Subgroups — #16854742

Order 3968.

464 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₂²⋊D₃₁	C₂	1	1	yes	maximal, characteristic
2	#14521124	C₂	1	2	yes	maximal, complemented
3	#14521123	C₂	1	2	yes	maximal, complemented
4	#14521120	C₂	1	2	yes	maximal, complemented
5	C₈×C₂²⋊D₃₁	C₂	1	2	yes	maximal, complemented
6	#14521103	C₂	1	1	yes	maximal, characteristic, complemented
7	#14521099	C₂	1	1	yes	maximal, characteristic, complemented
8	#14521013	C₂	1	1	yes	maximal, characteristic, complemented
9	#14521011	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²⋊C₈×D₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×C₂²⋊D₃₁	C₄	1	1	yes	characteristic
14	C₂²×C₄×D₃₁	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₃₁	C₂²	1	2	yes	complemented
16	C₂×C₂²⋊Dic₃₁	C₄	1	1	yes	characteristic
17	C₄×C₂²⋊D₃₁	C₄	1	1	yes	characteristic
18	C₄×C₂²⋊D₃₁	C₄	1	1	yes	characteristic
19	C₄×C₂²⋊D₃₁	C₄	1	1	yes	characteristic
20	C₄×C₂²⋊D₃₁	C₄	1	1	yes	characteristic
21	C₄×C₂²⋊D₃₁	C₂²	1	1	yes	characteristic
22	C₄×C₂²⋊D₃₁	C₂²	1	1	yes	characteristic
23	C₄×C₂²⋊D₃₁	C₂²	1	1	yes	characteristic
24	C₄×C₂²⋊D₃₁	C₂²	1	1	yes	characteristic
25	C₂×(C₂×D₃₁)⋊C₄	C₄	1	1	yes	characteristic
26	C₂×Dic₃₁⋊C₄	C₄	1	1	yes	characteristic
27	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
28	C₂×#92474	C₂²	1	2	yes	complemented
29	C₂×#92469	C₂²	1	2	yes	complemented
30	C₂×C₈×D₃₁	C₂²	1	2	yes	complemented
31	C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
32	C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
33	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	complemented
34	C₂²⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
35	C₂²⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
36	C₂²⋊(C₃₁⋊C₈)	C₂²	1	2	yes	complemented
37	(C₂×D₃₁)⋊C₈	C₂²	1	2	yes	complemented
38	(C₂×D₃₁)⋊C₈	C₂²	1	2	yes	complemented
39	#11758403	C₂²	1	2	yes	complemented
40	Dic₃₁⋊C₈	C₂²	1	2	yes	complemented
41	Dic₃₁⋊C₈	C₂²	1	2	yes	complemented
42	C₈×Dic₃₁	C₂²	1	2	yes	complemented
43	C₂⁴×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂³×D₃₁	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₃₁	—	4	2	no	—
46	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
52	C₂×C₂²⋊D₃₁	—	2	4	no	—
53	C₂×C₂²⋊D₃₁	C₂×C₄	1	1	yes	characteristic
54	C₂×C₂²⋊D₃₁	C₂×C₄	1	1	yes	characteristic
55	C₂×C₂²⋊D₃₁	C₂×C₄	1	1	yes	characteristic
56	C₂×C₂²⋊D₃₁	C₂×C₄	1	1	yes	characteristic
57	C₂²×Dic₃₁	C₂×C₄	1	1	yes	characteristic
58	C₂²×Dic₃₁	C₂×C₄	1	1	yes	characteristic
59	C₂²×Dic₃₁	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄×D₃₁	—	4	2	no	—
61	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
65	C₂×C₈×C₃₁	—	4	2	no	—
66	C₂×C₈×C₃₁	C₂³	1	2	yes	—
67	C₂×C₈×C₃₁	C₂³	1	2	yes	—
68	C₂²⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
69	C₂²⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
70	C₂²⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
71	C₂²⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
