Subgroups — C₂²×#11719182 #16769617

Order 3328.

12878 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#6127	C₂	1	2	yes	maximal, complemented
2	C₂×#11719182	C₂	1	48	yes	maximal, direct product factor
3	C₂²×#88123	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#88128	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#88129	C₂	1	2	yes	maximal, complemented
6	C₂²×#88184	C₂	1	4	yes	maximal, complemented
7	C₂²×C₄×D₅₂	C₂	1	4	yes	maximal, complemented
8	C₂³×C₄²×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁴×C₄×C₁₃	C₂²	1	2	yes	—
10	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
11	C₂³×C₂²⋊D₁₃	C₂²	1	4	yes	—
12	C₂²×#6127	—	2	32	no	—
13	C₂²×#6127	C₂²	1	48	yes	complemented
14	C₂³×D₅₂	C₂²	1	4	yes	—
15	C₂³×D₅₂	C₂²	1	2	yes	—
16	C₂³×C₄×D₁₃	C₂²	1	4	yes	—
17	C₂³×Dic₂₆	C₂²	1	2	yes	—
18	C₂²×C₄²×C₁₃	C₂²	1	2	yes	—
19	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
21	C₂²×C₄²×C₁₃	C₂²	1	24	yes	complemented
22	C₂×#88184	C₂²	1	96	yes	complemented
23	C₂²×#6103	C₂²	1	8	yes	—
24	C₂²×C₄⋊Dic₁₃	C₂²	1	4	yes	—
25	#11719182	C₂²	1	256	yes	direct product factor
26	C₂×#88129	C₂²	1	48	yes	complemented
27	C₂×#88128	C₂²	1	24	yes	complemented
28	C₂×C₄×D₅₂	C₂²	1	96	yes	complemented
29	C₂×#88123	C₂²	1	24	yes	complemented
30	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
31	C₂⁴×D₁₃	C₂³	1	4	yes	—
32	C₂³×C₄×C₁₃	—	2	16	no	—
33	C₂³×C₄×C₁₃	D₄	1	16	yes	complemented
34	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
35	C₂³×C₄×C₁₃	C₂³	1	24	yes	—
36	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
37	C₂³×C₄×C₁₃	C₂³	1	12	yes	—
38	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
39	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
40	C₂²×C₂²⋊D₁₃	—	2	32	no	—
41	C₂²×C₂²⋊D₁₃	C₂³	1	48	yes	—
42	C₂³×Dic₁₃	C₂³	1	4	yes	—
43	C₂×#6127	—	2	384	no	—
44	C₂×#6127	C₂³	1	128	yes	complemented
45	C₂²×D₅₂	—	2	16	no	—
46	C₂²×D₅₂	—	2	32	no	—
47	C₂²×D₅₂	C₂³	1	48	yes	—
48	C₂²×D₅₂	C₂³	1	24	yes	—
49	C₂²×C₄×D₁₃	—	2	32	no	—
50	C₂²×C₄×D₁₃	C₂³	1	48	yes	—
51	C₂²×Dic₂₆	—	2	16	no	—
52	C₂²×Dic₂₆	C₂³	1	24	yes	—
53	C₂×C₄²×C₁₃	C₂³	1	12	yes	—
54	C₂×C₄²×C₁₃	C₂³	1	24	yes	—
55	C₂×C₄²×C₁₃	C₂³	1	12	yes	—
56	C₂×C₄²×C₁₃	C₂³	1	64	yes	complemented
