Subgroups — #27173827

Order 12928.

668 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15692214	C₂	1	1	yes	maximal, characteristic, complemented
3	#20265080	C₂	1	1	yes	maximal, characteristic, complemented
4	#20265080	C₂	1	2	yes	maximal, complemented
5	#20265076	C₂	1	1	yes	maximal, characteristic, complemented
6	#20265070	C₂	1	2	yes	maximal, complemented
7	#20265057	C₂	1	2	yes	maximal, complemented
8	#20265052	C₂	1	2	yes	maximal, complemented
9	#20265037	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄⋊C₄×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#121×C₁₀₁	C₂	1	1	yes	maximal, characteristic
12	C₂³×Dic₁₀₁	C₂²	1	2	yes	—
13	D₄×Dic₁₀₁	—	2	4	no	—
14	#15692214	—	2	4	no	—
15	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
20	C₂×C₂²⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
22	C₂×C₂²⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
24	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
25	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
26	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
28	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
29	#15692162	C₂²	1	2	yes	—
30	#15692162	C₂²	1	2	yes	—
31	#15692162	C₂²	1	2	yes	—
32	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
33	C₂²×D₄×C₁₀₁	C₂²	1	2	yes	—
34	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
35	#121×C₁₀₁	C₄	1	8	yes	complemented
36	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
38	C₂²×Dic₁₀₁	—	2	4	no	—
39	C₂²×Dic₁₀₁	—	4	2	no	—
40	C₂²×Dic₁₀₁	—	4	2	no	—
41	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
42	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
43	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
44	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
45	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
46	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
47	C₄⋊Dic₁₀₁	—	2	2	no	—
48	C₄⋊Dic₁₀₁	—	2	2	no	—
49	C₄⋊Dic₁₀₁	—	2	2	no	—
50	C₄⋊Dic₁₀₁	—	2	2	no	—
51	C₄⋊Dic₁₀₁	—	4	2	no	—
52	C₂²⋊Dic₁₀₁	—	2	2	no	—
53	C₂²⋊Dic₁₀₁	—	2	2	no	—
54	C₂²⋊Dic₁₀₁	—	2	4	no	—
55	C₂²⋊Dic₁₀₁	—	4	2	no	—
56	C₂²⋊Dic₁₀₁	—	4	2	no	—
57	C₂²⋊Dic₁₀₁	—	4	2	no	—
58	C₄×Dic₁₀₁	—	2	2	no	—
59	C₄×Dic₁₀₁	—	4	2	no	—
60	C₄×Dic₁₀₁	D₄	1	4	yes	complemented
61	C₂⁴×C₁₀₁	C₂³	1	2	yes	—
62	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
63	C₂×D₄×C₁₀₁	—	2	2	no	—
64	C₂×D₄×C₁₀₁	—	2	2	no	—
65	C₂×D₄×C₁₀₁	—	4	2	no	—
66	C₂×D₄×C₁₀₁	—	4	2	no	—
67	C₂×D₄×C₁₀₁	C₂×C₄	1	4	yes	—
68	C₂×D₄×C₁₀₁	C₂×C₄	1	8	yes	complemented
69	C₂²×C₄×C₁₀₁	—	2	2	no	—
70	C₂²×C₄×C₁₀₁	—	2	2	no	—
71	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	complemented
72	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
73	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
74	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₁₀₁	C₂×C₄	1	4	yes	complemented