72	(C₂×D₃₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
73	(C₂×D₃₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
74	(C₂×D₃₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
75	(C₂×D₃₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
76	Dic₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
77	Dic₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
78	Dic₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
79	Dic₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
80	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
81	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
82	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
83	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
84	#92474	—	4	2	no	—
85	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
86	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
87	#92469	—	4	2	no	—
88	C₈×D₃₁	—	4	2	no	—
89	C₂³×C₃₁	—	2	2	no	—
90	C₂³×C₃₁	—	2	2	no	—
91	C₂³×C₃₁	—	4	1	no	—
92	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
93	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
94	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
95	C₂²×D₃₁	—	2	2	no	—
96	C₂²×D₃₁	—	2	2	no	—
97	C₂²×D₃₁	—	4	1	no	—
98	C₂²×D₃₁	C₂²×C₄	1	1	yes	characteristic
99	C₂²×D₃₁	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×C₃₁	—	2	2	no	—
101	C₂×C₄×C₃₁	—	2	2	no	—
102	C₂×C₄×C₃₁	—	2	2	no	—
103	C₂×C₄×C₃₁	—	4	1	no	—
104	C₂×C₄×C₃₁	—	4	1	no	—
105	C₂×C₄×C₃₁	—	4	1	no	—
106	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
110	C₂²⋊D₃₁	—	2	2	no	—
111	C₂²⋊D₃₁	—	2	2	no	—
112	C₂²⋊D₃₁	—	2	2	no	—
113	C₂²⋊D₃₁	M₁₆	1	4	yes	complemented
114	C₂×Dic₃₁	—	2	2	no	—
115	C₂×Dic₃₁	—	2	2	no	—
116	C₂×Dic₃₁	—	4	1	no	—
117	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
118	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
119	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
120	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
121	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
122	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
123	C₄×D₃₁	—	2	2	no	—
124	C₄×D₃₁	—	2	2	no	—
125	C₄×D₃₁	—	4	1	no	—
126	C₄×D₃₁	—	4	1	no	—
127	C₈×C₃₁	—	2	2	no	—
128	C₈×C₃₁	—	2	2	no	—
129	C₃₁⋊C₈	—	2	2	no	—
130	C₃₁⋊C₈	—	2	2	no	—
131	#2974	—	31	1	no	maximal, Sylow
132	C₂²×C₃₁	—	2	1	no	—
133	C₂²×C₃₁	—	2	1	no	—
134	C₂²×C₃₁	—	2	1	no	—
135	C₂²×C₃₁	—	2	1	no	—
136	C₂²×C₃₁	—	2	1	no	—
137	C₂²×C₃₁	—	2	1	no	—
138	C₂²×C₃₁	—	2	1	no	—
139	C₂²×C₃₁	—	4	4	no	—
140	C₂²×C₃₁	C₂×M₁₆	1	2	yes	—
141	C₂²×C₃₁	#131	1	2	yes	—
142	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
143	C₂×D₃₁	—	2	1	no	—
144	C₂×D₃₁	—	2	1	no	—
145	C₂×D₃₁	—	2	1	no	—
146	C₂×D₃₁	—	2	1	no	—
147	C₂×D₃₁	—	2	1	no	—
148	C₂×D₃₁	—	4	4	no	—
149	C₂×D₃₁	C₂×M₁₆	1	2	yes	—
150	C₄×C₃₁	—	2	1	no	—
151	C₄×C₃₁	—	2	1	no	—
152	C₄×C₃₁	—	2	1	no	—
153	C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
154	Dic₃₁	—	2	1	no	—
