Subgroups — C₂²×#11718498 #16726529

Order 3328.

3438 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×#6094	C₂	1	2	yes	maximal
2	C₂²×C₄²×D₁₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718498	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#88054	C₂	1	2	yes	maximal, complemented
6	C₂²×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₄×D₁₃	C₄	1	1	yes	characteristic
8	C₂³×C₄×D₁₃	C₄	1	1	yes	characteristic
9	C₂³×C₄×D₁₃	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
11	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
12	C₂²×C₄×Dic₁₃	C₄	1	1	yes	characteristic
13	C₂²×C₄×Dic₁₃	C₄	1	1	yes	characteristic
14	C₂²×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
15	C₂³×#1055	C₂²	1	2	yes	—
16	C₂²×#6094	—	2	16	no	—
17	C₂²×#6094	C₂²	1	24	yes	—
18	C₂×C₄²×D₁₃	C₄	1	12	yes	—
19	C₂×C₄²×D₁₃	C₂²	1	12	yes	—
20	C₂×C₁₃×C₄⋊C₈	C₂²	1	12	yes	complemented
21	C₂×#88054	C₂²	1	24	yes	complemented
22	C₂×C₄⋊#1055	C₂²	1	12	yes	complemented
23	#11718498	C₂²	1	64	yes	direct product factor
24	C₂⁴×D₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
28	C₂³×Dic₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂³×Dic₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂³×Dic₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×D₁₃	—	2	8	no	—
32	C₂²×C₄×D₁₃	—	2	8	no	—
33	C₂²×C₄×D₁₃	C₂×C₄	1	12	yes	—
34	C₂²×C₄×D₁₃	C₂×C₄	1	12	yes	—
35	C₂²×C₄×D₁₃	C₂×C₄	1	6	yes	—
36	C₂²×C₄×D₁₃	D₄	1	8	yes	—
37	C₂²×C₄×D₁₃	Q₈	1	8	yes	—
38	C₂²×C₄×D₁₃	C₂³	1	6	yes	—
39	C₂²×C₈×C₁₃	—	2	8	no	—
40	C₂²×C₈×C₁₃	C₂³	1	12	yes	—
41	C₂×C₄²×C₁₃	C₂×C₄	1	6	yes	—
42	C₂×C₄²×C₁₃	C₂³	1	6	yes	—
43	C₂×C₄×Dic₁₃	C₂×C₄	1	6	yes	—
44	C₂×C₄×Dic₁₃	C₂×C₄	1	12	yes	—
45	C₂×C₄×Dic₁₃	C₂×C₄	1	12	yes	—
46	C₂×C₄×Dic₁₃	C₂³	1	6	yes	—
47	C₂²×#1055	—	2	8	no	—
48	C₂²×#1055	C₂³	1	12	yes	—
49	C₂×#6094	—	2	96	no	—
50	C₂×#6094	C₂³	1	32	yes	—
51	C₄²×D₁₃	C₂×C₄	1	48	yes	—
52	C₄²×D₁₃	C₂³	1	16	yes	—
53	C₁₃×C₄⋊C₈	C₂³	1	16	yes	complemented
54	#88054	C₂³	1	32	yes	complemented
55	C₄⋊#1055	C₂³	1	16	yes	complemented
56	C₂²×#423	—	13	1	no	maximal, Sylow
57	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₂³×D₁₃	—	2	4	no	—
59	C₂³×D₁₃	—	2	4	no	—
