Subgroups — C₂²×#11719536 #16769855

Order 3328.

12142 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₈×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11719536	C₂	1	48	yes	maximal, direct product factor
3	C₂²×#88190	C₂	1	3	yes	maximal, complemented
4	C₂²×#88178	C₂	1	3	yes	maximal, complemented
5	C₂²×#88192	C₂	1	3	yes	maximal, complemented
6	C₂²×Q₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×C₂²⋊D₁₃	C₂	1	3	yes	maximal, complemented
8	C₂³×C₁₃×#41	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₁₃×#41	—	2	16	no	—
10	C₂²×C₁₃×#41	C₂²	1	24	yes	complemented
11	C₂³×Q₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂³×D₄×C₁₃	C₂²	1	3	yes	—
13	C₂⁴×C₄×C₁₃	C₂²	1	3	yes	—
14	C₂³×C₂²⋊D₁₃	C₂²	1	3	yes	—
15	C₂²×Q₈×D₁₃	—	2	16	no	—
16	C₂²×Q₈×D₁₃	C₂²	1	24	yes	complemented
17	C₂³×C₄×D₁₃	C₂²	1	3	yes	—
18	C₂³×Dic₂₆	C₂²	1	3	yes	—
19	C₂²×C₂²⋊Dic₁₃	C₂²	1	3	yes	—
20	C₂²×C₂²⋊Dic₁₃	C₂²	1	6	yes	—
21	#11719536	C₂²	1	256	yes	direct product factor
22	C₂×Q₈×Dic₁₃	C₂²	1	24	yes	complemented
23	C₂×#88192	C₂²	1	72	yes	complemented
24	C₂×#88190	C₂²	1	72	yes	complemented
25	C₂×C₄×C₂²⋊D₁₃	C₂²	1	72	yes	complemented
26	C₂²×#6103	C₂²	1	1	yes	characteristic
27	C₂²×C₄⋊Dic₁₃	C₂²	1	3	yes	—
28	C₂×#88178	C₂²	1	72	yes	complemented
29	C₂²×#6101	C₂²	1	3	yes	—
30	C₂²×C₄×Dic₁₃	C₂²	1	3	yes	—
31	C₂⁵×C₁₃	C₂³	1	3	yes	—
32	C₂⁴×D₁₃	C₂³	1	1	yes	characteristic
33	C₂×C₁₃×#41	—	2	192	no	—
34	C₂×C₁₃×#41	C₂³	1	64	yes	complemented
35	C₂²×Q₈×C₁₃	D₄	1	16	yes	complemented
36	C₂²×Q₈×C₁₃	C₂³	1	12	yes	—
37	C₂²×D₄×C₁₃	—	2	24	no	—
38	C₂²×D₄×C₁₃	C₂³	1	36	yes	—
39	C₂³×C₄×C₁₃	—	2	24	no	—
40	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
41	C₂³×C₄×C₁₃	C₂³	1	36	yes	—
42	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
43	C₂²×C₂²⋊D₁₃	—	2	24	no	—
44	C₂²×C₂²⋊D₁₃	C₂³	1	36	yes	—
45	C₂³×Dic₁₃	C₂³	1	1	yes	characteristic
46	C₂³×Dic₁₃	C₂³	1	3	yes	—
47	C₂³×Dic₁₃	C₂³	1	3	yes	—
48	C₂×Q₈×D₁₃	—	2	192	no	—
49	C₂×Q₈×D₁₃	C₂³	1	64	yes	complemented
50	C₂²×C₄×D₁₃	—	2	24	no	—
51	C₂²×C₄×D₁₃	C₂³	1	36	yes	—
52	C₂²×Dic₂₆	—	2	24	no	—
53	C₂²×Dic₂₆	C₂³	1	36	yes	—
54	C₂×C₂²⋊Dic₁₃	C₂³	1	72	yes	—
55	C₂×C₂²⋊Dic₁₃	C₂³	1	36	yes	—
56	Q₈×Dic₁₃	C₂³	1	64	yes	complemented
57	#88192	C₂³	1	192	yes	complemented
58	#88190	C₂³	1	192	yes	complemented
