Subgroups — #27174146

Order 12928.

788 conjugacy classes of subgroups.

The subgroup lattice has 332 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	C₂×#15692215	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄⋊Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#20265079	C₂	1	1	yes	maximal, characteristic, complemented
6	#20265075	C₂	1	1	yes	maximal, characteristic, complemented
7	#20265069	C₂	1	2	yes	maximal, complemented
8	#20265054	C₂	1	2	yes	maximal, complemented
9	#20265051	C₂	1	2	yes	maximal, complemented
10	C₄×C₄⋊Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₂²⋊C₄×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	C₂³×Dic₁₀₁	C₂²	1	2	yes	—
15	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
16	#15692229	C₄	1	8	yes	complemented
17	C₄×D₄₀₄	—	2	4	no	—
18	#15692215	—	2	8	no	—
19	C₂×C₄⋊Dic₁₀₁	—	2	4	no	—
20	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
21	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
24	C₂×#13108053	C₂²	1	2	yes	—
25	C₂×#13108051	C₂²	1	2	yes	—
26	C₂×#13108051	C₂²	1	2	yes	—
27	C₂×#13108051	C₂²	1	2	yes	—
28	C₂×#13108051	C₂²	1	2	yes	—
29	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
30	#15692162	C₂²	1	2	yes	—
31	#15692162	C₂²	1	2	yes	—
32	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
35	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
36	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
37	#95×C₁₀₁	C₂²	1	1	yes	characteristic
38	#95×C₁₀₁	C₂²	1	1	yes	characteristic
39	C₂³×D₁₀₁	C₂³	1	2	yes	—
40	C₂×D₄₀₄	—	2	2	no	—
41	C₂×D₄₀₄	—	2	2	no	—
42	C₂×D₄₀₄	C₂×C₄	1	4	yes	complemented
43	C₂×C₂²⋊D₁₀₁	—	2	4	no	—
44	C₂×C₂²⋊D₁₀₁	—	2	4	no	—
45	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	8	yes	complemented
46	C₂²×Dic₁₀₁	—	2	4	no	—
47	C₂²×Dic₁₀₁	—	2	4	no	—
48	C₂²×Dic₁₀₁	—	4	2	no	—
49	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
50	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
51	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
52	C₂×C₄×D₁₀₁	—	2	4	no	—
53	C₂×C₄×D₁₀₁	—	2	4	no	—
54	C₂×C₄×D₁₀₁	—	4	2	no	—
55	C₄⋊Dic₁₀₁	—	2	2	no	—
56	C₄⋊Dic₁₀₁	—	2	4	no	—
57	C₄⋊Dic₁₀₁	C₂×C₄	1	4	yes	complemented
58	C₄⋊Dic₁₀₁	D₄	1	4	yes	complemented
59	#13108053	—	2	4	no	—
60	#13108051	—	2	4	no	—
61	#13108051	—	2	4	no	—
62	#13108051	—	4	2	no	—
63	#13108051	C₂×C₄	1	8	yes	complemented
64	C₄×Dic₁₀₁	—	2	4	no	—
65	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₁₀₁	—	2	2	no	—
67	C₂²×C₄×C₁₀₁	—	2	2	no	—
68	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	complemented
69	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
70	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
71	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
72	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
73	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
74	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
75	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
76	C₂²⋊C₄×C₁₀₁	D₄	1	4	yes	complemented
77	C₄²×C₁₀₁	—	2	2	no	—
78	C₄²×C₁₀₁	—	2	2	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	C₂×D₄	1	2	yes	—
85	C₂³×C₁₀₁	#41	1	2	yes	—
86	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
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	—	2	2	no	—
92	C₂²×D₁₀₁	—	2	2	no	—
93	C₂²×D₁₀₁	C₂²×C₄	1	4	yes	—
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	1	no	—
99	C₂×C₄×C₁₀₁	—	2	1	no	—
100	C₂×C₄×C₁₀₁	—	2	8	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	—	2	1	no	—
106	C₂×C₄×C₁₀₁	—	2	1	no	—
107	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
108	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
109	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
111	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