60	C₂³×D₁₃	C₄⋊C₄	1	8	yes	—
61	C₂³×D₁₃	C₂²×C₄	1	6	yes	—
62	C₂²×C₄×C₁₃	—	2	4	no	—
63	C₂²×C₄×C₁₃	M₁₆	1	8	yes	complemented
64	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
67	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
68	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
69	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
70	C₂²×Dic₁₃	—	2	4	no	—
71	C₂²×Dic₁₃	C₄⋊C₄	1	8	yes	—
72	C₂²×Dic₁₃	M₁₆	1	8	yes	complemented
73	C₂²×Dic₁₃	C₂²×C₄	1	6	yes	—
74	C₂²×Dic₁₃	C₂²×C₄	1	6	yes	—
75	C₂²×Dic₁₃	C₂²×C₄	1	6	yes	—
76	C₂×C₄×D₁₃	—	2	48	no	—
77	C₂×C₄×D₁₃	—	2	48	no	—
78	C₂×C₄×D₁₃	C₄⋊C₄	1	48	yes	—
79	C₂×C₄×D₁₃	C₂²×C₄	1	16	yes	—
80	C₂×C₄×D₁₃	C₂²×C₄	1	16	yes	—
81	C₂×C₄×D₁₃	C₂²×C₄	1	12	yes	—
82	C₂×C₄×D₁₃	C₂×D₄	1	24	yes	—
83	C₂×C₄×D₁₃	C₂×Q₈	1	24	yes	—
84	C₂×C₄×D₁₃	C₂⁴	1	4	yes	—
85	C₂×C₈×C₁₃	—	2	24	no	—
86	C₂×C₈×C₁₃	C₂⁴	1	8	yes	—
87	C₄²×C₁₃	C₂²×C₄	1	12	yes	—
88	C₄²×C₁₃	C₂⁴	1	4	yes	—
89	C₄×Dic₁₃	C₂²×C₄	1	12	yes	—
90	C₄×Dic₁₃	C₂²×C₄	1	16	yes	—
91	C₄×Dic₁₃	C₂²×C₄	1	16	yes	—
92	C₄×Dic₁₃	C₂⁴	1	4	yes	—
93	C₂×#1055	—	2	24	no	—
94	C₂×#1055	C₂⁴	1	8	yes	—
95	#6094	—	2	64	no	—
96	C₂³×M₁₆	—	13	2	no	—
97	C₂³×C₄²	—	13	1	no	—
98	C₂×#423	—	13	24	no	—
99	C₂²×C₄⋊C₈	—	13	1	no	—
100	C₂²×C₄⋊C₈	—	13	2	no	—
101	C₂²×C₄⋊C₈	D₁₃	1	1	yes	characteristic, complemented
102	C₂³×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
103	C₂³×C₁₃	C₂×M₁₆	1	4	yes	—
104	C₂³×C₁₃	C₂×M₁₆	1	4	yes	—
105	C₂³×C₁₃	C₂³×C₄	1	3	yes	—
106	C₂²×D₁₃	—	2	12	no	—
107	C₂²×D₁₃	—	2	12	no	—
108	C₂²×D₁₃	—	2	32	no	—
109	C₂²×D₁₃	C₂×C₄⋊C₄	1	24	yes	—
110	C₂²×D₁₃	C₂³×C₄	1	4	yes	—
111	C₂×C₄×C₁₃	—	2	12	no	—
112	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	12	yes	—
113	C₂×C₄×C₁₃	C₂×M₁₆	1	24	yes	complemented
114	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
115	C₂×C₄×C₁₃	C₂³×C₄	1	3	yes	—
116	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
117	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
118	C₂×C₄×C₁₃	C₂²×Q₈	1	6	yes	—
119	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
120	C₂×Dic₁₃	—	2	12	no	—
121	C₂×Dic₁₃	C₂×C₄⋊C₄	1	24	yes	—
122	C₂×Dic₁₃	C₂×M₁₆	1	24	yes	complemented
123	C₂×Dic₁₃	C₂³×C₄	1	4	yes	—
