Subgroups — C₂²×#11718589 #16021266

Order 3328.

4894 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	C₂³×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	C₂³×#6101	C₂	1	2	yes	maximal
4	C₂×#11718589	C₂	1	24	yes	maximal, direct product factor
5	C₂²×#88071	C₂	1	2	yes	maximal
6	C₂³×C₄²×C₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
8	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
9	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
10	C₂⁴×Dic₁₃	C₂²	1	2	yes	—
11	C₂⁴×Dic₁₃	C₂²	1	2	yes	—
12	C₂²×C₄²×C₁₃	—	2	8	no	—
13	C₂²×C₄²×C₁₃	C₂²	1	12	yes	—
14	C₂²×C₄⋊Dic₁₃	—	2	8	no	—
15	C₂²×C₄⋊Dic₁₃	C₄	1	16	yes	complemented
16	C₂²×C₄⋊Dic₁₃	C₂²	1	12	yes	—
17	C₂²×C₄⋊Dic₁₃	C₂²	1	12	yes	—
18	C₂²×#6101	—	2	16	no	—
19	C₂²×#6101	C₂²	1	24	yes	—
20	C₂×#88071	C₂²	1	24	yes	—
21	#11718589	C₂²	1	64	yes	direct product factor
22	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
23	C₂³×C₄×C₁₃	—	2	4	no	—
24	C₂³×C₄×C₁₃	—	2	4	no	—
25	C₂³×C₄×C₁₃	—	2	4	no	—
26	C₂³×C₄×C₁₃	—	2	4	no	—
27	C₂³×C₄×C₁₃	C₂×C₄	1	8	yes	—
28	C₂³×C₄×C₁₃	D₄	1	8	yes	—
29	C₂³×C₄×C₁₃	D₄	1	8	yes	—
30	C₂³×C₄×C₁₃	Q₈	1	8	yes	—
31	C₂³×C₄×C₁₃	Q₈	1	8	yes	—
32	C₂³×C₄×C₁₃	C₂³	1	6	yes	—
33	C₂³×C₄×C₁₃	C₂³	1	6	yes	—
34	C₂³×C₄×C₁₃	C₂³	1	6	yes	—
35	C₂³×Dic₁₃	—	2	8	no	—
36	C₂³×Dic₁₃	—	2	8	no	—
37	C₂³×Dic₁₃	—	2	8	no	—
38	C₂³×Dic₁₃	—	2	8	no	—
39	C₂³×Dic₁₃	—	2	8	no	—
40	C₂³×Dic₁₃	C₂×C₄	1	16	yes	—
41	C₂³×Dic₁₃	C₂³	1	12	yes	—
42	C₂³×Dic₁₃	C₂³	1	12	yes	—
43	C₂×C₄²×C₁₃	—	2	48	no	—
44	C₂×C₄²×C₁₃	C₂³	1	16	yes	—
45	C₂×C₄⋊Dic₁₃	—	2	48	no	—
46	C₂×C₄⋊Dic₁₃	C₂×C₄	1	96	yes	complemented
47	C₂×C₄⋊Dic₁₃	C₂³	1	16	yes	—
48	C₂×C₄⋊Dic₁₃	C₂³	1	16	yes	—
49	C₂×#6101	—	2	96	no	—
50	C₂×#6101	C₂³	1	32	yes	—
51	#88071	C₂³	1	32	yes	—
52	C₂²×#389	—	13	1	no	maximal, Sylow
53	C₂⁴×C₁₃	C₂²×C₄	1	4	yes	—
54	C₂⁴×C₁₃	C₂×D₄	1	4	yes	—
55	C₂⁴×C₁₃	C₂×D₄	1	4	yes	—
56	C₂⁴×C₁₃	C₂×Q₈	1	4	yes	—
57	C₂⁴×C₁₃	C₂×Q₈	1	4	yes	—
58	C₂⁴×C₁₃	#41	1	4	yes	—
59	C₂⁴×C₁₃	#41	1	4	yes	—
60	C₂⁴×C₁₃	C₂⁴	1	3	yes	—
61	C₂²×C₄×C₁₃	—	2	32	no	—
62	C₂²×C₄×C₁₃	—	2	32	no	—
63	C₂²×C₄×C₁₃	—	2	12	no	—
64	C₂²×C₄×C₁₃	—	2	12	no	—
65	C₂²×C₄×C₁₃	—	2	12	no	—
66	C₂²×C₄×C₁₃	—	2	12	no	—
