Subgroups — C₄×#11719092 #16797192

Order 3328.

1182 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719092	C₂	1	2	yes	maximal
2	C₄×#88237	C₂	1	2	yes	maximal
3	C₄×#88102	C₂	1	4	yes	maximal, complemented
4	C₄×#88101	C₂	1	4	yes	maximal, complemented
5	#13130522	C₂	1	2	yes	maximal
6	C₂×C₄×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#88237	C₂²	1	4	yes	—
8	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
9	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
11	C₄×C₂²⋊#261	—	2	4	no	—
12	#11719092	C₄	1	16	yes	direct product factor
13	C₂×#88102	C₂²	1	4	yes	—
14	C₂×#88101	C₂²	1	4	yes	—
15	C₄×C₄⋊#261	C₂²	1	8	yes	—
16	#11719057	—	2	2	no	—
17	C₄²×#261	—	2	2	no	—
18	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic, complemented
19	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic, complemented
20	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic, complemented
21	C₂×C₄×#1055	C₄	1	1	yes	characteristic, complemented
22	C₄×C₈×D₁₃	C₄	1	2	yes	complemented
23	C₄×C₈×D₁₃	C₄	1	2	yes	complemented
24	C₄×C₈×D₁₃	C₂²	1	2	yes	—
25	C₄×C₈×D₁₃	C₂²	1	2	yes	—
26	#11718265	C₂²	1	8	yes	—
27	C₂²×C₄×D₁₃	D₄	1	2	yes	—
28	C₂²×C₄×D₁₃	C₂³	1	1	yes	characteristic
29	C₂×C₂²⋊#261	—	2	4	no	—
30	#88237	—	2	8	no	—
31	#88237	C₂×C₄	1	16	yes	—
32	C₂×C₄⋊#261	C₂³	1	8	yes	—
33	C₂×C₄×#261	—	2	4	no	—
34	C₂×C₄×#261	—	2	8	no	—
35	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×Dic₁₃	Q₈	1	2	yes	—
40	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
41	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
42	C₂×C₈×D₁₃	—	2	4	no	—
43	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
44	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₈×D₁₃	C₂×C₄	1	8	yes	—
47	C₂×C₈×D₁₃	C₂³	1	2	yes	—
48	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
49	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
50	C₄²×D₁₃	C₂×C₄	1	2	yes	—
51	C₄²×D₁₃	D₄	1	1	yes	characteristic
52	C₄²×D₁₃	D₄	1	1	yes	characteristic
53	C₄²×D₁₃	Q₈	1	1	yes	characteristic
54	C₄²×D₁₃	Q₈	1	1	yes	characteristic
55	C₄²×D₁₃	C₂³	1	2	yes	—
56	(C₂×#261)⋊C₄	—	2	8	no	—
57	#88102	C₂×C₄	1	16	yes	complemented
58	#88101	C₂×C₄	1	16	yes	complemented
59	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
61	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
62	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
63	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
64	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
65	C₄×#1055	C₂×C₄	1	2	yes	—
66	C₄×#1055	C₂×C₄	1	2	yes	—
67	C₄×#427	—	13	1	no	maximal, Sylow
68	C₂³×D₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×#261	—	2	8	no	—
70	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×Dic₁₃	C₄⋊C₄	1	2	yes	—
73	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
74	C₂×C₄×D₁₃	—	2	8	no	—
75	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
76	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
77	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×D₁₃	C₂²×C₄	1	4	yes	—