76	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	8	yes	complemented
77	C₂³×C₁₀₁	—	2	1	no	—
78	C₂³×C₁₀₁	—	2	2	no	—
79	C₂³×C₁₀₁	—	2	1	no	—
80	C₂³×C₁₀₁	—	2	2	no	—
81	C₂³×C₁₀₁	—	2	1	no	—
82	C₂³×C₁₀₁	—	2	1	no	—
83	C₂³×C₁₀₁	—	2	2	no	—
84	C₂³×C₁₀₁	—	2	2	no	—
85	C₂³×C₁₀₁	—	2	1	no	—
86	C₂³×C₁₀₁	—	2	2	no	—
87	C₂³×C₁₀₁	—	2	2	no	—
88	C₂³×C₁₀₁	—	2	1	no	—
89	C₂³×C₁₀₁	C₂²×C₄	1	4	yes	—
90	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
91	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
92	D₄×C₁₀₁	—	2	8	no	—
93	D₄×C₁₀₁	—	4	8	no	—
94	C₂×C₄×C₁₀₁	—	2	1	no	—
95	C₂×C₄×C₁₀₁	—	2	1	no	—
96	C₂×C₄×C₁₀₁	—	2	2	no	—
97	C₂×C₄×C₁₀₁	—	2	2	no	—
98	C₂×C₄×C₁₀₁	—	2	8	no	—
99	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
100	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
101	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
102	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
103	C₂×C₄×C₁₀₁	#41	1	2	yes	—
104	C₂×Dic₁₀₁	—	2	1	no	—
105	C₂×Dic₁₀₁	—	2	1	no	—
106	C₂×Dic₁₀₁	—	2	1	no	—
107	C₂×Dic₁₀₁	—	2	1	no	—
108	C₂×Dic₁₀₁	—	2	1	no	—
109	C₂×Dic₁₀₁	—	2	1	no	—
110	C₂×Dic₁₀₁	—	2	2	no	—
111	C₂×Dic₁₀₁	—	2	2	no	—
112	C₂×Dic₁₀₁	—	2	2	no	—
113	C₂×Dic₁₀₁	—	2	1	no	—
114	C₂×Dic₁₀₁	—	2	1	no	—
115	C₂×Dic₁₀₁	—	2	2	no	—
116	C₂×Dic₁₀₁	—	2	2	no	—
117	C₂×Dic₁₀₁	—	2	1	no	—
118	C₂×Dic₁₀₁	—	2	2	no	—
119	C₂×Dic₁₀₁	—	2	1	no	—
120	C₂×Dic₁₀₁	—	4	8	no	—
121	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
122	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
123	C₂²×C₁₀₁	—	2	2	no	—
124	C₂²×C₁₀₁	—	2	4	no	—
125	C₂²×C₁₀₁	—	2	2	no	—
126	C₂²×C₁₀₁	—	2	4	no	—
127	C₂²×C₁₀₁	—	2	2	no	—
128	C₂²×C₁₀₁	—	2	4	no	—
129	C₂²×C₁₀₁	—	4	2	no	—
130	C₂²×C₁₀₁	—	4	1	no	—
131	C₂²×C₁₀₁	—	4	1	no	—
132	C₂²×C₁₀₁	—	4	2	no	—
133	C₂²×C₁₀₁	—	4	1	no	—
134	C₂²×C₁₀₁	—	4	2	no	—
135	C₂²×C₁₀₁	—	4	1	no	—
136	C₂²×C₁₀₁	—	4	2	no	—
137	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
138	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
139	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
140	C₂²×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
141	C₂²×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
142	C₂²×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
143	C₂²×C₁₀₁	2₋¹⁺⁴	1	1	yes	characteristic
144	C₄×C₁₀₁	—	2	2	no	—
145	C₄×C₁₀₁	—	2	4	no	—
146	C₄×C₁₀₁	C₄×D₄	1	4	yes	—
147	Dic₁₀₁	—	2	2	no	—
148	Dic₁₀₁	—	2	2	no	—
149	Dic₁₀₁	—	4	1	no	—
150	Dic₁₀₁	—	4	2	no	—
151	Dic₁₀₁	—	4	1	no	—
152	Dic₁₀₁	—	4	2	no	—
153	C₂×C₁₀₁	—	4	2	no	—
154	C₂×C₁₀₁	—	4	4	no	—
155	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
156	C₂×C₁₀₁	#518	1	1	yes	characteristic
157	C₂×C₁₀₁	#520	1	1	yes	characteristic
158	C₂×C₁₀₁	#535	1	1	yes	characteristic
159	C₂×C₁₀₁	#537	1	1	yes	characteristic
160	C₂×C₁₀₁	#540	1	1	yes	characteristic
161	C₂×C₁₀₁	#542	1	1	yes	characteristic