59	C₄×C₂²⋊D₁₃	C₂³	1	192	yes	complemented
60	C₂×#6103	C₂³	1	12	yes	—
61	C₂×C₄⋊Dic₁₃	C₂³	1	36	yes	—
62	#88178	C₂³	1	192	yes	complemented
63	C₂×#6101	C₂³	1	36	yes	—
64	C₂×C₄×Dic₁₃	C₂³	1	36	yes	—
65	C₂²×#548	—	13	1	no	maximal, Sylow
66	C₂⁴×C₁₃	—	2	12	no	—
67	C₂⁴×C₁₃	—	2	12	no	—
68	C₂⁴×C₁₃	—	2	12	no	—
69	C₂⁴×C₁₃	C₂⁴	1	18	yes	—
70	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
71	C₂³×D₁₃	—	2	4	no	—
72	C₂³×D₁₃	—	2	4	no	—
73	C₂³×D₁₃	#41	1	8	yes	complemented
74	C₂³×D₁₃	C₂⁴	1	6	yes	—
75	C₁₃×#41	—	2	256	no	—
76	C₂×Q₈×C₁₃	C₂×D₄	1	96	yes	complemented
77	C₂×Q₈×C₁₃	C₂⁴	1	16	yes	—
78	C₂×D₄×C₁₃	—	2	144	no	—
79	C₂×D₄×C₁₃	C₂⁴	1	48	yes	—
80	C₂²×C₄×C₁₃	—	2	4	no	—
81	C₂²×C₄×C₁₃	—	2	144	no	—
82	C₂²×C₄×C₁₃	C₂×D₄	1	24	yes	—
83	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
84	C₂²×C₄×C₁₃	C₂⁴	1	48	yes	—
85	C₂²×C₄×C₁₃	C₂⁴	1	18	yes	—
86	C₂×C₂²⋊D₁₃	—	2	144	no	—
87	C₂×C₂²⋊D₁₃	C₂⁴	1	48	yes	—
88	C₂²×Dic₁₃	—	2	12	no	—
89	C₂²×Dic₁₃	—	2	12	no	—
90	C₂²×Dic₁₃	—	2	4	no	—
91	C₂²×Dic₁₃	C₂⁴	1	6	yes	—
92	C₂²×Dic₁₃	C₂⁴	1	18	yes	—
93	C₂²×Dic₁₃	C₂⁴	1	18	yes	—
94	Q₈×D₁₃	—	2	256	no	—
95	C₂×C₄×D₁₃	—	2	144	no	—
96	C₂×C₄×D₁₃	C₂⁴	1	48	yes	—
97	C₂×Dic₂₆	—	2	144	no	—
98	C₂×Dic₂₆	C₂⁴	1	48	yes	—
99	C₂²⋊Dic₁₃	C₂⁴	1	96	yes	—
100	C₂²⋊Dic₁₃	C₂⁴	1	48	yes	—
101	#6103	C₂⁴	1	16	yes	—
102	C₄⋊Dic₁₃	C₂⁴	1	48	yes	—
103	#6101	C₂⁴	1	48	yes	—
104	C₄×Dic₁₃	C₂⁴	1	48	yes	—
105	C₂³×#41	D₁₃	1	1	yes	characteristic, complemented
106	C₂⁴×Q₈	—	13	1	no	—
107	C₂×#548	—	13	48	no	—
108	C₂²×#124	—	13	3	no	—
109	C₂²×C₂²⋊Q₈	—	13	3	no	—
110	C₂²×#121	—	13	3	no	—
111	C₂²×C₄×Q₈	—	13	1	no	—
112	C₂²×C₄×D₄	—	13	3	no	—
113	C₂³×C₁₃	—	2	36	no	—
114	C₂³×C₁₃	—	2	36	no	—
115	C₂³×C₁₃	—	2	36	no	—
116	C₂³×C₁₃	—	4	48	no	—
117	C₂³×C₁₃	C₂²×D₄	1	4	yes	—
118	C₂³×C₁₃	C₂×#41	1	4	yes	—
119	C₂³×C₁₃	2₋¹⁺⁴	1	4	yes	—
120	C₂³×C₁₃	C₂⁵	1	12	yes	—
121	C₂³×C₁₃	C₂⁵	1	3	yes	—
122	C₂²×D₁₃	—	2	32	no	—
123	C₂²×D₁₃	—	2	12	no	—
124	C₂²×D₁₃	—	2	12	no	—
125	C₂²×D₁₃	C₂×#41	1	24	yes	complemented
126	C₂²×D₁₃	C₂⁵	1	4	yes	—
127	Q₈×C₁₃	C₂²×D₄	1	64	yes	complemented
128	D₄×C₁₃	—	2	96	no	—
129	C₂×C₄×C₁₃	—	2	12	no	—
130	C₂×C₄×C₁₃	—	2	96	no	—
131	C₂×C₄×C₁₃	C₂²×D₄	1	72	yes	—
132	C₂×C₄×C₁₃	C₂⁵	1	4	yes	—
133	C₂×C₄×C₁₃	C₂⁵	1	12	yes	—
134	C₂²⋊D₁₃	—	2	96	no	—
135	C₂×Dic₁₃	—	2	36	no	—