79	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
80	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
82	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
83	C₂×C₄×D₁₃	#41	1	4	yes	—
84	C₂×C₄×D₁₃	C₂⁴	1	1	yes	characteristic
85	C₂²⋊#261	—	2	16	no	—
86	C₄⋊#261	—	2	8	no	—
87	C₄⋊#261	C₂²×C₄	1	16	yes	—
88	C₄×#261	—	2	16	no	—
89	C₄×#261	—	4	16	no	—
90	C₂×C₈×C₁₃	—	2	2	no	—
91	C₂×C₈×C₁₃	C₄²	1	4	yes	complemented
92	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
96	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
97	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
98	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
99	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
100	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
101	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
102	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
103	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
104	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
105	C₂×#1055	—	2	2	no	—
106	C₂×#1055	C₄²	1	4	yes	complemented
107	C₂×#1055	C₂²×C₄	1	2	yes	—
108	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
109	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
110	C₈×D₁₃	—	2	4	no	—
111	C₈×D₁₃	—	2	4	no	—
112	C₈×D₁₃	C₄²	1	4	yes	complemented
113	C₈×D₁₃	C₄²	1	4	yes	complemented
114	C₈×D₁₃	C₂²×C₄	1	4	yes	—
115	C₈×D₁₃	C₂²×C₄	1	4	yes	—
116	C₂×#427	—	13	2	no	—
117	C₂²×C₄×C₈	—	13	1	no	—
118	C₄×#117	—	13	2	no	—
119	C₄×#107	—	13	4	no	—
120	C₄×#106	—	13	4	no	—
121	#1735	—	13	2	no	—
122	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂²×D₁₃	—	2	4	no	—
124	C₂²×D₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂²×D₁₃	C₄×D₄	1	4	yes	—
126	C₂²×D₁₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×#261	—	2	16	no	—
128	C₂×#261	—	4	16	no	—
129	C₂×C₄×C₁₃	—	2	4	no	—
130	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
131	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
132	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
134	C₂×C₄×C₁₃	#117	1	2	yes	—
135	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
136	C₂×Dic₁₃	—	2	4	no	—
137	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
138	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
140	C₂×Dic₁₃	C₄×Q₈	1	4	yes	—
141	C₂×Dic₁₃	C₂²×Q₈	1	1	yes	characteristic
142	C₄×D₁₃	—	2	4	no	—
143	C₄×D₁₃	—	2	4	no	—
144	C₄×D₁₃	—	2	4	no	—
145	C₄×D₁₃	C₂×C₄²	1	2	yes	—
146	C₄×D₁₃	#117	1	2	yes	—
147	C₄×D₁₃	C₄×D₄	1	4	yes	—
148	C₄×D₁₃	C₄×Q₈	1	4	yes	—
149	C₄×D₁₃	#135	1	8	yes	—
150	C₄×D₁₃	C₂³×C₄	1	2	yes	—
151	C₄×D₁₃	C₂×#41	1	2	yes	—
152	C₈×C₁₃	—	2	2	no	—
153	C₈×C₁₃	C₂×C₄²	1	2	yes	—
154	C₈×C₁₃	C₂×C₄²	1	2	yes	—
155	#1055	—	2	2	no	—
156	#1055	C₂×C₄²	1	2	yes	—
157	#1055	C₂×C₄²	1	2	yes	—
158	C₂³×C₈	—	13	1	no	—
159	C₂³×C₈	—	13	1	no	—
160	C₂×#117	—	13	4	no	—
161	C₂²×C₄²	—	13	1	no	—
162	#427	—	13	16	no	—
163	C₂×#107	—	13	4	no	—
164	C₂×#106	—	13	4	no	—
165	C₂×C₄×C₈	—	13	1	no	—
166	C₂×C₄×C₈	—	13	1	no	—
167	C₂×C₄×C₈	—	13	1	no	—
168	C₂×C₄×C₈	—	13	2	no	—
