Subgroups — #27187633

Order 12928.

540 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	(C₂×#87938)⋊Q₈	C₂	1	2	yes	maximal, complemented
2	#20266127	C₂	1	2	yes	maximal, complemented
3	#20266127	C₂	1	2	yes	maximal, complemented
4	#20266125	C₂	1	2	yes	maximal
5	#20266124	C₂	1	2	yes	maximal, complemented
6	#20266122	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₄⋊#87938	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²⋊C₄×#87938	C₂	1	2	yes	maximal, complemented
9	#124×D₁₀₁	C₂	1	1	yes	maximal, characteristic
10	C₂³×#87938	C₂²	1	2	yes	—
11	C₂×C₄⋊#87938	C₂²	1	2	yes	—
12	C₂×C₄⋊#87938	C₂²	1	2	yes	—
13	C₂×C₂²⋊#87938	C₂²	1	2	yes	—
14	C₂×C₂²⋊#87938	C₂²	1	2	yes	—
15	C₂×C₂²⋊#87938	C₂²	1	2	yes	—
16	C₂×C₄×#87938	C₂²	1	1	yes	characteristic
17	C₂×C₄×#87938	C₂²	1	2	yes	—
18	C₂×C₄×#87938	C₂²	1	1	yes	characteristic
19	(C₂×#87938)⋊C₄	C₂²	1	2	yes	—
20	(C₂×#87938)⋊C₄	C₂²	1	2	yes	—
21	(C₂×#87938)⋊C₄	C₂²	1	2	yes	—
22	(C₂×#87938)⋊C₄	C₂²	1	1	yes	characteristic
23	(C₂×#87938)⋊C₄	C₂²	1	2	yes	—
24	(C₂×#87938)⋊C₄	C₂²	1	1	yes	characteristic
25	(C₂×#87938)⋊C₄	C₂²	1	2	yes	—
26	C₂×Q₈×D₁₀₁	C₂²	1	1	yes	characteristic
27	C₂×D₄×D₁₀₁	C₂²	1	1	yes	characteristic
28	#15692244	C₄	1	1	yes	characteristic, complemented
29	#15692243	C₄	1	1	yes	characteristic, complemented
30	#15692242	C₄	1	1	yes	characteristic, complemented
31	#15692241	C₄	1	2	yes	complemented
32	#15692241	C₄	1	2	yes	complemented
33	C₂²⋊C₄×D₁₀₁	C₂²	1	2	yes	—
34	C₂²⋊C₄×D₁₀₁	C₂²	1	2	yes	—
35	C₄²×D₁₀₁	C₂²	1	1	yes	characteristic
36	#124×C₁₀₁	C₄	1	1	yes	characteristic, complemented
37	C₂²×#87938	—	2	4	no	—
38	C₂²×#87938	—	4	2	no	—
39	C₂²×#87938	—	4	2	no	—
40	C₂²×#87938	C₂³	1	2	yes	—
41	C₂²×#87938	C₂³	1	2	yes	—
42	C₂²×#87938	C₂³	1	2	yes	—
43	C₂²×#87938	C₂³	1	2	yes	—
44	C₄⋊#87938	—	2	4	no	—
45	C₄⋊#87938	—	4	2	no	—
46	C₂²⋊#87938	—	2	4	no	—
47	C₂²⋊#87938	—	4	2	no	—
48	C₂²⋊#87938	—	4	2	no	—
49	C₄×#87938	—	4	2	no	—
50	C₄×#87938	—	4	1	no	—
51	C₄×#87938	—	4	1	no	—
52	C₂³×D₁₀₁	C₂³	1	2	yes	—
53	C₂×Dic₂₀₂	C₂×C₄	1	2	yes	complemented
54	C₂×Dic₂₀₂	C₂×C₄	1	1	yes	characteristic, complemented
55	Q₈×D₁₀₁	—	4	2	no	—
56	C₂×D₄₀₄	C₂×C₄	1	1	yes	characteristic
57	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	2	yes	—
58	D₄×D₁₀₁	—	4	2	no	—
59	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
60	C₂×C₄×D₁₀₁	C₂³	1	2	yes	—
61	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
62	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
63	C₂²⋊Dic₁₀₁	C₂×C₄	1	2	yes	complemented
64	C₂²⋊Dic₁₀₁	C₂×C₄	1	2	yes	complemented
65	#13108051	C₂×C₄	1	2	yes	complemented
66	#13108051	C₂×C₄	1	2	yes	complemented
67	#13108051	C₂×C₄	1	2	yes	complemented
68	#13108051	C₂×C₄	1	2	yes	complemented
69	C₄×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic, complemented
70	C₄×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic, complemented
71	C₄×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic, complemented
72	C₂×Q₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic, complemented
73	C₂×D₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
74	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	complemented
75	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	complemented
