Subgroups — C₂²×#11718361 #16727211

Order 3328.

2750 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88158	C₂	1	1	yes	maximal, characteristic
2	C₂²×#88155	C₂	1	2	yes	maximal, complemented
3	C₂×#11718361	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#88059	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#88045	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#6127	C₂²	1	2	yes	—
8	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	C₂²×#6099	C₂²	1	2	yes	—
12	C₂²×Dic₅₂	C₄	1	1	yes	characteristic, complemented
13	C₂×#88158	C₄	1	12	yes	complemented
14	C₂×#88158	C₂²	1	12	yes	—
15	C₂²×D₁₀₄	C₄	1	1	yes	characteristic, complemented
16	C₂²×#6095	C₄	1	2	yes	complemented
17	C₂×#88155	—	2	12	no	—
18	C₂×#88155	C₂²	1	24	yes	complemented
19	C₂²×#6094	—	2	2	no	—
20	C₂²×C₈×D₁₃	—	2	2	no	—
21	C₂×C₄×C₈×C₁₃	C₂²	1	12	yes	complemented
22	C₂×#88059	C₂²	1	12	yes	complemented
23	C₂×#88045	C₂²	1	24	yes	complemented
24	#11718361	C₂²	1	64	yes	direct product factor
25	C₂³×C₄×C₁₃	—	2	1	no	—
26	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
27	C₂²×C₂²⋊D₁₃	—	2	2	no	—
28	C₂×#6127	—	2	6	no	—
29	C₂×#6127	—	2	6	no	—
30	C₂×#6127	C₂×C₄	1	12	yes	—
31	C₂×#6127	C₂³	1	12	yes	—
32	C₂²×D₅₂	C₂×C₄	1	2	yes	—
33	C₂²×C₄×D₁₃	—	2	2	no	—
34	C₂²×Dic₂₆	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	C₂×C₄	1	6	yes	—
38	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	D₄	1	6	yes	—
40	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	C₂³	1	6	yes	—
42	C₂²×C₈×C₁₃	C₂³	1	6	yes	—
43	C₂×C₄²×C₁₃	—	2	3	no	—
44	C₂×C₄²×C₁₃	C₂³	1	6	yes	—
45	C₂×#6099	—	2	6	no	—
46	C₂×#6099	C₂³	1	12	yes	—
47	C₂²×#1055	—	2	2	no	—
48	C₂×Dic₅₂	C₂×C₄	1	12	yes	complemented
49	#88158	C₂×C₄	1	48	yes	complemented
50	#88158	C₂³	1	16	yes	—
51	C₂×D₁₀₄	C₂×C₄	1	12	yes	complemented
52	C₂×#6095	C₂×C₄	1	24	yes	complemented
53	#88155	—	2	48	no	—
54	#88155	C₂³	1	32	yes	complemented
55	C₂×#6094	—	2	24	no	—
56	C₂×C₈×D₁₃	—	2	24	no	—
57	C₄×C₈×C₁₃	C₂³	1	16	yes	complemented
58	#88059	C₂³	1	16	yes	complemented
59	#88045	C₂³	1	32	yes	complemented
60	C₂²×#443	—	13	1	no	maximal, Sylow
61	C₂⁴×C₁₃	#41	1	1	yes	characteristic
62	C₂³×D₁₃	—	2	2	no	—
63	C₂²×C₄×C₁₃	—	2	8	no	—
64	C₂²×C₄×C₁₃	—	2	6	no	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
67	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
69	C₂²×C₄×C₁₃	#41	1	3	yes	—
70	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
71	C₂×C₂²⋊D₁₃	—	2	12	no	—
72	C₂×C₂²⋊D₁₃	—	2	12	no	—
73	C₂²×Dic₁₃	—	2	2	no	—
74	#6127	—	2	24	no	—
75	#6127	—	2	12	no	—
76	#6127	—	2	12	no	—
77	#6127	C₂²×C₄	1	24	yes	—
78	#6127	C₂⁴	1	8	yes	—
79	C₂×D₅₂	—	2	6	no	—
80	C₂×D₅₂	C₂²×C₄	1	12	yes	—
81	C₂×C₄×D₁₃	—	2	12	no	—
82	C₂×C₄×D₁₃	—	2	12	no	—
