Subgroups — C₂²×#11718744 #16729698

Order 3328.

3222 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#6106	C₂	1	2	yes	maximal
2	C₂²×C₄×D₅₂	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718744	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#88075	C₂	1	2	yes	maximal, complemented
6	C₂²×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×Q₈×C₁₃	C₂²	1	2	yes	—
8	C₂³×D₅₂	C₂²	1	1	yes	characteristic
9	C₂³×C₄×D₁₃	—	2	1	no	—
10	C₂×C₁₃×C₄⋊Q₈	—	2	6	no	—
11	C₂×C₁₃×C₄⋊Q₈	C₂²	1	12	yes	complemented
12	C₂²×C₁₃×C₄⋊C₄	—	2	1	no	—
13	C₂²×C₁₃×C₄⋊C₄	—	2	1	no	—
14	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
15	C₂²×#6106	—	2	16	no	—
16	C₂²×#6106	C₂²	1	24	yes	—
17	C₂²×#6103	—	2	1	no	—
18	C₂²×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
19	C₂³×#1055	C₂²	1	2	yes	—
20	C₂×C₄×D₅₂	—	2	6	no	—
21	C₂×C₄×D₅₂	C₂²	1	12	yes	—
22	C₂×#88075	C₂²	1	24	yes	complemented
23	#11718744	C₂²	1	64	yes	direct product factor
24	C₂×C₄⋊#1055	C₂²	1	12	yes	complemented
25	C₂⁴×D₁₃	—	2	1	no	—
26	C₂²×Q₈×C₁₃	—	2	6	no	—
27	C₂²×Q₈×C₁₃	—	2	8	no	—
28	C₂²×Q₈×C₁₃	—	2	8	no	—
29	C₂²×Q₈×C₁₃	C₂³	1	12	yes	—
30	C₂³×C₄×C₁₃	—	2	2	no	—
31	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
32	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
33	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
34	C₂³×Dic₁₃	—	2	1	no	—
35	C₂²×D₅₂	—	2	3	no	—
36	C₂²×D₅₂	—	2	4	no	—
37	C₂²×D₅₂	D₄	1	8	yes	—
38	C₂²×D₅₂	C₂³	1	6	yes	—
39	C₂²×C₄×D₁₃	—	2	12	no	—
40	C₂²×C₄×D₁₃	—	4	8	no	—
41	C₁₃×C₄⋊Q₈	—	2	24	no	—
42	C₁₃×C₄⋊Q₈	C₂³	1	16	yes	complemented
43	C₂×C₁₃×C₄⋊C₄	—	2	12	no	—
44	C₂×C₁₃×C₄⋊C₄	—	2	12	no	—
45	C₂×C₄²×C₁₃	D₄	1	6	yes	—
46	C₂×C₄²×C₁₃	C₂³	1	6	yes	—
47	C₂×#6106	—	2	96	no	—
48	C₂×#6106	C₂³	1	32	yes	—
49	C₂×#6103	—	2	12	no	—
50	C₂×C₄⋊Dic₁₃	—	2	3	no	—
51	C₂×C₄⋊Dic₁₃	C₂³	1	6	yes	—
52	C₂²×#1055	—	2	8	no	—
53	C₂²×#1055	C₂³	1	12	yes	—
54	C₄×D₅₂	—	2	24	no	—
55	C₄×D₅₂	C₂³	1	16	yes	—
56	#88075	C₂³	1	32	yes	complemented
57	C₄⋊#1055	C₂³	1	16	yes	complemented
58	C₂²×#461	—	13	1	no	maximal, Sylow
59	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂³×D₁₃	—	2	8	no	—
61	C₂³×D₁₃	—	2	6	no	—
62	C₂³×D₁₃	—	4	4	no	—
63	C₂³×D₁₃	—	4	4	no	—
64	C₂×Q₈×C₁₃	—	2	12	no	—
65	C₂×Q₈×C₁₃	—	2	24	no	—
66	C₂×Q₈×C₁₃	—	2	24	no	—
67	C₂×Q₈×C₁₃	—	4	24	no	—
68	C₂×Q₈×C₁₃	C₂⁴	1	8	yes	—
69	C₂²×C₄×C₁₃	—	2	4	no	—
70	C₂²×C₄×C₁₃	—	2	12	no	—
71	C₂²×C₄×C₁₃	—	4	8	no	—
72	C₂²×C₄×C₁₃	SD₁₆	1	8	yes	complemented
73	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
74	C₂²×C₄×C₁₃	C₂×D₄	1	6	yes	—
75	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
