Subgroups — #30622638

Order 15616.

772 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×#16840813	C₂	1	2	yes	maximal
2	#21461988	C₂	1	2	yes	maximal
3	#21461917	C₂	1	2	yes	maximal
4	#21461879	C₂	1	4	yes	maximal
5	#21461615	C₂	1	2	yes	maximal
6	C₄×C₄⋊#92420	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊#92420	C₂	1	1	yes	maximal, characteristic
8	C₄⋊Q₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
9	#16840813	—	2	8	no	—
10	C₂×#14519844	C₂²	1	4	yes	—
11	C₂×C₄⋊#92420	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊#92420	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊#92420	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊#92420	C₂²	1	1	yes	characteristic
15	C₂×#14519832	C₂²	1	4	yes	—
16	C₂×C₄×#92420	C₂²	1	2	yes	—
17	C₂×C₄×#92420	C₂²	1	2	yes	—
18	#16840628	C₂²	1	2	yes	—
19	#16840628	C₂²	1	2	yes	—
20	#16840628	C₂²	1	4	yes	—
21	#16840628	C₂²	1	4	yes	—
22	C₂×Q₈×Dic₆₁	C₂²	1	2	yes	—
23	#16839669	C₄	1	2	yes	—
24	#16839661	C₄	1	2	yes	—
25	#16839661	C₄	1	2	yes	—
26	#16839632	C₄	1	1	yes	characteristic
27	C₄⋊C₄×Dic₆₁	C₂²	1	2	yes	—
28	C₄⋊C₄×Dic₆₁	C₂²	1	2	yes	—
29	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊Q₈×C₆₁	C₄	1	1	yes	characteristic
31	C₂²×#92420	C₂³	1	2	yes	—
32	C₂²×#92420	C₂³	1	2	yes	—
33	C₂²×#92420	C₂³	1	2	yes	—
34	C₂²×#92420	C₂³	1	2	yes	—
35	#14519844	—	2	8	no	—
36	C₄⋊#92420	—	2	2	no	—
37	C₄⋊#92420	—	2	2	no	—
38	C₄⋊#92420	D₄	1	4	yes	—
39	C₄⋊#92420	Q₈	1	4	yes	complemented
40	#14519832	—	2	8	no	—
41	C₄×#92420	—	2	4	no	—
42	C₄×#92420	—	2	4	no	—
43	Q₈×Dic₆₁	—	2	4	no	—
44	Q₈×Dic₆₁	—	2	4	no	—
45	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊Dic₆₁	C₂×C₄	1	4	yes	—
47	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
48	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
49	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×Dic₆₁	C₂³	1	4	yes	—
53	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
54	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
55	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
56	#14519684	C₂×C₄	1	4	yes	—
57	#14519684	C₂×C₄	1	4	yes	—
58	C₂²×Q₈×C₆₁	C₂×C₄	1	2	yes	—
59	C₄⋊Q₈×C₆₁	—	2	4	no	—
60	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
61	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
62	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
63	C₂×Q₈×C₆₁	—	2	4	no	—
64	C₂×Q₈×C₆₁	—	2	4	no	—
65	C₂×Q₈×C₆₁	—	2	4	no	—
66	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
70	C₂²×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
71	C₂²×Dic₆₁	C₂²×C₄	1	4	yes	—
72	C₂²×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
