Subgroups — C₄×#11718855 #16016199

Order 3328.

690 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718855	C₂	1	1	yes	maximal, characteristic
2	C₄×#88158	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#88069	C₂	1	2	yes	maximal, complemented
4	C₄×#88068	C₂	1	2	yes	maximal, complemented
5	C₄×#88067	C₂	1	2	yes	maximal, complemented
6	C₄×#88066	C₂	1	2	yes	maximal, complemented
7	#13116618	C₂	1	1	yes	maximal, characteristic, complemented
8	#13116543	C₂	1	1	yes	maximal, characteristic, complemented
9	#13116541	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄×#6090	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₁₃×#135	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
13	C₄×C₁₃×#41	—	2	1	no	—
14	C₂×#88158	C₂²	1	1	yes	characteristic
15	C₄×#6127	—	2	1	no	—
16	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
17	C₄×C₁₃×SD₁₆	—	2	2	no	—
18	C₄×C₁₃×D₈	C₂²	1	2	yes	—
19	C₁₃×#427	C₂²	1	1	yes	characteristic
20	C₁₃×#416	—	2	1	no	—
21	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
22	#11718855	C₄	1	8	yes	direct product factor
23	C₂×#88069	C₂²	1	1	yes	characteristic
24	C₂×#88068	C₂²	1	1	yes	characteristic
25	C₂×#88067	C₂²	1	1	yes	characteristic
26	C₂×#88066	C₂²	1	1	yes	characteristic
27	#11718774	—	2	1	no	—
28	#11718767	C₂²	1	1	yes	characteristic
29	C₂²×#6090	C₂²	1	1	yes	characteristic
30	C₂²×#6090	C₂²	1	1	yes	characteristic
31	C₄×Dic₅₂	C₂²	1	2	yes	—
32	C₄×D₁₀₄	C₂²	1	2	yes	—
33	C₄×#6095	—	2	2	no	—
34	#11718156	C₂²	1	2	yes	—
35	#11718154	C₂²	1	2	yes	—
36	#11718084	C₂²	1	2	yes	—
37	#11718083	C₂²	1	2	yes	—
38	#11718082	C₂²	1	2	yes	—
39	#11718081	C₂²	1	2	yes	—
40	C₄×#6090	C₂²	1	2	yes	—
41	C₄×#6090	C₂²	1	2	yes	—
42	C₂×C₁₃×#41	—	2	1	no	—
43	C₂×#6127	—	2	1	no	—
44	C₁₃×#135	—	2	2	no	—
45	C₁₃×#135	C₂×C₄	1	4	yes	complemented
46	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
47	C₂×C₁₃×SD₁₆	—	2	1	no	—
48	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
51	C₄×Q₈×C₁₃	—	2	2	no	—
52	C₄×D₄×C₁₃	—	2	2	no	—
53	C₄×D₄×C₁₃	—	4	2	no	—
54	C₁₃×#117	—	2	1	no	—
55	C₁₃×#117	—	4	1	no	—
56	C₂×C₄²×C₁₃	—	4	1	no	—
57	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
58	C₄×C₂²⋊D₁₃	—	4	2	no	—
59	#88180	—	2	1	no	—
60	C₂×Dic₅₂	C₂³	1	1	yes	characteristic
61	#88158	—	2	2	no	—
62	#88158	C₂×C₄	1	4	yes	complemented
63	C₂×D₁₀₄	C₂³	1	1	yes	characteristic
64	C₂×#6095	—	2	1	no	—
65	C₄×D₅₂	—	2	2	no	—
66	#88126	—	4	1	no	—
67	C₄²×D₁₃	—	4	1	no	—
68	C₄×Dic₂₆	—	2	2	no	—
69	C₁₃×#107	C₂³	1	2	yes	—
70	C₁₃×#106	—	2	1	no	—