76	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic, complemented
77	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
78	C₂²×D₁₀₁	—	2	2	no	—
79	C₂²×D₁₀₁	—	2	2	no	—
80	C₂²×D₁₀₁	—	2	2	no	—
81	C₂²×D₁₀₁	—	2	2	no	—
82	C₂²×D₁₀₁	—	2	2	no	—
83	C₂²×D₁₀₁	—	2	2	no	—
84	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
85	C₂²×D₁₀₁	C₂⁴	1	1	yes	characteristic
86	C₂×#87938	—	2	2	no	—
87	C₂×#87938	—	2	2	no	—
88	C₂×#87938	—	2	2	no	—
89	C₂×#87938	—	2	2	no	—
90	C₂×#87938	—	2	2	no	—
91	C₂×#87938	—	2	2	no	—
92	C₂×#87938	—	2	2	no	—
93	C₂×#87938	—	2	2	no	—
94	C₂×#87938	—	2	2	no	—
95	C₂×#87938	—	2	2	no	—
96	C₂×#87938	—	2	2	no	—
97	C₂×#87938	—	4	8	no	—
98	C₂×#87938	#41	1	4	yes	—
99	Q₈×C₁₀₁	—	4	1	no	—
100	D₄×C₁₀₁	—	4	1	no	—
101	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
105	C₂²⋊D₁₀₁	—	4	2	no	—
106	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
107	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
108	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
109	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
110	D₄₀₄	—	4	1	no	—
111	C₄×D₁₀₁	—	2	2	no	—
112	C₄×D₁₀₁	—	2	2	no	—
113	C₄×D₁₀₁	—	2	2	no	—
114	C₄×D₁₀₁	—	2	2	no	—
115	C₄×D₁₀₁	—	2	1	no	—
116	C₄×D₁₀₁	—	2	1	no	—
117	Dic₂₀₂	—	4	1	no	—
118	Dic₂₀₂	—	4	2	no	—
119	C₂²×C₁₀₁	—	2	2	no	—
120	C₂²×C₁₀₁	—	2	2	no	—
121	C₂²×C₁₀₁	—	2	2	no	—
122	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
123	C₂×D₁₀₁	—	2	2	no	—
124	C₂×D₁₀₁	—	2	2	no	—
125	C₂×D₁₀₁	—	2	2	no	—
126	C₂×D₁₀₁	—	4	2	no	—
127	C₂×D₁₀₁	—	4	2	no	—
128	C₂×D₁₀₁	—	4	2	no	—
129	C₂×D₁₀₁	—	4	2	no	—
130	C₂×D₁₀₁	C₂×#41	1	2	yes	—
131	C₂×D₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂×D₁₀₁	H(5,𝔽₂)	1	2	yes	—
133	C₂×D₁₀₁	2₋¹⁺⁴	1	1	yes	characteristic
134	#87938	—	2	4	no	—
135	#87938	—	4	2	no	—
136	#87938	—	4	2	no	—
137	#87938	—	4	2	no	—
138	C₄×C₁₀₁	—	2	1	no	—
139	C₄×C₁₀₁	—	2	1	no	—
140	C₄×C₁₀₁	—	2	2	no	—
141	C₄×C₁₀₁	—	2	1	no	—
142	Dic₁₀₁	—	2	1	no	—
143	Dic₁₀₁	—	2	2	no	—
144	Dic₁₀₁	—	2	1	no	—
145	Dic₁₀₁	—	2	1	no	—
146	C₂×C₁₀₁	—	4	2	no	—
147	C₂×C₁₀₁	C₄×#41	1	1	yes	characteristic
148	C₂×C₁₀₁	#518	1	1	yes	characteristic
149	C₂×C₁₀₁	#520	1	1	yes	characteristic
150	D₁₀₁	—	4	2	no	—
151	D₁₀₁	#538	1	2	yes	—
152	D₁₀₁	#542	1	2	yes	—
153	#2337	—	101	1	no	maximal, Sylow
154	C₁₀₁	#2337	1	1	yes	characteristic, complemented, Sylow
155	C₂×#124	—	101	1	no	—
156	#391	—	101	2	no	—
157	#388	—	101	2	no	—
158	#388	—	101	2	no	—
159	#386	—	101	2	no	—
160	#385	—	101	2	no	—
161	#383	—	101	1	no	—
162	C₄×C₄⋊C₄	—	101	1	no	—
163	C₄×C₂²⋊C₄	—	101	2	no	—
164	C₂²×Q₈	—	101	1	no	—
165	C₂²×D₄	—	101	1	no	—
166	C₂³×C₄	—	101	2	no	—
167	#124	—	101	1	no	—
168	#124	—	101	1	no	—
169	#124	—	101	2	no	—
170	#124	—	101	2	no	—
171	#124	—	101	1	no	—
172	#124	#87938	1	1	yes	characteristic, complemented
173	C₂×C₄⋊C₄	—	101	2	no	—
174	C₂×C₄⋊C₄	—	101	2	no	—
175	C₂×C₂²⋊C₄	—	101	2	no	—
176	C₂×C₂²⋊C₄	—	101	2	no	—
177	C₂×C₂²⋊C₄	—	101	2	no	—
178	C₂×C₂²⋊C₄	—	101	2	no	—
179	C₂×C₂²⋊C₄	—	101	2	no	—
180	C₂×C₄²	—	101	1	no	—
181	C₂×C₄²	—	101	1	no	—
182	C₂×C₄²	—	101	2	no	—
183	C₂×C₄²	—	101	1	no	—
184	#95	—	101	2	no	—
185	#95	—	101	2	no	—
186	#95	—	101	2	no	—
187	#95	—	101	1	no	—