112	C₂×C₄×C₁₀₁	#41	1	2	yes	—
113	C₂²⋊D₁₀₁	—	2	16	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	—	2	2	no	—
121	C₂×Dic₁₀₁	—	4	8	no	—
122	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
123	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	—
124	D₄₀₄	—	2	8	no	—
125	C₄×D₁₀₁	—	4	8	no	—
126	C₂²×C₁₀₁	—	2	2	no	—
127	C₂²×C₁₀₁	—	2	2	no	—
128	C₂²×C₁₀₁	—	2	2	no	—
129	C₂²×C₁₀₁	—	2	2	no	—
130	C₂²×C₁₀₁	—	2	2	no	—
131	C₂²×C₁₀₁	—	2	2	no	—
132	C₂²×C₁₀₁	C₄×D₄	1	4	yes	complemented
133	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
134	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
135	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
136	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
137	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
138	C₂²×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
139	C₂²×C₁₀₁	2₋¹⁺⁴	1	1	yes	characteristic
140	C₂×D₁₀₁	—	2	4	no	—
141	C₂×D₁₀₁	—	2	4	no	—
142	C₂×D₁₀₁	—	2	4	no	—
143	C₂×D₁₀₁	—	4	2	no	—
144	C₂×D₁₀₁	—	4	2	no	—
145	C₂×D₁₀₁	—	4	2	no	—
146	C₂×D₁₀₁	—	4	2	no	—
147	C₄×C₁₀₁	—	2	2	no	—
148	C₄×C₁₀₁	—	2	2	no	—
149	C₄×C₁₀₁	—	2	2	no	—
150	C₄×C₁₀₁	—	4	1	no	—
151	C₄×C₁₀₁	—	4	1	no	—
152	C₄×C₁₀₁	C₄×D₄	1	4	yes	complemented
153	Dic₁₀₁	—	2	4	no	—
154	Dic₁₀₁	—	2	4	no	—
155	Dic₁₀₁	—	4	2	no	—
156	C₂×C₁₀₁	—	2	4	no	—
157	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
158	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
159	C₂×C₁₀₁	#520	1	1	yes	characteristic
160	C₂×C₁₀₁	D₄²	1	1	yes	characteristic
161	C₂×C₁₀₁	#547	1	1	yes	characteristic
162	C₂×C₁₀₁	#548	1	1	yes	characteristic
163	C₂×C₁₀₁	#553	1	1	yes	characteristic
164	D₁₀₁	—	4	4	no	—
165	#2361	—	101	1	no	maximal, Sylow
166	C₁₀₁	#2361	1	1	yes	characteristic, complemented, Sylow
167	C₂×#121	—	101	1	no	—
168	C₂×C₄×D₄	—	101	2	no	—
169	C₂×C₄×D₄	—	101	1	no	—
170	C₂²×C₄⋊C₄	—	101	1	no	—
171	#390	—	101	1	no	—
172	#389	—	101	1	no	—
173	#388	—	101	2	no	—
174	#386	—	101	2	no	—
175	#385	—	101	2	no	—
176	C₄×C₄⋊C₄	—	101	1	no	—
177	C₄×C₂²⋊C₄	D₁₀₁	1	1	yes	characteristic, complemented
178	C₂²×D₄	—	101	2	no	—
179	C₂²×D₄	—	101	1	no	—
180	C₂³×C₄	—	101	2	no	—
181	C₂³×C₄	—	101	2	no	—
182	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
183	#121	—	101	8	no	—
184	C₄×D₄	—	202	4	no	—
185	C₄×D₄	—	202	8	no	—
186	C₂×C₄⋊C₄	—	101	1	no	—
187	C₂×C₄⋊C₄	—	101	1	no	—
188	C₂×C₄⋊C₄	—	101	2	no	—
189	C₂×C₄⋊C₄	—	101	1	no	—
190	C₂×C₄⋊C₄	—	101	2	no	—
191	C₂×C₄⋊C₄	—	202	4	no	—
192	C₂×C₂²⋊C₄	—	101	2	no	—
193	C₂×C₂²⋊C₄	—	101	2	no	—
194	C₂×C₂²⋊C₄	—	101	2	no	—
195	C₂×C₂²⋊C₄	—	101	2	no	—
196	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
197	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
198	C₂×C₄²	—	101	2	no	—
199	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
200	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
201	#95	—	101	2	no	—
202	#95	—	101	2	no	—
203	#95	C₂×D₁₀₁	1	1	yes	characteristic
204	#95	C₂×D₁₀₁	1	1	yes	characteristic
205	C₂⁴	—	101	2	no	—
206	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
207	C₂×D₄	—	101	8	no	—
208	C₂×D₄	—	101	4	no	—
209	C₂×D₄	—	202	4	no	—
210	C₂×D₄	—	202	4	no	—
211	C₂×D₄	—	202	2	no	—
212	C₂×D₄	—	202	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	101	2	no	—
216	C₂²×C₄	—	101	2	no	—
217	C₂²×C₄	—	101	2	no	—
218	C₂²×C₄	—	404	2	no	—
219	C₂²×C₄	—	404	2	no	—
220	C₂²×C₄	—	202	4	no	—
221	C₂²×C₄	—	202	4	no	—
222	C₂²×C₄	—	202	4	no	—
223	C₂²×C₄	—	202	4	no	—
224	C₂²×C₄	C₄×D₁₀₁	1	4	yes	complemented
225	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
226	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
227	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
228	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
229	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
