Subgroups — C₂³×#88138 #16769555

Order 3328.

19998 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₅₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×C₂²⋊D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂⁴×C₂²⋊D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁴×C₄×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#88138	C₂	1	56	yes	maximal, direct product factor
6	C₂³×#6101	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×#6103	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₁₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁶×C₁₃	C₂²	1	1	yes	characteristic
10	C₂⁵×D₁₃	C₂²	1	1	yes	characteristic
11	C₂⁵×D₁₃	C₂²	1	1	yes	characteristic
12	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
13	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
14	C₂³×C₂²⋊D₁₃	—	2	16	no	—
15	C₂³×C₂²⋊D₁₃	—	2	16	no	—
16	C₂³×C₂²⋊D₁₃	C₂²	1	28	yes	complemented
17	C₂³×C₂²⋊D₁₃	C₂²	1	28	yes	complemented
18	C₂⁴×Dic₁₃	C₂²	1	1	yes	characteristic
19	C₂⁴×Dic₁₃	C₂²	1	1	yes	characteristic
20	C₂³×D₅₂	—	2	16	no	—
21	C₂³×D₅₂	C₂²	1	28	yes	complemented
22	C₂³×C₄×D₁₃	—	2	16	no	—
23	C₂³×C₄×D₁₃	C₂²	1	28	yes	complemented
24	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	28	yes	complemented
25	C₂²×#6103	C₂²	1	28	yes	complemented
26	C₂²×#6101	C₂²	1	28	yes	complemented
27	C₂×#88138	C₂²	1	448	yes	direct product factor
28	C₂⁵×C₁₃	—	2	8	no	—
29	C₂⁵×C₁₃	—	2	8	no	—
30	C₂⁵×C₁₃	—	2	8	no	—
31	C₂⁵×C₁₃	C₂³	1	14	yes	—
32	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
33	C₂⁴×D₁₃	—	2	8	no	—
34	C₂⁴×D₁₃	—	2	8	no	—
35	C₂⁴×D₁₃	—	2	8	no	—
36	C₂⁴×D₁₃	—	2	8	no	—
37	C₂⁴×D₁₃	—	2	8	no	—
38	C₂⁴×D₁₃	D₄	1	16	yes	complemented
39	C₂⁴×D₁₃	C₂³	1	14	yes	—
40	C₂⁴×D₁₃	C₂³	1	14	yes	—
41	C₂³×C₄×C₁₃	—	2	8	no	—
42	C₂³×C₄×C₁₃	—	2	8	no	—
43	C₂³×C₄×C₁₃	C₂³	1	14	yes	—
44	C₂³×C₄×C₁₃	C₂³	1	14	yes	—
45	C₂²×C₂²⋊D₁₃	—	2	224	no	—
46	C₂²×C₂²⋊D₁₃	—	2	224	no	—
47	C₂²×C₂²⋊D₁₃	C₂³	1	112	yes	complemented
48	C₂²×C₂²⋊D₁₃	C₂³	1	112	yes	complemented
49	C₂³×Dic₁₃	—	2	8	no	—
50	C₂³×Dic₁₃	D₄	1	16	yes	complemented
51	C₂³×Dic₁₃	C₂³	1	14	yes	—
52	C₂³×Dic₁₃	C₂³	1	14	yes	—
53	C₂²×D₅₂	—	2	224	no	—
54	C₂²×D₅₂	C₂³	1	112	yes	complemented
55	C₂²×C₄×D₁₃	—	2	224	no	—
56	C₂²×C₄×D₁₃	C₂³	1	112	yes	complemented
57	C₂×C₁₃×C₂²⋊C₄	C₂³	1	112	yes	complemented