67	C₂²×C₄×C₁₃	C₄⋊C₄	1	64	yes	complemented
68	C₂²×C₄×C₁₃	C₂²×C₄	1	24	yes	—
69	C₂²×C₄×C₁₃	C₂×D₄	1	24	yes	—
70	C₂²×C₄×C₁₃	C₂×D₄	1	24	yes	—
71	C₂²×C₄×C₁₃	C₂×Q₈	1	24	yes	—
72	C₂²×C₄×C₁₃	C₂×Q₈	1	24	yes	—
73	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
74	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
75	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
76	C₂²×Dic₁₃	—	2	64	no	—
77	C₂²×Dic₁₃	—	2	24	no	—
78	C₂²×Dic₁₃	—	2	24	no	—
79	C₂²×Dic₁₃	—	2	24	no	—
80	C₂²×Dic₁₃	—	2	24	no	—
81	C₂²×Dic₁₃	—	2	24	no	—
82	C₂²×Dic₁₃	—	4	32	no	—
83	C₂²×Dic₁₃	C₂²×C₄	1	48	yes	—
84	C₂²×Dic₁₃	C₂⁴	1	8	yes	—
85	C₂²×Dic₁₃	C₂⁴	1	8	yes	—
86	C₄²×C₁₃	—	2	32	no	—
87	C₄⋊Dic₁₃	—	2	32	no	—
88	C₄⋊Dic₁₃	C₂²×C₄	1	64	yes	complemented
89	#6101	—	2	64	no	—
90	C₂³×C₄⋊C₄	—	13	1	no	—
91	C₂³×C₄⋊C₄	—	13	2	no	—
92	C₂³×C₄⋊C₄	—	13	1	no	—
93	C₂³×C₄²	D₁₃	1	1	yes	characteristic
94	C₂×#389	—	13	24	no	—
95	C₂²×#95	—	13	2	no	—
96	C₂³×C₁₃	C₂×C₄⋊C₄	1	16	yes	—
97	C₂³×C₁₃	C₄×D₄	1	16	yes	—
98	C₂³×C₁₃	C₄×Q₈	1	16	yes	—
99	C₂³×C₁₃	#121	1	16	yes	—
100	C₂³×C₁₃	C₂²⋊Q₈	1	16	yes	—
101	C₂³×C₁₃	#125	1	16	yes	—
102	C₂³×C₁₃	C₄⋊Q₈	1	16	yes	—
103	C₂³×C₁₃	C₂³×C₄	1	6	yes	—
104	C₂³×C₁₃	C₂²×D₄	1	6	yes	—
105	C₂³×C₁₃	C₂²×D₄	1	6	yes	—
106	C₂³×C₁₃	C₂²×Q₈	1	6	yes	—
107	C₂³×C₁₃	C₂²×Q₈	1	6	yes	—
108	C₂³×C₁₃	C₂×#41	1	6	yes	—
109	C₂³×C₁₃	C₂×#41	1	6	yes	—
110	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
111	C₂×C₄×C₁₃	—	2	48	no	—
112	C₂×C₄×C₁₃	—	2	48	no	—
113	C₂×C₄×C₁₃	—	2	4	no	—
114	C₂×C₄×C₁₃	—	2	4	no	—
115	C₂×C₄×C₁₃	—	2	4	no	—
116	C₂×C₄×C₁₃	—	2	4	no	—
117	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	96	yes	complemented
118	C₂×C₄×C₁₃	C₂³×C₄	1	8	yes	—
119	C₂×C₄×C₁₃	C₂²×D₄	1	8	yes	—
120	C₂×C₄×C₁₃	C₂²×D₄	1	8	yes	—
121	C₂×C₄×C₁₃	C₂²×Q₈	1	8	yes	—
122	C₂×C₄×C₁₃	C₂²×Q₈	1	8	yes	—
123	C₂×Dic₁₃	—	2	96	no	—
124	C₂×Dic₁₃	—	2	8	no	—
125	C₂×Dic₁₃	—	2	8	no	—
126	C₂×Dic₁₃	—	2	8	no	—
127	C₂×Dic₁₃	—	2	8	no	—
128	C₂×Dic₁₃	—	2	8	no	—
129	C₂×Dic₁₃	—	4	48	no	—
130	C₂×Dic₁₃	C₂³×C₄	1	16	yes	—
131	C₂⁴×C₄	—	13	2	no	—
132	C₂⁴×C₄	—	13	2	no	—
133	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
