Subgroups — #107×Dic₁₂₂ #29698852

Order 15616.

838 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₄×Dic₁₂₂	C₂	1	2	yes	maximal
3	#21446577	C₂	1	2	yes	maximal
4	#21446576	C₂	1	4	yes	maximal
5	#21445979	C₂	1	1	yes	maximal, characteristic
6	#21445977	C₂	1	2	yes	maximal
7	#107×Dic₆₁	C₂	1	2	yes	maximal
8	C₄×#107×C₆₁	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₂₂	—	2	2	no	—
10	C₂×C₄×Dic₁₂₂	C₂²	1	1	yes	characteristic
11	C₈×Dic₁₂₂	C₄	1	8	yes	complemented
12	C₂×#14519711	C₂²	1	2	yes	—
13	C₂×#14519711	C₂²	1	2	yes	—
14	C₂×#14519710	C₂²	1	4	yes	—
15	C₂×C₄⋊#92418	C₂²	1	1	yes	characteristic
16	C₂×#14519706	C₂²	1	2	yes	—
17	C₂×C₈×Dic₆₁	C₂²	1	2	yes	—
18	#16839660	—	2	2	no	—
19	#16839659	—	2	4	no	—
20	#16839632	C₂²	1	2	yes	—
21	#16839631	C₂²	1	4	yes	—
22	C₄⋊C₄×Dic₆₁	C₂²	1	4	yes	—
23	#16839524	C₂²	1	4	yes	—
24	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
25	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
28	C₆₁×#340	C₂²	1	2	yes	—
29	C₂²×Dic₁₂₂	D₄	1	1	yes	characteristic
30	C₂²×#92418	C₂³	1	2	yes	—
31	C₄×Dic₁₂₂	—	4	8	no	—
32	C₄×Dic₁₂₂	C₂×C₄	1	4	yes	—
33	C₄×Dic₁₂₂	Q₈	1	4	yes	—
34	C₂×C₄⋊Dic₆₁	—	2	2	no	—
35	C₂×C₄⋊Dic₆₁	—	2	2	no	—
36	C₂×C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
37	C₂×C₄⋊Dic₆₁	C₂³	1	4	yes	—
38	C₂×#11758196	—	2	4	no	—
39	C₂×#11758196	—	2	4	no	—
40	C₂×#11758196	D₄	1	2	yes	—
41	C₂×C₄×Dic₆₁	—	2	4	no	—
42	C₂×C₄×Dic₆₁	C₂³	1	2	yes	—
43	#14519711	—	2	4	no	—
44	#14519711	—	4	2	no	—
45	#14519710	—	4	4	no	—
46	C₄⋊#92418	C₂×C₄	1	4	yes	—
47	#14519706	C₂×C₄	1	8	yes	—
48	C₈×Dic₆₁	C₂×C₄	1	8	yes	—
49	#14519684	—	2	4	no	—
50	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
52	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
53	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
54	C₂×C₄²×C₆₁	—	2	2	no	—
55	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
56	#107×C₆₁	—	2	2	no	—
57	#107×C₆₁	Q₈	1	4	yes	complemented
58	C₄×C₈×C₆₁	C₂×C₄	1	4	yes	—
59	#95×C₆₁	—	2	2	no	—
60	C₂²×C₄×C₆₁	—	2	2	no	—
61	C₂²×C₄×C₆₁	—	2	2	no	—
62	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
65	C₂²×Dic₆₁	—	2	4	no	—
66	C₂²×Dic₆₁	C₂×D₄	1	2	yes	—
67	C₂²×Dic₆₁	C₂×D₄	1	2	yes	—
68	C₂×Dic₁₂₂	C₄⋊C₄	1	4	yes	—
69	C₂×Dic₁₂₂	D₈	1	4	yes	—
70	C₂×Dic₁₂₂	Q₁₆	1	4	yes	—
71	C₂×C₈×C₆₁	—	2	1	no	—
72	C₂×C₈×C₆₁	—	2	1	no	—
73	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
76	C₂×C₈×C₆₁	#41	1	2	yes	—
77	C₄⋊C₄×C₆₁	—	2	2	no	—
78	C₄⋊C₄×C₆₁	—	2	2	no	—
79	C₄⋊C₄×C₆₁	—	2	2	no	—
80	C₄⋊C₄×C₆₁	C₂×Q₈	1	4	yes	—
81	C₄²×C₆₁	—	4	4	no	—
