Subgroups — C₄×#11719158 #16797457

Order 3328.

958 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719158	C₂	1	2	yes	maximal
2	#13130636	C₂	1	4	yes	maximal
3	#13130632	C₂	1	2	yes	maximal
4	C₄×#88112	C₂	1	2	yes	maximal
5	C₄×C₄⋊#1057	C₂	1	1	yes	maximal, characteristic
6	C₄×#88110	C₂	1	2	yes	maximal
7	C₄²×#1057	C₂	1	1	yes	maximal, characteristic
8	C₄×Q₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Q₈×C₁₃	C₄	1	1	yes	characteristic
10	C₂×Q₈×Dic₁₃	C₄	1	1	yes	characteristic
11	C₂×Q₈×Dic₁₃	C₂²	1	1	yes	characteristic
12	#11719158	C₄	1	16	yes	direct product factor
13	C₂×#88112	C₂²	1	4	yes	—
14	C₂×#88110	C₂²	1	4	yes	—
15	C₂×#88110	C₂²	1	4	yes	—
16	C₂×C₄⋊#1057	C₂²	1	2	yes	—
17	C₂×C₄×#1057	C₂²	1	2	yes	—
18	C₂×C₄×#1057	C₂²	1	1	yes	characteristic
19	C₂×C₄×#1057	C₂²	1	2	yes	—
20	C₂×C₄×#1057	C₂²	1	2	yes	—
21	C₂×C₄×#1057	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×Dic₁₃	C₄	1	2	yes	—
23	C₄⋊C₄×Dic₁₃	C₄	1	1	yes	characteristic
24	C₄⋊C₄×Dic₁₃	C₂²	1	1	yes	characteristic
25	C₄⋊C₄×Dic₁₃	C₂²	1	2	yes	—
26	C₄×C₄⋊Dic₁₃	C₄	1	2	yes	—
27	C₄×C₄⋊Dic₁₃	C₄	1	1	yes	characteristic
28	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
29	C₄²×Dic₁₃	C₂²	1	2	yes	—
30	#11718290	C₂²	1	2	yes	—
31	#11718290	C₂²	1	4	yes	—
32	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×#1057	C₂³	1	2	yes	—
34	C₂²×#1057	C₂³	1	4	yes	—
35	C₂²×#1057	C₂³	1	2	yes	—
36	C₄×Q₈×C₁₃	—	2	4	no	—
37	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
38	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
41	Q₈×Dic₁₃	—	2	4	no	—
42	Q₈×Dic₁₃	C₂×C₄	1	8	yes	—
43	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
45	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
48	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
49	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
51	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
54	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
55	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
56	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
57	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
58	#88112	C₂×C₄	1	16	yes	—
59	#88110	—	2	8	no	—
60	#88110	C₂×C₄	1	16	yes	—
61	C₄⋊#1057	C₂×C₄	1	8	yes	—
62	C₄×#1057	—	2	4	no	—
63	C₄×#1057	—	2	4	no	—
64	C₄×#1057	—	2	2	no	—
65	C₄×#1057	C₂×C₄	1	8	yes	—
66	C₄×#1057	Q₈	1	4	yes	complemented
67	#88071	C₂×C₄	1	2	yes	—
68	#88071	C₂×C₄	1	2	yes	—
69	#88071	C₂×C₄	1	2	yes	—
70	C₄×#446	—	13	1	no	maximal, Sylow
71	C₂×Q₈×C₁₃	—	2	2	no	—
72	C₂×Q₈×C₁₃	—	2	2	no	—
73	C₂×Q₈×C₁₃	C₄²	1	4	yes	—
74	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
80	C₂²×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
82	C₂²×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×Dic₁₃	C₂⁴	1	1	yes	characteristic
85	C₂×#1057	—	2	2	no	—
86	C₂×#1057	—	2	2	no	—
