Subgroups — #16854879

Order 3968.

346 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521577	C₂	1	1	yes	maximal, characteristic
2	#14521178	C₂	1	1	yes	maximal, characteristic, complemented
3	#14521176	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521126	C₂	1	1	yes	maximal, characteristic, complemented
5	#14521125	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521120	C₂	1	1	yes	maximal, characteristic, complemented
7	#14521120	C₂	1	1	yes	maximal, characteristic, complemented
8	#14521119	C₂	1	1	yes	maximal, characteristic, complemented
9	#14521118	C₂	1	1	yes	maximal, characteristic, complemented
10	#14521110	C₂	1	1	yes	maximal, characteristic, complemented
11	#14521109	C₂	1	1	yes	maximal, characteristic, complemented
12	#14521105	C₂	1	1	yes	maximal, characteristic, complemented
13	#14521074	C₂	1	1	yes	maximal, characteristic, complemented
14	#14521071	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#424	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#117	C₂²	1	1	yes	characteristic
19	C₄×C₂²⋊D₃₁	C₂²	1	1	yes	characteristic
20	C₄×C₂²⋊D₃₁	C₂²	1	1	yes	characteristic
21	C₂×#92469	C₂²	1	1	yes	characteristic
22	C₂×#92469	C₂²	1	1	yes	characteristic
23	C₂×#92469	C₂²	1	1	yes	characteristic
24	C₂×C₈×D₃₁	C₂²	1	1	yes	characteristic
25	#11758475	C₄	1	1	yes	characteristic
26	#11758473	C₄	1	1	yes	characteristic
27	#11758471	C₄	1	1	yes	characteristic
28	#11758466	C₄	1	1	yes	characteristic
29	#11758462	C₄	1	1	yes	characteristic
30	#11758461	C₄	1	1	yes	characteristic
31	#11758460	C₄	1	1	yes	characteristic
32	#11758456	C₄	1	1	yes	characteristic
33	C₄×D₁₂₄	C₂²	1	1	yes	characteristic
34	#11758449	C₂²	1	1	yes	characteristic
35	C₄²×D₃₁	C₂²	1	1	yes	characteristic
36	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
37	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
38	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
39	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
40	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
41	C₂²⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
42	C₂²⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
43	(C₂×D₃₁)⋊C₈	C₂²	1	1	yes	characteristic
44	(C₂×D₃₁)⋊C₈	C₂²	1	1	yes	characteristic
45	(C₂×D₃₁)⋊C₈	C₂²	1	1	yes	characteristic
46	(C₂×D₃₁)⋊C₈	C₂²	1	1	yes	characteristic
47	#11758403	C₂²	1	1	yes	characteristic
48	#11758403	C₂²	1	1	yes	characteristic
49	#11758403	C₂²	1	1	yes	characteristic
50	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
51	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
52	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
53	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
54	C₈×Dic₃₁	C₂²	1	1	yes	characteristic
55	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
56	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
57	#11758391	C₂²	1	1	yes	characteristic
58	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
60	C₂×C₂²⋊D₃₁	C₂×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₃₁	C₂×C₄	1	1	yes	characteristic
62	C₂×D₁₂₄	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
66	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
67	C₂×Dic₆₂	C₂×C₄	1	1	yes	characteristic
68	M₁₆×C₃₁	—	4	1	no	—
69	C₂×C₈×C₃₁	—	4	1	no	—
70	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
75	C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
76	C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
77	C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
78	C₄²×C₃₁	C₂³	1	1	yes	characteristic
79	C₄²×C₃₁	C₂³	1	1	yes	characteristic
80	C₂²⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
81	C₂²⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
82	(C₂×D₃₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
83	(C₂×D₃₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
84	(C₂×D₃₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
85	(C₂×D₃₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
86	(C₂×D₃₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
87	(C₂×D₃₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
88	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