76	C₂²×C₄×C₁₃	C₂×D₄	1	6	yes	—
77	C₂²×C₄×C₁₃	#41	1	4	yes	—
78	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
79	C₂²×Dic₁₃	—	2	6	no	—
80	C₂²×Dic₁₃	—	4	4	no	—
81	C₂×D₅₂	—	2	24	no	—
82	C₂×D₅₂	—	2	6	no	—
83	C₂×D₅₂	—	2	12	no	—
84	C₂×D₅₂	C₂×D₄	1	24	yes	—
85	C₂×D₅₂	C₂⁴	1	4	yes	—
86	C₂×C₄×D₁₃	—	2	16	no	—
87	C₂×C₄×D₁₃	—	4	48	no	—
88	C₁₃×C₄⋊C₄	—	2	16	no	—
89	C₁₃×C₄⋊C₄	—	2	16	no	—
90	C₄²×C₁₃	C₂×D₄	1	12	yes	—
91	C₄²×C₁₃	C₂⁴	1	4	yes	—
92	#6106	—	2	64	no	—
93	#6103	—	2	16	no	—
94	C₄⋊Dic₁₃	—	2	6	no	—
95	C₄⋊Dic₁₃	C₂⁴	1	4	yes	—
96	C₂×#1055	—	2	24	no	—
97	C₂×#1055	C₂⁴	1	8	yes	—
98	C₂³×SD₁₆	—	13	2	no	—
99	C₂²×C₄⋊Q₈	D₁₃	1	1	yes	characteristic, complemented
100	C₂²×C₄×D₄	—	13	1	no	—
101	C₂×#461	—	13	24	no	—
102	C₂²×C₄⋊C₈	—	13	1	no	—
103	C₂²×#103	—	13	2	no	—
104	C₂³×C₁₃	#121	1	4	yes	—
105	C₂³×C₁₃	C₂×SD₁₆	1	4	yes	—
106	C₂³×C₁₃	#137	1	4	yes	—
107	C₂³×C₁₃	C₂²×D₄	1	3	yes	—
108	C₂²×D₁₃	—	2	24	no	—
109	C₂²×D₁₃	—	2	4	no	—
110	C₂²×D₁₃	—	4	12	no	—
111	C₂²×D₁₃	—	4	12	no	—
112	C₂²×D₁₃	—	4	32	no	—
113	Q₈×C₁₃	—	2	8	no	—
114	Q₈×C₁₃	—	2	8	no	—
115	Q₈×C₁₃	—	4	24	no	—
116	C₂×C₄×C₁₃	—	2	12	no	—
117	C₂×C₄×C₁₃	—	2	8	no	—
118	C₂×C₄×C₁₃	—	4	24	no	—
119	C₂×C₄×C₁₃	#121	1	12	yes	—
120	C₂×C₄×C₁₃	C₂×SD₁₆	1	24	yes	complemented
121	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
122	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
123	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
124	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
125	C₂×C₄×C₁₃	C₂×#41	1	6	yes	—
126	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
127	C₂×Dic₁₃	—	2	4	no	—
128	C₂×Dic₁₃	—	4	12	no	—
129	D₅₂	—	2	24	no	—
130	D₅₂	—	2	4	no	—
131	D₅₂	C₂²×D₄	1	8	yes	—
132	C₄×D₁₃	—	4	32	no	—
133	#1055	—	2	8	no	—
134	C₂³×Q₈	C₂×D₁₃	1	2	yes	—
135	C₂³×D₄	—	13	1	no	—
136	C₂⁴×C₄	—	26	1	no	—
137	C₂²×SD₁₆	—	13	24	no	—
138	C₂²×SD₁₆	—	26	16	no	—
139	C₂³×C₈	—	13	2	no	—
140	C₂×C₄⋊Q₈	—	2	6	no	—
141	C₂×C₄⋊Q₈	C₂×D₁₃	1	12	yes	complemented
142	C₂×C₄×D₄	—	13	12	no	—
143	C₂×C₄×D₄	—	26	6	no	—
144	C₂²×C₄⋊C₄	—	2	1	no	—
145	C₂²×C₄⋊C₄	—	2	1	no	—
146	C₂²×C₄⋊C₄	—	13	1	no	—
147	C₂²×C₂²⋊C₄	—	26	1	no	—
148	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
149	#461	—	13	64	no	—
150	C₂×C₄⋊C₈	—	13	12	no	—
151	C₂×#103	—	13	24	no	—
152	C₂²×C₁₃	#461	1	16	yes	complemented
153	C₂²×C₁₃	C₂×#121	1	6	yes	—
154	C₂²×C₁₃	C₂²×SD₁₆	1	6	yes	—
155	C₂²×C₁₃	C₂×#137	1	6	yes	—
156	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
157	C₂×D₁₃	—	2	8	no	—
158	C₂×D₁₃	—	4	48	no	—
159	C₂×D₁₃	—	4	4	no	—
160	C₂×D₁₃	—	4	4	no	—
161	C₄×C₁₃	—	2	4	no	—
162	C₄×C₁₃	—	4	8	no	—