58	C₂×#6103	C₂³	1	112	yes	complemented
59	C₂×#6101	C₂³	1	112	yes	complemented
60	#88138	C₂³	1	512	yes	direct product factor
61	C₂³×#121	—	13	1	no	maximal, Sylow
62	C₂⁴×C₁₃	—	2	56	no	—
63	C₂⁴×C₁₃	—	2	56	no	—
64	C₂⁴×C₁₃	—	2	56	no	—
65	C₂⁴×C₁₃	—	4	64	no	—
66	C₂⁴×C₁₃	C₂×D₄	1	8	yes	—
67	C₂⁴×C₁₃	C₂×D₄	1	8	yes	—
68	C₂⁴×C₁₃	#41	1	8	yes	—
69	C₂⁴×C₁₃	C₂⁴	1	28	yes	—
70	C₂⁴×C₁₃	C₂⁴	1	7	yes	—
71	C₂³×D₁₃	—	2	128	no	—
72	C₂³×D₁₃	—	2	56	no	—
73	C₂³×D₁₃	—	2	56	no	—
74	C₂³×D₁₃	—	2	56	no	—
75	C₂³×D₁₃	—	2	56	no	—
76	C₂³×D₁₃	—	2	56	no	—
77	C₂³×D₁₃	—	4	64	no	—
78	C₂³×D₁₃	C₂×D₄	1	112	yes	complemented
79	C₂³×D₁₃	C₂⁴	1	28	yes	—
80	C₂³×D₁₃	C₂⁴	1	28	yes	—
81	C₂²×C₄×C₁₃	—	2	56	no	—
82	C₂²×C₄×C₁₃	—	2	56	no	—
83	C₂²×C₄×C₁₃	C₂⁴	1	28	yes	—
84	C₂²×C₄×C₁₃	C₂⁴	1	28	yes	—
85	C₂×C₂²⋊D₁₃	—	2	448	no	—
86	C₂×C₂²⋊D₁₃	—	2	448	no	—
87	C₂×C₂²⋊D₁₃	C₂⁴	1	64	yes	complemented
88	C₂×C₂²⋊D₁₃	C₂⁴	1	64	yes	complemented
89	C₂²×Dic₁₃	—	2	56	no	—
90	C₂²×Dic₁₃	C₂×D₄	1	112	yes	complemented
91	C₂²×Dic₁₃	C₂⁴	1	28	yes	—
92	C₂²×Dic₁₃	C₂⁴	1	28	yes	—
93	C₂×D₅₂	—	2	448	no	—
94	C₂×D₅₂	C₂⁴	1	64	yes	complemented
95	C₂×C₄×D₁₃	—	2	448	no	—
96	C₂×C₄×D₁₃	C₂⁴	1	64	yes	complemented
97	C₁₃×C₂²⋊C₄	C₂⁴	1	64	yes	complemented
98	#6103	C₂⁴	1	64	yes	complemented
99	#6101	C₂⁴	1	64	yes	complemented
100	C₂⁴×D₄	—	13	1	no	—
101	C₂⁴×D₄	—	13	1	no	—
102	C₂⁴×D₄	—	13	1	no	—
103	C₂⁵×C₄	—	13	1	no	—
104	C₂²×#121	—	13	56	no	—
105	C₂³×C₄⋊C₄	—	13	1	no	—
106	C₂³×C₂²⋊C₄	—	13	1	no	—
107	C₂³×C₂²⋊C₄	D₁₃	1	1	yes	characteristic, complemented
108	C₂³×C₁₃	—	2	56	no	—
109	C₂³×C₁₃	—	2	56	no	—
110	C₂³×C₁₃	—	2	56	no	—
111	C₂³×C₁₃	—	4	224	no	—
112	C₂³×C₁₃	#121	1	64	yes	complemented
113	C₂³×C₁₃	C₂²×D₄	1	28	yes	—
114	C₂³×C₁₃	C₂²×D₄	1	28	yes	—
115	C₂³×C₁₃	C₂×#41	1	28	yes	—
116	C₂³×C₁₃	C₂⁵	1	8	yes	—
117	C₂³×C₁₃	C₂⁵	1	7	yes	—
118	C₂²×D₁₃	—	2	448	no	—
119	C₂²×D₁₃	—	2	56	no	—
120	C₂²×D₁₃	—	2	56	no	—
121	C₂²×D₁₃	—	2	56	no	—
122	C₂²×D₁₃	—	2	56	no	—
123	C₂²×D₁₃	—	2	56	no	—
124	C₂²×D₁₃	—	4	224	no	—
125	C₂²×D₁₃	C₂²×D₄	1	112	yes	complemented
126	C₂²×D₁₃	C₂⁵	1	8	yes	—
127	C₂²×D₁₃	C₂⁵	1	8	yes	—
128	C₂×C₄×C₁₃	—	2	56	no	—
129	C₂×C₄×C₁₃	—	2	56	no	—
130	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
131	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
132	C₂²⋊D₁₃	—	2	128	no	—
133	C₂²⋊D₁₃	—	2	128	no	—
134	C₂×Dic₁₃	—	2	56	no	—