82	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
83	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
84	C₄⋊Dic₆₁	—	2	4	no	—
85	C₄⋊Dic₆₁	—	2	4	no	—
86	C₄⋊Dic₆₁	—	4	4	no	—
87	C₄⋊Dic₆₁	—	8	2	no	—
88	C₄⋊Dic₆₁	C₄⋊C₄	1	4	yes	—
89	#11758196	—	4	8	no	—
90	#11758196	—	8	4	no	—
91	#11758196	C₄⋊C₄	1	8	yes	—
92	C₄×Dic₆₁	—	4	8	no	—
93	C₄×Dic₆₁	C₂²×C₄	1	4	yes	—
94	C₄×Dic₆₁	C₂×Q₈	1	4	yes	—
95	C₂×#92418	—	2	2	no	—
96	C₂×#92418	—	2	2	no	—
97	C₂×#92418	C₂²×C₄	1	4	yes	—
98	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₆₁	—	2	1	no	—
100	C₂×C₄×C₆₁	—	2	1	no	—
101	C₂×C₄×C₆₁	—	2	4	no	—
102	C₂×C₄×C₆₁	—	4	2	no	—
103	C₂×C₄×C₆₁	—	4	2	no	—
104	C₂×C₄×C₆₁	—	4	2	no	—
105	C₂×C₄×C₆₁	—	4	2	no	—
106	C₂×C₄×C₆₁	—	4	2	no	—
107	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
110	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
111	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
114	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
115	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
116	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
117	C₂×Dic₆₁	—	2	2	no	—
118	C₂×Dic₆₁	—	2	2	no	—
119	C₂×Dic₆₁	—	4	4	no	—
120	C₂×Dic₆₁	—	4	4	no	—
121	C₂×Dic₆₁	—	4	4	no	—
122	C₂×Dic₆₁	C₂×C₄⋊C₄	1	4	yes	—
123	C₂×Dic₆₁	C₂×C₄⋊C₄	1	4	yes	—
124	C₂×Dic₆₁	C₂×D₈	1	4	yes	—
125	C₂×Dic₆₁	C₂×Q₁₆	1	4	yes	—
126	Dic₁₂₂	#107	1	16	yes	direct product factor
127	C₈×C₆₁	—	2	2	no	—
128	C₈×C₆₁	C₄×Q₈	1	4	yes	complemented
129	#92418	—	2	4	no	—
130	Q₈×#107	—	61	1	no	maximal, Sylow
131	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
132	C₂²×C₆₁	D₄×Q₈	1	1	yes	characteristic
133	C₂²×C₆₁	#550	1	1	yes	characteristic
134	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
135	C₂²×C₆₁	C₂²×Q₁₆	1	1	yes	characteristic
136	C₂²×C₆₁	#576	1	1	yes	characteristic
137	C₂²×C₆₁	#578	1	1	yes	characteristic
138	C₄×C₆₁	—	2	2	no	—
139	C₄×C₆₁	—	4	4	no	—
140	C₄×C₆₁	—	8	2	no	—
141	C₄×C₆₁	C₂×#107	1	4	yes	—
142	C₄×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
143	C₄×C₆₁	#520	1	1	yes	characteristic
144	C₄×C₆₁	#557	1	1	yes	characteristic
145	C₄×C₆₁	Q₈²	1	1	yes	characteristic
146	Dic₆₁	—	2	4	no	—
147	Dic₆₁	—	8	4	no	—
148	Dic₆₁	C₂×#107	1	8	yes	—
149	C₂×C₈×Q₈	—	61	1	no	—
150	Q₈×C₄⋊C₄	—	61	2	no	—
151	#1947	—	61	2	no	—
152	#1947	—	61	4	no	—
153	#1845	—	61	2	no	—
154	#1845	—	61	1	no	—
155	C₄×#107	—	61	2	no	—
156	C₄×#107	D₆₁	1	1	yes	characteristic
157	C₂×C₆₁	Q₈×C₄⋊C₄	1	1	yes	characteristic
158	C₂×C₆₁	C₂²×#107	1	1	yes	characteristic
159	C₂×C₆₁	#2913	1	1	yes	characteristic
160	C₂×C₆₁	Q₈×D₈	1	1	yes	characteristic