83	C₂×Dic₂₆	—	2	6	no	—
84	C₂×Dic₂₆	C₂²×C₄	1	12	yes	—
85	C₂×C₈×C₁₃	C₂²×C₄	1	6	yes	—
86	C₂×C₈×C₁₃	C₂²×C₄	1	6	yes	—
87	C₂×C₈×C₁₃	C₂²×C₄	1	12	yes	—
88	C₂×C₈×C₁₃	C₂×D₄	1	6	yes	—
89	C₂×C₈×C₁₃	C₂×D₄	1	6	yes	—
90	C₂×C₈×C₁₃	C₂×D₄	1	12	yes	—
91	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
92	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
93	C₄²×C₁₃	—	2	6	no	—
94	C₄²×C₁₃	C₂⁴	1	4	yes	—
95	#6099	—	2	12	no	—
96	#6099	C₂⁴	1	8	yes	—
97	C₂×#1055	—	2	12	no	—
98	Dic₅₂	C₂²×C₄	1	16	yes	complemented
99	D₁₀₄	C₂²×C₄	1	16	yes	complemented
100	#6095	C₂²×C₄	1	32	yes	complemented
101	#6094	—	2	32	no	—
102	C₈×D₁₃	—	2	32	no	—
103	C₂²×#135	—	13	1	no	—
104	C₂²×#131	—	13	2	no	—
105	C₂×#443	—	13	24	no	—
106	C₂²×#108	—	13	1	no	—
107	C₂²×#104	—	13	2	no	—
108	C₂²×C₄×C₈	D₁₃	1	1	yes	characteristic, complemented
109	C₂³×C₁₃	—	2	4	no	—
110	C₂³×C₁₃	C₄×D₄	1	3	yes	—
111	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
112	C₂³×C₁₃	C₂×#41	1	3	yes	—
113	C₂²×D₁₃	—	2	12	no	—
114	C₂²×D₁₃	—	4	8	no	—
115	C₂×C₄×C₁₃	—	2	24	no	—
116	C₂×C₄×C₁₃	—	2	4	no	—
117	C₂×C₄×C₁₃	C₄×D₄	1	3	yes	—
118	C₂×C₄×C₁₃	C₄×D₄	1	3	yes	—
119	C₂×C₄×C₁₃	C₄×D₄	1	6	yes	—
120	C₂×C₄×C₁₃	C₂³×C₄	1	3	yes	—
121	C₂×C₄×C₁₃	C₂³×C₄	1	3	yes	—
122	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
123	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
124	C₂×C₄×C₁₃	C₂×#41	1	3	yes	—
125	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
126	C₂²⋊D₁₃	—	2	24	no	—
127	C₂²⋊D₁₃	—	2	8	no	—
128	C₂×Dic₁₃	—	2	12	no	—
129	D₅₂	—	2	12	no	—
130	D₅₂	C₂³×C₄	1	8	yes	—
131	C₄×D₁₃	—	2	24	no	—
132	C₄×D₁₃	—	2	8	no	—
133	Dic₂₆	—	2	12	no	—
134	Dic₂₆	C₂³×C₄	1	8	yes	—
135	C₈×C₁₃	C₂³×C₄	1	4	yes	—
136	C₈×C₁₃	C₂³×C₄	1	4	yes	—
137	C₈×C₁₃	C₂²×D₄	1	4	yes	—
138	C₈×C₁₃	C₂²×D₄	1	4	yes	—
139	#1055	—	2	8	no	—
140	C₂²×#41	—	13	2	no	—
141	C₂×#135	—	13	12	no	—
142	C₂×#135	—	13	12	no	—
143	C₂²×Q₁₆	—	13	1	no	—
144	C₂²×SD₁₆	—	13	2	no	—
145	C₂²×D₈	—	13	1	no	—
146	C₂×#131	—	13	24	no	—
147	C₂×#131	—	26	12	no	—
148	C₂²×M₁₆	—	13	2	no	—
149	C₂²×M₁₆	—	26	2	no	—
150	C₂³×C₈	—	26	2	no	—
151	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
152	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
153	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
154	#443	—	13	64	no	—
155	C₂×#108	—	13	12	no	—
156	C₂×#104	—	13	24	no	—
157	C₂×C₄×C₈	C₂×D₁₃	1	12	yes	complemented
158	C₂²×C₁₃	—	2	6	no	—
159	C₂²×C₁₃	—	2	6	no	—
160	C₂²×C₁₃	#443	1	4	yes	complemented
161	C₂²×C₁₃	C₂×C₄×D₄	1	3	yes	—
162	C₂²×C₁₃	C₂×C₄×D₄	1	3	yes	—
163	C₂²×C₁₃	C₂²×#41	1	1	yes	characteristic
164	C₂×D₁₃	—	2	8	no	—
165	C₂×D₁₃	—	4	24	no	—
166	C₄×C₁₃	—	2	8	no	—
167	C₄×C₁₃	C₂×C₄×D₄	1	3	yes	—
168	C₄×C₁₃	C₂×C₄×D₄	1	3	yes	—
