Subgroups — #27173825

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×#15692229	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₄₀₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#20265079	C₂	1	1	yes	maximal, characteristic, complemented
4	#20265077	C₂	1	2	yes	maximal, complemented
5	#20265075	C₂	1	1	yes	maximal, characteristic, complemented
6	#20265067	C₂	1	2	yes	maximal, complemented
7	#20265053	C₂	1	2	yes	maximal, complemented
8	#20265048	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×#117×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄₀₄	C₂²	1	1	yes	characteristic
13	C₂²×C₂²⋊D₁₀₁	C₂²	1	2	yes	—
14	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
15	#15692229	C₄	1	8	yes	complemented
16	C₄×D₄₀₄	—	2	4	no	—
17	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
20	C₂×#13108053	C₂²	1	2	yes	—
21	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
22	C₂×#13108051	C₂²	1	2	yes	—
23	C₂×#13108051	C₂²	1	2	yes	—
24	C₂×#13108051	C₂²	1	2	yes	—
25	C₂×#13108051	C₂²	1	2	yes	—
26	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
27	#15692162	C₂²	1	2	yes	—
28	#15692162	C₂²	1	2	yes	—
29	#15692162	C₂²	1	2	yes	—
30	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
31	#117×C₁₀₁	—	2	4	no	—
32	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
36	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
37	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
38	C₂³×D₁₀₁	C₂³	1	2	yes	—
39	C₂×D₄₀₄	—	2	2	no	—
40	C₂×D₄₀₄	—	2	2	no	—
41	C₂×D₄₀₄	C₂×C₄	1	4	yes	—
42	C₂×C₂²⋊D₁₀₁	—	4	2	no	—
43	C₂×C₂²⋊D₁₀₁	—	4	2	no	—
44	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	8	yes	complemented
45	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
46	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
47	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
48	C₂×C₄×D₁₀₁	—	2	4	no	—
49	C₂×C₄×D₁₀₁	—	4	2	no	—
50	C₂×C₄×D₁₀₁	—	4	2	no	—
51	C₄⋊Dic₁₀₁	—	2	2	no	—
52	C₄⋊Dic₁₀₁	—	2	2	no	—
53	C₄⋊Dic₁₀₁	C₂×C₄	1	4	yes	complemented
54	#13108053	—	4	2	no	—
55	C₂²⋊Dic₁₀₁	—	4	2	no	—
56	#13108051	—	2	4	no	—
57	#13108051	—	4	2	no	—
58	#13108051	—	4	2	no	—
59	#13108051	C₂×C₄	1	8	yes	complemented
60	C₄×Dic₁₀₁	—	4	2	no	—
61	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₀₁	—	2	2	no	—
63	C₂²×C₄×C₁₀₁	—	2	2	no	—
64	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	complemented
65	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
68	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
70	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₁₀₁	—	2	2	no	—
72	C₄⋊C₄×C₁₀₁	—	2	2	no	—
73	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
74	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
75	C₄²×C₁₀₁	—	2	2	no	—
76	C₄²×C₁₀₁	D₄	1	4	yes	complemented
77	C₂³×C₁₀₁	—	2	1	no	—
78	C₂³×C₁₀₁	—	2	1	no	—
79	C₂³×C₁₀₁	—	2	1	no	—
80	C₂³×C₁₀₁	—	2	1	no	—
81	C₂³×C₁₀₁	—	2	1	no	—
82	C₂³×C₁₀₁	—	2	1	no	—
83	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
84	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
85	C₂²×D₁₀₁	—	2	2	no	—
86	C₂²×D₁₀₁	—	2	2	no	—
87	C₂²×D₁₀₁	—	2	2	no	—
88	C₂²×D₁₀₁	—	2	2	no	—
89	C₂²×D₁₀₁	—	2	2	no	—
90	C₂²×D₁₀₁	—	2	2	no	—
91	C₂²×D₁₀₁	C₂²×C₄	1	4	yes	—
92	C₂×C₄×C₁₀₁	—	2	1	no	—
93	C₂×C₄×C₁₀₁	—	2	1	no	—
94	C₂×C₄×C₁₀₁	—	2	1	no	—
95	C₂×C₄×C₁₀₁	—	2	1	no	—
96	C₂×C₄×C₁₀₁	—	2	1	no	—
97	C₂×C₄×C₁₀₁	—	2	1	no	—
98	C₂×C₄×C₁₀₁	—	2	8	no	—
99	C₂×C₄×C₁₀₁	—	2	1	no	—
100	C₂×C₄×C₁₀₁	—	2	1	no	—
101	C₂×C₄×C₁₀₁	—	2	1	no	—
102	C₂×C₄×C₁₀₁	—	2	1	no	—
103	C₂×C₄×C₁₀₁	—	2	1	no	—
104	C₂×C₄×C₁₀₁	—	2	1	no	—
