Subgroups — D₄×#3998 #13043536

Order 1280.

734 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#3998	C₂	1	2	yes	maximal, complemented
2	C₂×D₄×#168	C₂	1	1	yes	maximal, characteristic
3	#10604872	C₂	1	2	yes	maximal, complemented
4	#10604414	C₂	1	2	yes	maximal, complemented
5	#10604413	C₂	1	4	yes	maximal, complemented
6	D₄×#674	C₂	1	2	yes	maximal, complemented
7	#10604150	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#3998	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄×C₅×D₄	C₄	1	1	yes	characteristic
10	C₂³×#168	C₂²	1	2	yes	—
11	C₂×#3998	—	2	8	no	—
12	C₂×#3998	C₂²	1	1	yes	characteristic
13	C₂×#3998	C₂²	1	2	yes	—
14	C₂×#3998	C₂²	1	1	yes	characteristic
15	C₂×#3998	C₂²	1	4	yes	complemented
16	C₂²×#674	C₂²	1	4	yes	complemented
17	C₂×C₂²⋊#168	C₄	1	2	yes	—
18	D₄×#168	C₄	1	2	yes	—
19	D₄×#168	C₄	1	1	yes	characteristic
20	D₄×#168	C₄	1	1	yes	characteristic
21	D₄×#168	C₂²	1	1	yes	characteristic
22	D₄×#168	C₂²	1	2	yes	—
23	D₄×#168	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊#168	C₄	1	1	yes	characteristic
25	C₂×C₄×#168	C₂²	1	1	yes	characteristic
26	C₂²⋊#674	C₂²	1	4	yes	complemented
27	C₂²⋊#674	C₂²	1	4	yes	complemented
28	#64859	C₂²	1	2	yes	complemented
29	C₄⋊#674	C₂²	1	2	yes	complemented
30	C₄⋊#674	C₂²	1	2	yes	complemented
31	C₄×#674	C₂²	1	2	yes	complemented
32	D₄×M₃₂	—	5	1	no	maximal, Sylow
33	C₂²×C₅×D₄	C₈	1	1	yes	characteristic
34	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
35	C₄×C₅×D₄	C₈	1	2	yes	—
36	C₄×C₅×D₄	C₈	1	1	yes	characteristic
37	C₄×C₅×D₄	C₈	1	1	yes	characteristic
38	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
39	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic
40	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic
41	C₂×C₅×C₄⋊C₄	C₈	1	1	yes	characteristic
42	C₂×C₅×C₂²⋊C₄	C₈	1	2	yes	—
43	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
44	C₂²×#168	—	2	4	no	—
45	C₂²×#168	—	2	4	no	—
46	C₂²×#168	C₂×C₄	1	2	yes	—
47	C₂²×#168	C₂×C₄	1	2	yes	—
48	C₂²×#168	C₂×C₄	1	2	yes	—
49	C₂²×#168	C₂×C₄	1	1	yes	characteristic
50	C₂²×#168	C₂³	1	2	yes	—
51	C₂²×#168	C₂³	1	2	yes	—
52	C₂²×#168	C₂³	1	1	yes	characteristic
53	#3998	—	2	4	no	—
54	#3998	—	2	2	no	—
55	#3998	—	2	8	no	—
56	#3998	D₄	1	4	yes	direct product factor
57	C₂×#674	—	2	8	no	—
58	C₂×#674	C₂³	1	2	yes	—
59	C₂×#674	C₂³	1	4	yes	—
60	C₂×#674	C₂³	1	2	yes	—
61	C₂²⋊#168	C₂×C₄	1	2	yes	—
62	C₂²⋊#168	C₂×C₄	1	2	yes	—
63	C₂²⋊#168	C₂×C₄	1	2	yes	—
64	C₂²⋊#168	C₂×C₄	1	2	yes	—