169	C₂×C₄×C₈	—	13	2	no	—
170	C₂×C₄×C₈	—	13	2	no	—
171	C₂×C₄×C₈	—	13	2	no	—
172	C₂×C₄×C₈	#261	1	1	yes	characteristic, complemented
173	C₄×C₄⋊C₄	—	13	8	no	—
174	C₄×C₂²⋊C₄	—	26	4	no	—
175	#376	—	26	2	no	—
176	C₄³	—	26	2	no	—
177	#340	—	13	8	no	—
178	C₂²×C₁₃	—	2	2	no	—
179	C₂²×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
180	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
181	C₂×D₁₃	—	2	2	no	—
182	C₂×D₁₃	—	2	8	no	—
183	C₂×D₁₃	C₄×C₄⋊C₄	1	2	yes	—
184	C₂×D₁₃	C₂×C₄×D₄	1	2	yes	—
185	C₂×D₁₃	C₂×#135	1	4	yes	—
186	#261	—	4	16	no	—
187	C₄×C₁₃	—	2	2	no	—
188	C₄×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
189	C₄×C₁₃	#427	1	4	yes	complemented
190	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
191	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
192	Dic₁₃	—	2	2	no	—
193	Dic₁₃	C₄×C₄⋊C₄	1	2	yes	—
194	Dic₁₃	#427	1	4	yes	complemented
195	Dic₁₃	C₂×C₄×Q₈	1	2	yes	—
196	C₂³×C₄	—	13	1	no	—
197	C₂³×C₄	—	13	2	no	—
198	C₂²×C₈	—	13	1	no	—
199	C₂²×C₈	—	13	8	no	—
200	C₂²×C₈	—	13	1	no	—
201	C₂²×C₈	—	13	1	no	—
202	C₂²×C₈	—	13	1	no	—
203	C₂²×C₈	—	13	1	no	—
204	C₂²×C₈	—	13	1	no	—
205	C₂²×C₈	—	13	2	no	—
206	C₂²×C₈	—	13	2	no	—
207	C₂²×C₈	—	26	4	no	—
208	C₂²×C₈	C₂×#261	1	1	yes	characteristic
209	C₂²×C₈	C₂×#261	1	1	yes	characteristic
210	#117	—	13	16	no	—
211	#117	—	26	8	no	—
212	C₂×C₄⋊C₄	—	13	8	no	—
213	C₂×C₂²⋊C₄	—	26	4	no	—
214	C₂×C₄²	—	13	2	no	—
215	C₂×C₄²	—	13	1	no	—
216	C₂×C₄²	—	13	1	no	—
217	C₂×C₄²	—	13	1	no	—
218	C₂×C₄²	—	13	1	no	—
219	C₂×C₄²	—	13	1	no	—
220	C₂×C₄²	—	13	2	no	—
221	C₂×C₄²	—	13	2	no	—
222	C₂×C₄²	—	26	4	no	—
223	C₂×C₄²	—	26	8	no	—
224	C₂×C₄²	C₂×#261	1	1	yes	characteristic
225	#107	—	13	16	no	—
226	#106	—	13	16	no	—
227	C₄×C₈	—	13	2	no	—
228	C₄×C₈	—	13	2	no	—
229	C₄×C₈	—	13	2	no	—
230	C₄×C₈	—	13	2	no	—
231	C₄×C₈	—	13	2	no	—
232	C₄×C₈	—	13	2	no	—
233	C₄×C₈	C₂×#261	1	2	yes	—
234	C₄×C₈	C₂×#261	1	2	yes	—
235	#95	—	26	8	no	—
236	C₂×C₁₃	—	2	2	no	—
237	C₂×C₁₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
238	C₂×C₁₃	C₂×#427	1	2	yes	—
239	D₁₃	—	2	2	no	—
240	D₁₃	C₄×#135	1	4	yes	—
241	C₂⁴	—	13	1	no	—
242	C₂²×C₄	—	13	1	no	—
243	C₂²×C₄	—	13	4	no	—
244	C₂²×C₄	—	13	1	no	—
245	C₂²×C₄	—	13	2	no	—
246	C₂²×C₄	—	13	1	no	—
247	C₂²×C₄	—	13	4	no	—
248	C₂²×C₄	—	13	1	no	—
249	C₂²×C₄	—	13	1	no	—
250	C₂²×C₄	—	13	2	no	—
251	C₂²×C₄	—	13	2	no	—
252	C₂²×C₄	—	13	2	no	—
253	C₂²×C₄	—	13	2	no	—
254	C₂²×C₄	—	26	8	no	—
255	C₂²×C₄	—	26	8	no	—
256	C₂²×C₄	C₄⋊#261	1	2	yes	—
257	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	13	4	no	—
260	C₂×C₈	—	13	1	no	—
261	C₂×C₈	—	13	1	no	—
262	C₂×C₈	—	13	4	no	—
263	C₂×C₈	—	13	4	no	—
264	C₂×C₈	—	13	4	no	—
265	C₂×C₈	—	13	4	no	—
266	C₂×C₈	—	13	2	no	—
267	C₂×C₈	—	26	4	no	—
268	C₂×C₈	—	26	4	no	—
269	C₂×C₈	—	26	2	no	—
270	C₂×C₈	C₄×#261	1	4	yes	complemented
271	C₂×C₈	C₂²×#261	1	2	yes	—
272	C₂×C₈	C₂²×#261	1	1	yes	characteristic
273	C₂×C₈	C₂²×#261	1	1	yes	characteristic
274	C₄⋊C₄	—	13	16	no	—
275	C₄⋊C₄	—	26	8	no	—
276	C₂²⋊C₄	—	26	16	no	—
277	C₄²	—	13	1	no	—