71	C₁₃×#103	—	2	2	no	—
72	C₁₃×#102	—	2	2	no	—
73	C₄×C₈×C₁₃	D₄	1	2	yes	—
74	C₄×C₈×C₁₃	C₂³	1	2	yes	—
75	#88069	C₂×C₄	1	4	yes	complemented
76	#88068	C₂×C₄	1	4	yes	complemented
77	#88067	C₂×C₄	1	4	yes	complemented
78	#88066	C₂×C₄	1	4	yes	complemented
79	#88061	—	2	2	no	—
80	#88058	C₂³	1	2	yes	—
81	#88057	—	2	1	no	—
82	#88056	—	2	2	no	—
83	C₂×#6090	—	2	2	no	—
84	C₂×#6090	C₂×C₄	1	4	yes	complemented
85	C₂×#6090	C₂³	1	2	yes	—
86	C₂×#6090	C₂³	1	1	yes	characteristic
87	C₂×#6090	C₂³	1	1	yes	characteristic
88	C₄×#508	—	13	1	no	maximal, Sylow
89	C₁₃×#41	—	2	4	no	—
90	C₁₃×#41	—	4	2	no	—
91	C₂×Q₈×C₁₃	—	2	1	no	—
92	C₂×D₄×C₁₃	—	2	1	no	—
93	C₂×D₄×C₁₃	—	4	1	no	—
94	C₂²×C₄×C₁₃	—	4	1	no	—
95	C₂²×C₄×C₁₃	—	4	2	no	—
96	C₂²×C₄×C₁₃	D₈	1	2	yes	—
97	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₂²⋊D₁₃	—	4	1	no	—
99	#6127	—	2	4	no	—
100	#6127	—	4	2	no	—
101	C₂×D₅₂	—	2	1	no	—
102	C₂×C₄×D₁₃	—	4	1	no	—
103	C₂×C₄×D₁₃	—	4	2	no	—
104	C₂×Dic₂₆	—	2	1	no	—
105	C₁₃×Q₁₆	—	2	1	no	—
106	C₁₃×Q₁₆	C₂²×C₄	1	2	yes	—
107	C₁₃×SD₁₆	—	2	4	no	—
108	C₁₃×D₈	—	2	1	no	—
109	C₁₃×D₈	C₂²×C₄	1	2	yes	—
110	C₂×C₈×C₁₃	—	2	2	no	—
111	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
112	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
113	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₁₃	#41	1	2	yes	—
115	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
116	C₁₃×C₄⋊C₄	—	2	2	no	—
117	C₁₃×C₄⋊C₄	—	4	2	no	—
118	C₁₃×C₂²⋊C₄	—	4	1	no	—
119	C₁₃×C₂²⋊C₄	—	4	2	no	—
120	C₄²×C₁₃	—	4	1	no	—
121	C₄²×C₁₃	—	4	2	no	—
122	C₄²×C₁₃	D₈	1	1	yes	characteristic
123	C₄²×C₁₃	D₈	1	1	yes	characteristic
124	C₄²×C₁₃	C₂×D₄	1	2	yes	—
125	C₂²⋊Dic₁₃	—	4	1	no	—
126	#6103	—	4	2	no	—
127	C₄⋊Dic₁₃	—	2	2	no	—
128	#6101	—	4	2	no	—
129	C₄×Dic₁₃	—	4	2	no	—
130	C₄×Dic₁₃	—	4	1	no	—
131	Dic₅₂	—	2	1	no	—
132	Dic₅₂	C₂²×C₄	1	2	yes	—
133	D₁₀₄	—	2	1	no	—
134	D₁₀₄	C₂²×C₄	1	2	yes	—
135	#6095	—	2	4	no	—
136	#6090	—	2	2	no	—
137	#6090	C₂²×C₄	1	2	yes	—
138	#6090	C₂²×C₄	1	2	yes	—
139	C₂×#508	—	13	1	no	—
140	C₄×#135	—	13	1	no	—
141	C₄×#135	D₁₃	1	1	yes	characteristic, complemented
142	C₄×Q₃₂	—	13	2	no	—
143	C₄×SD₃₂	—	13	2	no	—
144	C₄×SD₃₂	—	13	2	no	—
145	C₄×D₁₆	—	13	2	no	—
146	#2158	—	13	1	no	—
147	#2138	—	13	1	no	—
148	#2138	—	13	1	no	—
149	C₂×C₄×C₁₆	—	13	1	no	—
150	C₂³×C₁₃	—	4	1	no	—
151	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
152	C₂²×D₁₃	—	4	1	no	—
153	Q₈×C₁₃	—	2	2	no	—
154	Q₈×C₁₃	—	4	1	no	—
155	D₄×C₁₃	—	2	2	no	—
156	D₄×C₁₃	—	4	4	no	—
157	D₄×C₁₃	—	4	1	no	—
158	C₂×C₄×C₁₃	—	2	4	no	—