161	C₂×C₆₁	Q₈×Q₁₆	1	1	yes	characteristic
162	C₂×C₆₁	#3391	1	1	yes	characteristic
163	C₂×C₆₁	#3395	1	1	yes	characteristic
164	C₂×C₄×Q₈	—	61	1	no	—
165	C₂×C₄×Q₈	—	122	2	no	—
166	C₈×Q₈	—	61	8	no	—
167	C₂×#107	—	61	2	no	—
168	C₂×#107	—	61	2	no	—
169	C₂×#107	—	61	4	no	—
170	C₂×#107	C₂×D₆₁	1	1	yes	characteristic
171	C₂×#107	C₂×D₆₁	1	1	yes	characteristic
172	C₂×C₄⋊C₈	—	61	2	no	—
173	C₂×C₄⋊C₈	—	61	1	no	—
174	C₂×C₄×C₈	—	61	2	no	—
175	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
176	#389	—	122	4	no	—
177	#389	—	122	2	no	—
178	#384	—	61	4	no	—
179	#384	—	61	2	no	—
180	C₄×C₄⋊C₄	—	61	4	no	—
181	C₄×C₄⋊C₄	C₂×D₆₁	1	2	yes	—
182	#340	—	61	4	no	—
183	#340	C₂×D₆₁	1	2	yes	—
184	C₆₁	Q₈×#107	1	1	yes	characteristic, complemented, Sylow
185	C₂²×Q₈	—	61	1	no	—
186	C₂²×C₈	—	61	2	no	—
187	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
188	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
189	C₄×Q₈	—	61	4	no	—
190	C₄×Q₈	—	61	4	no	—
191	C₄×Q₈	—	244	8	no	—
192	C₂×C₄⋊C₄	—	61	2	no	—
193	C₂×C₄⋊C₄	—	61	1	no	—
194	C₂×C₄⋊C₄	—	61	4	no	—
195	C₂×C₄⋊C₄	—	122	4	no	—
196	C₂×C₄⋊C₄	—	122	2	no	—
197	C₂×C₄⋊C₄	—	122	4	no	—
198	C₂×C₄⋊C₄	—	122	2	no	—
199	C₂×C₄⋊C₄	C₂²×D₆₁	1	2	yes	—
200	C₂×C₄⋊C₄	C₂²×D₆₁	1	2	yes	—
201	C₂×C₄²	—	2	2	no	—
202	C₂×C₄²	—	61	2	no	—
203	C₂×C₄²	—	122	4	no	—
204	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
205	#107	—	2	2	no	—
206	#107	—	122	4	no	—
207	#107	—	244	4	no	—
208	#107	—	244	2	no	—
209	#107	Dic₁₂₂	1	4	yes	direct product factor
210	C₄⋊C₈	—	61	8	no	—
211	C₄⋊C₈	—	61	4	no	—
212	C₄×C₈	—	61	8	no	—
213	C₄×C₈	C₄×D₆₁	1	4	yes	—
214	#95	—	2	2	no	—
215	#95	—	122	4	no	—
216	C₂×Q₈	—	61	4	no	—
217	C₂×Q₈	—	61	4	no	—
218	C₂×Q₈	—	61	4	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	61	2	no	—
222	C₂²×C₄	—	61	2	no	—
223	C₂²×C₄	—	122	4	no	—
224	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
225	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
226	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	61	4	no	—
230	C₂×C₈	—	122	2	no	—
231	C₂×C₈	—	122	2	no	—
232	C₂×C₈	C₂×Dic₁₂₂	1	2	yes	—
233	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
234	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
235	C₂×C₈	#11758222	1	2	yes	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	61	8	no	—
240	C₄⋊C₄	—	61	4	no	—
241	C₄⋊C₄	—	122	4	no	—
242	C₄⋊C₄	—	122	4	no	—
243	C₄⋊C₄	—	488	2	no	—
244	C₄⋊C₄	—	488	4	no	—
245	C₄⋊C₄	—	244	8	no	—
246	C₄⋊C₄	—	244	4	no	—
247	C₄⋊C₄	C₂×Dic₁₂₂	1	4	yes	—
248	C₄²	—	4	4	no	—