65	C₄⋊#168	C₂×C₄	1	1	yes	characteristic
66	C₄⋊#168	C₂×C₄	1	2	yes	—
67	C₄⋊#168	C₂×C₄	1	1	yes	characteristic
68	C₄×#168	C₂×C₄	1	1	yes	characteristic
69	C₄×#168	C₂×C₄	1	1	yes	characteristic
70	C₄×#168	C₂³	1	1	yes	characteristic
71	C₄×#168	C₂³	1	1	yes	characteristic
72	C₂²×M₃₂	—	5	2	no	—
73	C₂×C₈×D₄	—	5	1	no	—
74	#2169	—	5	2	no	—
75	#2168	—	5	4	no	—
76	D₄×C₁₆	—	5	2	no	—
77	#2150	—	5	1	no	—
78	#2112	—	5	2	no	—
79	C₄×M₃₂	—	5	1	no	—
80	C₂⁴×C₅	C₂×C₈	1	2	yes	—
81	C₂×C₅×D₄	—	2	4	no	—
82	C₂×C₅×D₄	C₂×C₈	1	2	yes	—
83	C₂×C₅×D₄	C₂×C₈	1	1	yes	characteristic
84	C₂×C₅×D₄	C₂×C₈	1	1	yes	characteristic
85	C₂²×C₄×C₅	—	2	4	no	—
86	C₂²×C₄×C₅	—	2	4	no	—
87	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
88	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
89	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
90	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
91	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
92	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
95	C₅×C₄⋊C₄	C₂×C₈	1	2	yes	—
96	C₅×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
97	C₅×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
98	C₅×C₂²⋊C₄	C₂×C₈	1	2	yes	—
99	C₅×C₂²⋊C₄	C₂×C₈	1	2	yes	—
100	C₅×C₂²⋊C₄	C₂×C₈	1	2	yes	—
101	C₅×C₂²⋊C₄	C₂×C₈	1	2	yes	—
102	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
103	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
104	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
106	C₂×#168	—	2	4	no	—
107	C₂×#168	—	2	2	no	—
108	C₂×#168	—	2	4	no	—
109	C₂×#168	—	2	4	no	—
110	C₂×#168	—	2	2	no	—
111	C₂×#168	—	2	2	no	—
112	C₂×#168	—	2	2	no	—
113	C₂×#168	—	2	2	no	—
114	C₂×#168	C₂²×C₄	1	1	yes	characteristic
115	C₂×#168	C₂²×C₄	1	1	yes	characteristic
116	C₂×#168	C₂²×C₄	1	2	yes	—
117	C₂×#168	C₂²×C₄	1	2	yes	—
118	C₂×#168	C₂²×C₄	1	1	yes	characteristic
119	C₂×#168	C₂×D₄	1	2	yes	—
120	C₂×#168	#41	1	2	yes	—
121	C₂×#168	C₂⁴	1	1	yes	characteristic
122	#674	—	2	4	no	—
123	#674	—	2	2	no	—
124	#674	C₂×D₄	1	4	yes	complemented
125	C₂³×C₈	—	5	2	no	—
126	C₂×C₄×D₄	AGL(1,𝔽₅)	1	1	yes	characteristic
127	C₂×M₃₂	—	5	1	no	—
128	C₂×M₃₂	—	5	1	no	—
129	C₂×M₃₂	—	5	2	no	—
130	C₂×M₃₂	—	5	4	no	—
131	C₂×M₃₂	—	10	8	no	—
132	C₂²×C₁₆	—	5	4	no	—
133	C₈×D₄	—	5	2	no	—
134	C₈×D₄	—	5	1	no	—
135	C₈×D₄	—	5	1	no	—
136	C₈×D₄	—	5	1	no	—
137	C₈×D₄	—	5	2	no	—
138	C₈×D₄	—	5	1	no	—
139	C₂×C₄⋊C₈	—	5	1	no	—
140	C₂×C₂²⋊C₈	—	5	2	no	—
141	C₂×C₄×C₈	—	5	1	no	—
142	C₄⋊C₁₆	—	5	2	no	—
143	C₄⋊C₁₆	—	5	2	no	—
144	C₂²⋊C₁₆	—	5	4	no	—
