₂²×#88106	1	1	yes	characteristic
334	C₂	C₂×#11719162	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×#261
commutator	C₂²×C₁₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂×C₄×D₄×C₁₃	C₄
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂²×D₄×C₁₃	C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	C₂×C₄×D₄	#261
hypocenter	C₁₃	C₂×#2168
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#88106	C₂	semidirect
2. C₂³×#6092	C₂	semidirect
3. #13131216	C₂	direct (N × H)
4. C₂×C₄⋊#6092	C₂	semidirect
5. C₂×C₂²⋊#6092	C₂	semidirect
6. C₂×C₂²⋊#6092	C₂	semidirect
7. C₂×#11718231	C₂	semidirect
13. C₂×#88106	C₂²	semidirect
19. C₂²×#6092	C₂²	semidirect
26. C₂²⋊#6092	C₂²	semidirect
27. C₂²⋊#6092	C₂²	semidirect
28. #11718231	C₂²	semidirect
29. C₄⋊#6092	C₂²	semidirect
56. C₂×#6092	D₄	semidirect
108. #6092	C₂×D₄	semidirect
122. C₂³×C₁₃	M₃₂	semidirect
181. C₂²×C₁₃	C₂×M₃₂	semidirect
229. C₂×C₁₃	#2168	semidirect
275. C₁₃	C₂×#2168	semidirect
281. C₂³	#88106	semidirect
318. C₂²	C₂×#88106	semidirect
331. C₂	#13131216	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×#88106	2
C₂³×#6092	2
#13131216	2
C₂×C₄⋊#6092	2
C₂×C₂²⋊#6092	2
C₂×#11718231	2
C₂×D₄×#1055	2
C₂×#2168	13

Subgroups — C2×#13131216 #16807122

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13131216 #16807122