159	C₂×C₄×C₁₃	—	4	1	no	—
160	C₂×C₄×C₁₃	—	4	4	no	—
161	C₂×C₄×C₁₃	—	4	2	no	—
162	C₂×C₄×C₁₃	—	8	4	no	—
163	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
164	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
165	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
166	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
167	C₂×C₄×C₁₃	#135	1	2	yes	—
168	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
169	C₂²⋊D₁₃	—	4	4	no	—
170	C₂×Dic₁₃	—	4	1	no	—
171	C₂×Dic₁₃	—	4	2	no	—
172	D₅₂	—	2	2	no	—
173	D₅₂	—	4	1	no	—
174	C₄×D₁₃	—	4	4	no	—
175	C₄×D₁₃	—	8	4	no	—
176	Dic₂₆	—	2	2	no	—
177	Dic₂₆	—	4	1	no	—
178	C₈×C₁₃	—	2	2	no	—
179	C₈×C₁₃	C₄×D₄	1	2	yes	—
180	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
181	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
182	C₂×#135	—	13	1	no	—
183	C₂×#135	C₂×D₁₃	1	1	yes	characteristic
184	C₄×#41	—	2	1	no	—
185	C₄×#41	—	26	1	no	—
186	#508	—	13	8	no	—
187	C₂×Q₃₂	—	13	1	no	—
188	C₂×SD₃₂	—	13	1	no	—
189	C₂×SD₃₂	—	13	1	no	—
190	C₂×D₁₆	—	13	1	no	—
191	C₂²×C₁₆	—	13	1	no	—
192	C₂²×C₁₆	—	13	1	no	—
193	C₄×Q₁₆	—	13	2	no	—
194	C₄×Q₁₆	C₂×D₁₃	1	2	yes	—
195	C₄×SD₁₆	—	2	2	no	—
196	C₄×SD₁₆	—	26	2	no	—
197	C₄×D₈	—	13	2	no	—
198	C₄×D₈	C₂×D₁₃	1	2	yes	—
199	#427	—	13	1	no	—
200	#427	C₂×D₁₃	1	1	yes	characteristic
201	#416	—	2	1	no	—
202	#416	—	26	1	no	—
203	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
204	#367	—	13	2	no	—
205	#366	—	13	2	no	—
206	#358	—	13	2	no	—
207	#358	—	13	2	no	—
208	#357	—	13	2	no	—
209	#357	—	13	2	no	—
210	C₄×C₁₆	—	13	2	no	—
211	C₄×C₁₆	—	13	2	no	—
212	C₂²×C₁₃	—	2	2	no	—
213	C₂²×C₁₃	—	4	2	no	—
214	C₂²×C₁₃	—	8	2	no	—
215	C₂²×C₁₃	C₄×D₈	1	2	yes	—
216	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
217	C₂×D₁₃	—	4	2	no	—
218	C₂×D₁₃	—	8	2	no	—
219	C₄×C₁₃	—	2	2	no	—
220	C₄×C₁₃	—	4	2	no	—
221	C₄×C₁₃	—	8	2	no	—
222	C₄×C₁₃	C₄×D₈	1	2	yes	—
223	C₄×C₁₃	#508	1	4	yes	complemented
224	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
225	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
226	Dic₁₃	—	4	2	no	—
227	Dic₁₃	—	8	2	no	—
228	C₂×#41	—	2	1	no	—
229	C₂×#41	—	26	1	no	—
230	#135	—	2	2	no	—
231	#135	—	13	4	no	—
232	#135	—	26	2	no	—
233	#135	C₄×D₁₃	1	4	yes	complemented
234	C₂×Q₁₆	—	13	1	no	—
235	C₂×Q₁₆	C₂²×D₁₃	1	1	yes	characteristic
236	C₂×SD₁₆	—	2	1	no	—
237	C₂×SD₁₆	—	26	1	no	—
238	C₂×D₈	—	13	1	no	—
239	C₂×D₈	C₂²×D₁₃	1	1	yes	characteristic
240	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
241	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