278	C₄²	—	13	2	no	—
279	C₄²	—	13	2	no	—
280	C₄²	—	13	1	no	—
281	C₄²	—	13	2	no	—
282	C₄²	—	13	2	no	—
283	C₄²	—	13	2	no	—
284	C₄²	—	26	16	no	—
285	C₄²	—	52	16	no	—
286	C₄²	C₄⋊#261	1	1	yes	characteristic
287	C₄²	C₄⋊#261	1	1	yes	characteristic
288	C₄²	C₂²×#261	1	2	yes	—
289	C₁₃	C₄×#427	1	1	yes	characteristic, complemented, Sylow
290	C₂³	—	13	1	no	—
291	C₂³	—	13	4	no	—
292	C₂³	—	13	1	no	—
293	C₂³	—	26	4	no	—
294	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	13	4	no	—
297	C₂×C₄	—	13	1	no	—
298	C₂×C₄	—	13	2	no	—
299	C₂×C₄	—	13	1	no	—
300	C₂×C₄	—	13	4	no	—
301	C₂×C₄	—	13	4	no	—
302	C₂×C₄	—	13	8	no	—
303	C₂×C₄	—	13	2	no	—
304	C₂×C₄	—	13	2	no	—
305	C₂×C₄	—	13	2	no	—
306	C₂×C₄	—	13	2	no	—
307	C₂×C₄	—	13	2	no	—
308	C₂×C₄	—	26	4	no	—
309	C₂×C₄	—	26	4	no	—
310	C₂×C₄	—	26	4	no	—
311	C₂×C₄	—	26	4	no	—
312	C₂×C₄	—	26	16	no	—
313	C₂×C₄	—	52	16	no	—
314	C₂×C₄	C₂×C₄×#261	1	2	yes	—
315	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
316	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
317	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
318	C₂×C₄	#88237	1	2	yes	—
319	C₂×C₄	C₂³×#261	1	1	yes	characteristic
320	C₈	—	2	2	no	—
321	C₈	—	13	2	no	—
322	C₈	—	13	2	no	—
323	C₈	—	26	2	no	—
324	C₈	C₂×C₄×#261	1	2	yes	—
325	C₈	C₂×C₄×#261	1	2	yes	—
326	C₂²	—	2	2	no	—
327	C₂²	—	13	2	no	—
328	C₂²	—	13	4	no	—
329	C₂²	—	13	2	no	—
330	C₂²	—	26	8	no	—
331	C₂²	—	26	2	no	—
332	C₂²	C₄×C₄⋊#261	1	2	yes	—
333	C₂²	C₂²×C₄⋊#261	1	1	yes	characteristic
334	C₄	—	2	2	no	—
335	C₄	—	13	2	no	—
336	C₄	—	13	2	no	—
337	C₄	—	13	4	no	—
338	C₄	—	26	2	no	—
339	C₄	—	52	16	no	—
340	C₄	C₄×C₄⋊#261	1	2	yes	—
341	C₄	#11719092	1	4	yes	direct product factor
342	C₄	C₂²×C₄×#261	1	1	yes	characteristic
343	C₄	C₂×#88237	1	1	yes	characteristic
344	C₂	—	2	2	no	—
345	C₂	—	13	4	no	—
346	C₂	—	26	2	no	—
347	C₂	C₂×C₄×C₄⋊#261	1	1	yes	characteristic
348	C₂	C₂×#11719092	1	2	yes	—

	subgroup	G/H
center	C₄²	C₄⋊#261
commutator	C₄×C₁₃	C₂²×C₄²
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₁₃	C₂×D₄
cosocle	C₂×C₄×D₁₃	C₂⁴
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	C₄×#427
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₄×#88102	C₂	semidirect
4. C₄×#88101	C₂	semidirect
12. #11719092	C₄	direct (N × H)
18. C₂×C₄×C₈×C₁₃	C₄	semidirect
19. C₂×C₈×Dic₁₃	C₄	semidirect
20. C₂×C₈×Dic₁₃	C₄	semidirect
21. C₂×C₄×#1055	C₄	semidirect
22. C₄×C₈×D₁₃	C₄	semidirect
23. C₄×C₈×D₁₃	C₄	semidirect
57. #88102	C₂×C₄	semidirect
58. #88101	C₂×C₄	semidirect
91. C₂×C₈×C₁₃	C₄²	semidirect
106. C₂×#1055	C₄²	semidirect
112. C₈×D₁₃	C₄²	semidirect
113. C₈×D₁₃	C₄²	semidirect
172. C₂×C₄×C₈	#261	semidirect
189. C₄×C₁₃	#427	semidirect
194. Dic₁₃	#427	semidirect
270. C₂×C₈	C₄×#261	semidirect
289. C₁₃	C₄×#427	semidirect
341. C₄	#11719092	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×#11719092	2
C₄×#88237	2
C₄×#88102	2
C₄×#88101	2
#13130522	2
C₂×C₄×C₈×D₁₃	2
C₄×#427	13

Subgroups — C4×#11719092 #16797192

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11719092 #16797192