155	Dic₃₁	—	2	1	no	—
156	Dic₃₁	—	2	1	no	—
157	Dic₃₁	—	2	1	no	—
158	Dic₃₁	—	2	1	no	—
159	Dic₃₁	C₂×M₁₆	1	2	yes	—
160	C₂×C₄×D₄	—	31	1	no	—
161	#436	—	31	2	no	—
162	#435	—	31	2	no	—
163	#435	—	31	2	no	—
164	C₈×D₄	—	31	2	no	—
165	#432	—	31	1	no	—
166	#431	—	31	1	no	—
167	#407	—	31	1	no	—
168	#407	—	31	1	no	—
169	C₂×C₂²⋊C₈	—	31	1	no	—
170	C₂×C₂²⋊C₈	D₃₁	1	1	yes	characteristic, complemented
171	C₂×C₃₁	—	2	2	no	—
172	C₂×C₃₁	—	2	2	no	—
173	C₂×C₃₁	—	4	1	no	—
174	C₂×C₃₁	#518	1	1	yes	characteristic
175	C₂×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
176	C₂×C₃₁	C₂×#131	1	1	yes	characteristic
177	D₃₁	—	2	2	no	—
178	D₃₁	—	4	1	no	—
179	C₂²×D₄	—	31	1	no	—
180	C₂³×C₄	—	31	1	no	—
181	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
182	C₂×M₁₆	—	31	2	no	—
183	C₂×M₁₆	—	31	2	no	—
184	C₂²×C₈	—	31	2	no	—
185	C₂²×C₈	C₂×D₃₁	1	2	yes	complemented
186	C₄×D₄	—	31	1	no	—
187	C₄×D₄	—	31	1	no	—
188	C₄×D₄	—	31	1	no	—
189	C₄×D₄	—	31	1	no	—
190	C₄×D₄	—	31	1	no	—
191	C₄×D₄	—	31	1	no	—
192	C₄×D₄	—	31	1	no	—
193	C₄×D₄	—	31	1	no	—
194	C₂×C₄⋊C₄	—	31	1	no	—
195	C₂×C₂²⋊C₄	—	31	1	no	—
196	C₂×C₂²⋊C₄	—	31	1	no	—
197	C₂×C₄²	—	31	1	no	—
198	C₄⋊C₈	—	31	2	no	—
199	C₄⋊C₈	—	31	2	no	—
200	C₂²⋊C₈	—	31	1	no	—
201	C₂²⋊C₈	—	31	1	no	—
202	C₂²⋊C₈	—	31	2	no	—
203	C₂²⋊C₈	—	31	2	no	—
204	C₂²⋊C₈	—	31	2	no	—
205	C₂²⋊C₈	C₂×D₃₁	1	1	yes	characteristic
206	C₂²⋊C₈	C₂×D₃₁	1	1	yes	characteristic
207	C₂²⋊C₈	C₂×D₃₁	1	2	yes	complemented
208	#97	—	31	2	no	—
209	C₄×C₈	—	31	2	no	—
210	C₃₁	#2974	1	1	yes	characteristic, complemented, Sylow
211	C₂⁴	—	31	1	no	—
212	C₂⁴	C₄×D₃₁	1	1	yes	characteristic
213	C₂×D₄	—	31	1	no	—
214	C₂×D₄	—	31	1	no	—
215	C₂×D₄	—	31	1	no	—
216	C₂×D₄	—	31	1	no	—
217	C₂×D₄	—	62	4	no	—
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	—	31	1	no	—
220	C₂²×C₄	—	31	1	no	—
221	C₂²×C₄	—	31	1	no	—
222	C₂²×C₄	—	31	1	no	—
223	C₂²×C₄	—	31	1	no	—
224	C₂²×C₄	—	31	1	no	—
225	C₂²×C₄	—	31	1	no	—
226	C₂²×C₄	—	124	2	no	—
227	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
228	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
229	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
230	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
231	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
232	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
233	M₁₆	—	124	2	no	—
234	M₁₆	—	124	2	no	—
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	—	31	2	no	—
237	C₂×C₈	—	31	2	no	—
238	C₂×C₈	—	124	2	no	—
239	C₂×C₈	C₂²×D₃₁	1	2	yes	—
240	C₂×C₈	C₂²×D₃₁	1	2	yes	—
241	C₄⋊C₄	—	31	1	no	—
242	C₄⋊C₄	—	31	1	no	—
243	C₄⋊C₄	—	31	1	no	—
244	C₄⋊C₄	—	31	1	no	—
245	C₂²⋊C₄	—	31	1	no	—
246	C₂²⋊C₄	—	31	1	no	—
247	C₂²⋊C₄	—	31	1	no	—
248	C₂²⋊C₄	—	31	1	no	—
249	C₂²⋊C₄	—	31	1	no	—
250	C₂²⋊C₄	—	31	1	no	—
251	C₂²⋊C₄	—	31	1	no	—
252	C₂²⋊C₄	—	31	1	no	—
253	C₄²	—	31	1	no	—
254	C₄²	—	31	1	no	—
255	C₄²	—	31	1	no	—
256	C₄²	—	31	1	no	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	4	1	no	—