230	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
231	C₄⋊C₄	—	101	4	no	—
232	C₄⋊C₄	—	101	4	no	—
233	C₄⋊C₄	—	202	4	no	—
234	C₄⋊C₄	—	202	2	no	—
235	C₄⋊C₄	—	202	4	no	—
236	C₂²⋊C₄	—	2	2	no	—
237	C₂²⋊C₄	—	101	8	no	—
238	C₂²⋊C₄	—	404	2	no	—
239	C₂²⋊C₄	—	202	4	no	—
240	C₂²⋊C₄	—	202	4	no	—
241	C₂²⋊C₄	D₄₀₄	1	4	yes	complemented
242	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	202	4	no	—
245	C₂³	—	2	1	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	2	1	no	—
249	C₂³	—	101	4	no	—
250	C₂³	—	202	2	no	—
251	C₂³	—	202	2	no	—
252	C₂³	—	202	2	no	—
253	C₂³	—	202	2	no	—
254	C₂³	—	202	2	no	—
255	C₂³	—	202	2	no	—
256	C₂³	C₂×C₄×D₁₀₁	1	2	yes	—
257	C₂³	C₂×D₄₀₄	1	2	yes	—
258	C₂³	#13108074	1	2	yes	—
259	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
260	D₄	—	202	16	no	—
261	D₄	—	202	8	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	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	8	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	101	4	no	—
276	C₂×C₄	—	101	4	no	—
277	C₂×C₄	—	404	8	no	—
278	C₂×C₄	—	404	8	no	—
279	C₂×C₄	—	202	2	no	—
280	C₂×C₄	—	202	2	no	—
281	C₂×C₄	—	202	2	no	—
282	C₂×C₄	—	202	2	no	—
283	C₂×C₄	—	202	2	no	—
284	C₂×C₄	—	202	2	no	—
285	C₂×C₄	—	202	2	no	—
286	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
287	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
288	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
289	C₂×C₄	C₂×D₄₀₄	1	2	yes	—
290	C₂×C₄	C₂×D₄₀₄	1	2	yes	—
291	C₂×C₄	#13108073	1	2	yes	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	404	2	no	—
299	C₂²	—	404	2	no	—
300	C₂²	—	404	2	no	—
301	C₂²	—	404	2	no	—
302	C₂²	—	202	4	no	—
303	C₂²	—	202	4	no	—
304	C₂²	—	202	4	no	—
305	C₂²	C₄×D₄₀₄	1	4	yes	complemented
306	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
307	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
308	C₂²	C₂²×D₄₀₄	1	1	yes	characteristic
309	C₂²	C₂×#13108073	1	1	yes	characteristic
310	C₂²	C₂×#13108074	1	1	yes	characteristic
311	C₂²	#15692301	1	1	yes	characteristic
312	C₂²	#15692302	1	1	yes	characteristic
313	C₄	—	2	2	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	404	2	no	—
319	C₄	—	202	4	no	—
320	C₄	—	202	4	no	—
321	C₄	#15692215	1	4	yes	complemented
322	C₂	—	2	4	no	—
323	C₂	—	404	4	no	—
324	C₂	C₂×#15692215	1	1	yes	characteristic
325	C₂	C₂×C₄×D₄₀₄	1	1	yes	characteristic
326	C₂	#20265657	1	1	yes	characteristic
327	C₂	D₄×D₄₀₄	1	1	yes	characteristic
328	C₂	#20265831	1	1	yes	characteristic
329	C₂	#20265836	1	1	yes	characteristic
330	C₂	#20265873	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Fitting	C₄×C₂²⋊C₄×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂×#15692229	C₂
hypercenter	C₄×C₂²⋊C₄	D₁₀₁
hypocenter	C₁₀₁	#2361
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₄₀₄	C₂	semidirect
3. C₂×#15692215	C₂	semidirect
4. C₂²×C₄⋊Dic₁₀₁	C₂	semidirect
5. #20265079	C₂	semidirect
6. #20265075	C₂	semidirect
7. #20265069	C₂	semidirect
8. #20265054	C₂	semidirect
9. #20265051	C₂	semidirect
10. C₄×C₄⋊Dic₁₀₁	C₂	semidirect
11. C₄×C₂²⋊C₄×C₁₀₁	C₂	semidirect
16. #15692229	C₄	semidirect
42. C₂×D₄₀₄	C₂×C₄	semidirect
45. C₂×C₂²⋊D₁₀₁	C₂×C₄	semidirect
57. C₄⋊Dic₁₀₁	C₂×C₄	semidirect
58. C₄⋊Dic₁₀₁	D₄	semidirect
63. #13108051	C₂×C₄	semidirect
68. C₂²×C₄×C₁₀₁	C₂×C₄	semidirect
76. C₂²⋊C₄×C₁₀₁	D₄	semidirect
132. C₂²×C₁₀₁	C₄×D₄	semidirect
152. C₄×C₁₀₁	C₄×D₄	semidirect
166. C₁₀₁	#2361	semidirect
177. C₄×C₂²⋊C₄	D₁₀₁	semidirect
224. C₂²×C₄	C₄×D₁₀₁	semidirect
241. C₂²⋊C₄	D₄₀₄	semidirect
305. C₂²	C₄×D₄₀₄	semidirect
321. C₄	#15692215	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
C₂×#15692215	2
C₂²×C₄⋊Dic₁₀₁	2
#20265079	2
#20265075	2
#20265069	2
#20265054	2
#20265051	2
C₄×C₄⋊Dic₁₀₁	2
C₄×C₂²⋊C₄×C₁₀₁	2
#2361	101