57	#88184	C₂³	1	256	yes	complemented
58	C₂×#6103	C₂³	1	96	yes	—
59	C₂×C₄⋊Dic₁₃	C₂³	1	48	yes	—
60	#88129	C₂³	1	128	yes	complemented
61	#88128	C₂³	1	64	yes	complemented
62	C₄×D₅₂	C₂³	1	256	yes	complemented
63	#88123	C₂³	1	64	yes	complemented
64	C₂²×#532	—	13	1	no	maximal, Sylow
65	C₂⁴×C₁₃	—	2	8	no	—
66	C₂⁴×C₁₃	C₂×D₄	1	8	yes	—
67	C₂⁴×C₁₃	C₂⁴	1	6	yes	—
68	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
69	C₂³×D₁₃	—	2	16	no	—
70	C₂³×D₁₃	—	2	32	no	—
71	C₂³×D₁₃	C₂⁴	1	24	yes	—
72	C₂²×C₄×C₁₃	—	2	8	no	—
73	C₂²×C₄×C₁₃	—	2	96	no	—
74	C₂²×C₄×C₁₃	C₂×D₄	1	8	yes	—
75	C₂²×C₄×C₁₃	C₂×D₄	1	8	yes	—
76	C₂²×C₄×C₁₃	C₂×D₄	1	96	yes	complemented
77	C₂²×C₄×C₁₃	#41	1	16	yes	complemented
78	C₂²×C₄×C₁₃	C₂⁴	1	12	yes	—
79	C₂²×C₄×C₁₃	C₂⁴	1	32	yes	—
80	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
81	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
82	C₂²×C₄×C₁₃	C₂⁴	1	16	yes	—
83	C₂²×C₄×C₁₃	C₂⁴	1	12	yes	—
84	C₂×C₂²⋊D₁₃	—	2	192	no	—
85	C₂×C₂²⋊D₁₃	C₂⁴	1	64	yes	—
86	C₂²×Dic₁₃	—	2	16	no	—
87	C₂²×Dic₁₃	C₂⁴	1	24	yes	—
88	#6127	—	2	512	no	—
89	C₂×D₅₂	—	2	96	no	—
90	C₂×D₅₂	—	2	192	no	—
91	C₂×D₅₂	C₂⁴	1	64	yes	—
92	C₂×D₅₂	C₂⁴	1	32	yes	—
93	C₂×C₄×D₁₃	—	2	192	no	—
94	C₂×C₄×D₁₃	C₂⁴	1	64	yes	—
95	C₂×Dic₂₆	—	2	96	no	—
96	C₂×Dic₂₆	C₂⁴	1	32	yes	—
97	C₄²×C₁₃	C₂⁴	1	16	yes	—
98	C₄²×C₁₃	C₂⁴	1	32	yes	—
99	C₄²×C₁₃	C₂⁴	1	16	yes	—
100	#6103	C₂⁴	1	128	yes	—
101	C₄⋊Dic₁₃	C₂⁴	1	64	yes	—
102	C₂³×#41	—	13	2	no	—
103	C₂×#532	—	13	48	no	—
104	C₂²×C₄⋊Q₈	—	13	1	no	—
105	C₂²×#127	—	13	1	no	—
106	C₂²×#124	—	13	2	no	—
107	C₂²×#121	—	13	4	no	—
108	C₂²×C₄×D₄	—	13	4	no	—
109	C₂³×C₄²	D₁₃	1	1	yes	characteristic, complemented
110	C₂³×C₁₃	—	2	32	no	—
111	C₂³×C₁₃	—	2	24	no	—
112	C₂³×C₁₃	C₂²×D₄	1	24	yes	—
113	C₂³×C₁₃	C₂²×D₄	1	4	yes	—
114	C₂³×C₁₃	C₂×#41	1	8	yes	—
115	C₂³×C₁₃	C₂⁵	1	4	yes	—
116	C₂³×C₁₃	C₂⁵	1	3	yes	—
117	C₂²×D₁₃	—	2	48	no	—
118	C₂²×D₁₃	—	2	96	no	—
119	C₂²×D₁₃	—	4	64	no	—
120	C₂²×D₁₃	C₂⁵	1	16	yes	—
121	C₂×C₄×C₁₃	—	2	24	no	—
122	C₂×C₄×C₁₃	—	2	64	no	—
123	C₂×C₄×C₁₃	C₂²×D₄	1	24	yes	—
124	C₂×C₄×C₁₃	C₂²×D₄	1	24	yes	—
125	C₂×C₄×C₁₃	C₂²×D₄	1	64	yes	complemented
126	C₂×C₄×C₁₃	C₂×#41	1	48	yes	complemented
127	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
128	C₂×C₄×C₁₃	C₂⁵	1	4	yes	—