134	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
135	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
136	C₂²×C₄⋊C₄	—	13	16	no	—
137	C₂²×C₄⋊C₄	—	13	12	no	—
138	C₂²×C₄⋊C₄	—	13	24	no	—
139	C₂²×C₄⋊C₄	—	13	12	no	—
140	C₂²×C₄⋊C₄	—	26	16	no	—
141	C₂²×C₄⋊C₄	—	26	8	no	—
142	C₂²×C₄²	—	2	8	no	—
143	C₂²×C₄²	C₂×D₁₃	1	12	yes	—
144	#389	—	13	64	no	—
145	C₂×#95	—	13	24	no	—
146	C₂²×C₁₃	#389	1	64	yes	complemented
147	C₂²×C₁₃	C₂²×C₄⋊C₄	1	12	yes	—
148	C₂²×C₁₃	C₂×C₄×D₄	1	12	yes	—
149	C₂²×C₁₃	C₂×C₄×Q₈	1	12	yes	—
150	C₂²×C₁₃	C₂×#121	1	12	yes	—
151	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	12	yes	—
152	C₂²×C₁₃	C₂×#125	1	12	yes	—
153	C₂²×C₁₃	C₂×C₄⋊Q₈	1	12	yes	—
154	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
155	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
156	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
157	C₂²×C₁₃	C₂³×Q₈	1	1	yes	characteristic
158	C₂²×C₁₃	C₂³×Q₈	1	1	yes	characteristic
159	C₂²×C₁₃	C₂²×#41	1	1	yes	characteristic
160	C₂²×C₁₃	C₂²×#41	1	1	yes	characteristic
161	C₄×C₁₃	—	2	8	no	—
162	C₄×C₁₃	—	2	8	no	—
163	C₄×C₁₃	C₂²×C₄⋊C₄	1	16	yes	complemented
164	Dic₁₃	—	2	16	no	—
165	Dic₁₃	—	4	8	no	—
166	C₂⁵	C₂²×D₁₃	1	1	yes	characteristic
167	C₂³×C₄	—	2	4	no	—
168	C₂³×C₄	—	2	4	no	—
169	C₂³×C₄	—	2	4	no	—
170	C₂³×C₄	—	2	4	no	—
171	C₂³×C₄	—	13	16	no	—
172	C₂³×C₄	—	13	12	no	—
173	C₂³×C₄	—	13	12	no	—
174	C₂³×C₄	—	26	8	no	—
175	C₂³×C₄	—	26	8	no	—
176	C₂³×C₄	—	26	8	no	—
177	C₂³×C₄	—	26	8	no	—
178	C₂³×C₄	—	26	8	no	—
179	C₂³×C₄	Dic₂₆	1	8	yes	—
180	C₂³×C₄	Dic₂₆	1	8	yes	—
181	C₂³×C₄	C₄×D₁₃	1	8	yes	—
182	C₂³×C₄	D₅₂	1	8	yes	—
183	C₂³×C₄	C₂²⋊D₁₃	1	8	yes	—
184	C₂³×C₄	C₂²×D₁₃	1	6	yes	—
185	C₂³×C₄	C₂²×D₁₃	1	6	yes	—
186	C₂³×C₄	C₂²×D₁₃	1	6	yes	—
187	C₂×C₄⋊C₄	—	13	96	no	—
188	C₂×C₄⋊C₄	—	13	16	no	—
189	C₂×C₄⋊C₄	—	13	32	no	—
190	C₂×C₄⋊C₄	—	13	16	no	—
191	C₂×C₄⋊C₄	—	26	96	no	—
192	C₂×C₄⋊C₄	—	26	48	no	—
193	C₂×C₄²	—	2	48	no	—
194	C₂×C₄²	C₂²×D₁₃	1	16	yes	—
195	#95	—	13	32	no	—
196	C₂×C₁₃	C₂×#389	1	24	yes	complemented
197	C₂×C₁₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
198	C₂×C₁₃	C₂²×C₄×D₄	1	1	yes	characteristic