169	C₄×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
170	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
171	Dic₁₃	—	2	8	no	—
172	C₂×#41	—	13	12	no	—
173	C₂×#41	—	13	12	no	—
174	C₂×#41	—	26	6	no	—
175	C₂×#41	—	26	6	no	—
176	C₂²×Q₈	—	13	2	no	—
177	C₂²×D₄	—	13	2	no	—
178	C₂²×D₄	—	26	2	no	—
179	C₂³×C₄	—	2	1	no	—
180	C₂³×C₄	—	26	2	no	—
181	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
182	#135	—	13	48	no	—
183	#135	—	13	16	no	—
184	C₂×Q₁₆	—	13	12	no	—
185	C₂×SD₁₆	—	13	24	no	—
186	C₂×D₈	—	13	12	no	—
187	#131	—	13	32	no	—
188	#131	—	26	48	no	—
189	C₂×M₁₆	—	13	12	no	—
190	C₂×M₁₆	—	26	6	no	—
191	C₂×M₁₆	—	26	24	no	—
192	C₂²×C₈	—	26	2	no	—
193	C₂²×C₈	—	26	24	no	—
194	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
195	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
196	C₂²×C₈	C₄×D₁₃	1	6	yes	—
197	C₂²×C₈	D₅₂	1	1	yes	characteristic
198	C₂²×C₈	D₅₂	1	6	yes	—
199	C₂²×C₈	D₅₂	1	1	yes	characteristic
200	C₂²×C₈	C₂²×D₁₃	1	6	yes	—
201	C₂²×C₈	C₂²×D₁₃	1	6	yes	—
202	C₂×C₄²	—	2	3	no	—
203	C₂×C₄²	C₂²×D₁₃	1	6	yes	—
204	#108	—	13	16	no	—
205	#104	—	13	32	no	—
206	C₄×C₈	C₂²×D₁₃	1	16	yes	complemented
207	C₂×C₁₃	—	2	1	no	—
208	C₂×C₁₃	—	2	3	no	—
209	C₂×C₁₃	C₂×#443	1	6	yes	complemented
210	C₂×C₁₃	C₂²×C₄×D₄	1	1	yes	characteristic
211	D₁₃	—	4	8	no	—
212	C₂⁴	—	26	2	no	—
213	C₂⁴	#6127	1	1	yes	characteristic
214	#41	—	13	24	no	—
215	#41	—	13	8	no	—
216	#41	—	26	24	no	—
217	#41	—	26	12	no	—
218	#41	—	26	12	no	—
219	C₂×Q₈	—	13	12	no	—
220	C₂×Q₈	—	26	6	no	—
221	C₂×D₄	—	13	12	no	—
222	C₂×D₄	—	26	12	no	—
223	C₂×D₄	—	26	12	no	—
224	C₂×D₄	—	26	6	no	—
225	C₂²×C₄	—	2	8	no	—
226	C₂²×C₄	—	2	6	no	—
227	C₂²×C₄	—	26	2	no	—
228	C₂²×C₄	—	26	12	no	—
229	C₂²×C₄	—	26	12	no	—
230	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₄×D₁₃	1	3	yes	—
232	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₅₂	1	3	yes	—
234	C₂²×C₄	#6127	1	3	yes	—
235	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
236	Q₁₆	—	13	16	no	—
237	SD₁₆	—	13	32	no	—
238	D₈	—	13	16	no	—
239	M₁₆	—	13	8	no	—
240	M₁₆	—	26	12	no	—
241	M₁₆	—	26	32	no	—
242	C₂×C₈	—	26	12	no	—
243	C₂×C₈	—	26	32	no	—
244	C₂×C₈	C₂×C₄×D₁₃	1	6	yes	—
245	C₂×C₈	C₂×C₄×D₁₃	1	6	yes	—
246	C₂×C₈	C₂×C₄×D₁₃	1	12	yes	—
247	C₂×C₈	C₂×D₅₂	1	6	yes	—
248	C₂×C₈	C₂×D₅₂	1	6	yes	—
249	C₂×C₈	C₂×D₅₂	1	12	yes	—
250	C₂×C₈	C₂³×D₁₃	1	4	yes	—
251	C₂×C₈	C₂³×D₁₃	1	4	yes	—
252	C₄²	—	2	6	no	—
253	C₄²	C₂³×D₁₃	1	4	yes	—
254	C₁₃	C₂²×#443	1	1	yes	characteristic, complemented, Sylow
255	C₂³	—	2	4	no	—
256	C₂³	—	26	12	no	—
257	C₂³	—	52	8	no	—
258	C₂³	C₄×D₅₂	1	3	yes	—
259	C₂³	C₄×D₅₂	1	1	yes	characteristic
260	C₂³	C₂×#6127	1	3	yes	—
261	Q₈	—	13	8	no	—
262	Q₈	—	26	12	no	—
263	D₄	—	13	8	no	—
264	D₄	—	26	24	no	—