129	C₂×C₄×C₁₃	C₂⁵	1	4	yes	—
130	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
131	C₂²⋊D₁₃	—	2	128	no	—
132	C₂×Dic₁₃	—	2	48	no	—
133	C₂×Dic₁₃	C₂⁵	1	16	yes	—
134	D₅₂	—	2	64	no	—
135	D₅₂	—	2	128	no	—
136	C₄×D₁₃	—	2	128	no	—
137	Dic₂₆	—	2	64	no	—
138	C₂²×#41	—	13	48	no	—
139	C₂²×#41	—	26	32	no	—
140	C₂³×Q₈	—	13	2	no	—
141	C₂³×D₄	—	13	4	no	—
142	C₂³×D₄	—	13	2	no	—
143	C₂³×D₄	—	13	4	no	—
144	C₂⁴×C₄	—	13	4	no	—
145	C₂⁴×C₄	C₂×D₁₃	1	2	yes	—
146	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
147	#532	—	13	256	no	—
148	C₂×C₄⋊Q₈	—	13	24	no	—
149	C₂×#127	—	13	24	no	—
150	C₂×#124	—	13	48	no	—
151	C₂×#121	—	13	96	no	—
152	C₂×C₄×D₄	—	13	96	no	—
153	C₂²×C₄⋊C₄	—	13	4	no	—
154	C₂²×C₂²⋊C₄	—	13	8	no	—
155	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
156	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
157	C₂²×C₄²	C₂×D₁₃	1	2	yes	—
158	C₂²×C₄²	C₂×D₁₃	1	24	yes	complemented
159	C₂²×C₁₃	—	2	8	no	—
160	C₂²×C₁₃	—	2	48	no	—
161	C₂²×C₁₃	#532	1	16	yes	complemented
162	C₂²×C₁₃	C₂³×D₄	1	8	yes	—
163	C₂²×C₁₃	C₂³×D₄	1	6	yes	—
164	C₂²×C₁₃	C₂²×#41	1	12	yes	—
165	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
166	C₂×D₁₃	—	2	16	no	—
167	C₂×D₁₃	—	2	32	no	—
168	C₂×D₁₃	—	4	96	no	—
169	C₄×C₁₃	—	2	8	no	—
170	C₄×C₁₃	C₂³×D₄	1	8	yes	—
171	C₄×C₁₃	C₂³×D₄	1	8	yes	—
172	C₄×C₁₃	C₂²×#41	1	16	yes	complemented
173	Dic₁₃	—	2	16	no	—
174	C₂⁵	—	13	4	no	—
175	C₂⁵	C₂²×D₁₃	1	1	yes	characteristic
176	C₂×#41	—	13	128	no	—
177	C₂×#41	—	26	384	no	—
178	C₂²×Q₈	—	13	24	no	—
179	C₂²×Q₈	—	26	16	no	—
180	C₂²×D₄	—	13	48	no	—
181	C₂²×D₄	—	13	48	no	—
182	C₂²×D₄	—	13	24	no	—
183	C₂²×D₄	—	26	32	no	—
184	C₂²×D₄	—	26	16	no	—
185	C₂²×D₄	—	26	32	no	—
186	C₂³×C₄	—	2	16	no	—
187	C₂³×C₄	—	13	4	no	—
188	C₂³×C₄	—	13	48	no	—
189	C₂³×C₄	—	26	32	no	—
190	C₂³×C₄	D₅₂	1	16	yes	complemented
191	C₂³×C₄	C₂²×D₁₃	1	2	yes	—
192	C₂³×C₄	C₂²×D₁₃	1	24	yes	—
193	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
194	C₂³×C₄	C₂²×D₁₃	1	12	yes	—
195	C₂³×C₄	C₂²×D₁₃	1	2	yes	—
196	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
197	C₄⋊Q₈	—	13	64	no	—
198	#127	—	13	64	no	—
199	#124	—	13	128	no	—
200	#121	—	13	256	no	—
201	C₄×D₄	—	13	256	no	—
202	C₂×C₄⋊C₄	—	13	48	no	—
203	C₂×C₂²⋊C₄	—	13	96	no	—