162	#2324	—	101	1	no	maximal, Sylow
163	C₁₀₁	#2324	1	1	yes	characteristic, complemented, Sylow
164	C₂×#121	D₁₀₁	1	1	yes	characteristic
165	C₂×C₄×D₄	—	101	1	no	—
166	C₂×#117	—	101	1	no	—
167	#390	—	101	2	no	—
168	#390	—	101	1	no	—
169	#389	—	101	1	no	—
170	#388	—	101	2	no	—
171	#386	—	101	2	no	—
172	#385	—	101	2	no	—
173	#380	—	101	1	no	—
174	C₄×C₄⋊C₄	—	101	1	no	—
175	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
176	C₂²×D₄	C₂×D₁₀₁	1	2	yes	—
177	C₂³×C₄	—	101	2	no	—
178	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
179	#121	Dic₁₀₁	1	8	yes	complemented
180	C₄×D₄	—	202	4	no	—
181	#117	—	202	4	no	—
182	C₂×C₄⋊C₄	—	101	2	no	—
183	C₂×C₄⋊C₄	—	101	1	no	—
184	C₂×C₄⋊C₄	—	101	1	no	—
185	C₂×C₄⋊C₄	—	101	1	no	—
186	C₂×C₄⋊C₄	—	101	1	no	—
187	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
188	C₂×C₂²⋊C₄	—	101	2	no	—
189	C₂×C₂²⋊C₄	—	101	2	no	—
190	C₂×C₂²⋊C₄	—	101	2	no	—
191	C₂×C₂²⋊C₄	—	101	1	no	—
192	C₂×C₂²⋊C₄	—	101	1	no	—
193	C₂×C₂²⋊C₄	—	101	2	no	—
194	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
195	C₂×C₄²	—	101	1	no	—
196	C₂×C₄²	—	101	2	no	—
197	C₂×C₄²	—	101	1	no	—
198	#95	—	101	2	no	—
199	#95	—	101	2	no	—
200	#95	—	101	2	no	—
201	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
202	C₂⁴	C₂²×D₁₀₁	1	2	yes	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	—	4	2	no	—
206	C₂×D₄	—	4	2	no	—
207	C₂×D₄	C₂×Dic₁₀₁	1	4	yes	—
208	C₂×D₄	C₂×Dic₁₀₁	1	8	yes	complemented
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	101	1	no	—
212	C₂²×C₄	—	101	1	no	—
213	C₂²×C₄	—	101	1	no	—
214	C₂²×C₄	—	101	2	no	—
215	C₂²×C₄	—	101	2	no	—
216	C₂²×C₄	—	101	1	no	—
217	C₂²×C₄	—	404	2	no	—
218	C₂²×C₄	—	404	2	no	—
219	C₂²×C₄	—	202	4	no	—
220	C₂²×C₄	C₂×Dic₁₀₁	1	4	yes	complemented
221	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
222	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
223	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
224	C₄⋊C₄	—	404	2	no	—
225	C₄⋊C₄	—	202	2	no	—
226	C₄⋊C₄	—	202	2	no	—
227	C₄⋊C₄	—	202	2	no	—
228	C₄⋊C₄	—	202	2	no	—
229	C₄⋊C₄	C₂×Dic₁₀₁	1	4	yes	complemented
230	C₂²⋊C₄	—	404	2	no	—
231	C₂²⋊C₄	—	404	2	no	—
232	C₂²⋊C₄	—	404	2	no	—
233	C₂²⋊C₄	—	202	2	no	—
234	C₂²⋊C₄	—	202	2	no	—
235	C₂²⋊C₄	—	202	4	no	—
236	C₂²⋊C₄	C₂×Dic₁₀₁	1	8	yes	complemented
237	C₄²	—	101	4	no	—
238	C₄²	—	404	2	no	—
239	C₄²	—	202	2	no	—
240	C₂³	—	2	1	no	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	1	no	—
243	C₂³	—	2	2	no	—
244	C₂³	—	2	2	no	—
245	C₂³	—	2	2	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	2	1	no	—
249	C₂³	—	2	2	no	—
250	C₂³	—	2	1	no	—
251	C₂³	—	2	2	no	—
252	C₂³	C₂²×Dic₁₀₁	1	4	yes	—
253	C₂³	C₂²×Dic₁₀₁	1	2	yes	—
254	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
255	D₄	—	2	8	no	—
256	D₄	—	4	8	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	101	2	no	—
263	C₂×C₄	—	101	2	no	—
264	C₂×C₄	—	404	8	no	—