242	C₄×Q₈	—	2	2	no	—
243	C₄×Q₈	—	26	2	no	—
244	C₄×D₄	—	2	2	no	—
245	C₄×D₄	—	4	2	no	—
246	C₄×D₄	—	26	2	no	—
247	C₄×D₄	—	52	2	no	—
248	#117	—	2	1	no	—
249	#117	—	4	1	no	—
250	#117	—	26	1	no	—
251	#117	—	52	1	no	—
252	C₂×C₄²	—	4	1	no	—
253	C₂×C₄²	—	52	1	no	—
254	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
255	Q₃₂	—	13	4	no	—
256	SD₃₂	—	13	4	no	—
257	SD₃₂	—	13	4	no	—
258	D₁₆	—	13	4	no	—
259	C₂×C₁₆	—	13	4	no	—
260	C₂×C₁₆	—	13	1	no	—
261	C₂×C₁₆	—	13	1	no	—
262	C₂×C₁₆	—	13	2	no	—
263	C₂×C₁₆	—	26	2	no	—
264	#107	—	13	2	no	—
265	#107	C₂²×D₁₃	1	2	yes	—
266	#106	—	2	1	no	—
267	#106	—	26	1	no	—
268	#103	—	2	2	no	—
269	#103	—	26	2	no	—
270	#102	—	2	2	no	—
271	#102	—	26	2	no	—
272	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
273	C₄×C₈	C₂²×D₁₃	1	2	yes	—
274	C₂×C₁₃	—	2	2	no	—
275	C₂×C₁₃	—	8	2	no	—
276	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
277	C₂×C₁₃	C₂×#508	1	2	yes	—
278	D₁₃	—	8	2	no	—
279	#41	—	2	4	no	—
280	#41	—	4	2	no	—
281	#41	—	26	4	no	—
282	#41	—	52	2	no	—
283	C₂×Q₈	—	2	1	no	—
284	C₂×Q₈	—	26	1	no	—
285	C₂×D₄	—	2	1	no	—
286	C₂×D₄	—	4	1	no	—
287	C₂×D₄	—	26	1	no	—
288	C₂×D₄	—	52	1	no	—
289	C₂²×C₄	—	4	1	no	—
290	C₂²×C₄	—	4	2	no	—
291	C₂²×C₄	—	52	1	no	—
292	C₂²×C₄	—	52	2	no	—
293	C₂²×C₄	#6104	1	2	yes	—
294	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
295	Q₁₆	—	2	1	no	—
296	Q₁₆	—	13	2	no	—
297	Q₁₆	—	26	1	no	—
298	Q₁₆	C₂×C₄×D₁₃	1	2	yes	—
299	SD₁₆	—	2	4	no	—
300	SD₁₆	—	26	4	no	—
301	D₈	—	2	1	no	—
302	D₈	—	13	2	no	—
303	D₈	—	26	1	no	—
304	D₈	C₂×C₄×D₁₃	1	2	yes	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
307	C₂×C₈	#6127	1	2	yes	—
308	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
309	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
310	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	—	4	2	no	—
313	C₄⋊C₄	—	26	2	no	—
314	C₄⋊C₄	—	52	2	no	—
315	C₂²⋊C₄	—	4	1	no	—
316	C₂²⋊C₄	—	4	2	no	—
317	C₂²⋊C₄	—	52	1	no	—
318	C₂²⋊C₄	—	52	2	no	—
319	C₄²	—	4	1	no	—
320	C₄²	—	4	2	no	—
321	C₄²	—	52	1	no	—
322	C₄²	—	52	2	no	—
323	C₄²	#6104	1	1	yes	characteristic
324	C₄²	#6104	1	1	yes	characteristic
325	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
326	C₁₆	—	13	2	no	—
327	C₁₆	—	13	2	no	—
328	C₁₆	—	26	2	no	—
329	C₁₃	C₄×#508	1	1	yes	characteristic, complemented, Sylow
330	C₂³	—	4	1	no	—
331	C₂³	—	52	1	no	—
332	C₂³	C₂×#6104	1	1	yes	characteristic
333	Q₈	—	2	2	no	—
334	Q₈	—	4	1	no	—
335	Q₈	—	26	2	no	—
336	Q₈	—	52	1	no	—
337	D₄	—	2	2	no	—
338	D₄	—	4	4	no	—
339	D₄	—	4	1	no	—