87	C₂×#1057	—	2	4	no	—
88	C₂×#1057	—	2	4	no	—
89	C₂×#1057	C₂²×C₄	1	4	yes	—
90	C₂×#1057	C₂²×C₄	1	8	yes	—
91	C₂×#1057	C₂²×C₄	1	4	yes	—
92	C₂×#1057	C₂×Q₈	1	4	yes	—
93	C₂×#1057	#41	1	4	yes	—
94	C₁₃×C₄⋊C₄	—	2	4	no	—
95	C₁₃×C₄⋊C₄	—	2	2	no	—
96	C₄²×C₁₃	—	2	4	no	—
97	C₄²×C₁₃	M₁₆	1	4	yes	—
98	C₄⋊Dic₁₃	—	2	2	no	—
99	C₄⋊Dic₁₃	—	2	4	no	—
100	C₄⋊Dic₁₃	C₄²	1	8	yes	—
101	C₄⋊Dic₁₃	C₄²	1	4	yes	—
102	C₄×Dic₁₃	—	2	2	no	—
103	C₄×Dic₁₃	—	2	4	no	—
104	C₄×Dic₁₃	—	2	4	no	—
105	C₄×Dic₁₃	—	2	2	no	—
106	C₄×Dic₁₃	—	2	4	no	—
107	C₄×Dic₁₃	—	2	2	no	—
108	C₄×Dic₁₃	—	2	4	no	—
109	C₄×Dic₁₃	M₁₆	1	4	yes	—
110	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
111	C₄×Dic₁₃	C₂²×C₄	1	8	yes	—
112	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
113	C₄×Dic₁₃	#41	1	2	yes	—
114	C₂×#446	—	13	2	no	—
115	C₄²×Q₈	—	13	1	no	—
116	#1771	—	13	4	no	—
117	#1770	—	13	2	no	—
118	C₄×C₄⋊C₈	—	13	1	no	—
119	C₄×C₄⋊C₈	—	13	2	no	—
120	C₄×#97	—	13	2	no	—
121	C₄²×C₈	—	13	1	no	—
122	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
123	Q₈×C₁₃	—	2	8	no	—
124	C₂×C₄×C₁₃	—	2	1	no	—
125	C₂×C₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	—	2	1	no	—
128	C₂×C₄×C₁₃	—	2	2	no	—
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	—	2	2	no	—
131	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
132	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
133	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
134	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
135	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
136	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
137	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
138	C₂×C₄×C₁₃	#131	1	2	yes	—
139	C₂×Dic₁₃	—	2	1	no	—
140	C₂×Dic₁₃	—	2	1	no	—
141	C₂×Dic₁₃	—	2	2	no	—
142	C₂×Dic₁₃	—	2	1	no	—
143	C₂×Dic₁₃	—	2	2	no	—
144	C₂×Dic₁₃	—	2	1	no	—
145	C₂×Dic₁₃	—	2	2	no	—
146	C₂×Dic₁₃	—	2	2	no	—
147	C₂×Dic₁₃	—	2	2	no	—
148	C₂×Dic₁₃	C₂×C₄²	1	2	yes	—
149	C₂×Dic₁₃	C₂×C₄²	1	4	yes	—
150	C₂×Dic₁₃	C₄×Q₈	1	2	yes	—
151	C₂×Dic₁₃	C₂×M₁₆	1	2	yes	—
152	C₂×Dic₁₃	C₂×M₁₆	1	2	yes	—
153	C₂×Dic₁₃	#131	1	2	yes	—
154	C₂×Dic₁₃	C₂³×C₄	1	2	yes	—
155	C₂×Dic₁₃	C₂²×Q₈	1	1	yes	characteristic
156	C₂×Dic₁₃	C₂×#41	1	2	yes	—
157	C₂×Dic₁₃	C₂×#41	1	1	yes	characteristic
158	#1057	—	2	4	no	—
159	#1057	—	2	8	no	—
160	#1057	C₄×Q₈	1	8	yes	complemented
161	C₂×C₄×Q₈	—	13	1	no	—
162	C₂×C₄×Q₈	—	13	1	no	—
163	C₂×C₄×Q₈	#261	1	1	yes	characteristic
164	#446	—	13	16	no	—
165	C₂×C₄⋊C₈	—	13	2	no	—
166	C₂×C₄⋊C₈	—	13	4	no	—
167	C₂×#97	—	13	4	no	—
168	C₂×#97	—	13	4	no	—
169	C₂×C₄×C₈	—	13	2	no	—
170	C₂×C₄×C₈	—	13	1	no	—
171	C₂×C₄×C₈	—	13	1	no	—
172	C₂×C₄×C₈	—	13	2	no	—
173	C₂×C₄×C₈	—	13	2	no	—
174	C₄×C₄⋊C₄	—	13	1	no	—
175	C₄×C₄⋊C₄	—	13	2	no	—
176	C₄×C₄⋊C₄	—	13	1	no	—
177	C₄×C₄⋊C₄	—	13	2	no	—
178	C₄×C₄⋊C₄	—	13	2	no	—
179	C₄×C₄⋊C₄	—	13	1	no	—