199	C₂×C₁₃	C₂²×C₄×Q₈	1	1	yes	characteristic
200	C₂×C₁₃	C₂²×#121	1	1	yes	characteristic
201	C₂×C₁₃	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
202	C₂×C₁₃	C₂²×#125	1	1	yes	characteristic
203	C₂×C₁₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
204	C₂⁴	C₂×Dic₂₆	1	4	yes	—
205	C₂⁴	C₂×Dic₂₆	1	4	yes	—
206	C₂⁴	C₂×C₄×D₁₃	1	4	yes	—
207	C₂⁴	C₂×D₅₂	1	4	yes	—
208	C₂⁴	#6127	1	4	yes	—
209	C₂⁴	#6127	1	4	yes	—
210	C₂⁴	C₂×C₂²⋊D₁₃	1	4	yes	—
211	C₂⁴	C₂³×D₁₃	1	3	yes	—
212	C₂²×C₄	—	2	32	no	—
213	C₂²×C₄	—	2	32	no	—
214	C₂²×C₄	—	2	12	no	—
215	C₂²×C₄	—	2	12	no	—
216	C₂²×C₄	—	2	12	no	—
217	C₂²×C₄	—	2	12	no	—
218	C₂²×C₄	—	13	48	no	—
219	C₂²×C₄	—	13	8	no	—
220	C₂²×C₄	—	13	8	no	—
221	C₂²×C₄	—	26	64	no	—
222	C₂²×C₄	—	26	24	no	—
223	C₂²×C₄	—	26	24	no	—
224	C₂²×C₄	—	26	24	no	—
225	C₂²×C₄	—	26	24	no	—
226	C₂²×C₄	—	26	24	no	—
227	C₂²×C₄	—	52	32	no	—
228	C₂²×C₄	#6101	1	64	yes	complemented
229	C₂²×C₄	C₂×Dic₂₆	1	24	yes	—
230	C₂²×C₄	C₂×Dic₂₆	1	24	yes	—
231	C₂²×C₄	C₂×C₄×D₁₃	1	24	yes	—
232	C₂²×C₄	C₂×D₅₂	1	24	yes	—
233	C₂²×C₄	C₂×C₂²⋊D₁₃	1	24	yes	—
234	C₂²×C₄	C₂³×D₁₃	1	4	yes	—
235	C₂²×C₄	C₂³×D₁₃	1	4	yes	—
236	C₂²×C₄	C₂³×D₁₃	1	4	yes	—
237	C₄⋊C₄	—	13	64	no	—
238	C₄⋊C₄	—	26	64	no	—
239	C₄⋊C₄	—	26	32	no	—
240	C₄²	—	2	32	no	—
241	C₁₃	C₂²×#389	1	1	yes	characteristic, complemented, Sylow
242	C₂³	C₄×Dic₂₆	1	16	yes	—
243	C₂³	#88123	1	16	yes	—
244	C₂³	#88124	1	16	yes	—
245	C₂³	C₄×D₅₂	1	16	yes	—
246	C₂³	C₂×#6101	1	16	yes	—
247	C₂³	#88178	1	16	yes	—
248	C₂³	#88184	1	16	yes	—
249	C₂³	C₂²×Dic₂₆	1	6	yes	—
250	C₂³	C₂²×Dic₂₆	1	6	yes	—
251	C₂³	C₂²×C₄×D₁₃	1	6	yes	—
252	C₂³	C₂²×D₅₂	1	6	yes	—
253	C₂³	C₂×#6127	1	6	yes	—
254	C₂³	C₂×#6127	1	6	yes	—
255	C₂³	C₂²×C₂²⋊D₁₃	1	6	yes	—
256	C₂³	C₂⁴×D₁₃	1	1	yes	characteristic
257	C₂×C₄	—	2	48	no	—
258	C₂×C₄	—	2	48	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	13	16	no	—
264	C₂×C₄	—	26	96	no	—
265	C₂×C₄	—	26	8	no	—
266	C₂×C₄	—	26	8	no	—
267	C₂×C₄	—	26	8	no	—
268	C₂×C₄	—	26	8	no	—
269	C₂×C₄	—	26	8	no	—
270	C₂×C₄	—	52	48	no	—
271	C₂×C₄	C₂×#6101	1	96	yes	complemented
272	C₂×C₄	C₂²×Dic₂₆	1	8	yes	—
273	C₂×C₄	C₂²×Dic₂₆	1	8	yes	—
274	C₂×C₄	C₂²×C₄×D₁₃	1	8	yes	—
