Subgroups — #27180443

Order 12928.

712 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692302	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15692285	C₂	1	1	yes	maximal, characteristic, complemented
3	#20265755	C₂	1	1	yes	maximal, characteristic, complemented
4	#20265336	C₂	1	2	yes	maximal, complemented
5	#20265331	C₂	1	2	yes	maximal, complemented
6	#20265319	C₂	1	2	yes	maximal, complemented
7	#20265314	C₂	1	2	yes	maximal, complemented
8	#20265178	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15692180	C₂	1	1	yes	maximal, characteristic, complemented
10	#20265143	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#135×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	#15692302	—	2	4	no	—
13	#15692302	—	2	4	no	—
14	C₂×#13108074	—	2	1	no	—
15	C₂×#13108074	C₂²	1	1	yes	characteristic
16	C₂×#13108073	—	2	2	no	—
17	C₂×#13108073	C₂²	1	2	yes	—
18	C₂²×Dic₂₀₂	—	2	1	no	—
19	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
20	C₂×Q₈×D₁₀₁	C₂²	1	2	yes	—
21	#15692285	—	2	4	no	—
22	C₂×#13108064	C₂²	1	2	yes	—
23	C₂×#13108062	C₂²	1	2	yes	—
24	C₂×#13108058	C₂²	1	1	yes	characteristic
25	#15692234	—	2	1	no	—
26	#15692231	C₂²	1	2	yes	—
27	#15692229	—	2	1	no	—
28	#15692227	—	2	2	no	—
29	#15692226	C₂²	1	2	yes	—
30	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
31	#15692180	C₂²	1	2	yes	—
32	#15692180	C₂²	1	2	yes	—
33	#15692179	C₂²	1	2	yes	—
34	#15692176	C₂²	1	2	yes	—
35	#15692167	C₂²	1	2	yes	—
36	#15692167	C₂²	1	2	yes	—
37	C₂×#41×C₁₀₁	C₂²	1	1	yes	characteristic
38	C₂×#41×C₁₀₁	C₂²	1	1	yes	characteristic
39	#135×C₁₀₁	—	2	4	no	—
40	C₂×Q₁₆×C₁₀₁	C₂²	1	1	yes	characteristic
41	C₂×SD₁₆×C₁₀₁	C₂²	1	1	yes	characteristic
42	C₂×SD₁₆×C₁₀₁	C₂²	1	1	yes	characteristic
43	C₂×D₈×C₁₀₁	C₂²	1	1	yes	characteristic
44	C₂²×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
45	#13108074	—	2	8	no	—
46	#13108074	—	2	2	no	—
47	#13108074	—	2	2	no	—
48	#13108073	—	2	16	no	—
49	#13108073	—	2	4	no	—
50	#13108073	—	2	4	no	—
51	C₂×Dic₂₀₂	—	2	8	no	—
52	C₂×Dic₂₀₂	—	2	1	no	—
53	C₂×Dic₂₀₂	—	2	2	no	—
54	C₂×Dic₂₀₂	—	2	2	no	—
55	C₂×Dic₂₀₂	D₄	1	1	yes	characteristic
56	C₂×Dic₂₀₂	D₄	1	4	yes	complemented
57	C₂×Dic₂₀₂	C₂³	1	2	yes	—
58	Q₈×D₁₀₁	—	2	4	no	—
59	Q₈×D₁₀₁	—	2	4	no	—
60	C₂×D₄₀₄	D₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₀₁	—	2	2	no	—
62	C₂×C₂²⋊D₁₀₁	—	2	2	no	—
63	C₂×C₂²⋊D₁₀₁	D₄	1	2	yes	—
64	C₂²×Dic₁₀₁	—	2	1	no	—
65	C₂²×Dic₁₀₁	—	2	2	no	—
66	C₂²×Dic₁₀₁	—	2	1	no	—
67	C₂×C₄×D₁₀₁	—	2	2	no	—
68	C₂×C₄×D₁₀₁	C₂³	1	2	yes	—
69	#13108064	—	2	4	no	—
70	#13108062	—	2	4	no	—
71	#13108058	—	2	2	no	—
72	C₂×#11717579	C₂³	1	2	yes	—
73	C₄⋊Dic₁₀₁	—	2	1	no	—
74	C₄⋊Dic₁₀₁	C₂³	1	2	yes	—
75	#13108053	—	2	2	no	—