135	C₂×Dic₁₃	C₂²×D₄	1	112	yes	complemented
136	C₂×Dic₁₃	C₂⁵	1	8	yes	—
137	C₂×Dic₁₃	C₂⁵	1	8	yes	—
138	D₅₂	—	2	128	no	—
139	C₄×D₁₃	—	2	128	no	—
140	C₂⁶	—	13	1	no	—
141	C₂⁶	—	13	1	no	—
142	C₂⁶	C₂×D₁₃	1	1	yes	characteristic
143	C₂³×D₄	—	13	28	no	—
144	C₂³×D₄	—	13	28	no	—
145	C₂³×D₄	—	13	28	no	—
146	C₂³×D₄	—	26	16	no	—
147	C₂³×D₄	—	26	16	no	—
148	C₂³×D₄	—	26	16	no	—
149	C₂⁴×C₄	—	13	1	no	—
150	C₂⁴×C₄	—	13	1	no	—
151	C₂⁴×C₄	—	13	28	no	—
152	C₂⁴×C₄	—	26	16	no	—
153	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
154	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
155	C₂×#121	—	13	448	no	—
156	C₂²×C₄⋊C₄	—	13	28	no	—
157	C₂²×C₂²⋊C₄	—	13	28	no	—
158	C₂²×C₂²⋊C₄	C₂×D₁₃	1	28	yes	complemented
159	C₂²×C₁₃	—	2	8	no	—
160	C₂²×C₁₃	—	2	8	no	—
161	C₂²×C₁₃	—	2	8	no	—
162	C₂²×C₁₃	—	4	112	no	—
163	C₂²×C₁₃	C₂×#121	1	112	yes	complemented
164	C₂²×C₁₃	C₂³×D₄	1	14	yes	—
165	C₂²×C₁₃	C₂³×D₄	1	14	yes	—
166	C₂²×C₁₃	C₂²×#41	1	14	yes	—
167	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
168	C₂×D₁₃	—	2	8	no	—
169	C₂×D₁₃	—	2	8	no	—
170	C₂×D₁₃	—	2	224	no	—
171	C₂×D₁₃	—	2	8	no	—
172	C₂×D₁₃	—	2	8	no	—
173	C₂×D₁₃	—	2	8	no	—
174	C₂×D₁₃	—	4	112	no	—
175	C₂×D₁₃	C₂³×D₄	1	16	yes	complemented
176	C₄×C₁₃	—	2	8	no	—
177	C₄×C₁₃	—	2	8	no	—
178	Dic₁₃	—	2	8	no	—
179	Dic₁₃	C₂³×D₄	1	16	yes	complemented
180	C₂⁵	—	2	8	no	—
181	C₂⁵	—	2	8	no	—
182	C₂⁵	—	2	8	no	—
183	C₂⁵	—	13	16	no	—
184	C₂⁵	—	13	14	no	—
185	C₂⁵	—	13	14	no	—
186	C₂⁵	—	26	8	no	—
187	C₂⁵	—	26	8	no	—
188	C₂⁵	—	26	8	no	—
189	C₂⁵	—	26	8	no	—
190	C₂⁵	—	26	8	no	—
191	C₂⁵	C₂²×D₁₃	1	14	yes	—
192	C₂⁵	C₂²×D₁₃	1	1	yes	characteristic
193	C₂²×D₄	—	13	112	no	—
194	C₂²×D₄	—	13	112	no	—
195	C₂²×D₄	—	13	112	no	—
196	C₂²×D₄	—	26	224	no	—
197	C₂²×D₄	—	26	224	no	—
198	C₂²×D₄	—	26	224	no	—
199	C₂³×C₄	—	2	8	no	—
200	C₂³×C₄	—	2	8	no	—
201	C₂³×C₄	—	13	16	no	—
202	C₂³×C₄	—	13	14	no	—
203	C₂³×C₄	—	13	14	no	—
204	C₂³×C₄	—	13	112	no	—
205	C₂³×C₄	—	26	224	no	—
206	C₂³×C₄	—	26	8	no	—
207	C₂³×C₄	C₂²×D₁₃	1	14	yes	—
208	C₂³×C₄	C₂²×D₁₃	1	14	yes	—
209	#121	—	13	512	no	—
210	C₂×C₄⋊C₄	—	13	112	no	—
211	C₂×C₂²⋊C₄	—	13	112	no	—
212	C₂×C₂²⋊C₄	C₂²×D₁₃	1	112	yes	complemented
213	C₂×C₁₃	—	4	8	no	—
214	C₂×C₁₃	C₂²×#121	1	28	yes	complemented
215	C₂×C₁₃	C₂⁴×D₄	1	1	yes	characteristic
216	C₂×C₁₃	C₂⁴×D₄	1	1	yes	characteristic