265	D₄	—	26	8	no	—
266	D₄	—	26	12	no	—
267	C₂×C₄	—	2	24	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	26	12	no	—
270	C₂×C₄	—	26	24	no	—
271	C₂×C₄	—	26	8	no	—
272	C₂×C₄	C₄×D₅₂	1	3	yes	—
273	C₂×C₄	C₄×D₅₂	1	3	yes	—
274	C₂×C₄	C₄×D₅₂	1	6	yes	—
275	C₂×C₄	C₂²×C₄×D₁₃	1	3	yes	—
276	C₂×C₄	C₂²×C₄×D₁₃	1	3	yes	—
277	C₂×C₄	C₂²×D₅₂	1	3	yes	—
278	C₂×C₄	C₂²×D₅₂	1	3	yes	—
279	C₂×C₄	C₂×#6127	1	3	yes	—
280	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
281	C₈	—	26	8	no	—
282	C₈	C₂²×C₄×D₁₃	1	4	yes	—
283	C₈	C₂²×C₄×D₁₃	1	4	yes	—
284	C₈	C₂²×D₅₂	1	4	yes	—
285	C₈	C₂²×D₅₂	1	4	yes	—
286	C₂²	—	2	6	no	—
287	C₂²	—	2	6	no	—
288	C₂²	—	26	8	no	—
289	C₂²	—	52	24	no	—
290	C₂²	#11718361	1	4	yes	direct product factor
291	C₂²	C₂×C₄×D₅₂	1	3	yes	—
292	C₂²	C₂×C₄×D₅₂	1	3	yes	—
293	C₂²	C₂²×#6127	1	1	yes	characteristic
294	C₄	—	2	8	no	—
295	C₄	—	26	8	no	—
296	C₄	C₂×C₄×D₅₂	1	3	yes	—
297	C₄	C₂×C₄×D₅₂	1	3	yes	—
298	C₄	C₂³×C₄×D₁₃	1	1	yes	characteristic
299	C₄	C₂³×D₅₂	1	1	yes	characteristic
300	C₂	—	2	1	no	—
301	C₂	—	2	3	no	—
302	C₂	—	52	8	no	—
303	C₂	C₂×#11718361	1	6	yes	direct product factor
304	C₂	C₂²×C₄×D₅₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	D₅₂
commutator	C₄×C₁₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂²×C₄×C₈×C₁₃	C₂
socle	C₂³×C₁₃	C₄×D₄
prime radical	C₂²×#88158	C₂
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂²×C₄×C₈	D₁₃
hypocenter	C₁₃	C₂²×#443
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×#88155	C₂	semidirect
3. C₂×#11718361	C₂	direct (N × H)
4. C₂²×#88059	C₂	semidirect
5. C₂²×#88045	C₂	semidirect
6. C₂²×C₄×C₈×C₁₃	C₂	semidirect
12. C₂²×Dic₅₂	C₄	semidirect
13. C₂×#88158	C₄	semidirect
15. C₂²×D₁₀₄	C₄	semidirect
16. C₂²×#6095	C₄	semidirect
18. C₂×#88155	C₂²	semidirect
21. C₂×C₄×C₈×C₁₃	C₂²	semidirect
22. C₂×#88059	C₂²	semidirect
23. C₂×#88045	C₂²	semidirect
24. #11718361	C₂²	direct (N × H)
48. C₂×Dic₅₂	C₂×C₄	semidirect
49. #88158	C₂×C₄	semidirect
51. C₂×D₁₀₄	C₂×C₄	semidirect
52. C₂×#6095	C₂×C₄	semidirect
54. #88155	C₂³	semidirect
57. C₄×C₈×C₁₃	C₂³	semidirect
58. #88059	C₂³	semidirect
59. #88045	C₂³	semidirect
98. Dic₅₂	C₂²×C₄	semidirect
99. D₁₀₄	C₂²×C₄	semidirect
100. #6095	C₂²×C₄	semidirect
108. C₂²×C₄×C₈	D₁₃	semidirect
157. C₂×C₄×C₈	C₂×D₁₃	semidirect
160. C₂²×C₁₃	#443	semidirect
206. C₄×C₈	C₂²×D₁₃	semidirect
209. C₂×C₁₃	C₂×#443	semidirect
254. C₁₃	C₂²×#443	semidirect
290. C₂²	#11718361	direct (N × H)
303. C₂	C₂×#11718361	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×#88158	2
C₂²×#88155	2
C₂×#11718361	2
C₂²×#88059	2
C₂²×#88045	2
C₂²×C₄×C₈×C₁₃	2
C₂²×#443	13

Subgroups — C22×#11718361 #16727211

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718361 #16727211