105	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
106	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
107	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
108	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₁₀₁	#41	1	2	yes	—
111	C₂²⋊D₁₀₁	—	4	8	no	—
112	C₂×Dic₁₀₁	—	2	2	no	—
113	C₂×Dic₁₀₁	—	2	2	no	—
114	C₂×Dic₁₀₁	—	2	2	no	—
115	C₂×Dic₁₀₁	—	2	2	no	—
116	C₂×Dic₁₀₁	—	2	2	no	—
117	C₂×Dic₁₀₁	—	2	2	no	—
118	C₂×Dic₁₀₁	—	2	2	no	—
119	C₂×Dic₁₀₁	—	2	2	no	—
120	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
121	D₄₀₄	—	2	8	no	—
122	C₄×D₁₀₁	—	4	8	no	—
123	C₂²×C₁₀₁	—	2	2	no	—
124	C₂²×C₁₀₁	—	2	2	no	—
125	C₂²×C₁₀₁	—	2	2	no	—
126	C₂²×C₁₀₁	—	4	1	no	—
127	C₂²×C₁₀₁	—	4	1	no	—
128	C₂²×C₁₀₁	—	4	1	no	—
129	C₂²×C₁₀₁	—	4	1	no	—
130	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
131	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
132	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
133	C₂²×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
134	C₂²×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
135	C₂²×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
136	C₂²×C₁₀₁	2₋¹⁺⁴	1	1	yes	characteristic
137	C₂×D₁₀₁	—	2	4	no	—
138	C₂×D₁₀₁	—	2	4	no	—
139	C₂×D₁₀₁	—	2	4	no	—
140	C₂×D₁₀₁	—	4	2	no	—
141	C₂×D₁₀₁	—	4	2	no	—
142	C₂×D₁₀₁	—	4	2	no	—
143	C₂×D₁₀₁	—	4	2	no	—
144	C₄×C₁₀₁	—	2	2	no	—
145	C₄×C₁₀₁	—	2	2	no	—
146	C₄×C₁₀₁	—	2	2	no	—
147	C₄×C₁₀₁	—	4	1	no	—
148	C₄×C₁₀₁	—	4	1	no	—
149	C₄×C₁₀₁	C₄×D₄	1	4	yes	—
150	Dic₁₀₁	—	2	4	no	—
151	Dic₁₀₁	—	4	2	no	—
152	Dic₁₀₁	—	4	2	no	—
153	C₂×C₁₀₁	—	4	2	no	—
154	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
155	C₂×C₁₀₁	#518	1	1	yes	characteristic
156	C₂×C₁₀₁	#520	1	1	yes	characteristic
157	C₂×C₁₀₁	#535	1	1	yes	characteristic
158	C₂×C₁₀₁	#537	1	1	yes	characteristic
159	C₂×C₁₀₁	#540	1	1	yes	characteristic
160	C₂×C₁₀₁	#542	1	1	yes	characteristic
161	D₁₀₁	—	4	4	no	—
162	#2324	—	101	1	no	maximal, Sylow
163	C₁₀₁	#2324	1	1	yes	characteristic, complemented, Sylow
164	C₂×#121	—	101	1	no	—
165	C₂×C₄×D₄	—	101	1	no	—
166	C₂×#117	D₁₀₁	1	1	yes	characteristic, complemented
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	—	101	2	no	—
176	C₂²×D₄	—	101	1	no	—
177	C₂³×C₄	—	101	2	no	—
178	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
179	#121	—	101	8	no	—
180	C₄×D₄	—	202	4	no	—
181	#117	—	2	4	no	—
182	C₂×C₄⋊C₄	—	101	1	no	—
183	C₂×C₄⋊C₄	—	101	2	no	—
184	C₂×C₄⋊C₄	—	101	1	no	—
185	C₂×C₄⋊C₄	—	101	1	no	—
186	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
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	2	no	—
192	C₂×C₂²⋊C₄	—	101	2	no	—
193	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
194	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
195	C₂×C₄²	—	101	2	no	—
196	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
197	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
198	#95	—	101	2	no	—
199	#95	—	101	2	no	—
200	#95	—	101	2	no	—
201	C₂⁴	—	101	2	no	—
202	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
203	C₂×D₄	—	101	8	no	—
204	C₂×D₄	—	101	4	no	—
205	C₂×D₄	—	404	2	no	—
206	C₂×D₄	—	404	2	no	—
207	C₂×D₄	—	202	2	no	—
208	C₂×D₄	—	202	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	101	2	no	—
212	C₂²×C₄	—	101	2	no	—
213	C₂²×C₄	—	101	2	no	—
214	C₂²×C₄	—	404	2	no	—
215	C₂²×C₄	—	404	2	no	—
216	C₂²×C₄	—	202	4	no	—
217	C₂²×C₄	C₄×D₁₀₁	1	4	yes	complemented
218	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
219	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
220	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