145	C₂²⋊C₁₆	—	5	4	no	—
146	#346	—	5	2	no	—
147	C₄×C₁₆	—	5	2	no	—
148	C₂³×C₅	—	2	4	no	—
149	C₂³×C₅	—	2	4	no	—
150	C₂³×C₅	—	4	2	no	—
151	C₂³×C₅	C₂²×C₈	1	2	yes	—
152	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
153	C₂³×C₅	C₂²×C₈	1	2	yes	—
154	C₅×D₄	—	2	4	no	—
155	C₅×D₄	—	2	2	no	—
156	C₅×D₄	M₃₂	1	4	yes	complemented
157	C₂×C₄×C₅	—	2	4	no	—
158	C₂×C₄×C₅	—	2	2	no	—
159	C₂×C₄×C₅	—	2	2	no	—
160	C₂×C₄×C₅	—	2	2	no	—
161	C₂×C₄×C₅	—	2	2	no	—
162	C₂×C₄×C₅	—	2	4	no	—
163	C₂×C₄×C₅	—	2	4	no	—
164	C₂×C₄×C₅	—	2	2	no	—
165	C₂×C₄×C₅	—	4	1	no	—
166	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
167	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
168	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
169	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
170	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
171	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
172	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
173	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
174	C₂×C₄×C₅	#131	1	2	yes	—
175	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
176	#168	—	2	2	no	—
177	#168	—	2	2	no	—
178	#168	—	2	1	no	—
179	#168	—	2	1	no	—
180	#168	C₄×D₄	1	2	yes	—
181	#168	C₂²×D₄	1	1	yes	characteristic
182	#168	C₂×#41	1	1	yes	characteristic
183	C₂²×D₄	#170	1	1	yes	characteristic
184	C₂³×C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
185	C₂²×C₈	—	5	1	no	—
186	C₂²×C₈	—	5	2	no	—
187	C₂²×C₈	—	5	2	no	—
188	C₂²×C₈	—	5	2	no	—
189	C₂²×C₈	—	5	2	no	—
190	C₂²×C₈	—	5	2	no	—
191	C₂²×C₈	—	5	1	no	—
192	C₂²×C₈	—	10	4	no	—
193	C₂²×C₈	—	10	4	no	—
194	C₄×D₄	#170	1	2	yes	—
195	C₄×D₄	#170	1	1	yes	characteristic
196	C₄×D₄	#170	1	1	yes	characteristic
197	C₄×D₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
198	C₄×D₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
199	C₄×D₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
200	C₂×C₄⋊C₄	#170	1	1	yes	characteristic
201	C₂×C₂²⋊C₄	#170	1	2	yes	—
202	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
203	M₃₂	—	5	4	no	—
204	M₃₂	—	10	8	no	—
205	M₃₂	—	10	2	no	—
206	M₃₂	—	10	4	no	—
207	C₂×C₁₆	—	5	2	no	—
208	C₂×C₁₆	—	5	2	no	—
209	C₂×C₁₆	—	5	4	no	—
210	C₂×C₁₆	—	10	8	no	—
211	C₄⋊C₈	—	5	1	no	—
212	C₄⋊C₈	—	5	2	no	—
213	C₄⋊C₈	—	5	1	no	—
214	C₂²⋊C₈	—	5	2	no	—
215	C₂²⋊C₈	—	5	2	no	—
216	C₂²⋊C₈	—	5	2	no	—
217	C₂²⋊C₈	—	5	2	no	—
218	C₄×C₈	—	5	1	no	—
219	C₄×C₈	—	5	1	no	—
220	C₄×C₈	—	5	1	no	—
221	C₄×C₈	—	5	1	no	—
222	C₂²×C₅	—	2	1	no	—
