Subgroups — #29566582

Order 15616.

352 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21445971	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#92418	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#92418	—	2	1	no	—
5	C₂×C₄⋊#92418	—	2	1	no	—
6	C₂×C₄⋊#92418	C₄	1	2	yes	—
7	C₂×C₄⋊#92418	C₄	1	2	yes	—
8	C₂×C₄×#92418	—	2	1	no	—
9	#16839517	—	2	1	no	—
10	C₂×C₄⋊C₈×C₆₁	—	2	1	no	—
11	C₂×C₄×C₈×C₆₁	—	2	1	no	—
12	C₄³×C₆₁	C₂²	1	1	yes	characteristic
13	C₆₁×#336	C₄	1	2	yes	—
14	C₂²×#92418	—	2	2	no	—
15	C₂²×#92418	—	2	2	no	—
16	C₄⋊#92418	—	2	2	no	—
17	C₄⋊#92418	—	4	1	no	—
18	C₄⋊#92418	—	4	1	no	—
19	C₄⋊#92418	—	4	1	no	—
20	C₄⋊#92418	—	4	1	no	—
21	C₄⋊#92418	—	4	2	no	—
22	C₄⋊#92418	C₈	1	4	yes	complemented
23	C₄×#92418	—	4	1	no	—
24	C₄×#92418	—	4	1	no	—
25	C₂²×C₈×C₆₁	—	2	2	no	—
26	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₆₁	Q₈	1	1	yes	characteristic
33	C₄⋊C₈×C₆₁	—	2	4	no	—
34	C₄×C₈×C₆₁	—	4	1	no	—
35	C₄×C₈×C₆₁	—	4	1	no	—
36	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
37	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₆₁	—	2	4	no	—
44	C₂×C₈×C₆₁	—	4	2	no	—
45	C₂×C₈×C₆₁	—	4	2	no	—
46	C₄²×C₆₁	—	2	1	no	—
47	C₄²×C₆₁	—	2	1	no	—
48	C₄²×C₆₁	—	2	2	no	—
49	C₄²×C₆₁	—	2	2	no	—
50	C₄²×C₆₁	—	4	1	no	—
51	C₄²×C₆₁	C₂×C₈	1	2	yes	—
52	C₄²×C₆₁	M₁₆	1	2	yes	—
53	C₄²×C₆₁	D₈	1	1	yes	characteristic
54	C₄²×C₆₁	D₈	1	1	yes	characteristic
55	C₄²×C₆₁	SD₁₆	1	1	yes	characteristic
56	C₄²×C₆₁	SD₁₆	1	1	yes	characteristic
57	C₄²×C₆₁	SD₁₆	1	1	yes	characteristic
58	C₄²×C₆₁	SD₁₆	1	1	yes	characteristic
59	C₄²×C₆₁	Q₁₆	1	1	yes	characteristic
60	C₄²×C₆₁	Q₁₆	1	1	yes	characteristic
61	C₂×#92418	—	2	4	no	—
62	C₂×#92418	—	4	2	no	—
63	C₂×#92418	—	4	2	no	—
64	C₂×#92418	—	4	2	no	—
65	C₂×#92418	—	4	2	no	—
66	C₂×#92418	—	4	2	no	—
67	C₂³×C₆₁	#95	1	1	yes	characteristic
68	C₂×C₄×C₆₁	—	2	1	no	—
69	C₂×C₄×C₆₁	—	2	1	no	—
70	C₂×C₄×C₆₁	—	2	1	no	—
71	C₂×C₄×C₆₁	—	2	1	no	—
72	C₂×C₄×C₆₁	—	2	1	no	—
73	C₂×C₄×C₆₁	—	2	1	no	—
74	C₂×C₄×C₆₁	—	2	1	no	—
75	C₂×C₄×C₆₁	—	2	1	no	—
76	C₂×C₄×C₆₁	—	2	1	no	—
77	C₂×C₄×C₆₁	C₄×C₈	1	1	yes	characteristic
78	C₂×C₄×C₆₁	#97	1	1	yes	characteristic
79	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
80	C₂×C₄×C₆₁	#100	1	1	yes	characteristic
81	C₂×C₄×C₆₁	#101	1	1	yes	characteristic
82	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
83	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
84	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
85	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
86	C₂×C₄×C₆₁	#103	1	1	yes	characteristic
87	C₂×C₄×C₆₁	#103	1	1	yes	characteristic
88	C₂×C₄×C₆₁	#103	1	1	yes	characteristic
89	C₂×C₄×C₆₁	#103	1	1	yes	characteristic
90	C₂×C₄×C₆₁	#104	1	2	yes	—
91	C₂×C₄×C₆₁	C₄⋊C₈	1	2	yes	—
92	C₂×C₄×C₆₁	#106	1	1	yes	characteristic
93	C₂×C₄×C₆₁	#106	1	1	yes	characteristic
94	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