124	C₂×Dic₁₃	C₂³×C₄	1	4	yes	—
125	C₂×Dic₁₃	C₂³×C₄	1	4	yes	—
126	C₄×D₁₃	—	2	32	no	—
127	C₄×D₁₃	—	2	32	no	—
128	C₄×D₁₃	C₂×C₄⋊C₄	1	48	yes	—
129	C₄×D₁₃	C₂²×D₄	1	8	yes	—
130	C₄×D₁₃	C₂²×Q₈	1	8	yes	—
131	C₈×C₁₃	—	2	8	no	—
132	#1055	—	2	8	no	—
133	C₂⁴×C₄	—	13	1	no	—
134	C₂⁴×C₄	—	13	1	no	—
135	C₂⁴×C₄	—	13	1	no	—
136	C₂²×M₁₆	—	13	24	no	—
137	C₂²×M₁₆	—	26	16	no	—
138	C₂³×C₈	—	13	2	no	—
139	C₂³×C₈	C₂×D₁₃	1	2	yes	—
140	C₂²×C₄²	—	13	1	no	—
141	C₂²×C₄²	—	13	1	no	—
142	C₂²×C₄²	—	13	1	no	—
143	C₂²×C₄²	—	13	12	no	—
144	C₂²×C₄²	—	13	12	no	—
145	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
146	#423	—	13	64	no	—
147	C₂×C₄⋊C₈	—	13	24	no	—
148	C₂×C₄⋊C₈	—	13	12	no	—
149	C₂×C₄⋊C₈	C₂×D₁₃	1	12	yes	complemented
150	C₂²×C₁₃	#423	1	16	yes	complemented
151	C₂²×C₁₃	C₂²×C₄⋊C₄	1	6	yes	—
152	C₂²×C₁₃	C₂²×M₁₆	1	6	yes	—
153	C₂²×C₁₃	C₂²×M₁₆	1	6	yes	—
154	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
155	C₂×D₁₃	—	2	48	no	—
156	C₂×D₁₃	—	2	4	no	—
157	C₂×D₁₃	—	2	4	no	—
158	C₂×D₁₃	C₂²×C₄⋊C₄	1	8	yes	—
159	C₄×C₁₃	—	2	4	no	—
160	C₄×C₁₃	C₂²×C₄⋊C₄	1	6	yes	—
161	C₄×C₁₃	C₂²×M₁₆	1	8	yes	complemented
162	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
163	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
164	Dic₁₃	—	2	4	no	—
165	Dic₁₃	C₂²×C₄⋊C₄	1	8	yes	—
166	Dic₁₃	C₂²×M₁₆	1	8	yes	complemented
167	C₂⁵	—	13	1	no	—
168	C₂³×C₄	—	13	1	no	—
169	C₂³×C₄	—	13	1	no	—
170	C₂³×C₄	—	13	6	no	—
171	C₂³×C₄	—	13	8	no	—
172	C₂³×C₄	—	13	8	no	—
173	C₂³×C₄	—	13	6	no	—
174	C₂³×C₄	—	13	1	no	—
175	C₂³×C₄	—	13	12	no	—
176	C₂³×C₄	—	13	12	no	—
177	C₂³×C₄	—	26	8	no	—
178	C₂³×C₄	—	26	8	no	—
179	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
180	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
181	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
182	C₂×M₁₆	—	13	32	no	—
183	C₂×M₁₆	—	26	96	no	—
184	C₂²×C₈	—	2	8	no	—
185	C₂²×C₈	—	13	12	no	—
186	C₂²×C₈	—	26	8	no	—
187	C₂²×C₈	C₂²×D₁₃	1	12	yes	—
188	C₂×C₄²	—	13	12	no	—
189	C₂×C₄²	—	13	12	no	—
190	C₂×C₄²	—	13	6	no	—
191	C₂×C₄²	—	13	6	no	—
192	C₂×C₄²	—	13	48	no	—
193	C₂×C₄²	—	13	16	no	—
194	C₂×C₄²	C₄×D₁₃	1	6	yes	—
195	C₂×C₄²	C₂²×D₁₃	1	6	yes	—
196	C₄⋊C₈	—	13	32	no	—
197	C₄⋊C₈	—	13	16	no	—