180	C₄³	—	13	1	no	—
181	C₄³	—	13	2	no	—
182	#336	—	13	4	no	—
183	#336	—	13	2	no	—
184	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
185	C₂²×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
186	C₂²×C₁₃	C₄×#41	1	1	yes	characteristic
187	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
188	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
189	C₂²×C₁₃	C₂×#131	1	1	yes	characteristic
190	C₂²×C₁₃	C₂×#131	1	1	yes	characteristic
191	C₄×C₁₃	—	2	4	no	—
192	C₄×C₁₃	—	2	2	no	—
193	C₄×C₁₃	—	4	2	no	—
194	C₄×C₁₃	C₄×M₁₆	1	4	yes	—
195	C₄×C₁₃	#446	1	4	yes	complemented
196	Dic₁₃	—	2	2	no	—
197	Dic₁₃	—	2	4	no	—
198	Dic₁₃	—	2	2	no	—
199	Dic₁₃	—	4	2	no	—
200	Dic₁₃	#446	1	4	yes	complemented
201	Dic₁₃	C₂×C₄×Q₈	1	2	yes	—
202	Dic₁₃	C₄×#41	1	2	yes	—
203	C₂²×Q₈	C₂×#261	1	1	yes	characteristic
204	C₂²×C₈	—	13	2	no	—
205	C₂²×C₈	—	13	2	no	—
206	C₂²×C₈	—	13	4	no	—
207	C₄×Q₈	—	2	4	no	—
208	C₄×Q₈	—	13	8	no	—
209	C₄×Q₈	—	26	4	no	—
210	C₂×C₄⋊C₄	—	13	2	no	—
211	C₂×C₄⋊C₄	—	13	1	no	—
212	C₂×C₄⋊C₄	—	13	2	no	—
213	C₂×C₄⋊C₄	—	13	1	no	—
214	C₂×C₄⋊C₄	C₂×#261	1	1	yes	characteristic
215	C₂×C₄⋊C₄	C₂×#261	1	2	yes	—
216	C₂×C₄²	—	13	1	no	—
217	C₂×C₄²	—	13	1	no	—
218	C₂×C₄²	—	13	2	no	—
219	C₂×C₄²	—	13	1	no	—
220	C₂×C₄²	—	13	2	no	—
221	C₂×C₄²	—	13	1	no	—
222	C₂×C₄²	—	13	2	no	—
223	C₂×C₄²	—	13	2	no	—
224	C₂×C₄²	—	13	1	no	—
225	C₂×C₄²	—	13	1	no	—
226	C₂×C₄²	—	13	2	no	—
227	C₂×C₄²	C₂×#261	1	2	yes	—
228	C₂×C₄²	C₂×#261	1	1	yes	characteristic
229	C₄⋊C₈	—	13	8	no	—
230	C₄⋊C₈	—	13	16	no	—
231	#97	—	13	16	no	—
232	#97	—	26	8	no	—
233	C₄×C₈	—	13	8	no	—
234	C₄×C₈	—	13	4	no	—
235	C₄×C₈	—	26	2	no	—
236	C₄×C₈	—	26	4	no	—
237	C₄×C₈	—	26	4	no	—
238	#95	—	13	2	no	—
239	#95	—	13	2	no	—
240	#95	—	13	2	no	—
241	C₂×C₁₃	C₄²×Q₈	1	1	yes	characteristic
242	C₂×C₁₃	C₂×C₄×M₁₆	1	1	yes	characteristic
243	C₂×C₁₃	C₄×#131	1	1	yes	characteristic
244	C₂×C₁₃	C₂×#446	1	1	yes	characteristic
245	C₂×C₁₃	C₂×#446	1	1	yes	characteristic
246	C₂×C₁₃	#2965	1	1	yes	characteristic
247	C₂×C₁₃	#2965	1	1	yes	characteristic
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	2	2	no	—
250	C₂×Q₈	C₄×#261	1	4	yes	—
251	C₂²×C₄	—	13	1	no	—
252	C₂²×C₄	—	13	1	no	—
253	C₂²×C₄	—	13	1	no	—
254	C₂²×C₄	—	13	1	no	—
255	C₂²×C₄	—	13	2	no	—
256	C₂²×C₄	—	13	2	no	—
257	C₂²×C₄	C₂²×#261	1	2	yes	—
258	C₂²×C₄	C₂²×#261	1	2	yes	—
259	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
260	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
261	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
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	8	no	—
267	C₂×C₈	—	26	2	no	—
268	C₂×C₈	—	26	4	no	—
269	C₂×C₈	—	26	2	no	—
270	C₂×C₈	—	26	4	no	—
271	C₄⋊C₄	—	2	4	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	13	8	no	—
274	C₄⋊C₄	—	13	4	no	—
275	C₄⋊C₄	—	26	4	no	—
276	C₄⋊C₄	—	26	2	no	—