76	C₂²⋊Dic₁₀₁	—	2	2	no	—
77	#13108051	—	2	2	no	—
78	#41×C₁₀₁	—	2	2	no	—
79	#41×C₁₀₁	—	2	2	no	—
80	#41×C₁₀₁	—	2	2	no	—
81	#41×C₁₀₁	D₄	1	4	yes	complemented
82	C₂×Q₈×C₁₀₁	C₂³	1	1	yes	characteristic
83	C₂×Q₈×C₁₀₁	C₂³	1	1	yes	characteristic
84	C₂×D₄×C₁₀₁	—	2	1	no	—
85	C₂×D₄×C₁₀₁	—	2	1	no	—
86	C₂×D₄×C₁₀₁	C₂³	1	1	yes	characteristic
87	C₂×D₄×C₁₀₁	C₂³	1	1	yes	characteristic
88	C₂²×C₄×C₁₀₁	—	2	1	no	—
89	C₂²×C₄×C₁₀₁	—	2	1	no	—
90	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
91	Q₁₆×C₁₀₁	—	2	2	no	—
92	SD₁₆×C₁₀₁	—	2	2	no	—
93	SD₁₆×C₁₀₁	—	2	2	no	—
94	D₈×C₁₀₁	—	2	2	no	—
95	C₂×C₈×C₁₀₁	—	2	2	no	—
96	C₂×C₈×C₁₀₁	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₁₀₁	C₂³	1	1	yes	characteristic
98	C₂³×C₁₀₁	—	2	1	no	—
99	C₂³×C₁₀₁	—	2	1	no	—
100	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
101	C₂²×D₁₀₁	C₂×D₄	1	2	yes	—
102	Q₈×C₁₀₁	—	2	1	no	—
103	Q₈×C₁₀₁	—	2	1	no	—
104	Q₈×C₁₀₁	—	2	1	no	—
105	Q₈×C₁₀₁	C₂×D₄	1	2	yes	—
106	D₄×C₁₀₁	—	2	4	no	—
107	D₄×C₁₀₁	—	2	1	no	—
108	D₄×C₁₀₁	—	2	1	no	—
109	D₄×C₁₀₁	—	2	1	no	—
110	D₄×C₁₀₁	—	4	2	no	—
111	D₄×C₁₀₁	C₂×D₄	1	2	yes	—
112	C₂×C₄×C₁₀₁	—	2	1	no	—
113	C₂×C₄×C₁₀₁	—	2	4	no	—
114	C₂×C₄×C₁₀₁	—	2	1	no	—
115	C₂×C₄×C₁₀₁	—	4	2	no	—
116	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
118	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
119	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
120	C₂²⋊D₁₀₁	—	2	8	no	—
121	C₂²⋊D₁₀₁	—	2	4	no	—
122	C₂²⋊D₁₀₁	—	8	2	no	—
123	C₂×Dic₁₀₁	—	2	8	no	—
124	C₂×Dic₁₀₁	—	2	4	no	—
125	C₂×Dic₁₀₁	—	2	2	no	—
126	C₂×Dic₁₀₁	—	2	2	no	—
127	C₂×Dic₁₀₁	—	8	1	no	—
128	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
129	D₄₀₄	—	2	2	no	—
130	C₄×D₁₀₁	—	2	8	no	—
131	C₄×D₁₀₁	—	2	2	no	—
132	C₄×D₁₀₁	—	2	2	no	—
133	Dic₂₀₂	—	2	8	no	—
134	Dic₂₀₂	—	2	2	no	—
135	Dic₂₀₂	—	2	4	no	—
136	Dic₂₀₂	—	2	2	no	—
137	Dic₂₀₂	C₂×D₄	1	4	yes	—
138	C₈×C₁₀₁	—	2	1	no	—
139	C₈×C₁₀₁	—	2	1	no	—
140	#11717579	—	2	2	no	—
141	C₂²×C₁₀₁	—	2	1	no	—
142	C₂²×C₁₀₁	—	2	1	no	—
143	C₂²×C₁₀₁	—	2	2	no	—
144	C₂²×C₁₀₁	—	4	1	no	—
145	C₂²×C₁₀₁	—	4	1	no	—
146	C₂²×C₁₀₁	—	4	1	no	—
147	C₂²×C₁₀₁	—	4	1	no	—
148	C₂²×C₁₀₁	—	4	1	no	—
149	C₂²×C₁₀₁	#120	1	2	yes	—
150	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
151	C₂×D₁₀₁	—	2	2	no	—
152	C₂×D₁₀₁	—	2	2	no	—
153	C₂×D₁₀₁	—	2	2	no	—
154	C₄×C₁₀₁	—	2	2	no	—
155	C₄×C₁₀₁	—	4	1	no	—
156	C₄×C₁₀₁	#120	1	2	yes	—
157	C₄×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
158	C₄×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
159	Dic₁₀₁	—	2	2	no	—
160	Dic₁₀₁	—	2	4	no	—
161	Dic₁₀₁	—	4	2	no	—
162	C₂×C₁₀₁	—	2	2	no	—
163	C₂×C₁₀₁	—	4	2	no	—
164	C₂×C₁₀₁	—	8	1	no	—
165	C₂×C₁₀₁	C₂×#120	1	1	yes	characteristic
166	C₂×C₁₀₁	#577	1	1	yes	characteristic