198	C₄⋊C₈	C₂²×D₁₃	1	16	yes	complemented
199	C₂×C₁₃	C₂×#423	1	12	yes	complemented
200	C₂×C₁₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
201	C₂×C₁₃	C₂³×M₁₆	1	1	yes	characteristic
202	C₂×C₁₃	C₂³×M₁₆	1	1	yes	characteristic
203	D₁₃	—	2	8	no	—
204	C₂⁴	—	13	8	no	—
205	C₂⁴	—	13	6	no	—
206	C₂⁴	—	26	4	no	—
207	C₂⁴	—	26	4	no	—
208	C₂⁴	C₂×C₄×D₁₃	1	1	yes	characteristic
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	13	8	no	—
211	C₂²×C₄	—	13	8	no	—
212	C₂²×C₄	—	13	6	no	—
213	C₂²×C₄	—	13	6	no	—
214	C₂²×C₄	—	13	48	no	—
215	C₂²×C₄	—	13	12	no	—
216	C₂²×C₄	—	13	24	no	—
217	C₂²×C₄	—	13	24	no	—
218	C₂²×C₄	—	13	4	no	—
219	C₂²×C₄	—	13	6	no	—
220	C₂²×C₄	—	13	16	no	—
221	C₂²×C₄	—	13	16	no	—
222	C₂²×C₄	—	26	48	no	—
223	C₂²×C₄	—	26	4	no	—
224	C₂²×C₄	—	26	48	no	—
225	C₂²×C₄	#6094	1	8	yes	complemented
226	C₂²×C₄	C₂×C₄×D₁₃	1	6	yes	—
227	C₂²×C₄	C₂×C₄×D₁₃	1	3	yes	—
228	C₂²×C₄	C₂×C₄×D₁₃	1	6	yes	—
229	C₂²×C₄	D₄×D₁₃	1	4	yes	—
230	C₂²×C₄	Q₈×D₁₃	1	4	yes	—
231	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
232	M₁₆	—	26	64	no	—
233	C₂×C₈	—	2	24	no	—
234	C₂×C₈	—	13	8	no	—
235	C₂×C₈	—	26	24	no	—
236	C₂×C₈	C₂³×D₁₃	1	8	yes	—
237	C₄²	—	13	16	no	—
238	C₄²	—	13	16	no	—
239	C₄²	—	13	12	no	—
240	C₄²	—	13	4	no	—
241	C₄²	C₂×C₄×D₁₃	1	12	yes	—
242	C₄²	C₂³×D₁₃	1	4	yes	—
243	C₁₃	C₂²×#423	1	1	yes	characteristic, complemented, Sylow
244	C₂³	—	13	24	no	—
245	C₂³	—	13	4	no	—
246	C₂³	—	26	12	no	—
247	C₂³	—	26	12	no	—
248	C₂³	—	26	32	no	—
249	C₂³	C₄⋊C₄×D₁₃	1	4	yes	—
250	C₂³	C₂×#6094	1	4	yes	—
251	C₂³	M₁₆×D₁₃	1	4	yes	—
252	C₂³	C₂²×C₄×D₁₃	1	3	yes	—
253	C₂×C₄	—	2	12	no	—
254	C₂×C₄	—	13	24	no	—
255	C₂×C₄	—	13	24	no	—
256	C₂×C₄	—	13	4	no	—
257	C₂×C₄	—	13	4	no	—
258	C₂×C₄	—	13	48	no	—
259	C₂×C₄	—	13	8	no	—
260	C₂×C₄	—	13	8	no	—
261	C₂×C₄	—	13	4	no	—
262	C₂×C₄	—	26	32	no	—
263	C₂×C₄	—	26	12	no	—
264	C₂×C₄	—	26	32	no	—
265	C₂×C₄	C₄⋊C₄×D₁₃	1	12	yes	—
266	C₂×C₄	C₂×#6094	1	24	yes	complemented
267	C₂×C₄	C₂²×C₄×D₁₃	1	4	yes	—
268	C₂×C₄	C₂²×C₄×D₁₃	1	4	yes	—
269	C₂×C₄	C₂²×C₄×D₁₃	1	3	yes	—
270	C₂×C₄	C₂×D₄×D₁₃	1	6	yes	—
271	C₂×C₄	C₂×Q₈×D₁₃	1	6	yes	—
272	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
273	C₈	—	2	8	no	—