217	C₂×C₁₃	C₂³×#41	1	1	yes	characteristic
218	D₁₃	—	2	16	no	—
219	D₁₃	—	4	8	no	—
220	C₂⁴	—	2	56	no	—
221	C₂⁴	—	2	56	no	—
222	C₂⁴	—	2	56	no	—
223	C₂⁴	—	4	64	no	—
224	C₂⁴	—	13	112	no	—
225	C₂⁴	—	13	28	no	—
226	C₂⁴	—	13	28	no	—
227	C₂⁴	—	26	128	no	—
228	C₂⁴	—	26	56	no	—
229	C₂⁴	—	26	56	no	—
230	C₂⁴	—	26	56	no	—
231	C₂⁴	—	26	56	no	—
232	C₂⁴	—	26	56	no	—
233	C₂⁴	—	52	64	no	—
234	C₂⁴	#6127	1	8	yes	—
235	C₂⁴	D₄×D₁₃	1	8	yes	—
236	C₂⁴	D₄×D₁₃	1	8	yes	—
237	C₂⁴	C₂³×D₁₃	1	28	yes	—
238	C₂⁴	C₂³×D₁₃	1	7	yes	—
239	C₂×D₄	—	13	64	no	—
240	C₂×D₄	—	13	64	no	—
241	C₂×D₄	—	13	64	no	—
242	C₂×D₄	—	26	448	no	—
243	C₂×D₄	—	26	448	no	—
244	C₂×D₄	—	26	448	no	—
245	C₂²×C₄	—	2	56	no	—
246	C₂²×C₄	—	2	56	no	—
247	C₂²×C₄	—	13	112	no	—
248	C₂²×C₄	—	13	28	no	—
249	C₂²×C₄	—	13	28	no	—
250	C₂²×C₄	—	13	64	no	—
251	C₂²×C₄	—	26	448	no	—
252	C₂²×C₄	—	26	56	no	—
253	C₂²×C₄	C₂³×D₁₃	1	28	yes	—
254	C₂²×C₄	C₂³×D₁₃	1	28	yes	—
255	C₄⋊C₄	—	13	64	no	—
256	C₂²⋊C₄	—	13	64	no	—
257	C₂²⋊C₄	C₂³×D₁₃	1	64	yes	complemented
258	C₁₃	C₂³×#121	1	1	yes	characteristic, complemented, Sylow
259	C₂³	—	2	56	no	—
260	C₂³	—	2	56	no	—
261	C₂³	—	2	56	no	—
262	C₂³	—	4	224	no	—
263	C₂³	—	13	112	no	—
264	C₂³	—	13	8	no	—
265	C₂³	—	13	8	no	—
266	C₂³	—	26	448	no	—
267	C₂³	—	26	56	no	—
268	C₂³	—	26	56	no	—
269	C₂³	—	26	56	no	—
270	C₂³	—	26	56	no	—
271	C₂³	—	26	56	no	—
272	C₂³	—	52	224	no	—
273	C₂³	#88138	1	64	yes	direct product factor
274	C₂³	C₂×#6127	1	28	yes	—
275	C₂³	C₂×D₄×D₁₃	1	28	yes	—
276	C₂³	C₂×D₄×D₁₃	1	28	yes	—
277	C₂³	C₂⁴×D₁₃	1	8	yes	—
278	C₂³	C₂⁴×D₁₃	1	7	yes	—
279	D₄	—	26	128	no	—
280	D₄	—	26	128	no	—
281	D₄	—	26	128	no	—
282	C₂×C₄	—	2	56	no	—
283	C₂×C₄	—	2	56	no	—
284	C₂×C₄	—	13	112	no	—
285	C₂×C₄	—	13	8	no	—
286	C₂×C₄	—	13	8	no	—
287	C₂×C₄	—	26	128	no	—
288	C₂×C₄	—	26	56	no	—
289	C₂×C₄	C₂⁴×D₁₃	1	8	yes	—
290	C₂×C₄	C₂⁴×D₁₃	1	8	yes	—
291	C₂²	—	2	8	no	—
292	C₂²	—	2	8	no	—
293	C₂²	—	2	8	no	—
294	C₂²	—	4	112	no	—
295	C₂²	—	13	16	no	—
296	C₂²	—	26	224	no	—
297	C₂²	—	26	8	no	—
298	C₂²	—	26	8	no	—
299	C₂²	—	26	8	no	—
300	C₂²	—	26	8	no	—
301	C₂²	—	26	8	no	—
302	C₂²	—	52	112	no	—
303	C₂²	C₂×#88138	1	112	yes	direct product factor
304	C₂²	C₂²×#6127	1	14	yes	—
305	C₂²	C₂²×D₄×D₁₃	1	14	yes	—
306	C₂²	C₂²×D₄×D₁₃	1	14	yes	—
307	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