204	C₂×C₄²	C₂²×D₁₃	1	24	yes	—
205	C₂×C₄²	C₂²×D₁₃	1	12	yes	—
206	C₂×C₄²	C₂²×D₁₃	1	12	yes	—
207	C₂×C₄²	C₂²×D₁₃	1	64	yes	complemented
208	C₂×C₁₃	—	2	8	no	—
209	C₂×C₁₃	C₂×#532	1	12	yes	complemented
210	C₂×C₁₃	C₂⁴×D₄	1	1	yes	characteristic
211	C₂×C₁₃	C₂³×#41	1	2	yes	—
212	D₁₃	—	4	16	no	—
213	C₂⁴	—	2	8	no	—
214	C₂⁴	—	13	24	no	—
215	C₂⁴	—	26	16	no	—
216	C₂⁴	—	26	32	no	—
217	C₂⁴	C₂×D₅₂	1	8	yes	—
218	C₂⁴	C₂³×D₁₃	1	6	yes	—
219	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
220	#41	—	26	512	no	—
221	C₂×Q₈	—	13	32	no	—
222	C₂×Q₈	—	26	96	no	—
223	C₂×D₄	—	13	64	no	—
224	C₂×D₄	—	13	64	no	—
225	C₂×D₄	—	13	32	no	—
226	C₂×D₄	—	26	192	no	—
227	C₂×D₄	—	26	96	no	—
228	C₂×D₄	—	26	192	no	—
229	C₂²×C₄	—	2	8	no	—
230	C₂²×C₄	—	2	96	no	—
231	C₂²×C₄	—	13	24	no	—
232	C₂²×C₄	—	13	64	no	—
233	C₂²×C₄	—	26	16	no	—
234	C₂²×C₄	—	26	192	no	—
235	C₂²×C₄	C₂×D₅₂	1	8	yes	—
236	C₂²×C₄	C₂×D₅₂	1	8	yes	—
237	C₂²×C₄	C₂×D₅₂	1	96	yes	complemented
238	C₂²×C₄	#6127	1	16	yes	complemented
239	C₂²×C₄	C₂³×D₁₃	1	12	yes	—
240	C₂²×C₄	C₂³×D₁₃	1	32	yes	—
241	C₂²×C₄	C₂³×D₁₃	1	6	yes	—
242	C₂²×C₄	C₂³×D₁₃	1	6	yes	—
243	C₂²×C₄	C₂³×D₁₃	1	16	yes	—
244	C₂²×C₄	C₂³×D₁₃	1	12	yes	—
245	C₄⋊C₄	—	13	64	no	—
246	C₂²⋊C₄	—	13	128	no	—
247	C₄²	C₂³×D₁₃	1	32	yes	—
248	C₄²	C₂³×D₁₃	1	16	yes	—
249	C₄²	C₂³×D₁₃	1	16	yes	—
250	C₁₃	C₂²×#532	1	1	yes	characteristic, complemented, Sylow
251	C₂³	—	2	32	no	—
252	C₂³	—	2	24	no	—
253	C₂³	—	13	16	no	—
254	C₂³	—	26	48	no	—
255	C₂³	—	26	96	no	—
256	C₂³	—	52	64	no	—
257	C₂³	C₂²×D₅₂	1	24	yes	—
258	C₂³	C₂²×D₅₂	1	4	yes	—
259	C₂³	C₂×#6127	1	8	yes	—
260	C₂³	C₂⁴×D₁₃	1	4	yes	—
261	C₂³	C₂⁴×D₁₃	1	3	yes	—
262	Q₈	—	26	64	no	—
263	D₄	—	26	128	no	—
264	D₄	—	26	64	no	—
265	D₄	—	26	128	no	—
266	C₂×C₄	—	2	24	no	—
267	C₂×C₄	—	2	64	no	—
268	C₂×C₄	—	13	16	no	—
269	C₂×C₄	—	26	48	no	—
270	C₂×C₄	—	26	128	no	—
271	C₂×C₄	C₂²×D₅₂	1	24	yes	—
272	C₂×C₄	C₂²×D₅₂	1	24	yes	—
273	C₂×C₄	C₂²×D₅₂	1	64	yes	complemented
274	C₂×C₄	C₂×#6127	1	48	yes	complemented
275	C₂×C₄	C₂⁴×D₁₃	1	8	yes	—
276	C₂×C₄	C₂⁴×D₁₃	1	4	yes	—
277	C₂×C₄	C₂⁴×D₁₃	1	4	yes	—
278	C₂×C₄	C₂⁴×D₁₃	1	8	yes	—
279	C₂²	—	2	8	no	—
280	C₂²	—	2	48	no	—
281	C₂²	—	26	16	no	—
282	C₂²	—	26	32	no	—
283	C₂²	—	52	96	no	—
284	C₂²	#11719182	1	16	yes	direct product factor