249	C₄²	—	61	4	no	—
250	C₄²	—	61	4	no	—
251	C₄²	—	244	8	no	—
252	C₄²	C₂×C₄×D₆₁	1	2	yes	—
253	C₄²	Q₈×D₆₁	1	2	yes	—
254	C₂³	C₂×D₄×D₆₁	1	1	yes	characteristic
255	Q₈	—	61	16	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	61	4	no	—
265	C₂×C₄	—	61	4	no	—
266	C₂×C₄	—	61	4	no	—
267	C₂×C₄	—	61	4	no	—
268	C₂×C₄	—	122	2	no	—
269	C₂×C₄	—	122	2	no	—
270	C₂×C₄	—	244	4	no	—
271	C₂×C₄	—	244	4	no	—
272	C₂×C₄	—	244	4	no	—
273	C₂×C₄	C₄⋊C₄×D₆₁	1	2	yes	—
274	C₂×C₄	C₄⋊C₄×D₆₁	1	2	yes	—
275	C₂×C₄	D₈×D₆₁	1	2	yes	—
276	C₂×C₄	Q₁₆×D₆₁	1	2	yes	—
277	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
278	C₂×C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
279	C₂×C₄	C₂²×Dic₁₂₂	1	1	yes	characteristic
280	C₂×C₄	C₂×#11758222	1	1	yes	characteristic
281	C₂×C₄	#14519823	1	1	yes	characteristic
282	C₂×C₄	#14519824	1	1	yes	characteristic
283	C₈	—	2	2	no	—
284	C₈	—	122	4	no	—
285	C₈	C₄×Dic₁₂₂	1	4	yes	complemented
286	C₂²	C₂×C₄⋊C₄×D₆₁	1	1	yes	characteristic
287	C₂²	D₄×Dic₁₂₂	1	1	yes	characteristic
288	C₂²	#16840433	1	1	yes	characteristic
289	C₂²	C₂×D₈×D₆₁	1	1	yes	characteristic
290	C₂²	C₂×Q₁₆×D₆₁	1	1	yes	characteristic
291	C₂²	#16840564	1	1	yes	characteristic
292	C₂²	#16840576	1	1	yes	characteristic
293	C₄	—	2	2	no	—
294	C₄	—	4	4	no	—
295	C₄	—	8	2	no	—
296	C₄	—	61	8	no	—
297	C₄	—	122	4	no	—
298	C₄	—	488	4	no	—
299	C₄	#107×D₆₁	1	4	yes	—
300	C₄	C₂×C₄×Dic₁₂₂	1	1	yes	characteristic
301	C₄	#16840242	1	1	yes	characteristic
302	C₄	#16840490	1	1	yes	characteristic
303	C₄	Q₈×Dic₁₂₂	1	1	yes	characteristic
304	C₂	C₄⋊C₄×Dic₁₂₂	1	1	yes	characteristic
305	C₂	C₂×#107×D₆₁	1	1	yes	characteristic
306	C₂	#21454416	1	1	yes	characteristic
307	C₂	D₈×Dic₁₂₂	1	1	yes	characteristic
308	C₂	Q₁₆×Dic₁₂₂	1	1	yes	characteristic
309	C₂	#21458793	1	1	yes	characteristic
310	C₂	#21458824	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₁
Fitting	C₄×#107×C₆₁	C₂
socle, prime radical	C₂³×C₆₁	C₂²×D₄
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	C₄×#107	D₆₁
hypocenter	C₆₁	Q₈×#107
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
11. C₈×Dic₁₂₂	C₄	semidirect
57. #107×C₆₁	Q₈	semidirect
126. Dic₁₂₂	#107	direct (N × H)
128. C₈×C₆₁	C₄×Q₈	semidirect
184. C₆₁	Q₈×#107	semidirect
209. #107	Dic₁₂₂	direct (N × H)
285. C₈	C₄×Dic₁₂₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₂₂	2
C₄⋊C₄×Dic₁₂₂	2
#21446577	2
#21446576	2
#21445979	2
#21445977	2
#107×Dic₆₁	2
C₄×#107×C₆₁	2
Q₈×#107	61

Subgroups — #107×Dic122 #29698852

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #107×Dic₁₂₂ #29698852