188	#95	—	101	2	no	—
189	#95	—	101	1	no	—
190	#95	—	101	2	no	—
191	C₂⁴	—	101	2	no	—
192	C₂×Q₈	—	101	2	no	—
193	C₂×Q₈	—	101	1	no	—
194	C₂×Q₈	—	404	2	no	—
195	C₂×Q₈	C₂×#87938	1	1	yes	characteristic, complemented
196	C₂×D₄	—	101	2	no	—
197	C₂×D₄	—	101	1	no	—
198	C₂×D₄	—	404	2	no	—
199	C₂×D₄	C₂×#87938	1	1	yes	characteristic
200	C₂²×C₄	—	101	2	no	—
201	C₂²×C₄	—	101	1	no	—
202	C₂²×C₄	—	101	1	no	—
203	C₂²×C₄	—	101	1	no	—
204	C₂²×C₄	—	101	2	no	—
205	C₂²×C₄	—	101	2	no	—
206	C₂²×C₄	—	101	2	no	—
207	C₂²×C₄	—	101	2	no	—
208	C₂²×C₄	—	404	2	no	—
209	C₂²×C₄	—	404	2	no	—
210	C₂²×C₄	—	202	4	no	—
211	C₄⋊C₄	—	404	2	no	—
212	C₄⋊C₄	—	202	4	no	—
213	C₂²⋊C₄	—	101	2	no	—
214	C₂²⋊C₄	—	101	2	no	—
215	C₂²⋊C₄	—	101	2	no	—
216	C₂²⋊C₄	—	101	2	no	—
217	C₂²⋊C₄	—	101	2	no	—
218	C₂²⋊C₄	—	101	2	no	—
219	C₂²⋊C₄	—	404	2	no	—
220	C₂²⋊C₄	—	404	2	no	—
221	C₂²⋊C₄	—	202	4	no	—
222	C₂²⋊C₄	C₂×#87938	1	2	yes	complemented
223	C₂²⋊C₄	C₂×#87938	1	2	yes	complemented
224	C₄²	—	101	1	no	—
225	C₄²	—	101	1	no	—
226	C₄²	—	101	1	no	—
227	C₄²	—	404	1	no	—
228	C₄²	—	404	1	no	—
229	C₄²	—	404	2	no	—
230	C₄²	C₂×#87938	1	1	yes	characteristic, complemented
231	C₂³	—	101	1	no	—
232	C₂³	—	101	2	no	—
233	C₂³	—	202	2	no	—
234	C₂³	—	202	2	no	—
235	C₂³	—	202	2	no	—
236	C₂³	—	202	2	no	—
237	C₂³	—	202	2	no	—
238	C₂³	—	202	2	no	—
239	C₂³	C₂²×#87938	1	2	yes	—
240	Q₈	—	4	1	no	—
241	Q₈	—	404	1	no	—
242	Q₈	—	404	2	no	—
243	D₄	—	4	1	no	—
244	D₄	—	404	1	no	—
245	D₄	—	404	2	no	—
246	C₂×C₄	—	101	4	no	—
247	C₂×C₄	—	101	1	no	—
248	C₂×C₄	—	101	1	no	—
249	C₂×C₄	—	101	1	no	—
250	C₂×C₄	—	101	2	no	—
251	C₂×C₄	—	404	8	no	—
252	C₂×C₄	—	202	1	no	—
253	C₂×C₄	—	202	2	no	—
254	C₂×C₄	—	202	2	no	—
255	C₂×C₄	—	202	2	no	—
256	C₂×C₄	—	202	2	no	—
257	C₂×C₄	—	202	1	no	—
258	C₂×C₄	—	202	2	no	—
259	C₂×C₄	—	202	2	no	—
260	C₂×C₄	—	202	2	no	—
261	C₂×C₄	—	202	2	no	—
262	C₂×C₄	—	202	2	no	—
263	C₂×C₄	—	202	2	no	—
264	C₂×C₄	—	202	2	no	—
265	C₂×C₄	—	202	2	no	—
266	C₂×C₄	—	202	2	no	—
267	C₂×C₄	—	202	2	no	—
268	C₂×C₄	—	202	2	no	—
269	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
270	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
271	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
272	C₂×C₄	C₂²×#87938	1	2	yes	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	101	2	no	—
277	C₂²	—	101	2	no	—
278	C₂²	—	101	1	no	—
279	C₂²	—	101	1	no	—
280	C₂²	—	404	2	no	—
281	C₂²	—	404	2	no	—
282	C₂²	—	404	2	no	—
283	C₂²	—	404	2	no	—
284	C₂²	—	202	2	no	—
285	C₂²	—	202	2	no	—
286	C₂²	—	202	2	no	—
287	C₂²	C₂³×#87938	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	1	no	—
292	C₄	—	404	2	no	—
293	C₄	—	404	2	no	—
294	C₄	—	404	2	no	—
295	C₄	—	202	4	no	—
296	C₄	—	202	1	no	—
297	C₄	—	202	1	no	—
298	C₄	—	202	2	no	—
299	C₄	—	202	1	no	—
300	C₂	—	4	2	no	—
301	C₂	—	101	2	no	—
302	C₂	—	101	2	no	—
303	C₂	—	404	2	no	—
304	C₂	#41×#87938	1	1	yes	characteristic
305	C₂	#20266244	1	1	yes	characteristic
306	C₂	#20266246	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#87938
commutator, socle	C₂²×C₁₀₁	C₂³×C₄
Fitting	#124×C₁₀₁	C₄
prime radical	C₂×D₄×D₁₀₁	C₂²
cosocle	C₂²×D₁₀₁	C₂⁴
hypercenter	#124	#87938
hypocenter	C₁₀₁	#2337
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ #124