221	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
222	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
223	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	101	4	no	—
227	C₄⋊C₄	—	404	2	no	—
228	C₄⋊C₄	—	202	2	no	—
229	C₄⋊C₄	—	202	2	no	—
230	C₂²⋊C₄	—	2	2	no	—
231	C₂²⋊C₄	—	2	2	no	—
232	C₂²⋊C₄	—	101	8	no	—
233	C₂²⋊C₄	—	404	2	no	—
234	C₂²⋊C₄	—	404	2	no	—
235	C₂²⋊C₄	—	404	2	no	—
236	C₂²⋊C₄	—	202	4	no	—
237	C₄²	—	2	2	no	—
238	C₄²	—	404	2	no	—
239	C₄²	C₂²⋊D₁₀₁	1	4	yes	complemented
240	C₂³	—	2	1	no	—
241	C₂³	—	2	1	no	—
242	C₂³	—	2	1	no	—
243	C₂³	—	2	1	no	—
244	C₂³	—	2	1	no	—
245	C₂³	—	2	1	no	—
246	C₂³	—	101	4	no	—
247	C₂³	—	202	2	no	—
248	C₂³	—	202	2	no	—
249	C₂³	—	202	2	no	—
250	C₂³	—	202	2	no	—
251	C₂³	—	202	2	no	—
252	C₂³	—	202	2	no	—
253	C₂³	C₂×C₄×D₁₀₁	1	2	yes	—
254	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
255	D₄	—	404	8	no	—
256	D₄	—	202	8	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	101	4	no	—
271	C₂×C₄	—	404	8	no	—
272	C₂×C₄	—	202	2	no	—
273	C₂×C₄	—	202	2	no	—
274	C₂×C₄	—	202	2	no	—
275	C₂×C₄	—	202	2	no	—
276	C₂×C₄	—	202	2	no	—
277	C₂×C₄	—	202	2	no	—
278	C₂×C₄	—	202	2	no	—
279	C₂×C₄	—	202	2	no	—
280	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
281	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
282	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
283	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
284	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
285	C₂×C₄	#13108074	1	2	yes	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	4	1	no	—
292	C₂²	—	4	1	no	—
293	C₂²	—	404	2	no	—
294	C₂²	—	404	2	no	—
295	C₂²	—	404	2	no	—
296	C₂²	—	404	2	no	—
297	C₂²	—	202	4	no	—
298	C₂²	—	202	4	no	—
299	C₂²	—	202	4	no	—
300	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
301	C₂²	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
302	C₂²	C₂×#13108074	1	1	yes	characteristic
303	C₂²	#15692301	1	1	yes	characteristic
304	C₂²	#15692301	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	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	4	1	no	—
311	C₄	—	4	1	no	—
312	C₄	—	404	2	no	—
313	C₄	—	404	2	no	—
314	C₄	—	202	4	no	—
315	C₄	C₄×C₂²⋊D₁₀₁	1	4	yes	—
316	C₂	—	4	2	no	—
317	C₂	—	404	4	no	—
318	C₂	C₂×C₄×C₂²⋊D₁₀₁	1	1	yes	characteristic
319	C₂	#20265651	1	1	yes	characteristic
320	C₂	#20265657	1	1	yes	characteristic
321	C₂	#20265741	1	1	yes	characteristic
322	C₂	#20265755	1	1	yes	characteristic
323	C₂	#20265782	1	1	yes	characteristic
324	C₂	#20265798	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Fitting	C₂×#117×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂×#15692229	C₂
hypercenter	C₂×#117	D₁₀₁
hypocenter	C₁₀₁	#2324
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₄₀₄	C₂	semidirect
3. #20265079	C₂	semidirect
4. #20265077	C₂	semidirect
5. #20265075	C₂	semidirect
6. #20265067	C₂	semidirect
7. #20265053	C₂	semidirect
8. #20265048	C₂	semidirect
9. #20265037	C₂	semidirect
10. C₄×C₄⋊Dic₁₀₁	C₂	semidirect
11. C₂×#117×C₁₀₁	C₂	semidirect
15. #15692229	C₄	semidirect
44. C₂×C₂²⋊D₁₀₁	C₂×C₄	semidirect
53. C₄⋊Dic₁₀₁	C₂×C₄	semidirect
59. #13108051	C₂×C₄	semidirect
64. C₂²×C₄×C₁₀₁	C₂×C₄	semidirect
76. C₄²×C₁₀₁	D₄	semidirect
163. C₁₀₁	#2324	semidirect
166. C₂×#117	D₁₀₁	semidirect
217. C₂²×C₄	C₄×D₁₀₁	semidirect
239. C₄²	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×#15692229	2
C₂×C₄×D₄₀₄	2
#20265079	2
#20265077	2
#20265075	2
#20265067	2
#20265053	2
#20265048	2
#20265037	2
C₄×C₄⋊Dic₁₀₁	2
C₂×#117×C₁₀₁	2
#2324	101