136	C₂×Dic₁₃	—	2	36	no	—
137	C₂×Dic₁₃	—	2	12	no	—
138	C₂×Dic₁₃	C₂⁵	1	4	yes	—
139	C₂×Dic₁₃	C₂⁵	1	12	yes	—
140	C₂×Dic₁₃	C₂⁵	1	12	yes	—
141	C₄×D₁₃	—	2	96	no	—
142	Dic₂₆	—	2	96	no	—
143	C₂²×#41	—	2	16	no	—
144	C₂²×#41	C₂×D₁₃	1	24	yes	complemented
145	C₂³×Q₈	—	13	24	no	—
146	C₂³×Q₈	—	13	3	no	—
147	C₂³×Q₈	—	26	16	no	—
148	C₂³×Q₈	C₂×D₁₃	1	1	yes	characteristic
149	C₂³×D₄	—	13	3	no	—
150	C₂³×D₄	C₂×D₁₃	1	3	yes	—
151	C₂⁴×C₄	—	13	3	no	—
152	C₂⁴×C₄	C₂×D₁₃	1	3	yes	—
153	#548	—	13	256	no	—
154	C₂×#124	—	13	72	no	—
155	C₂×C₂²⋊Q₈	—	13	72	no	—
156	C₂×#121	—	13	72	no	—
157	C₂×C₄×Q₈	—	13	24	no	—
158	C₂×C₄×D₄	—	13	72	no	—
159	C₂²×C₄⋊C₄	—	13	3	no	—
160	C₂²×C₄⋊C₄	—	13	3	no	—
161	C₂²×C₂²⋊C₄	—	13	3	no	—
162	C₂²×C₂²⋊C₄	—	13	6	no	—
163	C₂²×C₂²⋊C₄	—	13	1	no	—
164	C₂²×C₄²	—	13	3	no	—
165	C₂²×C₁₃	—	2	12	no	—
166	C₂²×C₁₃	—	2	12	no	—
167	C₂²×C₁₃	—	2	12	no	—
168	C₂²×C₁₃	—	4	72	no	—
169	C₂²×C₁₃	#548	1	16	yes	complemented
170	C₂²×C₁₃	C₂³×D₄	1	6	yes	—
171	C₂²×C₁₃	C₂²×#41	1	6	yes	—
172	C₂²×C₁₃	C₂×2₋¹⁺⁴	1	6	yes	—
173	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
174	C₂×D₁₃	—	2	4	no	—
175	C₂×D₁₃	—	2	4	no	—
176	C₂×D₁₃	—	2	48	no	—
177	C₂×D₁₃	C₂²×#41	1	8	yes	complemented
178	C₄×C₁₃	—	2	4	no	—
179	C₄×C₁₃	C₂³×D₄	1	24	yes	—
180	Dic₁₃	—	2	12	no	—
181	Dic₁₃	—	2	12	no	—
182	Dic₁₃	—	2	4	no	—
183	C₂⁵	—	13	1	no	—
184	C₂⁵	C₂²×D₁₃	1	3	yes	—
185	C₂×#41	—	2	192	no	—
186	C₂×#41	C₂²×D₁₃	1	64	yes	complemented
187	C₂²×Q₈	—	13	64	no	—
188	C₂²×Q₈	—	13	36	no	—
189	C₂²×Q₈	—	26	192	no	—
190	C₂²×Q₈	—	26	24	no	—
191	C₂²×Q₈	C₂²⋊D₁₃	1	16	yes	complemented
192	C₂²×Q₈	C₂²×D₁₃	1	12	yes	—
193	C₂²×D₄	—	2	24	no	—
194	C₂²×D₄	—	13	36	no	—
195	C₂²×D₄	—	26	24	no	—
196	C₂²×D₄	C₂²×D₁₃	1	36	yes	—
197	C₂³×C₄	—	2	24	no	—
198	C₂³×C₄	—	13	1	no	—
199	C₂³×C₄	—	13	3	no	—
200	C₂³×C₄	—	13	3	no	—
201	C₂³×C₄	—	13	36	no	—
202	C₂³×C₄	—	26	24	no	—
203	C₂³×C₄	C₂²×D₁₃	1	36	yes	—
204	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
205	C₂³×C₄	C₂²×D₁₃	1	3	yes	—
206	#124	—	13	192	no	—
207	C₂²⋊Q₈	—	13	192	no	—
208	#121	—	13	192	no	—
209	C₄×Q₈	—	13	64	no	—
210	C₄×D₄	—	13	192	no	—
211	C₂×C₄⋊C₄	—	13	36	no	—
212	C₂×C₄⋊C₄	—	13	36	no	—
213	C₂×C₂²⋊C₄	—	13	36	no	—
214	C₂×C₂²⋊C₄	—	13	72	no	—
215	C₂×C₂²⋊C₄	—	13	12	no	—
216	C₂×C₄²	—	13	36	no	—
217	C₂×C₁₃	—	4	12	no	—