167	C₂×C₁₀₁	#578	1	1	yes	characteristic
168	D₁₀₁	—	4	2	no	—
169	#3013	—	101	1	no	maximal, Sylow
170	C₁₀₁	#3013	1	1	yes	characteristic, complemented, Sylow
171	C₂×2₋¹⁺⁴	—	101	1	no	—
172	C₂×#137	—	101	1	no	—
173	C₂×#135	D₁₀₁	1	1	yes	characteristic, complemented
174	#537	—	101	1	no	—
175	#452	—	101	2	no	—
176	#451	—	101	2	no	—
177	#449	—	101	2	no	—
178	#448	—	101	2	no	—
179	#417	—	101	1	no	—
180	C₂×#103	—	101	1	no	—
181	#408	—	101	1	no	—
182	2₋¹⁺⁴	—	202	4	no	—
183	2₋¹⁺⁴	—	202	4	no	—
184	C₂×#41	—	101	1	no	—
185	C₂×#41	—	101	2	no	—
186	C₂×#41	—	202	1	no	—
187	C₂×#41	—	202	2	no	—
188	C₂×#41	C₂×D₁₀₁	1	1	yes	characteristic
189	C₂×#41	C₂×D₁₀₁	1	1	yes	characteristic
190	C₂²×Q₈	—	101	2	no	—
191	C₂²×Q₈	—	101	1	no	—
192	C₂²×Q₈	—	202	1	no	—
193	#137	—	202	4	no	—
194	#135	—	2	4	no	—
195	C₂×Q₁₆	—	101	2	no	—
196	C₂×Q₁₆	C₂×D₁₀₁	1	1	yes	characteristic
197	C₂×SD₁₆	—	101	2	no	—
198	C₂×SD₁₆	C₂×D₁₀₁	1	1	yes	characteristic
199	C₂×SD₁₆	C₂×D₁₀₁	1	1	yes	characteristic
200	C₂×D₈	C₂×D₁₀₁	1	1	yes	characteristic
201	C₂×M₁₆	—	101	1	no	—
202	C₂²×C₈	C₂×D₁₀₁	1	1	yes	characteristic
203	C₂²⋊Q₈	—	101	2	no	—
204	C₂²⋊Q₈	—	202	1	no	—
205	#121	—	101	2	no	—
206	#121	—	202	1	no	—
207	#121	—	202	2	no	—
208	C₂×C₄⋊C₄	—	101	1	no	—
209	#103	—	101	2	no	—
210	#103	—	101	2	no	—
211	#103	—	101	2	no	—
212	#102	—	101	2	no	—
213	C₂²⋊C₈	—	101	2	no	—
214	C₂²⋊C₈	—	101	2	no	—
215	#41	—	2	2	no	—
216	#41	—	2	2	no	—
217	#41	—	2	2	no	—
218	#41	—	202	8	no	—
219	#41	—	202	4	no	—
220	#41	—	202	16	no	—
221	#41	—	202	4	no	—
222	#41	—	202	2	no	—
223	#41	—	202	2	no	—
224	#41	C₂²⋊D₁₀₁	1	4	yes	complemented
225	C₂×Q₈	—	101	1	no	—
226	C₂×Q₈	—	101	4	no	—
227	C₂×Q₈	—	101	2	no	—
228	C₂×Q₈	—	202	2	no	—
229	C₂×Q₈	—	202	8	no	—
230	C₂×Q₈	—	202	4	no	—
231	C₂×Q₈	—	202	4	no	—
232	C₂×Q₈	—	202	1	no	—
233	C₂×Q₈	—	202	2	no	—
234	C₂×Q₈	C₂²×D₁₀₁	1	1	yes	characteristic
235	C₂×Q₈	C₂²×D₁₀₁	1	1	yes	characteristic
236	C₂×D₄	—	2	1	no	—
237	C₂×D₄	—	2	1	no	—
238	C₂×D₄	—	101	2	no	—
239	C₂×D₄	—	101	1	no	—
240	C₂×D₄	—	202	2	no	—
241	C₂×D₄	—	202	2	no	—
242	C₂×D₄	C₂²×D₁₀₁	1	1	yes	characteristic
243	C₂×D₄	C₂²×D₁₀₁	1	1	yes	characteristic
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	101	2	no	—
247	C₂²×C₄	—	202	2	no	—
248	C₂²×C₄	—	202	1	no	—
249	C₂²×C₄	—	202	2	no	—
250	C₂²×C₄	—	202	1	no	—
251	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
252	Q₁₆	—	2	2	no	—
253	Q₁₆	—	202	4	no	—
254	SD₁₆	—	2	2	no	—
255	SD₁₆	—	2	2	no	—
256	SD₁₆	—	202	4	no	—
257	D₈	—	2	2	no	—
258	M₁₆	—	202	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	101	2	no	—
261	C₂×C₈	C₂²×D₁₀₁	1	1	yes	characteristic
262	C₂×C₈	C₂²×D₁₀₁	1	1	yes	characteristic
263	C₄⋊C₄	—	101	2	no	—
