Subgroups — C₂²×#11719388 #16769791

Order 3328.

10246 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	C₂×#11719388	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#88201	C₂	1	2	yes	maximal, complemented
3	C₂²×#88129	C₂	1	2	yes	maximal, complemented
4	C₂²×#88139	C₂	1	4	yes	maximal, complemented
5	C₂²×#88136	C₂	1	4	yes	maximal, complemented
6	C₂²×#88191	C₂	1	2	yes	maximal, complemented
7	C₂²×C₁₃×#124	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂⁵×D₁₃	C₂²	1	2	yes	complemented
9	C₂³×Q₈×C₁₃	C₂²	1	1	yes	characteristic, complemented
10	C₂³×D₄×C₁₃	C₂²	1	1	yes	characteristic
11	C₂³×C₂²⋊D₁₃	C₂²	1	4	yes	—
12	C₂³×D₅₂	C₂²	1	2	yes	—
13	C₂³×D₅₂	C₂²	1	2	yes	—
14	C₂³×Dic₂₆	C₂²	1	2	yes	complemented
15	C₂×C₁₃×#124	C₂²	1	24	yes	complemented
16	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	4	yes	complemented
17	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic, complemented
18	C₂²×C₂²⋊Dic₁₃	C₂²	1	2	yes	complemented
19	C₂×#88201	C₂²	1	48	yes	complemented
20	C₂×#88191	C₂²	1	48	yes	complemented
21	C₂²×#6103	C₂²	1	4	yes	complemented
22	C₂²×#6103	C₂²	1	4	yes	complemented
23	C₂²×#6103	C₂²	1	4	yes	complemented
24	C₂²×C₄×Dic₁₃	C₂²	1	2	yes	complemented
25	#11719388	C₂²	1	256	yes	direct product factor
26	C₂×#88139	C₂²	1	96	yes	complemented
27	C₂×#88136	C₂²	1	96	yes	complemented
28	C₂×#88129	C₂²	1	48	yes	complemented
29	C₂⁵×C₁₃	C₂³	1	2	yes	—
30	C₂⁴×D₁₃	—	4	8	no	—
31	C₂⁴×D₁₃	—	4	16	no	—
32	C₂⁴×D₁₃	C₂³	1	4	yes	—
33	C₂⁴×D₁₃	C₂³	1	24	yes	complemented
34	C₂²×Q₈×C₁₃	—	4	4	no	—
35	C₂²×Q₈×C₁₃	C₂³	1	12	yes	complemented
36	C₂²×D₄×C₁₃	—	4	4	no	—
37	C₂²×D₄×C₁₃	C₂³	1	12	yes	—
38	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
39	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
40	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
41	C₂²×C₂²⋊D₁₃	—	4	16	no	—
42	C₂²×C₂²⋊D₁₃	C₂³	1	48	yes	—
43	C₂³×Dic₁₃	C₂³	1	4	yes	—
44	C₂²×D₅₂	—	4	8	no	—
45	C₂²×D₅₂	—	4	8	no	—
46	C₂²×D₅₂	C₂³	1	24	yes	—
47	C₂²×D₅₂	C₂³	1	24	yes	—
48	C₂²×Dic₂₆	—	4	8	no	—
49	C₂²×Dic₂₆	C₂³	1	24	yes	complemented
50	C₁₃×#124	C₂³	1	64	yes	complemented
51	C₂×C₁₃×C₂²⋊C₄	C₂³	1	48	yes	complemented
52	C₂×C₄²×C₁₃	C₂³	1	12	yes	complemented
53	C₂×C₂²⋊Dic₁₃	C₂³	1	24	yes	complemented
54	#88201	C₂³	1	128	yes	complemented
55	#88191	C₂³	1	128	yes	complemented
56	C₂×#6103	C₂³	1	48	yes	complemented
57	C₂×#6103	C₂³	1	48	yes	complemented
58	C₂×#6103	C₂³	1	48	yes	complemented