89	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
90	Dic₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
91	Dic₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
92	Dic₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
93	Dic₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
94	Dic₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
95	Dic₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
96	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
97	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
98	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
99	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
100	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
101	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
102	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
103	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
104	#92469	—	4	1	no	—
105	#92469	—	4	1	no	—
106	#92469	—	4	1	no	—
107	C₈×D₃₁	—	4	1	no	—
108	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
109	C₂²×D₃₁	C₂²×C₄	1	1	yes	characteristic
110	C₂²×D₃₁	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×C₃₁	—	2	1	no	—
112	C₂×C₄×C₃₁	—	2	1	no	—
113	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
119	C₂²⋊D₃₁	—	4	1	no	—
120	C₂²⋊D₃₁	—	4	1	no	—
121	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
122	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
123	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
124	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
125	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
126	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
127	D₁₂₄	—	4	1	no	—
128	C₄×D₃₁	—	2	1	no	—
129	C₄×D₃₁	—	2	1	no	—
130	C₄×D₃₁	—	2	1	no	—
131	C₄×D₃₁	—	2	1	no	—
132	C₄×D₃₁	—	4	1	no	—
133	C₄×D₃₁	—	4	1	no	—
134	Dic₆₂	—	4	1	no	—
135	C₈×C₃₁	—	2	1	no	—
136	C₈×C₃₁	—	2	1	no	—
137	C₈×C₃₁	—	2	1	no	—
138	C₈×C₃₁	—	2	1	no	—
139	C₃₁⋊C₈	—	2	1	no	—
140	C₃₁⋊C₈	—	2	1	no	—
141	C₃₁⋊C₈	—	2	1	no	—
142	C₃₁⋊C₈	—	2	1	no	—
143	#2995	—	31	1	no	maximal, Sylow
144	C₂²×C₃₁	—	2	1	no	—
145	C₂²×C₃₁	—	2	1	no	—
146	C₂²×C₃₁	—	2	1	no	—
147	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
148	C₂×D₃₁	—	2	1	no	—
149	C₂×D₃₁	—	2	1	no	—
150	C₂×D₃₁	—	2	1	no	—
151	C₂×D₃₁	—	2	1	no	—
152	C₂×D₃₁	—	2	1	no	—
153	C₂×D₃₁	—	2	1	no	—
154	C₄×C₃₁	—	2	1	no	—
155	C₄×C₃₁	—	2	1	no	—
156	C₄×C₃₁	—	2	1	no	—
157	C₄×C₃₁	—	2	1	no	—
158	C₄×C₃₁	—	2	1	no	—
159	C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
160	C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
161	Dic₃₁	—	2	1	no	—
162	Dic₃₁	—	2	1	no	—
163	Dic₃₁	—	2	1	no	—
164	Dic₃₁	—	2	1	no	—
165	Dic₃₁	—	2	1	no	—
166	Dic₃₁	—	2	1	no	—
167	#529	—	31	1	no	—
168	#446	—	31	1	no	—
169	#446	—	31	1	no	—
170	#436	—	31	1	no	—
171	#436	—	31	1	no	—
172	#435	—	31	1	no	—
173	#435	—	31	1	no	—
174	#435	—	31	1	no	—
175	#435	—	31	1	no	—
176	#433	—	31	1	no	—
177	#433	—	31	1	no	—
178	#432	—	31	1	no	—
179	#424	—	31	1	no	—
180	#424	D₃₁	1	1	yes	characteristic, complemented
181	#423	—	31	1	no	—
182	C₂×C₃₁	—	4	1	no	—
183	C₂×C₃₁	#520	1	1	yes	characteristic
184	C₂×C₃₁	#568	1	1	yes	characteristic
185	C₂×C₃₁	#568	1	1	yes	characteristic
186	D₃₁	—	4	1	no	—
187	D₃₁	—	4	1	no	—
188	C₂×M₁₆	—	31	1	no	—
189	C₂×M₁₆	—	31	1	no	—
190	C₂×M₁₆	—	31	1	no	—
191	C₂×M₁₆	C₂×D₃₁	1	1	yes	characteristic
192	C₂²×C₈	—	31	1	no	—
193	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
194	#126	—	31	1	no	—
195	#126	—	31	1	no	—
196	#125	—	31	1	no	—
197	#124	—	31	1	no	—
198	#123	—	31	1	no	—
199	#123	—	31	1	no	—
200	C₂²⋊Q₈	—	31	1	no	—
201	#121	—	31	1	no	—
202	C₄×Q₈	—	31	1	no	—
203	C₄×D₄	—	31	1	no	—
204	C₄×D₄	—	31	1	no	—
205	C₄×D₄	—	31	1	no	—
206	#117	—	31	1	no	—
207	#117	C₂×D₃₁	1	1	yes	characteristic
208	C₂×C₄²	—	31	1	no	—
209	C₄⋊C₈	—	31	1	no	—
210	C₄⋊C₈	—	31	1	no	—
211	C₄⋊C₈	—	31	1	no	—
212	C₄⋊C₈	—	31	1	no	—
213	C₄⋊C₈	—	31	1	no	—
214	C₄⋊C₈	—	31	1	no	—
215	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic
216	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic
217	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic
218	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic
219	C₂²⋊C₈	—	31	1	no	—
220	C₂²⋊C₈	—	31	1	no	—
221	C₂²⋊C₈	—	31	1	no	—
222	C₂²⋊C₈	—	31	1	no	—
223	C₂²⋊C₈	—	31	1	no	—
224	C₂²⋊C₈	—	31	1	no	—
225	#97	—	31	1	no	—
226	#97	—	31	1	no	—
227	#97	—	31	1	no	—
228	#97	—	31	1	no	—
229	C₄×C₈	—	31	1	no	—
230	C₄×C₈	—	31	1	no	—
231	C₃₁	#2995	1	1	yes	characteristic, complemented, Sylow
232	C₂×Q₈	—	31	1	no	—
233	C₂×D₄	—	31	1	no	—
234	C₂×D₄	—	31	1	no	—
235	C₂×D₄	—	31	1	no	—
236	C₂²×C₄	—	31	1	no	—
237	C₂²×C₄	—	31	1	no	—
238	C₂²×C₄	—	31	1	no	—
239	C₂²×C₄	—	31	1	no	—
240	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
241	M₁₆	—	4	1	no	—
242	M₁₆	—	124	1	no	—
243	M₁₆	—	124	1	no	—
244	M₁₆	—	124	1	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	—	31	1	no	—
247	C₂×C₈	—	31	1	no	—
248	C₂×C₈	—	31	1	no	—
249	C₂×C₈	—	31	1	no	—
250	C₂×C₈	—	124	1	no	—
251	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
252	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
253	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
254	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
255	C₄⋊C₄	—	31	1	no	—
256	C₄⋊C₄	—	31	1	no	—
257	C₄⋊C₄	—	31	1	no	—
258	C₄⋊C₄	—	31	1	no	—
259	C₄⋊C₄	—	31	1	no	—
260	C₄⋊C₄	—	31	1	no	—
261	C₄⋊C₄	—	31	1	no	—
262	C₄⋊C₄	—	31	1	no	—
263	C₄⋊C₄	C₄×D₃₁	1	1	yes	characteristic
264	C₄⋊C₄	C₄×D₃₁	1	1	yes	characteristic
265	C₂²⋊C₄	—	31	1	no	—
266	C₂²⋊C₄	—	31	1	no	—
267	C₂²⋊C₄	—	31	1	no	—
268	C₂²⋊C₄	—	31	1	no	—
269	C₂²⋊C₄	—	31	1	no	—
270	C₂²⋊C₄	—	31	1	no	—
271	C₂²⋊C₄	—	31	1	no	—
272	C₂²⋊C₄	—	31	1	no	—
273	C₂²⋊C₄	C₄×D₃₁	1	1	yes	characteristic
274	C₂²⋊C₄	C₄×D₃₁	1	1	yes	characteristic
275	C₄²	—	31	1	no	—
276	C₄²	—	31	1	no	—
277	C₄²	—	31	1	no	—
278	C₄²	—	31	1	no	—
279	C₄²	C₂²×D₃₁	1	1	yes	characteristic
280	C₄²	C₂²×D₃₁	1	1	yes	characteristic
281	C₂³	—	31	1	no	—
282	C₂³	—	31	1	no	—
283	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
284	Q₈	—	124	1	no	—
285	D₄	—	124	1	no	—
286	D₄	—	124	1	no	—
287	D₄	—	124	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	31	1	no	—
291	C₂×C₄	—	31	1	no	—
292	C₂×C₄	—	31	1	no	—
293	C₂×C₄	—	31	1	no	—
294	C₂×C₄	—	31	1	no	—
295	C₂×C₄	—	31	1	no	—
296	C₂×C₄	—	62	1	no	—
297	C₂×C₄	—	62	1	no	—
298	C₂×C₄	—	62	1	no	—
299	C₂×C₄	—	62	1	no	—
300	C₂×C₄	—	124	1	no	—
301	C₂×C₄	—	124	1	no	—
302	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
303	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
304	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
305	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
306	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	62	1	no	—
313	C₈	—	62	1	no	—
314	C₈	—	62	1	no	—
315	C₈	—	62	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	62	1	no	—
320	C₂²	—	62	1	no	—
321	C₂²	—	62	1	no	—
322	C₂²	—	62	1	no	—
323	C₂²	—	62	1	no	—
324	C₂²	—	62	1	no	—
325	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	62	1	no	—
332	C₄	—	62	1	no	—
333	C₄	—	62	1	no	—
334	C₄	—	62	1	no	—
335	C₄	—	62	1	no	—
336	C₄	—	62	1	no	—
337	C₄	#11758561	1	1	yes	characteristic
338	C₄	#11758564	1	1	yes	characteristic
339	C₂	—	4	1	no	—
340	C₂	—	124	1	no	—
341	C₂	—	124	1	no	—
342	C₂	#14521530	1	1	yes	characteristic
343	C₂	#14521814	1	1	yes	characteristic
344	C₂	#14521814	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₁
commutator, socle	C₂²×C₃₁	C₂³×C₄
Fitting	C₃₁×#424	C₂
prime radical	#11758461	C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	#424	D₃₁
hypocenter	C₃₁	#2995
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
2. #14521178	C₂	semidirect
3. #14521176	C₂	semidirect
4. #14521126	C₂	semidirect
5. #14521125	C₂	semidirect
6. #14521120	C₂	semidirect
7. #14521120	C₂	semidirect
8. #14521119	C₂	semidirect
9. #14521118	C₂	semidirect
10. #14521110	C₂	semidirect
11. #14521109	C₂	semidirect
12. #14521105	C₂	semidirect
13. #14521074	C₂	semidirect
14. #14521071	C₂	semidirect
15. C₃₁×#424	C₂	semidirect
180. #424	D₃₁	semidirect
231. C₃₁	#2995	semidirect

Maximal subgroups

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

subgroup	index
#14521577	2
#14521178	2
#14521176	2
#14521126	2
#14521125	2
#14521120	2
#14521119	2
#14521118	2
#14521110	2
#14521109	2
#14521105	2
#14521074	2
#14521071	2
C₃₁×#424	2
#2995	31