264	C₄⋊C₄	—	202	2	no	—
265	C₄⋊C₄	—	202	1	no	—
266	C₂²⋊C₄	—	202	2	no	—
267	C₂²⋊C₄	—	202	2	no	—
268	C₂³	—	2	1	no	—
269	C₂³	—	2	1	no	—
270	C₂³	—	101	2	no	—
271	C₂³	D₄×D₁₀₁	1	1	yes	characteristic
272	Q₈	—	2	1	no	—
273	Q₈	—	2	1	no	—
274	Q₈	—	2	1	no	—
275	Q₈	—	101	4	no	—
276	Q₈	—	202	8	no	—
277	Q₈	—	202	2	no	—
278	Q₈	—	202	2	no	—
279	Q₈	—	202	4	no	—
280	Q₈	C₂×C₂²⋊D₁₀₁	1	2	yes	—
281	D₄	—	2	4	no	—
282	D₄	—	2	1	no	—
283	D₄	—	2	1	no	—
284	D₄	—	2	1	no	—
285	D₄	—	4	2	no	—
286	D₄	—	808	2	no	—
287	D₄	—	202	8	no	—
288	D₄	—	202	4	no	—
289	D₄	—	202	2	no	—
290	D₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	808	1	no	—
296	C₂×C₄	—	101	2	no	—
297	C₂×C₄	—	202	8	no	—
298	C₂×C₄	—	202	2	no	—
299	C₂×C₄	—	202	8	no	—
300	C₂×C₄	—	202	2	no	—
301	C₂×C₄	—	202	4	no	—
302	C₂×C₄	—	202	2	no	—
303	C₂×C₄	—	202	2	no	—
304	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
305	C₂×C₄	D₄×D₁₀₁	1	1	yes	characteristic
306	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
307	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	202	2	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	4	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	202	2	no	—
320	C₂²	—	202	2	no	—
321	C₂²	—	202	2	no	—
322	C₂²	#15692223	1	2	yes	—
323	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
324	C₄	—	2	2	no	—
325	C₄	—	4	1	no	—
326	C₄	—	404	2	no	—
327	C₄	—	202	2	no	—
328	C₄	—	202	4	no	—
329	C₄	#15692223	1	2	yes	—
330	C₄	C₂×D₄×D₁₀₁	1	1	yes	characteristic
331	C₄	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
332	C₂	—	2	2	no	—
333	C₂	—	4	2	no	—
334	C₂	—	8	1	no	—
335	C₂	—	404	2	no	—
336	C₂	C₂×#15692223	1	1	yes	characteristic
337	C₂	#20265984	1	1	yes	characteristic
338	C₂	#20265991	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₁
commutator, cosocle	C₂×C₄×C₁₀₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₁
Fitting	C₂×#135×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂²×D₄
hypercenter	C₂×#135	D₁₀₁
hypocenter	C₁₀₁	#3013
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
1. C₂×#15692302	C₂	semidirect
2. C₂×#15692285	C₂	semidirect
3. #20265755	C₂	semidirect
4. #20265336	C₂	semidirect
5. #20265331	C₂	semidirect
6. #20265319	C₂	semidirect
7. #20265314	C₂	semidirect
8. #20265178	C₂	semidirect
9. C₂×#15692180	C₂	semidirect
10. #20265143	C₂	semidirect
11. C₂×#135×C₁₀₁	C₂	semidirect
56. C₂×Dic₂₀₂	D₄	semidirect
81. #41×C₁₀₁	D₄	semidirect
170. C₁₀₁	#3013	semidirect
173. C₂×#135	D₁₀₁	semidirect
224. #41	C₂²⋊D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15692302	2
C₂×#15692285	2
#20265755	2
#20265336	2
#20265331	2
#20265319	2
#20265314	2
#20265178	2
C₂×#15692180	2
#20265143	2
C₂×#135×C₁₀₁	2
#3013	101