223	C₂²×C₅	—	2	1	no	—
224	C₂²×C₅	—	2	2	no	—
225	C₂²×C₅	—	2	2	no	—
226	C₂²×C₅	—	2	2	no	—
227	C₂²×C₅	—	2	2	no	—
228	C₂²×C₅	—	2	2	no	—
229	C₂²×C₅	—	4	8	no	—
230	C₂²×C₅	C₈×D₄	1	2	yes	—
231	C₂²×C₅	C₂×M₃₂	1	4	yes	complemented
232	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
233	C₄×C₅	—	2	1	no	—
234	C₄×C₅	—	2	1	no	—
235	C₄×C₅	—	2	1	no	—
236	C₄×C₅	—	2	2	no	—
237	C₄×C₅	—	2	2	no	—
238	C₄×C₅	C₈×D₄	1	2	yes	—
239	C₄×C₅	C₂×M₃₂	1	2	yes	complemented
240	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
241	C₄×C₅	C₂×#131	1	1	yes	characteristic
242	C₂⁴	C₂×#170	1	2	yes	—
243	C₂×D₄	—	2	4	no	—
244	C₂×D₄	C₂×#170	1	2	yes	—
245	C₂×D₄	C₂×#170	1	1	yes	characteristic
246	C₂×D₄	C₂×#170	1	1	yes	characteristic
247	C₂²×C₄	—	2	4	no	—
248	C₂²×C₄	—	2	4	no	—
249	C₂²×C₄	C₂×#170	1	1	yes	characteristic
250	C₂²×C₄	C₂×#170	1	2	yes	—
251	C₂²×C₄	C₂×#170	1	2	yes	—
252	C₂²×C₄	C₂×#170	1	2	yes	—
253	C₂²×C₄	C₂×#170	1	1	yes	characteristic
254	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	2	yes	—
255	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	2	yes	—
256	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
257	C₂×C₈	—	5	2	no	—
258	C₂×C₈	—	5	2	no	—
259	C₂×C₈	—	5	1	no	—
260	C₂×C₈	—	5	1	no	—
261	C₂×C₈	—	5	1	no	—
262	C₂×C₈	—	5	1	no	—
263	C₂×C₈	—	5	2	no	—
264	C₂×C₈	—	5	2	no	—
265	C₂×C₈	—	10	2	no	—
266	C₂×C₈	—	10	2	no	—
267	C₂×C₈	—	10	2	no	—
268	C₂×C₈	—	10	2	no	—
269	C₂×C₈	—	10	4	no	—
270	C₂×C₈	—	10	4	no	—
271	C₂×C₈	—	10	2	no	—
272	C₂×C₈	—	10	4	no	—
273	C₄⋊C₄	C₂×#170	1	2	yes	—
274	C₄⋊C₄	C₂×#170	1	1	yes	characteristic
275	C₄⋊C₄	C₂×#170	1	1	yes	characteristic
276	C₂²⋊C₄	C₂×#170	1	2	yes	—
277	C₂²⋊C₄	C₂×#170	1	2	yes	—
278	C₂²⋊C₄	C₂×#170	1	2	yes	—
279	C₂²⋊C₄	C₂×#170	1	2	yes	—
280	C₄²	C₂×#170	1	1	yes	characteristic
281	C₄²	C₂×#170	1	1	yes	characteristic
282	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
283	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
284	C₁₆	—	5	4	no	—
285	C₁₆	—	10	4	no	—
286	C₁₆	—	10	2	no	—
287	C₂×C₅	—	2	2	no	—
288	C₂×C₅	—	2	4	no	—
289	C₂×C₅	—	4	2	no	—
290	C₂×C₅	C₂×C₈×D₄	1	1	yes	characteristic
291	C₂×C₅	C₂²×M₃₂	1	1	yes	characteristic
292	C₂×C₅	#3407	1	1	yes	characteristic
293	C₂³	—	2	4	no	—
294	C₂³	—	2	4	no	—
295	C₂³	—	4	2	no	—
296	C₂³	C₂²×#170	1	2	yes	—
297	C₂³	C₂²×#170	1	2	yes	—
298	C₂³	C₂²×#170	1	1	yes	characteristic
299	D₄	—	2	4	no	—
300	D₄	—	2	2	no	—
301	D₄	#3998	1	4	yes	direct product factor
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	4	no	—