59	C₂×C₄×Dic₁₃	C₂³	1	24	yes	complemented
60	#88139	C₂³	1	256	yes	complemented
61	#88136	C₂³	1	256	yes	complemented
62	#88129	C₂³	1	128	yes	complemented
63	C₂²×H(3,𝔽₄)	—	13	1	no	maximal, Sylow
64	C₂⁴×C₁₃	—	2	16	no	—
65	C₂⁴×C₁₃	—	2	8	no	—
66	C₂⁴×C₁₃	C₂⁴	1	12	yes	—
67	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
68	C₂³×D₁₃	—	2	16	no	—
69	C₂³×D₁₃	—	2	16	no	—
70	C₂³×D₁₃	—	2	16	no	—
71	C₂³×D₁₃	—	4	48	no	—
72	C₂³×D₁₃	—	4	96	no	—
73	C₂³×D₁₃	—	4	128	no	—
74	C₂³×D₁₃	C₂⁴	1	24	yes	—
75	C₂³×D₁₃	C₂⁴	1	32	yes	complemented
76	C₂×Q₈×C₁₃	—	4	24	no	—
77	C₂×Q₈×C₁₃	C₂⁴	1	16	yes	complemented
78	C₂×D₄×C₁₃	—	4	24	no	—
79	C₂×D₄×C₁₃	C₂⁴	1	16	yes	—
80	C₂²×C₄×C₁₃	—	2	8	no	—
81	C₂²×C₄×C₁₃	—	2	8	no	—
82	C₂²×C₄×C₁₃	—	2	4	no	—
83	C₂²×C₄×C₁₃	C₂⁴	1	12	yes	—
84	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
85	C₂²×C₄×C₁₃	C₂⁴	1	12	yes	—
86	C₂×C₂²⋊D₁₃	—	4	96	no	—
87	C₂×C₂²⋊D₁₃	C₂⁴	1	64	yes	—
88	C₂²×Dic₁₃	—	2	16	no	—
89	C₂²×Dic₁₃	C₂⁴	1	24	yes	—
90	C₂×D₅₂	—	4	48	no	—
91	C₂×D₅₂	—	4	48	no	—
92	C₂×D₅₂	C₂⁴	1	32	yes	—
93	C₂×D₅₂	C₂⁴	1	32	yes	—
94	C₂×Dic₂₆	—	4	48	no	—
95	C₂×Dic₂₆	C₂⁴	1	32	yes	complemented
96	C₁₃×C₂²⋊C₄	C₂⁴	1	64	yes	complemented
97	C₄²×C₁₃	C₂⁴	1	16	yes	complemented
98	C₂²⋊Dic₁₃	C₂⁴	1	32	yes	complemented
99	#6103	C₂⁴	1	64	yes	complemented
100	#6103	C₂⁴	1	64	yes	complemented
101	#6103	C₂⁴	1	64	yes	complemented
102	C₄×Dic₁₃	C₂⁴	1	32	yes	complemented
103	C₂×H(3,𝔽₄)	—	13	48	no	—
104	C₂²×#124	—	13	2	no	—
105	C₂²×#124	—	13	2	no	—
106	C₂²×#124	—	13	4	no	—
107	C₂²×#124	D₁₃	1	1	yes	characteristic, complemented
108	C₂²×#120	—	13	2	no	—
109	C₂²×#120	—	13	4	no	—
110	C₂³×C₁₃	—	2	48	no	—
111	C₂³×C₁₃	—	2	24	no	—
112	C₂³×C₁₃	—	4	32	no	—
113	C₂³×C₁₃	H(5,𝔽₂)	1	4	yes	—
114	C₂³×C₁₃	H(5,𝔽₂)	1	8	yes	—
115	C₂³×C₁₃	C₂⁵	1	8	yes	—
116	C₂³×C₁₃	C₂⁵	1	3	yes	—
117	C₂²×D₁₃	—	2	48	no	—
118	C₂²×D₁₃	—	2	48	no	—
119	C₂²×D₁₃	—	2	48	no	—
120	C₂²×D₁₃	—	4	32	no	—
121	C₂²×D₁₃	—	4	64	no	—
122	C₂²×D₁₃	—	4	384	no	—
123	C₂²×D₁₃	—	4	64	no	—
124	C₂²×D₁₃	C₂⁵	1	16	yes	—
125	Q₈×C₁₃	—	4	16	no	—
126	D₄×C₁₃	—	4	16	no	—
127	C₂×C₄×C₁₃	—	2	24	no	—
128	C₂×C₄×C₁₃	—	2	24	no	—
129	C₂×C₄×C₁₃	—	2	12	no	—
130	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
131	C₂×C₄×C₁₃	C₂⁵	1	4	yes	—
132	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
133	C₂²⋊D₁₃	—	4	64	no	—
134	C₂×Dic₁₃	—	2	48	no	—
135	C₂×Dic₁₃	C₂⁵	1	16	yes	—
136	D₅₂	—	4	32	no	—
137	D₅₂	—	4	32	no	—