277	C₄²	—	2	4	no	—
278	C₄²	—	13	4	no	—
279	C₄²	—	13	4	no	—
280	C₄²	—	13	8	no	—
281	C₄²	—	13	2	no	—
282	C₄²	—	13	2	no	—
283	C₄²	—	26	2	no	—
284	C₄²	—	26	4	no	—
285	C₄²	—	26	4	no	—
286	C₄²	—	26	2	no	—
287	C₄²	—	26	4	no	—
288	C₄²	—	26	2	no	—
289	C₄²	—	26	4	no	—
290	C₄²	#6122	1	4	yes	—
291	C₁₃	C₄×#446	1	1	yes	characteristic, complemented, Sylow
292	C₂³	C₂³×#261	1	1	yes	characteristic
293	Q₈	—	2	8	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	13	2	no	—
302	C₂×C₄	—	13	2	no	—
303	C₂×C₄	—	13	1	no	—
304	C₂×C₄	—	13	1	no	—
305	C₂×C₄	—	13	2	no	—
306	C₂×C₄	—	13	2	no	—
307	C₂×C₄	—	13	2	no	—
308	C₂×C₄	—	13	2	no	—
309	C₂×C₄	—	13	2	no	—
310	C₂×C₄	—	13	4	no	—
311	C₂×C₄	—	26	1	no	—
312	C₂×C₄	—	26	1	no	—
313	C₂×C₄	—	26	2	no	—
314	C₂×C₄	—	26	1	no	—
315	C₂×C₄	—	26	2	no	—
316	C₂×C₄	—	26	1	no	—
317	C₂×C₄	—	26	2	no	—
318	C₂×C₄	—	26	2	no	—
319	C₂×C₄	—	26	2	no	—
320	C₂×C₄	D₁₃⋊M₁₆	1	2	yes	—
321	C₂×C₄	C₂×C₄×#261	1	4	yes	—
322	C₂×C₄	C₂×C₄×#261	1	2	yes	—
323	C₂×C₄	#88240	1	2	yes	—
324	C₂×C₄	Q₈×#261	1	2	yes	—
325	C₂×C₄	C₂×#6122	1	2	yes	—
326	C₂×C₄	C₂×#6122	1	2	yes	—
327	C₂×C₄	C₂×#6122	1	2	yes	—
328	C₈	—	13	8	no	—
329	C₈	—	26	4	no	—
330	C₈	—	26	8	no	—
331	C₂²	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
332	C₂²	C₂²×C₄×#261	1	1	yes	characteristic
333	C₂²	C₂×#88240	1	1	yes	characteristic
334	C₂²	C₂×Q₈×#261	1	1	yes	characteristic
335	C₂²	D₁₃⋊#131	1	1	yes	characteristic
336	C₂²	#41×#261	1	1	yes	characteristic
337	C₂²	C₂²×#6122	1	1	yes	characteristic
338	C₄	—	2	2	no	—
339	C₄	—	2	4	no	—
340	C₄	—	4	2	no	—
341	C₄	—	13	2	no	—
342	C₄	—	13	2	no	—
343	C₄	—	13	4	no	—
344	C₄	—	26	2	no	—
345	C₄	—	26	2	no	—
346	C₄	—	26	4	no	—
347	C₄	—	52	2	no	—
348	C₄	C₄×#6122	1	4	yes	—
349	C₄	#11719158	1	4	yes	direct product factor
350	C₂	C₄×Q₈×#261	1	1	yes	characteristic
351	C₂	C₂×C₄×#6122	1	1	yes	characteristic
352	C₂	C₄×#88240	1	1	yes	characteristic
353	C₂	C₂×#11719158	1	1	yes	characteristic
354	C₂	D₁₃⋊#446	1	1	yes	characteristic
355	C₂	#13131644	1	1	yes	characteristic
356	C₂	#13131645	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×#261
commutator	C₂²×C₁₃	C₂²×C₄²
Frattini	C₂³	C₂³×#261
Fitting	C₂×C₄×Q₈×C₁₃	C₄
socle, prime radical	C₂³×C₁₃	C₂³×C₄
cosocle	C₂²×Dic₁₃	C₂⁴
hypercenter	C₂×C₄×Q₈	#261
hypocenter	C₁₃	C₄×#446
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
12. #11719158	C₄	direct (N × H)
66. C₄×#1057	Q₈	semidirect
160. #1057	C₄×Q₈	semidirect
195. C₄×C₁₃	#446	semidirect
200. Dic₁₃	#446	semidirect
291. C₁₃	C₄×#446	semidirect
349. C₄	#11719158	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×#11719158	2
#13130636	2
#13130632	2
C₄×#88112	2
C₄×C₄⋊#1057	2
C₄×#88110	2
C₄²×#1057	2
C₄×Q₈×Dic₁₃	2
C₄×#446	13

Subgroups — C4×#11719158 #16797457

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11719158 #16797457