275	C₂×C₄	C₂²×D₅₂	1	8	yes	—
276	C₂×C₄	C₂²×C₂²⋊D₁₃	1	8	yes	—
277	C₂²	#11718589	1	64	yes	direct product factor
278	C₂²	C₂×C₄×Dic₂₆	1	12	yes	—
279	C₂²	C₂×#88123	1	12	yes	—
280	C₂²	C₂×#88124	1	12	yes	—
281	C₂²	C₂×C₄×D₅₂	1	12	yes	—
282	C₂²	C₂²×#6101	1	12	yes	—
283	C₂²	C₂×#88178	1	12	yes	—
284	C₂²	C₂×#88184	1	12	yes	—
285	C₂²	C₂³×Dic₂₆	1	1	yes	characteristic
286	C₂²	C₂³×Dic₂₆	1	1	yes	characteristic
287	C₂²	C₂³×C₄×D₁₃	1	1	yes	characteristic
288	C₂²	C₂³×D₅₂	1	1	yes	characteristic
289	C₂²	C₂²×#6127	1	1	yes	characteristic
290	C₂²	C₂²×#6127	1	1	yes	characteristic
291	C₂²	C₂³×C₂²⋊D₁₃	1	1	yes	characteristic
292	C₄	—	2	8	no	—
293	C₄	—	2	8	no	—
294	C₄	—	26	16	no	—
295	C₄	—	52	8	no	—
296	C₄	C₂²×#6101	1	16	yes	complemented
297	C₂	C₂×#11718589	1	24	yes	direct product factor
298	C₂	C₂³×#6101	1	1	yes	characteristic
299	C₂	C₂²×C₄×D₅₂	1	1	yes	characteristic
300	C₂	C₂²×C₄×Dic₂₆	1	1	yes	characteristic
301	C₂	C₂²×#88184	1	1	yes	characteristic
302	C₂	C₂²×#88178	1	1	yes	characteristic
303	C₂	C₂²×#88124	1	1	yes	characteristic
304	C₂	C₂²×#88123	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₁₃
commutator	C₂²×C₁₃	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₁₃
Fitting	C₂³×C₄²×C₁₃	C₂
socle, prime radical	C₂⁵×C₁₃	C₂³
cosocle	C₂³×C₁₃	C₂⁵
hypercenter	C₂³×C₄²	D₁₃
hypocenter	C₁₃	C₂²×#389
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
4. C₂×#11718589	C₂	direct (N × H)
15. C₂²×C₄⋊Dic₁₃	C₄	semidirect
21. #11718589	C₂²	direct (N × H)
46. C₂×C₄⋊Dic₁₃	C₂×C₄	semidirect
67. C₂²×C₄×C₁₃	C₄⋊C₄	semidirect
88. C₄⋊Dic₁₃	C₂²×C₄	semidirect
117. C₂×C₄×C₁₃	C₂×C₄⋊C₄	semidirect
146. C₂²×C₁₃	#389	semidirect
163. C₄×C₁₃	C₂²×C₄⋊C₄	semidirect
196. C₂×C₁₃	C₂×#389	semidirect
228. C₂²×C₄	#6101	semidirect
241. C₁₃	C₂²×#389	semidirect
271. C₂×C₄	C₂×#6101	semidirect
277. C₂²	#11718589	direct (N × H)
296. C₄	C₂²×#6101	semidirect
297. C₂	C₂×#11718589	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×C₄⋊Dic₁₃	2
C₂³×#6101	2
C₂×#11718589	2
C₂²×#88071	2
C₂³×C₄²×C₁₃	2
C₂²×#389	13

Subgroups — C22×#11718589 #16021266

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718589 #16021266