138	Dic₂₆	—	4	32	no	—
139	C₂⁶	—	13	2	no	—
140	C₂³×Q₈	—	13	2	no	—
141	C₂³×Q₈	C₂×D₁₃	1	1	yes	characteristic, complemented
142	C₂³×D₄	—	13	4	no	—
143	C₂³×D₄	—	13	2	no	—
144	C₂³×D₄	—	13	2	no	—
145	C₂³×D₄	C₂×D₁₃	1	1	yes	characteristic
146	H(3,𝔽₄)	—	13	256	no	—
147	C₂×#124	—	13	48	no	—
148	C₂×#124	—	13	96	no	—
149	C₂×#124	—	13	48	no	—
150	C₂×#124	C₂×D₁₃	1	24	yes	complemented
151	C₂×#120	—	13	48	no	—
152	C₂×#120	—	13	96	no	—
153	C₂²×C₂²⋊C₄	—	13	2	no	—
154	C₂²×C₂²⋊C₄	—	13	4	no	—
155	C₂²×C₂²⋊C₄	—	13	4	no	—
156	C₂²×C₂²⋊C₄	—	13	4	no	—
157	C₂²×C₂²⋊C₄	C₂×D₁₃	1	4	yes	complemented
158	C₂²×C₄²	—	13	2	no	—
159	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic, complemented
160	C₂²×C₁₃	—	2	16	no	—
161	C₂²×C₁₃	—	2	8	no	—
162	C₂²×C₁₃	—	4	48	no	—
163	C₂²×C₁₃	H(3,𝔽₄)	1	16	yes	complemented
164	C₂²×C₁₃	C₂×H(5,𝔽₂)	1	6	yes	—
165	C₂²×C₁₃	C₂×H(5,𝔽₂)	1	12	yes	—
166	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
167	C₂×D₁₃	—	2	16	no	—
168	C₂×D₁₃	—	2	16	no	—
169	C₂×D₁₃	—	2	16	no	—
170	C₂×D₁₃	—	4	128	no	—
171	C₂×D₁₃	—	4	96	no	—
172	C₄×C₁₃	—	2	8	no	—
173	C₄×C₁₃	—	2	8	no	—
174	C₄×C₁₃	—	2	4	no	—
175	Dic₁₃	—	2	16	no	—
176	C₂⁵	—	13	4	no	—
177	C₂⁵	—	13	24	no	—
178	C₂⁵	—	52	8	no	—
179	C₂⁵	—	52	16	no	—
180	C₂⁵	C₂²×D₁₃	1	2	yes	—
181	C₂²×Q₈	—	4	4	no	—
182	C₂²×Q₈	—	13	24	no	—
183	C₂²×Q₈	—	52	8	no	—
184	C₂²×Q₈	C₂²×D₁₃	1	12	yes	complemented
185	C₂²×D₄	—	4	4	no	—
186	C₂²×D₄	—	13	48	no	—
187	C₂²×D₄	—	13	24	no	—
188	C₂²×D₄	—	13	24	no	—
189	C₂²×D₄	—	52	8	no	—
190	C₂²×D₄	—	52	8	no	—
191	C₂²×D₄	—	52	16	no	—
192	C₂²×D₄	C₂²×D₁₃	1	12	yes	—
193	C₂³×C₄	—	13	4	no	—
194	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
195	C₂³×C₄	C₂²×D₁₃	1	2	yes	—
196	C₂³×C₄	C₂²×D₁₃	1	2	yes	—
197	#124	—	13	128	no	—
198	#124	—	13	256	no	—
199	#124	—	13	128	no	—
200	#124	C₂²×D₁₃	1	64	yes	complemented
201	#120	—	13	128	no	—
202	#120	—	13	256	no	—
203	C₂×C₂²⋊C₄	—	13	24	no	—
204	C₂×C₂²⋊C₄	—	13	48	no	—
205	C₂×C₂²⋊C₄	—	13	48	no	—
206	C₂×C₂²⋊C₄	—	13	48	no	—
207	C₂×C₂²⋊C₄	C₂²×D₁₃	1	48	yes	complemented
208	C₂×C₄²	—	13	24	no	—
209	C₂×C₄²	C₂²×D₁₃	1	12	yes	complemented
210	C₂×C₁₃	—	4	8	no	—
211	C₂×C₁₃	C₂×H(3,𝔽₄)	1	12	yes	complemented
212	C₂×C₁₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
213	C₂×C₁₃	C₂²×H(5,𝔽₂)	1	2	yes	—
214	D₁₃	—	4	16	no	—
215	C₂⁴	—	2	8	no	—
216	C₂⁴	—	2	16	no	—
217	C₂⁴	—	13	24	no	—
218	C₂⁴	—	13	32	no	—
219	C₂⁴	—	26	16	no	—
220	C₂⁴	—	26	16	no	—