95	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
96	C₂×C₄×C₆₁	#108	1	2	yes	—
97	C₈×C₆₁	—	4	4	no	—
98	#92418	—	4	4	no	—
99	#92418	—	8	2	no	—
100	#7031	—	61	1	no	maximal, Sylow
101	C₂²×C₆₁	#336	1	1	yes	characteristic
102	C₂²×C₆₁	#339	1	1	yes	characteristic
103	C₂²×C₆₁	#340	1	1	yes	characteristic
104	C₂²×C₆₁	#340	1	1	yes	characteristic
105	C₂²×C₆₁	#341	1	1	yes	characteristic
106	C₂²×C₆₁	#343	1	1	yes	characteristic
107	C₂²×C₆₁	#344	1	1	yes	characteristic
108	C₄×C₆₁	—	2	1	no	—
109	C₄×C₆₁	—	2	1	no	—
110	C₄×C₆₁	—	2	1	no	—
111	C₄×C₆₁	—	2	1	no	—
112	C₄×C₆₁	—	2	1	no	—
113	C₄×C₆₁	—	2	1	no	—
114	C₄×C₆₁	—	4	1	no	—
115	C₄×C₆₁	D₄⋊C₈	1	2	yes	—
116	C₄×C₆₁	Q₈⋊C₈	1	2	yes	—
117	C₄×C₆₁	#331	1	1	yes	characteristic
118	C₄×C₆₁	#332	1	1	yes	characteristic
119	C₄×C₆₁	#332	1	1	yes	characteristic
120	C₄×C₆₁	#333	1	1	yes	characteristic
121	C₄×C₆₁	#334	1	2	yes	—
122	C₄×C₆₁	#335	1	2	yes	—
123	#1842	—	61	1	no	—
124	C₄×C₄⋊C₈	—	61	1	no	—
125	C₄×C₄⋊C₈	D₆₁	1	1	yes	characteristic
126	C₂×C₆₁	#1277	1	1	yes	characteristic
127	C₂×C₆₁	#1279	1	1	yes	characteristic
128	C₂×C₆₁	#1283	1	1	yes	characteristic
129	C₂×C₆₁	#1296	1	1	yes	characteristic
130	C₂×C₆₁	#1299	1	1	yes	characteristic
131	C₂×C₆₁	#1301	1	1	yes	characteristic
132	C₂×C₆₁	#1303	1	1	yes	characteristic
133	C₂×C₄⋊C₈	—	2	1	no	—
134	C₂×C₄⋊C₈	—	61	2	no	—
135	C₂×C₄⋊C₈	—	61	2	no	—
136	C₂×C₄⋊C₈	—	122	1	no	—
137	C₂×C₄⋊C₈	—	122	1	no	—
138	C₂×C₄×C₈	—	2	1	no	—
139	C₂×C₄×C₈	—	122	1	no	—
140	C₄³	C₂×D₆₁	1	1	yes	characteristic
141	#336	—	122	1	no	—
142	#336	Dic₆₁	1	2	yes	—
143	C₆₁	#7031	1	1	yes	characteristic, complemented, Sylow
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	—	122	2	no	—
146	C₂²×C₈	—	122	2	no	—
147	C₂×C₄²	Dic₁₂₂	1	1	yes	characteristic
148	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
149	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
150	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
151	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
152	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
153	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
154	C₄⋊C₈	—	2	4	no	—
155	C₄⋊C₈	—	61	4	no	—
156	C₄⋊C₈	—	122	2	no	—
157	C₄⋊C₈	—	244	2	no	—
158	C₄⋊C₈	—	244	1	no	—
159	C₄⋊C₈	—	244	1	no	—
160	C₄⋊C₈	—	244	1	no	—
161	C₄⋊C₈	—	244	1	no	—
162	C₄×C₈	—	4	1	no	—
163	C₄×C₈	—	4	1	no	—
164	C₄×C₈	—	244	1	no	—
165	C₄×C₈	—	244	1	no	—
166	C₂²×C₄	C₄×Dic₆₁	1	1	yes	characteristic
167	C₂²×C₄	#11758196	1	1	yes	characteristic
168	C₂²×C₄	#11758196	1	1	yes	characteristic
169	C₂²×C₄	C₄⋊Dic₆₁	1	1	yes	characteristic
170	C₂²×C₄	#11758198	1	1	yes	characteristic
171	C₂²×C₄	#11758198	1	1	yes	characteristic
172	C₂²×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	4	2	no	—
175	C₂×C₈	—	4	2	no	—
176	C₂×C₈	—	122	4	no	—
177	C₂×C₈	—	244	2	no	—
178	C₂×C₈	—	244	2	no	—
179	C₂×C₈	—	244	2	no	—
180	C₂×C₈	—	244	2	no	—
181	C₂×C₈	—	244	2	no	—
182	C₄²	—	2	1	no	—