163	C₄×C₁₃	C₂×#121	1	6	yes	—
164	C₄×C₁₃	C₂²×SD₁₆	1	8	yes	complemented
165	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
166	C₄×C₁₃	C₂²×#41	1	1	yes	characteristic
167	Dic₁₃	—	4	4	no	—
168	C₂⁵	—	26	1	no	—
169	C₂²×Q₈	—	2	6	no	—
170	C₂²×Q₈	—	2	8	no	—
171	C₂²×Q₈	—	2	8	no	—
172	C₂²×Q₈	C₂²×D₁₃	1	12	yes	—
173	C₂²×D₄	—	13	8	no	—
174	C₂²×D₄	—	13	6	no	—
175	C₂²×D₄	—	26	4	no	—
176	C₂²×D₄	—	26	3	no	—
177	C₂³×C₄	—	2	2	no	—
178	C₂³×C₄	—	26	1	no	—
179	C₂³×C₄	—	26	12	no	—
180	C₂³×C₄	—	52	8	no	—
181	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
182	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
183	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
184	C₂×SD₁₆	—	13	32	no	—
185	C₂×SD₁₆	—	26	96	no	—
186	C₂²×C₈	—	13	12	no	—
187	C₂²×C₈	—	26	8	no	—
188	C₄⋊Q₈	—	2	24	no	—
189	C₄⋊Q₈	C₂²×D₁₃	1	16	yes	complemented
190	C₄×D₄	—	13	16	no	—
191	C₄×D₄	—	26	24	no	—
192	C₂×C₄⋊C₄	—	2	12	no	—
193	C₂×C₄⋊C₄	—	2	12	no	—
194	C₂×C₄⋊C₄	—	13	6	no	—
195	C₂×C₄⋊C₄	—	26	3	no	—
196	C₂×C₂²⋊C₄	—	26	12	no	—
197	C₂×C₄²	C₂²⋊D₁₃	1	6	yes	—
198	C₂×C₄²	C₂²×D₁₃	1	6	yes	—
199	C₄⋊C₈	—	13	16	no	—
200	#103	—	13	32	no	—
201	C₂×C₁₃	C₂×#461	1	12	yes	complemented
202	C₂×C₁₃	C₂²×#121	1	1	yes	characteristic
203	C₂×C₁₃	C₂³×SD₁₆	1	1	yes	characteristic
204	C₂×C₁₃	C₂²×#137	1	1	yes	characteristic
205	D₁₃	—	4	8	no	—
206	C₂⁴	—	26	8	no	—
207	C₂⁴	—	26	6	no	—
208	C₂⁴	—	52	4	no	—
209	C₂⁴	—	52	4	no	—
210	C₂⁴	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
211	C₂×Q₈	—	2	12	no	—
212	C₂×Q₈	—	2	24	no	—
213	C₂×Q₈	—	2	24	no	—
214	C₂×Q₈	—	4	24	no	—
215	C₂×Q₈	C₂³×D₁₃	1	8	yes	—
216	C₂×D₄	—	13	24	no	—
217	C₂×D₄	—	13	4	no	—
218	C₂×D₄	—	26	24	no	—
219	C₂×D₄	—	26	12	no	—
220	C₂×D₄	—	26	6	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	2	12	no	—
223	C₂²×C₄	—	4	8	no	—
224	C₂²×C₄	—	26	6	no	—
225	C₂²×C₄	—	26	16	no	—
226	C₂²×C₄	—	52	4	no	—
227	C₂²×C₄	—	52	48	no	—
228	C₂²×C₄	#6106	1	8	yes	complemented
229	C₂²×C₄	D₄×D₁₃	1	4	yes	—
230	C₂²×C₄	#6129	1	4	yes	—
231	C₂²×C₄	C₂×C₂²⋊D₁₃	1	6	yes	—
232	C₂²×C₄	C₂×C₂²⋊D₁₃	1	3	yes	—
233	C₂²×C₄	C₂×C₂²⋊D₁₃	1	6	yes	—
234	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
235	SD₁₆	—	26	64	no	—
236	C₂×C₈	—	13	8	no	—
237	C₂×C₈	—	26	24	no	—
238	C₄⋊C₄	—	2	16	no	—
239	C₄⋊C₄	—	2	16	no	—
240	C₄⋊C₄	—	13	4	no	—
241	C₄⋊C₄	—	26	6	no	—
242	C₂²⋊C₄	—	26	16	no	—
243	C₄²	C₂×C₂²⋊D₁₃	1	12	yes	—
244	C₄²	C₂³×D₁₃	1	4	yes	—
245	C₁₃	C₂²×#461	1	1	yes	characteristic, complemented, Sylow
246	C₂³	—	26	24	no	—
247	C₂³	—	26	4	no	—
248	C₂³	—	52	32	no	—
249	C₂³	—	52	12	no	—
250	C₂³	—	52	12	no	—
251	C₂³	#88192	1	4	yes	—