221	C₂⁴	—	26	16	no	—
222	C₂⁴	—	52	48	no	—
223	C₂⁴	—	52	96	no	—
224	C₂⁴	—	52	128	no	—
225	C₂⁴	C₂³×D₁₃	1	12	yes	—
226	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
227	C₂×Q₈	—	4	24	no	—
228	C₂×Q₈	—	13	32	no	—
229	C₂×Q₈	—	52	48	no	—
230	C₂×Q₈	C₂³×D₁₃	1	16	yes	complemented
231	C₂×D₄	—	4	24	no	—
232	C₂×D₄	—	13	64	no	—
233	C₂×D₄	—	13	32	no	—
234	C₂×D₄	—	13	32	no	—
235	C₂×D₄	—	52	48	no	—
236	C₂×D₄	—	52	48	no	—
237	C₂×D₄	—	52	96	no	—
238	C₂×D₄	C₂³×D₁₃	1	16	yes	—
239	C₂²×C₄	—	2	4	no	—
240	C₂²×C₄	—	2	8	no	—
241	C₂²×C₄	—	2	8	no	—
242	C₂²×C₄	—	13	24	no	—
243	C₂²×C₄	—	26	16	no	—
244	C₂²×C₄	C₂³×D₁₃	1	12	yes	—
245	C₂²×C₄	C₂³×D₁₃	1	6	yes	—
246	C₂²×C₄	C₂³×D₁₃	1	12	yes	—
247	C₂²⋊C₄	—	13	32	no	—
248	C₂²⋊C₄	—	13	64	no	—
249	C₂²⋊C₄	—	13	64	no	—
250	C₂²⋊C₄	—	13	64	no	—
251	C₂²⋊C₄	C₂³×D₁₃	1	64	yes	complemented
252	C₄²	—	13	32	no	—
253	C₄²	C₂³×D₁₃	1	16	yes	complemented
254	C₁₃	C₂²×H(3,𝔽₄)	1	1	yes	characteristic, complemented, Sylow
255	C₂³	—	2	24	no	—
256	C₂³	—	2	48	no	—
257	C₂³	—	4	32	no	—
258	C₂³	—	13	16	no	—
259	C₂³	—	26	48	no	—
260	C₂³	—	26	48	no	—
261	C₂³	—	26	48	no	—
262	C₂³	—	52	32	no	—
263	C₂³	—	52	64	no	—
264	C₂³	—	52	384	no	—
265	C₂³	—	52	64	no	—
266	C₂³	#88251	1	4	yes	—
267	C₂³	#88256	1	8	yes	—
268	C₂³	C₂⁴×D₁₃	1	8	yes	—
269	C₂³	C₂⁴×D₁₃	1	3	yes	—
270	Q₈	—	4	16	no	—
271	Q₈	—	52	32	no	—
272	D₄	—	4	16	no	—
273	D₄	—	52	32	no	—
274	D₄	—	52	32	no	—
275	D₄	—	52	64	no	—
276	C₂×C₄	—	2	12	no	—
277	C₂×C₄	—	2	24	no	—
278	C₂×C₄	—	2	24	no	—
279	C₂×C₄	—	13	16	no	—
280	C₂×C₄	—	26	48	no	—
281	C₂×C₄	C₂⁴×D₁₃	1	8	yes	—
282	C₂×C₄	C₂⁴×D₁₃	1	4	yes	—
283	C₂×C₄	C₂⁴×D₁₃	1	8	yes	—
284	C₂²	—	2	8	no	—
285	C₂²	—	2	16	no	—
286	C₂²	—	4	48	no	—
287	C₂²	—	26	16	no	—
288	C₂²	—	26	16	no	—
289	C₂²	—	26	16	no	—
290	C₂²	—	52	128	no	—
291	C₂²	—	52	96	no	—
292	C₂²	#11719388	1	16	yes	direct product factor
293	C₂²	C₂×#88251	1	6	yes	—
294	C₂²	C₂×#88256	1	12	yes	—
295	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
296	C₄	—	2	4	no	—
297	C₄	—	2	8	no	—
298	C₄	—	2	8	no	—
299	C₄	—	26	16	no	—
300	C₂	—	4	8	no	—
301	C₂	—	52	16	no	—
302	C₂	C₂×#11719388	1	12	yes	direct product factor
303	C₂	C₂²×#88251	1	1	yes	characteristic
304	C₂	C₂²×#88256	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁶
Frattini	C₂²	C₂⁵×D₁₃
Fitting	C₂²×C₁₃×#124	C₂
socle	C₂⁴×C₁₃	C₂⁴
hypercenter	C₂²×#124	D₁₃
hypocenter	C₁₃	C₂²×H(3,𝔽₄)
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#124