340	D₄	—	26	2	no	—
341	D₄	—	52	4	no	—
342	D₄	—	52	1	no	—
343	C₂×C₄	—	2	4	no	—
344	C₂×C₄	—	4	1	no	—
345	C₂×C₄	—	4	4	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	8	4	no	—
348	C₂×C₄	—	52	1	no	—
349	C₂×C₄	—	52	4	no	—
350	C₂×C₄	—	52	2	no	—
351	C₂×C₄	—	104	4	no	—
352	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
353	C₂×C₄	C₂×#6104	1	2	yes	—
354	C₂×C₄	C₂×#6104	1	1	yes	characteristic
355	C₂×C₄	C₂×#6104	1	2	yes	—
356	C₂×C₄	#88204	1	2	yes	—
357	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
358	C₈	—	2	2	no	—
359	C₈	C₄×C₂²⋊D₁₃	1	2	yes	—
360	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
361	C₈	C₂×#6127	1	1	yes	characteristic
362	C₂²	—	2	2	no	—
363	C₂²	—	4	2	no	—
364	C₂²	—	8	2	no	—
365	C₂²	—	52	2	no	—
366	C₂²	—	104	2	no	—
367	C₂²	C₄×#6104	1	2	yes	—
368	C₂²	C₂²×#6104	1	1	yes	characteristic
369	C₄	—	2	2	no	—
370	C₄	—	4	2	no	—
371	C₄	—	8	2	no	—
372	C₄	—	52	2	no	—
373	C₄	—	104	2	no	—
374	C₄	C₄×#6104	1	2	yes	—
375	C₄	#11718855	1	4	yes	direct product factor
376	C₄	C₂×C₄×C₂²⋊D₁₃	1	1	yes	characteristic
377	C₄	C₂×#88204	1	1	yes	characteristic
378	C₂	—	2	2	no	—
379	C₂	—	8	2	no	—
380	C₂	—	104	2	no	—
381	C₂	C₂×C₄×#6104	1	1	yes	characteristic
382	C₂	C₂×#11718855	1	2	yes	—

	subgroup	G/H
center	C₄²	#6104
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₄×C₁₃×#135	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂×#11718855	C₂
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₄×#135	D₁₃
hypocenter	C₁₃	C₄×#508
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
2. C₄×#88158	C₂	semidirect
3. C₄×#88069	C₂	semidirect
4. C₄×#88068	C₂	semidirect
5. C₄×#88067	C₂	semidirect
6. C₄×#88066	C₂	semidirect
7. #13116618	C₂	semidirect
8. #13116543	C₂	semidirect
9. #13116541	C₂	semidirect
10. C₂×C₄×#6090	C₂	semidirect
11. C₄×C₁₃×#135	C₂	semidirect
22. #11718855	C₄	direct (N × H)
45. C₁₃×#135	C₂×C₄	semidirect
62. #88158	C₂×C₄	semidirect
75. #88069	C₂×C₄	semidirect
76. #88068	C₂×C₄	semidirect
77. #88067	C₂×C₄	semidirect
78. #88066	C₂×C₄	semidirect
84. C₂×#6090	C₂×C₄	semidirect
141. C₄×#135	D₁₃	semidirect
223. C₄×C₁₃	#508	semidirect
233. #135	C₄×D₁₃	semidirect
329. C₁₃	C₄×#508	semidirect
375. C₄	#11718855	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×#11718855	2
C₄×#88158	2
C₄×#88069	2
C₄×#88068	2
C₄×#88067	2
C₄×#88066	2
#13116618	2
#13116543	2
#13116541	2
C₂×C₄×#6090	2
C₄×C₁₃×#135	2
C₄×#508	13

Subgroups — C4×#11718855 #16016199

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11718855 #16016199