265	C₂×C₄	—	202	1	no	—
266	C₂×C₄	—	202	1	no	—
267	C₂×C₄	—	202	2	no	—
268	C₂×C₄	—	202	2	no	—
269	C₂×C₄	—	202	2	no	—
270	C₂×C₄	—	202	1	no	—
271	C₂×C₄	—	202	1	no	—
272	C₂×C₄	—	202	2	no	—
273	C₂×C₄	—	202	1	no	—
274	C₂×C₄	—	202	2	no	—
275	C₂×C₄	—	202	2	no	—
276	C₂×C₄	—	202	1	no	—
277	C₂×C₄	—	202	1	no	—
278	C₂×C₄	—	202	1	no	—
279	C₂×C₄	—	202	1	no	—
280	C₂×C₄	—	202	1	no	—
281	C₂×C₄	C₂²×Dic₁₀₁	1	2	yes	—
282	C₂×C₄	C₂²×Dic₁₀₁	1	4	yes	—
283	C₂×C₄	C₂²×Dic₁₀₁	1	2	yes	—
284	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
285	C₂×C₄	#13108073	1	2	yes	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	4	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	4	no	—
292	C₂²	—	4	2	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	1	no	—
295	C₂²	—	4	2	no	—
296	C₂²	—	4	1	no	—
297	C₂²	—	4	2	no	—
298	C₂²	—	4	1	no	—
299	C₂²	—	4	2	no	—
300	C₂²	C₂³×Dic₁₀₁	1	1	yes	characteristic
301	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
302	C₂²	C₂×#13108073	1	1	yes	characteristic
303	C₂²	#15692300	1	1	yes	characteristic
304	C₂²	#15692300	1	1	yes	characteristic
305	C₂²	#15692301	1	1	yes	characteristic
306	C₂²	#15692302	1	1	yes	characteristic
307	C₄	—	2	2	no	—
308	C₄	—	2	4	no	—
309	C₄	—	404	2	no	—
310	C₄	—	404	1	no	—
311	C₄	—	404	1	no	—
312	C₄	—	404	2	no	—
313	C₄	—	202	2	no	—
314	C₄	—	202	2	no	—
315	C₄	D₄×Dic₁₀₁	1	4	yes	—
316	C₂	—	4	2	no	—
317	C₂	—	4	4	no	—
318	C₂	C₂×D₄×Dic₁₀₁	1	1	yes	characteristic
319	C₂	#20265652	1	1	yes	characteristic
320	C₂	#20265660	1	1	yes	characteristic
321	C₂	#20265743	1	1	yes	characteristic
322	C₂	#20265756	1	1	yes	characteristic
323	C₂	#20265780	1	1	yes	characteristic
324	C₂	#20265794	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Fitting, prime radical	C₂×#121×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
hypercenter	C₂×#121	D₁₀₁
hypocenter	C₁₀₁	#2324
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×#121

Complements

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

N	H ≅ G/N	type
1. C₂×D₄×Dic₁₀₁	C₂	semidirect
2. C₂×#15692214	C₂	semidirect
3. #20265080	C₂	semidirect
4. #20265080	C₂	semidirect
5. #20265076	C₂	semidirect
6. #20265070	C₂	semidirect
7. #20265057	C₂	semidirect
8. #20265052	C₂	semidirect
9. #20265037	C₂	semidirect
10. C₄⋊C₄×Dic₁₀₁	C₂	semidirect
35. #121×C₁₀₁	C₄	semidirect
60. C₄×Dic₁₀₁	D₄	semidirect
68. C₂×D₄×C₁₀₁	C₂×C₄	semidirect
71. C₂²×C₄×C₁₀₁	C₂×C₄	semidirect
75. C₄⋊C₄×C₁₀₁	C₂×C₄	semidirect
76. C₂²⋊C₄×C₁₀₁	C₂×C₄	semidirect
163. C₁₀₁	#2324	semidirect
179. #121	Dic₁₀₁	semidirect
208. C₂×D₄	C₂×Dic₁₀₁	semidirect
220. C₂²×C₄	C₂×Dic₁₀₁	semidirect
229. C₄⋊C₄	C₂×Dic₁₀₁	semidirect
236. C₂²⋊C₄	C₂×Dic₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×Dic₁₀₁	2
C₂×#15692214	2
#20265080	2
#20265076	2
#20265070	2
#20265057	2
#20265052	2
#20265037	2
C₄⋊C₄×Dic₁₀₁	2
C₂×#121×C₁₀₁	2
#2324	101