Complements

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

N	H ≅ G/N	type
1. C₂×#11719388	C₂	direct (N × H)
2. C₂²×#88201	C₂	semidirect
3. C₂²×#88129	C₂	semidirect
4. C₂²×#88139	C₂	semidirect
5. C₂²×#88136	C₂	semidirect
6. C₂²×#88191	C₂	semidirect
7. C₂²×C₁₃×#124	C₂	semidirect
8. C₂⁵×D₁₃	C₂²	semidirect
9. C₂³×Q₈×C₁₃	C₂²	semidirect
14. C₂³×Dic₂₆	C₂²	semidirect
15. C₂×C₁₃×#124	C₂²	semidirect
16. C₂²×C₁₃×C₂²⋊C₄	C₂²	semidirect
17. C₂²×C₄²×C₁₃	C₂²	semidirect
18. C₂²×C₂²⋊Dic₁₃	C₂²	semidirect
19. C₂×#88201	C₂²	semidirect
20. C₂×#88191	C₂²	semidirect
21. C₂²×#6103	C₂²	semidirect
22. C₂²×#6103	C₂²	semidirect
23. C₂²×#6103	C₂²	semidirect
24. C₂²×C₄×Dic₁₃	C₂²	semidirect
25. #11719388	C₂²	direct (N × H)
26. C₂×#88139	C₂²	semidirect
27. C₂×#88136	C₂²	semidirect
28. C₂×#88129	C₂²	semidirect
33. C₂⁴×D₁₃	C₂³	semidirect
35. C₂²×Q₈×C₁₃	C₂³	semidirect
49. C₂²×Dic₂₆	C₂³	semidirect
50. C₁₃×#124	C₂³	semidirect
51. C₂×C₁₃×C₂²⋊C₄	C₂³	semidirect
52. C₂×C₄²×C₁₃	C₂³	semidirect
53. C₂×C₂²⋊Dic₁₃	C₂³	semidirect
54. #88201	C₂³	semidirect
55. #88191	C₂³	semidirect
56. C₂×#6103	C₂³	semidirect
57. C₂×#6103	C₂³	semidirect
58. C₂×#6103	C₂³	semidirect
59. C₂×C₄×Dic₁₃	C₂³	semidirect
60. #88139	C₂³	semidirect
61. #88136	C₂³	semidirect
62. #88129	C₂³	semidirect
75. C₂³×D₁₃	C₂⁴	semidirect
77. C₂×Q₈×C₁₃	C₂⁴	semidirect
95. C₂×Dic₂₆	C₂⁴	semidirect
96. C₁₃×C₂²⋊C₄	C₂⁴	semidirect
97. C₄²×C₁₃	C₂⁴	semidirect
98. C₂²⋊Dic₁₃	C₂⁴	semidirect
99. #6103	C₂⁴	semidirect
100. #6103	C₂⁴	semidirect
101. #6103	C₂⁴	semidirect
102. C₄×Dic₁₃	C₂⁴	semidirect
107. C₂²×#124	D₁₃	semidirect
141. C₂³×Q₈	C₂×D₁₃	semidirect
150. C₂×#124	C₂×D₁₃	semidirect
157. C₂²×C₂²⋊C₄	C₂×D₁₃	semidirect
159. C₂²×C₄²	C₂×D₁₃	semidirect
163. C₂²×C₁₃	H(3,𝔽₄)	semidirect
184. C₂²×Q₈	C₂²×D₁₃	semidirect
200. #124	C₂²×D₁₃	semidirect
207. C₂×C₂²⋊C₄	C₂²×D₁₃	semidirect
209. C₂×C₄²	C₂²×D₁₃	semidirect
211. C₂×C₁₃	C₂×H(3,𝔽₄)	semidirect
230. C₂×Q₈	C₂³×D₁₃	semidirect
251. C₂²⋊C₄	C₂³×D₁₃	semidirect
253. C₄²	C₂³×D₁₃	semidirect
254. C₁₃	C₂²×H(3,𝔽₄)	semidirect
292. C₂²	#11719388	direct (N × H)
302. C₂	C₂×#11719388	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×#11719388	2
C₂²×#88201	2
C₂²×#88129	2
C₂²×#88139	2
C₂²×#88136	2
C₂²×#88191	2
C₂²×C₁₃×#124	2
C₂²×H(3,𝔽₄)	13

Subgroups — C22×#11719388 #16769791

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11719388 #16769791