73	C₂²×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
74	C₂²×Dic₆₁	C₂⁴	1	1	yes	characteristic
75	C₂×#92420	—	2	2	no	—
76	C₂×#92420	—	2	2	no	—
77	C₂×#92420	—	2	2	no	—
78	C₂×#92420	—	2	2	no	—
79	C₂×#92420	—	2	2	no	—
80	C₂×#92420	—	2	2	no	—
81	C₂×#92420	—	2	2	no	—
82	C₂×#92420	—	2	2	no	—
83	C₂×#92420	—	2	2	no	—
84	C₂×#92420	—	2	2	no	—
85	C₂×#92420	C₂×D₄	1	4	yes	—
86	C₂×#92420	C₂×Q₈	1	4	yes	—
87	C₄⋊C₄×C₆₁	—	2	4	no	—
88	C₄⋊C₄×C₆₁	—	2	4	no	—
89	C₄²×C₆₁	M₁₆	1	4	yes	—
90	C₄⋊Dic₆₁	—	2	4	no	—
91	C₄⋊Dic₆₁	—	2	4	no	—
92	C₄⋊Dic₆₁	—	2	8	no	—
93	C₄⋊Dic₆₁	—	4	2	no	—
94	C₄×Dic₆₁	—	2	2	no	—
95	C₄×Dic₆₁	—	2	2	no	—
96	C₄×Dic₆₁	—	2	4	no	—
97	C₄×Dic₆₁	—	2	4	no	—
98	C₄×Dic₆₁	—	2	1	no	—
99	C₄×Dic₆₁	—	2	1	no	—
100	C₄×Dic₆₁	—	4	1	no	—
101	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
102	C₄×Dic₆₁	C₂×Q₈	1	2	yes	—
103	C₄×Dic₆₁	#41	1	2	yes	—
104	C₄×Dic₆₁	#41	1	2	yes	—
105	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
106	Q₈×C₆₁	—	4	8	no	—
107	C₂×C₄×C₆₁	—	2	1	no	—
108	C₂×C₄×C₆₁	—	2	1	no	—
109	C₂×C₄×C₆₁	—	2	4	no	—
110	C₂×C₄×C₆₁	—	2	4	no	—
111	C₂×C₄×C₆₁	—	2	4	no	—
112	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
114	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
115	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
116	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
117	C₂×C₄×C₆₁	#131	1	2	yes	—
118	C₂×C₄×C₆₁	#131	1	2	yes	—
119	C₂×Dic₆₁	—	2	1	no	—
120	C₂×Dic₆₁	—	2	1	no	—
121	C₂×Dic₆₁	—	2	1	no	—
122	C₂×Dic₆₁	—	2	1	no	—
123	C₂×Dic₆₁	—	2	4	no	—
124	C₂×Dic₆₁	—	2	1	no	—
125	C₂×Dic₆₁	—	2	1	no	—
126	C₂×Dic₆₁	—	2	1	no	—
127	C₂×Dic₆₁	—	2	4	no	—
128	C₂×Dic₆₁	—	2	4	no	—
129	C₂×Dic₆₁	C₄×D₄	1	2	yes	—
130	C₂×Dic₆₁	C₄×Q₈	1	2	yes	—
131	C₂×Dic₆₁	C₂²×D₄	1	1	yes	characteristic
132	C₂×Dic₆₁	C₂²×Q₈	1	1	yes	characteristic
133	C₂×Dic₆₁	C₂×#41	1	1	yes	characteristic
134	C₂×Dic₆₁	C₂×#41	1	1	yes	characteristic
135	C₂×Dic₆₁	2₋¹⁺⁴	1	1	yes	characteristic
136	C₂×Dic₆₁	2₋¹⁺⁴	1	1	yes	characteristic
137	#92420	—	2	4	no	—
138	#92420	—	2	4	no	—
139	#92420	—	4	2	no	—
140	#92420	—	4	2	no	—
141	#23153	—	61	1	no	maximal, Sylow
142	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
143	C₂²×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
144	C₂²×C₆₁	#519	1	1	yes	characteristic
145	C₂²×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
146	C₂²×C₆₁	C₂×#131	1	1	yes	characteristic
147	C₂²×C₆₁	C₂×#131	1	1	yes	characteristic
148	C₂²×C₆₁	#568	1	1	yes	characteristic
149	C₄×C₆₁	—	2	2	no	—
150	C₄×C₆₁	—	4	4	no	—
151	C₄×C₆₁	#436	1	4	yes	—
152	C₄×C₆₁	#446	1	4	yes	complemented
153	Dic₆₁	—	2	2	no	—
154	Dic₆₁	—	2	2	no	—
155	Dic₆₁	—	4	4	no	—
156	Dic₆₁	—	4	1	no	—
157	Dic₆₁	#547	1	1	yes	characteristic