218	C₂×C₁₃	C₂×#548	1	12	yes	complemented
219	C₂×C₁₃	C₂⁴×D₄	1	1	yes	characteristic
220	C₂×C₁₃	C₂³×#41	1	1	yes	characteristic
221	C₂×C₁₃	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
222	D₁₃	—	2	8	no	—
223	C₂⁴	—	2	12	no	—
224	C₂⁴	—	2	12	no	—
225	C₂⁴	—	2	12	no	—
226	C₂⁴	—	13	8	no	—
227	C₂⁴	—	13	6	no	—
228	C₂⁴	—	26	4	no	—
229	C₂⁴	—	26	4	no	—
230	C₂⁴	C₂³×D₁₃	1	18	yes	—
231	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
232	#41	—	2	256	no	—
233	C₂×Q₈	—	13	48	no	—
234	C₂×Q₈	—	26	256	no	—
235	C₂×Q₈	—	26	144	no	—
236	C₂×Q₈	C₂×C₂²⋊D₁₃	1	96	yes	complemented
237	C₂×Q₈	C₂³×D₁₃	1	16	yes	—
238	C₂×D₄	—	2	144	no	—
239	C₂×D₄	—	13	48	no	—
240	C₂×D₄	—	26	144	no	—
241	C₂×D₄	C₂³×D₁₃	1	48	yes	—
242	C₂²×C₄	—	2	4	no	—
243	C₂²×C₄	—	2	144	no	—
244	C₂²×C₄	—	13	6	no	—
245	C₂²×C₄	—	13	18	no	—
246	C₂²×C₄	—	13	18	no	—
247	C₂²×C₄	—	13	48	no	—
248	C₂²×C₄	—	26	12	no	—
249	C₂²×C₄	—	26	4	no	—
250	C₂²×C₄	—	26	12	no	—
251	C₂²×C₄	—	26	144	no	—
252	C₂²×C₄	C₂×C₂²⋊D₁₃	1	24	yes	—
253	C₂²×C₄	C₂³×D₁₃	1	6	yes	—
254	C₂²×C₄	C₂³×D₁₃	1	48	yes	—
255	C₂²×C₄	C₂³×D₁₃	1	18	yes	—
256	C₄⋊C₄	—	13	48	no	—
257	C₄⋊C₄	—	13	48	no	—
258	C₂²⋊C₄	—	13	48	no	—
259	C₂²⋊C₄	—	13	96	no	—
260	C₂²⋊C₄	—	13	16	no	—
261	C₄²	—	13	48	no	—
262	C₁₃	C₂²×#548	1	1	yes	characteristic, complemented, Sylow
263	C₂³	—	2	36	no	—
264	C₂³	—	2	36	no	—
265	C₂³	—	2	36	no	—
266	C₂³	—	4	48	no	—
267	C₂³	—	13	24	no	—
268	C₂³	—	13	4	no	—
269	C₂³	—	26	32	no	—
270	C₂³	—	26	12	no	—
271	C₂³	—	26	12	no	—
272	C₂³	#41×D₁₃	1	4	yes	—
273	C₂³	#88257	1	4	yes	—
274	C₂³	C₂²×C₂²⋊D₁₃	1	4	yes	—
275	C₂³	C₂⁴×D₁₃	1	12	yes	—
276	C₂³	C₂⁴×D₁₃	1	3	yes	—
277	Q₈	—	26	96	no	—
278	Q₈	C₂²×C₂²⋊D₁₃	1	64	yes	complemented
279	D₄	—	2	96	no	—
280	D₄	—	26	96	no	—
281	C₂×C₄	—	2	12	no	—
282	C₂×C₄	—	2	96	no	—
283	C₂×C₄	—	13	4	no	—
284	C₂×C₄	—	13	12	no	—
285	C₂×C₄	—	13	12	no	—
286	C₂×C₄	—	26	36	no	—
287	C₂×C₄	—	26	12	no	—
288	C₂×C₄	—	26	36	no	—
289	C₂×C₄	—	26	96	no	—
290	C₂×C₄	C₂²×C₂²⋊D₁₃	1	72	yes	—
291	C₂×C₄	C₂⁴×D₁₃	1	4	yes	—
292	C₂×C₄	C₂⁴×D₁₃	1	12	yes	—
293	C₂²	—	2	12	no	—
294	C₂²	—	2	12	no	—
295	C₂²	—	2	12	no	—
296	C₂²	—	4	72	no	—
297	C₂²	—	13	8	no	—
298	C₂²	—	26	48	no	—
299	C₂²	—	26	4	no	—
300	C₂²	—	26	4	no	—
301	C₂²	#11719536	1	16	yes	direct product factor
302	C₂²	C₂×#41×D₁₃	1	6	yes	—