274	C₈	—	26	8	no	—
275	C₂²	—	13	8	no	—
276	C₂²	—	26	48	no	—
277	C₂²	—	26	4	no	—
278	C₂²	—	26	4	no	—
279	C₂²	#11718498	1	16	yes	direct product factor
280	C₂²	C₂×C₄⋊C₄×D₁₃	1	6	yes	—
281	C₂²	C₂²×#6094	1	6	yes	—
282	C₂²	C₂×M₁₆×D₁₃	1	6	yes	—
283	C₂²	C₂³×C₄×D₁₃	1	1	yes	characteristic
284	C₄	—	2	4	no	—
285	C₄	—	13	8	no	—
286	C₄	—	13	8	no	—
287	C₄	—	26	4	no	—
288	C₄	C₂×C₄⋊C₄×D₁₃	1	6	yes	—
289	C₄	C₂²×#6094	1	8	yes	complemented
290	C₄	C₂²×D₄×D₁₃	1	1	yes	characteristic
291	C₄	C₂²×Q₈×D₁₃	1	1	yes	characteristic
292	C₂	—	26	8	no	—
293	C₂	C₂×#11718498	1	12	yes	direct product factor
294	C₂	C₂²×C₄⋊C₄×D₁₃	1	1	yes	characteristic
295	C₂	C₂²×M₁₆×D₁₃	1	1	yes	characteristic
296	C₂	C₂³×#6094	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²×D₁₃
commutator	C₂²×C₁₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂²×C₁₃×C₄⋊C₈	C₂
socle	C₂⁴×C₁₃	C₂²×C₄
prime radical	C₂⁴×D₁₃	C₂×C₄
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂²×C₄⋊C₈	D₁₃
hypocenter	C₁₃	C₂²×#423
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. C₂×#11718498	C₂	direct (N × H)
4. C₂²×C₄⋊#1055	C₂	semidirect
5. C₂²×#88054	C₂	semidirect
6. C₂²×C₁₃×C₄⋊C₈	C₂	semidirect
20. C₂×C₁₃×C₄⋊C₈	C₂²	semidirect
21. C₂×#88054	C₂²	semidirect
22. C₂×C₄⋊#1055	C₂²	semidirect
23. #11718498	C₂²	direct (N × H)
53. C₁₃×C₄⋊C₈	C₂³	semidirect
54. #88054	C₂³	semidirect
55. C₄⋊#1055	C₂³	semidirect
63. C₂²×C₄×C₁₃	M₁₆	semidirect
72. C₂²×Dic₁₃	M₁₆	semidirect
101. C₂²×C₄⋊C₈	D₁₃	semidirect
113. C₂×C₄×C₁₃	C₂×M₁₆	semidirect
122. C₂×Dic₁₃	C₂×M₁₆	semidirect
149. C₂×C₄⋊C₈	C₂×D₁₃	semidirect
150. C₂²×C₁₃	#423	semidirect
161. C₄×C₁₃	C₂²×M₁₆	semidirect
166. Dic₁₃	C₂²×M₁₆	semidirect
198. C₄⋊C₈	C₂²×D₁₃	semidirect
199. C₂×C₁₃	C₂×#423	semidirect
225. C₂²×C₄	#6094	semidirect
243. C₁₃	C₂²×#423	semidirect
266. C₂×C₄	C₂×#6094	semidirect
279. C₂²	#11718498	direct (N × H)
289. C₄	C₂²×#6094	semidirect
293. C₂	C₂×#11718498	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×#6094	2
C₂²×C₄²×D₁₃	2
C₂×#11718498	2
C₂²×C₄⋊#1055	2
C₂²×#88054	2
C₂²×C₁₃×C₄⋊C₈	2
C₂²×#423	13

Subgroups — C22×#11718498 #16726529

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718498 #16726529