Complements

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

N	H ≅ G/N	type
1. (C₂×#87938)⋊Q₈	C₂	semidirect
2. #20266127	C₂	semidirect
3. #20266127	C₂	semidirect
5. #20266124	C₂	semidirect
6. #20266122	C₂	semidirect
7. C₄×C₄⋊#87938	C₂	semidirect
8. C₂²⋊C₄×#87938	C₂	semidirect
28. #15692244	C₄	semidirect
29. #15692243	C₄	semidirect
30. #15692242	C₄	semidirect
31. #15692241	C₄	semidirect
32. #15692241	C₄	semidirect
36. #124×C₁₀₁	C₄	semidirect
53. C₂×Dic₂₀₂	C₂×C₄	semidirect
54. C₂×Dic₂₀₂	C₂×C₄	semidirect
63. C₂²⋊Dic₁₀₁	C₂×C₄	semidirect
64. C₂²⋊Dic₁₀₁	C₂×C₄	semidirect
65. #13108051	C₂×C₄	semidirect
66. #13108051	C₂×C₄	semidirect
67. #13108051	C₂×C₄	semidirect
68. #13108051	C₂×C₄	semidirect
69. C₄×Dic₁₀₁	C₂×C₄	semidirect
70. C₄×Dic₁₀₁	C₂×C₄	semidirect
71. C₄×Dic₁₀₁	C₂×C₄	semidirect
72. C₂×Q₈×C₁₀₁	C₂×C₄	semidirect
74. C₂²⋊C₄×C₁₀₁	C₂×C₄	semidirect
75. C₂²⋊C₄×C₁₀₁	C₂×C₄	semidirect
76. C₄²×C₁₀₁	C₂×C₄	semidirect
154. C₁₀₁	#2337	semidirect
172. #124	#87938	semidirect
195. C₂×Q₈	C₂×#87938	semidirect
222. C₂²⋊C₄	C₂×#87938	semidirect
223. C₂²⋊C₄	C₂×#87938	semidirect
230. C₄²	C₂×#87938	semidirect

Maximal subgroups

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

subgroup	index
(C₂×#87938)⋊Q₈	2
#20266127	2
#20266125	2
#20266124	2
#20266122	2
C₄×C₄⋊#87938	2
C₂²⋊C₄×#87938	2
#124×D₁₀₁	2
#2337	101