158	Dic₆₁	D₄×Q₈	1	1	yes	characteristic
159	Dic₆₁	#556	1	1	yes	characteristic
160	Dic₆₁	#557	1	1	yes	characteristic
161	C₂×#446	—	61	2	no	—
162	C₄×C₄⋊Q₈	—	61	1	no	—
163	#1984	—	61	2	no	—
164	#1939	—	61	2	no	—
165	#1917	—	61	4	no	—
166	#1770	—	61	2	no	—
167	C₄×C₄⋊C₈	—	61	1	no	—
168	C₄×C₄⋊C₈	—	61	1	no	—
169	C₂×C₆₁	#2365	1	1	yes	characteristic
170	C₂×C₆₁	C₂×#436	1	1	yes	characteristic
171	C₂×C₆₁	#2935	1	1	yes	characteristic
172	C₂×C₆₁	C₂×#446	1	1	yes	characteristic
173	C₂×C₆₁	#2962	1	1	yes	characteristic
174	C₂×C₆₁	#2983	1	1	yes	characteristic
175	C₂×C₆₁	#2986	1	1	yes	characteristic
176	C₂×C₄⋊Q₈	#6923	1	1	yes	characteristic
177	C₂×C₄×Q₈	—	61	2	no	—
178	#446	—	122	8	no	—
179	C₂×C₄⋊C₈	—	61	1	no	—
180	C₂×C₄⋊C₈	—	61	1	no	—
181	C₂×C₄⋊C₈	—	61	1	no	—
182	C₂×C₄⋊C₈	—	61	1	no	—
183	C₂×C₄⋊C₈	—	61	4	no	—
184	C₂×#97	—	61	4	no	—
185	C₂×C₄×C₈	—	61	2	no	—
186	C₂×C₄×C₈	—	61	2	no	—
187	#391	—	61	2	no	—
188	#389	—	61	2	no	—
189	#389	—	61	2	no	—
190	#384	—	61	1	no	—
191	C₄×C₄⋊C₄	—	61	2	no	—
192	C₄×C₄⋊C₄	—	61	2	no	—
193	C₄³	—	61	1	no	—
194	#336	—	61	2	no	—
195	#336	—	61	2	no	—
196	#336	—	61	4	no	—
197	#336	—	61	4	no	—
198	C₆₁	#23153	1	1	yes	characteristic, complemented, Sylow
199	C₂²×Q₈	C₂×#6923	1	2	yes	—
200	C₂²×C₈	—	61	2	no	—
201	C₂²×C₈	—	61	2	no	—
202	C₂²×C₈	—	61	2	no	—
203	C₂²×C₈	—	61	2	no	—
204	C₄⋊Q₈	—	2	4	no	—
205	C₄×Q₈	—	122	4	no	—
206	C₄×Q₈	—	122	4	no	—
207	C₂×C₄⋊C₄	—	61	2	no	—
208	C₂×C₄⋊C₄	—	61	4	no	—
209	C₂×C₄⋊C₄	—	61	2	no	—
210	C₂×C₄⋊C₄	—	61	2	no	—
211	C₂×C₄⋊C₄	C₂×#6923	1	2	yes	—
212	C₂×C₄⋊C₄	C₂×#6923	1	2	yes	—
213	C₂×C₄²	—	61	1	no	—
214	C₂×C₄²	—	61	1	no	—
215	C₂×C₄²	—	61	4	no	—
216	C₂×C₄²	—	61	1	no	—
217	C₂×C₄²	—	61	1	no	—
218	C₂×C₄²	—	61	1	no	—
219	C₂×C₄²	—	61	1	no	—
220	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
221	C₄⋊C₈	—	61	4	no	—
222	C₄⋊C₈	—	61	4	no	—
223	C₄⋊C₈	—	122	2	no	—
224	C₄⋊C₈	—	122	2	no	—
225	C₄⋊C₈	—	122	8	no	—
226	#97	—	122	8	no	—
227	C₄×C₈	—	122	4	no	—
228	C₄×C₈	—	122	4	no	—
229	#95	—	61	4	no	—
230	#95	—	61	4	no	—
231	C₂×Q₈	—	2	4	no	—
232	C₂×Q₈	—	2	4	no	—
233	C₂×Q₈	—	2	4	no	—
234	C₂²×C₄	—	61	1	no	—
235	C₂²×C₄	—	61	1	no	—
236	C₂²×C₄	—	61	1	no	—
237	C₂²×C₄	—	61	1	no	—
238	C₂²×C₄	—	61	4	no	—
239	C₂²×C₄	C₂²×#6923	1	4	yes	—
240	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
241	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
242	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
243	C₂×C₈	—	61	4	no	—
244	C₂×C₈	—	61	4	no	—
245	C₂×C₈	—	122	2	no	—
246	C₂×C₈	—	122	2	no	—
247	C₂×C₈	—	122	2	no	—
248	C₂×C₈	—	122	2	no	—
249	C₂×C₈	—	122	2	no	—
250	C₂×C₈	—	122	2	no	—
251	C₂×C₈	—	122	2	no	—
252	C₂×C₈	—	122	2	no	—