260	C₂³	—	31	1	no	—
261	C₂³	—	31	1	no	—
262	C₂³	—	62	2	no	—
263	C₂³	—	62	2	no	—
264	C₂³	—	124	1	no	—
265	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
266	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
267	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
268	D₄	—	31	4	no	—
269	D₄	—	62	2	no	—
270	D₄	—	62	2	no	—
271	D₄	—	62	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	31	1	no	—
279	C₂×C₄	—	31	1	no	—
280	C₂×C₄	—	31	1	no	—
281	C₂×C₄	—	31	1	no	—
282	C₂×C₄	—	31	1	no	—
283	C₂×C₄	—	31	1	no	—
284	C₂×C₄	—	62	2	no	—
285	C₂×C₄	—	62	2	no	—
286	C₂×C₄	—	62	2	no	—
287	C₂×C₄	—	62	2	no	—
288	C₂×C₄	—	124	1	no	—
289	C₂×C₄	—	124	1	no	—
290	C₂×C₄	—	124	1	no	—
291	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
292	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
293	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
294	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
295	C₈	—	2	2	no	—
296	C₈	—	2	2	no	—
297	C₈	—	62	2	no	—
298	C₈	—	62	2	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	4	4	no	—
307	C₂²	—	31	2	no	—
308	C₂²	—	62	1	no	—
309	C₂²	—	62	1	no	—
310	C₂²	—	62	1	no	—
311	C₂²	—	62	1	no	—
312	C₂²	—	62	1	no	—
313	C₂²	—	124	4	no	—
314	C₂²	#11758478	1	2	yes	—
315	C₂²	M₁₆×D₃₁	1	2	yes	—
316	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	31	2	no	—
321	C₄	—	62	1	no	—
322	C₄	—	62	1	no	—
323	C₄	—	62	1	no	—
324	C₄	—	62	1	no	—
325	C₄	—	62	1	no	—
326	C₄	#11758561	1	2	yes	—
327	C₂	—	2	2	no	—
328	C₂	—	2	2	no	—
329	C₂	—	4	1	no	—
330	C₂	—	62	2	no	—
331	C₂	—	124	1	no	—
332	C₂	#14521522	1	1	yes	characteristic
333	C₂	C₂×M₁₆×D₃₁	1	1	yes	characteristic
334	C₂	C₂×#11758478	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₁
commutator, socle	C₂²×C₃₁	C₂³×C₄
Fitting	C₂×C₃₁×C₂²⋊C₈	C₂
prime radical	C₂²×C₂²⋊D₃₁	C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₂×C₂²⋊C₈	D₃₁
hypocenter	C₃₁	#2974
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. #14521124	C₂	semidirect
3. #14521123	C₂	semidirect
4. #14521120	C₂	semidirect
5. C₈×C₂²⋊D₃₁	C₂	semidirect
6. #14521103	C₂	semidirect
7. #14521099	C₂	semidirect
8. #14521013	C₂	semidirect
9. #14521011	C₂	semidirect
10. C₂²⋊C₈×D₃₁	C₂	semidirect
11. C₂×C₃₁×C₂²⋊C₈	C₂	semidirect
15. C₂²×C₈×C₃₁	C₂²	semidirect
28. C₂×#92474	C₂²	semidirect
29. C₂×#92469	C₂²	semidirect
30. C₂×C₈×D₃₁	C₂²	semidirect
33. C₃₁×C₂²⋊C₈	C₂²	semidirect
36. C₂²⋊(C₃₁⋊C₈)	C₂²	semidirect
37. (C₂×D₃₁)⋊C₈	C₂²	semidirect
38. (C₂×D₃₁)⋊C₈	C₂²	semidirect
39. #11758403	C₂²	semidirect
40. Dic₃₁⋊C₈	C₂²	semidirect
41. Dic₃₁⋊C₈	C₂²	semidirect
42. C₈×Dic₃₁	C₂²	semidirect
113. C₂²⋊D₃₁	M₁₆	semidirect
170. C₂×C₂²⋊C₈	D₃₁	semidirect
185. C₂²×C₈	C₂×D₃₁	semidirect
207. C₂²⋊C₈	C₂×D₃₁	semidirect
210. C₃₁	#2974	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₂²⋊D₃₁	2
#14521124	2
#14521123	2
#14521120	2
C₈×C₂²⋊D₃₁	2
#14521103	2
#14521099	2
#14521013	2
#14521011	2
C₂²⋊C₈×D₃₁	2
C₂×C₃₁×C₂²⋊C₈	2
#2974	31