303	C₂²	C₂×#88257	1	6	yes	—
304	C₂²	C₂³×C₂²⋊D₁₃	1	6	yes	—
305	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
306	C₄	—	2	4	no	—
307	C₄	—	26	12	no	—
308	C₄	—	26	12	no	—
309	C₄	—	26	4	no	—
310	C₄	C₂³×C₂²⋊D₁₃	1	24	yes	—
311	C₂	—	4	12	no	—
312	C₂	—	26	8	no	—
313	C₂	C₂×#11719536	1	12	yes	direct product factor
314	C₂	C₂⁴×C₂²⋊D₁₃	1	1	yes	characteristic
315	C₂	C₂²×#41×D₁₃	1	1	yes	characteristic
316	C₂	C₂²×#88257	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁶
Frattini	C₂²	C₂⁵×D₁₃
Fitting	C₂³×C₁₃×#41	C₂
socle	C₂⁴×C₁₃	C₂⁴
hypercenter	C₂³×#41	D₁₃
hypocenter	C₁₃	C₂²×#548
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×#41

Complements

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

N	H ≅ G/N	type
1. C₂³×Q₈×D₁₃	C₂	semidirect
2. C₂×#11719536	C₂	direct (N × H)
3. C₂²×#88190	C₂	semidirect
4. C₂²×#88178	C₂	semidirect
5. C₂²×#88192	C₂	semidirect
6. C₂²×Q₈×Dic₁₃	C₂	semidirect
7. C₂²×C₄×C₂²⋊D₁₃	C₂	semidirect
8. C₂³×C₁₃×#41	C₂	semidirect
10. C₂²×C₁₃×#41	C₂²	semidirect
16. C₂²×Q₈×D₁₃	C₂²	semidirect
21. #11719536	C₂²	direct (N × H)
22. C₂×Q₈×Dic₁₃	C₂²	semidirect
23. C₂×#88192	C₂²	semidirect
24. C₂×#88190	C₂²	semidirect
25. C₂×C₄×C₂²⋊D₁₃	C₂²	semidirect
28. C₂×#88178	C₂²	semidirect
34. C₂×C₁₃×#41	C₂³	semidirect
35. C₂²×Q₈×C₁₃	D₄	semidirect
49. C₂×Q₈×D₁₃	C₂³	semidirect
56. Q₈×Dic₁₃	C₂³	semidirect
57. #88192	C₂³	semidirect
58. #88190	C₂³	semidirect
59. C₄×C₂²⋊D₁₃	C₂³	semidirect
62. #88178	C₂³	semidirect
73. C₂³×D₁₃	#41	semidirect
76. C₂×Q₈×C₁₃	C₂×D₄	semidirect
105. C₂³×#41	D₁₃	semidirect
125. C₂²×D₁₃	C₂×#41	semidirect
127. Q₈×C₁₃	C₂²×D₄	semidirect
144. C₂²×#41	C₂×D₁₃	semidirect
169. C₂²×C₁₃	#548	semidirect
177. C₂×D₁₃	C₂²×#41	semidirect
186. C₂×#41	C₂²×D₁₃	semidirect
191. C₂²×Q₈	C₂²⋊D₁₃	semidirect
218. C₂×C₁₃	C₂×#548	semidirect
236. C₂×Q₈	C₂×C₂²⋊D₁₃	semidirect
262. C₁₃	C₂²×#548	semidirect
278. Q₈	C₂²×C₂²⋊D₁₃	semidirect
301. C₂²	#11719536	direct (N × H)
313. C₂	C₂×#11719536	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×Q₈×D₁₃	2
C₂×#11719536	2
C₂²×#88190	2
C₂²×#88178	2
C₂²×#88192	2
C₂²×Q₈×Dic₁₃	2
C₂²×C₄×C₂²⋊D₁₃	2
C₂³×C₁₃×#41	2
C₂²×#548	13

Subgroups — C22×#11719536 #16769855

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11719536 #16769855