253	C₂×C₈	—	122	2	no	—
254	C₂×C₈	—	122	2	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	122	8	no	—
258	C₄⋊C₄	—	122	4	no	—
259	C₄⋊C₄	—	122	4	no	—
260	C₄⋊C₄	—	244	2	no	—
261	C₄²	—	61	2	no	—
262	C₄²	—	61	2	no	—
263	C₄²	—	61	2	no	—
264	C₄²	—	61	2	no	—
265	C₄²	—	122	4	no	—
266	C₄²	—	122	1	no	—
267	C₄²	—	122	1	no	—
268	C₄²	—	122	4	no	—
269	C₄²	—	122	2	no	—
270	C₄²	—	122	2	no	—
271	C₄²	—	244	1	no	—
272	C₄²	#11758217	1	4	yes	—
273	C₂³	C₂³×#6923	1	1	yes	characteristic
274	Q₈	—	4	8	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	61	1	no	—
281	C₂×C₄	—	61	1	no	—
282	C₂×C₄	—	61	1	no	—
283	C₂×C₄	—	61	1	no	—
284	C₂×C₄	—	61	1	no	—
285	C₂×C₄	—	61	1	no	—
286	C₂×C₄	—	61	2	no	—
287	C₂×C₄	—	61	2	no	—
288	C₂×C₄	—	122	1	no	—
289	C₂×C₄	—	122	1	no	—
290	C₂×C₄	—	122	1	no	—
291	C₂×C₄	—	122	1	no	—
292	C₂×C₄	—	122	4	no	—
293	C₂×C₄	—	122	1	no	—
294	C₂×C₄	—	122	1	no	—
295	C₂×C₄	—	122	1	no	—
296	C₂×C₄	—	122	4	no	—
297	C₂×C₄	—	122	4	no	—
298	C₂×C₄	D₄×#6923	1	2	yes	—
299	C₂×C₄	Q₈×#6923	1	2	yes	—
300	C₂×C₄	C₂×#11758217	1	2	yes	—
301	C₂×C₄	C₂×#11758217	1	2	yes	—
302	C₂×C₄	C₂×#11758217	1	2	yes	—
303	C₂×C₄	#14519864	1	2	yes	—
304	C₂×C₄	#14519865	1	2	yes	—
305	C₈	—	122	4	no	—
306	C₈	—	122	4	no	—
307	C₈	—	244	2	no	—
308	C₈	—	244	2	no	—
309	C₂²	C₂×D₄×#6923	1	1	yes	characteristic
310	C₂²	C₂×Q₈×#6923	1	1	yes	characteristic
311	C₂²	#16840830	1	1	yes	characteristic
312	C₂²	C₂²×#11758217	1	1	yes	characteristic
313	C₂²	C₂×#14519864	1	1	yes	characteristic
314	C₂²	C₂×#14519865	1	1	yes	characteristic
315	C₂²	#16840842	1	1	yes	characteristic
316	C₄	—	2	2	no	—
317	C₄	—	4	4	no	—
318	C₄	—	61	1	no	—
319	C₄	—	61	1	no	—
320	C₄	—	61	1	no	—
321	C₄	—	61	1	no	—
322	C₄	—	122	2	no	—
323	C₄	—	122	2	no	—
324	C₄	—	244	4	no	—
325	C₄	—	244	1	no	—
326	C₄	#16840797	1	4	yes	—
327	C₄	#16840813	1	4	yes	complemented
328	C₂	#21462348	1	1	yes	characteristic
329	C₂	C₂×#16840797	1	1	yes	characteristic
330	C₂	#21462565	1	1	yes	characteristic
331	C₂	C₂×#16840813	1	1	yes	characteristic
332	C₂	#21462613	1	1	yes	characteristic
333	C₂	#21462715	1	1	yes	characteristic
334	C₂	#21462730	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×#6923
commutator, socle, prime radical	C₂³×C₆₁	C₂³×C₄
Fitting	C₂×C₄⋊Q₈×C₆₁	C₄
cosocle	C₂²×Dic₆₁	C₂⁴
hypercenter	C₂×C₄⋊Q₈	#6923
hypocenter	C₆₁	#23153
LCS	G ▷ C₂³×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
39. C₄⋊#92420	Q₈	semidirect
152. C₄×C₆₁	#446	semidirect
198. C₆₁	#23153	semidirect
327. C₄	#16840813	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16840813	2
#21461988	2
#21461917	2
#21461879	2
#21461615	2
C₄×C₄⋊#92420	2
C₄⋊Q₈×Dic₆₁	2
#23153	61