285	C₂²	C₂³×D₅₂	1	8	yes	—
286	C₂²	C₂³×D₅₂	1	6	yes	—
287	C₂²	C₂²×#6127	1	12	yes	—
288	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
289	C₄	—	2	8	no	—
290	C₄	—	26	16	no	—
291	C₄	C₂³×D₅₂	1	8	yes	—
292	C₄	C₂³×D₅₂	1	8	yes	—
293	C₄	C₂²×#6127	1	16	yes	complemented
294	C₂	—	2	8	no	—
295	C₂	—	52	16	no	—
296	C₂	C₂×#11719182	1	12	yes	direct product factor
297	C₂	C₂⁴×D₅₂	1	1	yes	characteristic
298	C₂	C₂³×#6127	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁶
Frattini	C₂²	C₂⁵×D₁₃
Fitting	C₂³×C₄²×C₁₃	C₂
socle	C₂⁴×C₁₃	C₂⁴
hypercenter	C₂³×C₄²	D₁₃
hypocenter	C₁₃	C₂²×#532
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂³×#6127	C₂	semidirect
2. C₂×#11719182	C₂	direct (N × H)
3. C₂²×#88123	C₂	semidirect
4. C₂²×#88128	C₂	semidirect
5. C₂²×#88129	C₂	semidirect
6. C₂²×#88184	C₂	semidirect
7. C₂²×C₄×D₅₂	C₂	semidirect
8. C₂³×C₄²×C₁₃	C₂	semidirect
13. C₂²×#6127	C₂²	semidirect
21. C₂²×C₄²×C₁₃	C₂²	semidirect
22. C₂×#88184	C₂²	semidirect
25. #11719182	C₂²	direct (N × H)
26. C₂×#88129	C₂²	semidirect
27. C₂×#88128	C₂²	semidirect
28. C₂×C₄×D₅₂	C₂²	semidirect
29. C₂×#88123	C₂²	semidirect
33. C₂³×C₄×C₁₃	D₄	semidirect
44. C₂×#6127	C₂³	semidirect
56. C₂×C₄²×C₁₃	C₂³	semidirect
57. #88184	C₂³	semidirect
60. #88129	C₂³	semidirect
61. #88128	C₂³	semidirect
62. C₄×D₅₂	C₂³	semidirect
63. #88123	C₂³	semidirect
76. C₂²×C₄×C₁₃	C₂×D₄	semidirect
77. C₂²×C₄×C₁₃	#41	semidirect
109. C₂³×C₄²	D₁₃	semidirect
125. C₂×C₄×C₁₃	C₂²×D₄	semidirect
126. C₂×C₄×C₁₃	C₂×#41	semidirect
158. C₂²×C₄²	C₂×D₁₃	semidirect
161. C₂²×C₁₃	#532	semidirect
172. C₄×C₁₃	C₂²×#41	semidirect
190. C₂³×C₄	D₅₂	semidirect
207. C₂×C₄²	C₂²×D₁₃	semidirect
209. C₂×C₁₃	C₂×#532	semidirect
237. C₂²×C₄	C₂×D₅₂	semidirect
238. C₂²×C₄	#6127	semidirect
250. C₁₃	C₂²×#532	semidirect
273. C₂×C₄	C₂²×D₅₂	semidirect
274. C₂×C₄	C₂×#6127	semidirect
284. C₂²	#11719182	direct (N × H)
293. C₄	C₂²×#6127	semidirect
296. C₂	C₂×#11719182	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×#6127	2
C₂×#11719182	2
C₂²×#88123	2
C₂²×#88128	2
C₂²×#88129	2
C₂²×#88184	2
C₂²×C₄×D₅₂	2
C₂³×C₄²×C₁₃	2
C₂²×#532	13

Subgroups — C22×#11719182 #16769617

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11719182 #16769617