309	C₂×C₄	—	2	4	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	#4131	1	2	yes	—
312	C₂×C₄	D₄×AGL(1,𝔽₅)	1	2	yes	—
313	C₂×C₄	C₂²×#170	1	1	yes	characteristic
314	C₂×C₄	C₂²×#170	1	1	yes	characteristic
315	C₂×C₄	C₂²×#170	1	1	yes	characteristic
316	C₂×C₄	C₂²×#170	1	2	yes	—
317	C₂×C₄	C₂²×#170	1	2	yes	—
318	C₂×C₄	C₂²×#170	1	1	yes	characteristic
319	C₂×C₄	C₂²×#170	1	1	yes	characteristic
320	C₂×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
321	C₈	—	5	1	no	—
322	C₈	—	5	1	no	—
323	C₈	—	5	2	no	—
324	C₈	—	10	1	no	—
325	C₈	—	10	1	no	—
326	C₈	—	10	2	no	—
327	C₈	—	10	2	no	—
328	C₅	D₄×M₃₂	1	1	yes	characteristic, complemented, Sylow
329	C₂²	—	2	1	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	2	no	—
333	C₂²	—	2	2	no	—
334	C₂²	—	2	2	no	—
335	C₂²	—	2	2	no	—
336	C₂²	—	4	8	no	—
337	C₂²	C₂×#3998	1	4	yes	complemented
338	C₂²	D₄×#170	1	2	yes	—
339	C₂²	C₂³×#170	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	2	2	no	—
342	C₄	—	2	2	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	C₂×#3998	1	2	yes	complemented
346	C₄	D₄×#170	1	2	yes	—
347	C₄	C₂×#4131	1	1	yes	characteristic
348	C₄	C₂×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
349	C₂	—	2	2	no	—
350	C₂	—	2	4	no	—
351	C₂	—	4	2	no	—
352	C₂	C₂²×#3998	1	1	yes	characteristic
353	C₂	C₂×D₄×#170	1	1	yes	characteristic
354	C₂	#10617530	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂³×C₈
Fitting	C₂×C₄×C₅×D₄	C₄
prime radical	C₂²×C₅×D₄	C₈
cosocle	C₂×#168	C₂⁴
hypercenter	C₂×C₄×D₄	AGL(1,𝔽₅)
hypocenter	C₅	D₄×M₃₂
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#3998	C₂	semidirect
3. #10604872	C₂	semidirect
4. #10604414	C₂	semidirect
5. #10604413	C₂	semidirect
6. D₄×#674	C₂	semidirect
7. #10604150	C₂	semidirect
8. C₄×#3998	C₂	semidirect
15. C₂×#3998	C₂²	semidirect
16. C₂²×#674	C₂²	semidirect
26. C₂²⋊#674	C₂²	semidirect
27. C₂²⋊#674	C₂²	semidirect
28. #64859	C₂²	semidirect
29. C₄⋊#674	C₂²	semidirect
30. C₄⋊#674	C₂²	semidirect
31. C₄×#674	C₂²	semidirect
56. #3998	D₄	direct (N × H)
124. #674	C₂×D₄	semidirect
156. C₅×D₄	M₃₂	semidirect
231. C₂²×C₅	C₂×M₃₂	semidirect
239. C₄×C₅	C₂×M₃₂	semidirect
301. D₄	#3998	direct (N × H)
328. C₅	D₄×M₃₂	semidirect
337. C₂²	C₂×#3998	semidirect
345. C₄	C₂×#3998	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#3998	2
C₂×D₄×#168	2
#10604872	2
#10604414	2
#10604413	2
D₄×#674	2
#10604150	2
C₄×#3998	2
D₄×M₃₂	5

Subgroups — D4×#3998 #13043536

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×#3998 #13043536