308	C₄	—	2	8	no	—
309	C₄	—	2	8	no	—
310	C₄	—	13	16	no	—
311	C₄	—	26	8	no	—
312	C₂	—	4	8	no	—
313	C₂	—	26	16	no	—
314	C₂	—	52	8	no	—
315	C₂	C₂²×#88138	1	28	yes	direct product factor
316	C₂	C₂³×D₄×D₁₃	1	1	yes	characteristic
317	C₂	C₂³×D₄×D₁₃	1	1	yes	characteristic
318	C₂	C₂³×#6127	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁶
Frattini	C₂²	C₂⁵×D₁₃
Fitting	C₂³×C₁₃×C₂²⋊C₄	C₂
socle	C₂⁵×C₁₃	C₂³
hypercenter	C₂³×C₂²⋊C₄	D₁₃
hypocenter	C₁₃	C₂³×#121
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂⁴×D₅₂	C₂	semidirect
2. C₂⁴×C₂²⋊D₁₃	C₂	semidirect
3. C₂⁴×C₂²⋊D₁₃	C₂	semidirect
4. C₂⁴×C₄×D₁₃	C₂	semidirect
5. C₂²×#88138	C₂	direct (N × H)
6. C₂³×#6101	C₂	semidirect
7. C₂³×#6103	C₂	semidirect
8. C₂³×C₁₃×C₂²⋊C₄	C₂	semidirect
16. C₂³×C₂²⋊D₁₃	C₂²	semidirect
17. C₂³×C₂²⋊D₁₃	C₂²	semidirect
21. C₂³×D₅₂	C₂²	semidirect
23. C₂³×C₄×D₁₃	C₂²	semidirect
24. C₂²×C₁₃×C₂²⋊C₄	C₂²	semidirect
25. C₂²×#6103	C₂²	semidirect
26. C₂²×#6101	C₂²	semidirect
27. C₂×#88138	C₂²	direct (N × H)
38. C₂⁴×D₁₃	D₄	semidirect
47. C₂²×C₂²⋊D₁₃	C₂³	semidirect
48. C₂²×C₂²⋊D₁₃	C₂³	semidirect
50. C₂³×Dic₁₃	D₄	semidirect
54. C₂²×D₅₂	C₂³	semidirect
56. C₂²×C₄×D₁₃	C₂³	semidirect
57. C₂×C₁₃×C₂²⋊C₄	C₂³	semidirect
58. C₂×#6103	C₂³	semidirect
59. C₂×#6101	C₂³	semidirect
60. #88138	C₂³	direct (N × H)
78. C₂³×D₁₃	C₂×D₄	semidirect
87. C₂×C₂²⋊D₁₃	C₂⁴	semidirect
88. C₂×C₂²⋊D₁₃	C₂⁴	semidirect
90. C₂²×Dic₁₃	C₂×D₄	semidirect
94. C₂×D₅₂	C₂⁴	semidirect
96. C₂×C₄×D₁₃	C₂⁴	semidirect
97. C₁₃×C₂²⋊C₄	C₂⁴	semidirect
98. #6103	C₂⁴	semidirect
99. #6101	C₂⁴	semidirect
107. C₂³×C₂²⋊C₄	D₁₃	semidirect
112. C₂³×C₁₃	#121	semidirect
125. C₂²×D₁₃	C₂²×D₄	semidirect
135. C₂×Dic₁₃	C₂²×D₄	semidirect
158. C₂²×C₂²⋊C₄	C₂×D₁₃	semidirect
163. C₂²×C₁₃	C₂×#121	semidirect
175. C₂×D₁₃	C₂³×D₄	semidirect
179. Dic₁₃	C₂³×D₄	semidirect
212. C₂×C₂²⋊C₄	C₂²×D₁₃	semidirect
214. C₂×C₁₃	C₂²×#121	semidirect
257. C₂²⋊C₄	C₂³×D₁₃	semidirect
258. C₁₃	C₂³×#121	semidirect
273. C₂³	#88138	direct (N × H)
303. C₂²	C₂×#88138	direct (N × H)
315. C₂	C₂²×#88138	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂⁴×D₅₂	2
C₂⁴×C₂²⋊D₁₃	2
C₂⁴×C₄×D₁₃	2
C₂²×#88138	2
C₂³×#6101	2
C₂³×#6103	2
C₂³×C₁₃×C₂²⋊C₄	2
C₂³×#121	13

Subgroups — C23×#88138 #16769555

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#88138 #16769555