252	C₂³	C₂×#6106	1	4	yes	—
253	C₂³	#88196	1	4	yes	—
254	C₂³	C₂²×C₂²⋊D₁₃	1	3	yes	—
255	Q₈	—	2	8	no	—
256	Q₈	—	2	8	no	—
257	Q₈	—	4	24	no	—
258	D₄	—	13	8	no	—
259	D₄	—	26	24	no	—
260	D₄	—	26	4	no	—
261	C₂×C₄	—	2	12	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	4	24	no	—
264	C₂×C₄	—	26	4	no	—
265	C₂×C₄	—	52	12	no	—
266	C₂×C₄	—	52	32	no	—
267	C₂×C₄	#88192	1	12	yes	—
268	C₂×C₄	C₂×#6106	1	24	yes	complemented
269	C₂×C₄	C₂×D₄×D₁₃	1	6	yes	—
270	C₂×C₄	C₂×#6129	1	6	yes	—
271	C₂×C₄	C₂²×C₂²⋊D₁₃	1	4	yes	—
272	C₂×C₄	C₂²×C₂²⋊D₁₃	1	3	yes	—
273	C₂×C₄	C₂²×C₂²⋊D₁₃	1	4	yes	—
274	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
275	C₈	—	26	8	no	—
276	C₂²	—	26	8	no	—
277	C₂²	—	52	4	no	—
278	C₂²	—	52	4	no	—
279	C₂²	—	52	48	no	—
280	C₂²	#11718744	1	16	yes	direct product factor
281	C₂²	C₂×#88192	1	6	yes	—
282	C₂²	C₂²×#6106	1	6	yes	—
283	C₂²	C₂×#88196	1	6	yes	—
284	C₂²	C₂³×C₂²⋊D₁₃	1	1	yes	characteristic
285	C₄	—	2	4	no	—
286	C₄	—	4	8	no	—
287	C₄	—	52	4	no	—
288	C₄	C₂×#88192	1	6	yes	—
289	C₄	C₂²×#6106	1	8	yes	complemented
290	C₄	C₂²×D₄×D₁₃	1	1	yes	characteristic
291	C₄	C₂²×#6129	1	1	yes	characteristic
292	C₂	—	52	8	no	—
293	C₂	C₂×#11718744	1	12	yes	direct product factor
294	C₂	C₂²×#88192	1	1	yes	characteristic
295	C₂	C₂³×#6106	1	1	yes	characteristic
296	C₂	C₂²×#88196	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂²×C₁₃×C₄⋊Q₈	C₂
socle	C₂⁴×C₁₃	C₂×D₄
prime radical	C₂³×D₅₂	C₂²
hypercenter	C₂²×C₄⋊Q₈	D₁₃
hypocenter	C₁₃	C₂²×#461
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
3. C₂×#11718744	C₂	direct (N × H)
4. C₂²×C₄⋊#1055	C₂	semidirect
5. C₂²×#88075	C₂	semidirect
6. C₂²×C₁₃×C₄⋊Q₈	C₂	semidirect
11. C₂×C₁₃×C₄⋊Q₈	C₂²	semidirect
22. C₂×#88075	C₂²	semidirect
23. #11718744	C₂²	direct (N × H)
24. C₂×C₄⋊#1055	C₂²	semidirect
42. C₁₃×C₄⋊Q₈	C₂³	semidirect
56. #88075	C₂³	semidirect
57. C₄⋊#1055	C₂³	semidirect
72. C₂²×C₄×C₁₃	SD₁₆	semidirect
99. C₂²×C₄⋊Q₈	D₁₃	semidirect
120. C₂×C₄×C₁₃	C₂×SD₁₆	semidirect
141. C₂×C₄⋊Q₈	C₂×D₁₃	semidirect
152. C₂²×C₁₃	#461	semidirect
164. C₄×C₁₃	C₂²×SD₁₆	semidirect
189. C₄⋊Q₈	C₂²×D₁₃	semidirect
201. C₂×C₁₃	C₂×#461	semidirect
228. C₂²×C₄	#6106	semidirect
245. C₁₃	C₂²×#461	semidirect
268. C₂×C₄	C₂×#6106	semidirect
280. C₂²	#11718744	direct (N × H)
289. C₄	C₂²×#6106	semidirect
293. C₂	C₂×#11718744	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×#6106	2
C₂²×C₄×D₅₂	2
C₂×#11718744	2
C₂²×C₄⋊#1055	2
C₂²×#88075	2
C₂²×C₁₃×C₄⋊Q₈	2
C₂²×#461	13

Subgroups — C22×#11718744 #16729698

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718744 #16729698