183	C₄²	—	2	1	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	4	1	no	—
187	C₄²	C₈×D₆₁	1	2	yes	—
188	C₄²	#11758189	1	2	yes	—
189	C₄²	#11758190	1	1	yes	characteristic
190	C₄²	#11758190	1	1	yes	characteristic
191	C₄²	D₄₈₈	1	1	yes	characteristic
192	C₄²	Dic₂₄₄	1	1	yes	characteristic
193	C₄²	#11758199	1	1	yes	characteristic
194	C₄²	#11758200	1	1	yes	characteristic
195	C₄²	#11758201	1	1	yes	characteristic
196	C₄²	#11758202	1	1	yes	characteristic
197	C₂³	#14519684	1	1	yes	characteristic
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	C₈×Dic₆₁	1	1	yes	characteristic
208	C₂×C₄	#14519687	1	1	yes	characteristic
209	C₂×C₄	#14519689	1	2	yes	—
210	C₂×C₄	#14519693	1	1	yes	characteristic
211	C₂×C₄	#14519695	1	1	yes	characteristic
212	C₂×C₄	#14519697	1	1	yes	characteristic
213	C₂×C₄	#14519698	1	1	yes	characteristic
214	C₂×C₄	#14519699	1	1	yes	characteristic
215	C₂×C₄	#14519699	1	1	yes	characteristic
216	C₂×C₄	#14519700	1	1	yes	characteristic
217	C₂×C₄	#14519701	1	1	yes	characteristic
218	C₂×C₄	#14519702	1	1	yes	characteristic
219	C₂×C₄	#14519702	1	1	yes	characteristic
220	C₂×C₄	#14519705	1	2	yes	—
221	C₂×C₄	#14519706	1	2	yes	—
222	C₂×C₄	#14519708	1	1	yes	characteristic
223	C₂×C₄	#14519709	1	1	yes	characteristic
224	C₂×C₄	#14519710	1	1	yes	characteristic
225	C₂×C₄	#14519711	1	1	yes	characteristic
226	C₂×C₄	#14519712	1	2	yes	—
227	C₈	—	4	4	no	—
228	C₈	—	488	2	no	—
229	C₈	—	244	4	no	—
230	C₂²	#16839516	1	1	yes	characteristic
231	C₂²	#16839522	1	1	yes	characteristic
232	C₂²	#16839524	1	1	yes	characteristic
233	C₂²	#16839525	1	1	yes	characteristic
234	C₂²	#16839526	1	1	yes	characteristic
235	C₂²	#16839530	1	1	yes	characteristic
236	C₂²	#16839532	1	1	yes	characteristic
237	C₄	—	2	1	no	—
238	C₄	—	2	1	no	—
239	C₄	—	2	1	no	—
240	C₄	—	2	1	no	—
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	4	1	no	—
244	C₄	#16839491	1	2	yes	—
245	C₄	#16839494	1	2	yes	—
246	C₄	#16839505	1	1	yes	characteristic
247	C₄	#16839507	1	1	yes	characteristic
248	C₄	#16839509	1	1	yes	characteristic
249	C₄	#16839510	1	1	yes	characteristic
250	C₄	#16839512	1	2	yes	—
251	C₄	#16839514	1	2	yes	—
252	C₂	#21443653	1	1	yes	characteristic
253	C₂	#21443659	1	1	yes	characteristic
254	C₂	#21443668	1	1	yes	characteristic
255	C₂	#21443703	1	1	yes	characteristic
256	C₂	#21443709	1	1	yes	characteristic
257	C₂	#21443715	1	1	yes	characteristic
258	C₂	#21443722	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#11758198
commutator	C₂×C₄×C₆₁	C₄×C₈
Frattini	C₄³	C₂×D₆₁
Fitting	C₄×C₄⋊C₈×C₆₁	C₂
socle, prime radical	C₂³×C₆₁	#95
cosocle	C₄³×C₆₁	C₂²
hypercenter	C₄×C₄⋊C₈	D₆₁
hypocenter	C₆₁	#7031
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
22. C₄⋊#92418	C₈	semidirect
143. C₆₁	#7031	semidirect

Maximal subgroups

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

subgroup	index
#21445971	2
C₄×C₄⋊#92418	2
C₄×C₄⋊C₈×C₆₁	2
#7031	61