Subgroups — #27174169

Order 12928.

688 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692230	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₄₀₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15692213	C₂	1	1	yes	maximal, characteristic, complemented
4	#20265078	C₂	1	1	yes	maximal, characteristic, complemented
5	#20265078	C₂	1	2	yes	maximal, complemented
6	#20265067	C₂	1	2	yes	maximal, complemented
7	#20265055	C₂	1	2	yes	maximal, complemented
8	C₄⋊C₄×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#13108051	C₂	1	1	yes	maximal, characteristic, complemented
10	#20265024	C₂	1	2	yes	maximal, complemented
11	#389×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄₀₄	C₂²	1	1	yes	characteristic
13	C₂²×D₄₀₄	C₂²	1	2	yes	—
14	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
15	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
16	#15692230	C₄	1	8	yes	complemented
17	C₄×D₄₀₄	—	2	4	no	—
18	#15692213	—	2	4	no	—
19	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
20	C₂×#13108051	C₂²	1	2	yes	—
21	C₂×#13108051	C₂²	1	1	yes	characteristic
22	C₂×#13108051	C₂²	1	1	yes	characteristic
23	C₂×#13108051	C₂²	1	2	yes	—
24	C₂×#13108051	C₂²	1	2	yes	—
25	C₂×#13108051	C₂²	1	2	yes	—
26	C₂×#13108051	C₂²	1	2	yes	—
27	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
28	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
29	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
30	#15692162	C₂²	1	1	yes	characteristic
31	#15692162	C₂²	1	1	yes	characteristic
32	#15692162	C₂²	1	2	yes	—
33	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
35	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
36	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
37	#95×C₁₀₁	C₂²	1	2	yes	—
38	C₂³×D₁₀₁	C₂³	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	—	4	2	no	—
43	C₂×D₄₀₄	—	4	2	no	—
44	C₂×D₄₀₄	C₂×C₄	1	4	yes	—
45	C₂×D₄₀₄	C₂×C₄	1	8	yes	complemented
46	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
47	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
48	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
49	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
50	C₂×C₄×D₁₀₁	—	2	2	no	—
51	C₂×C₄×D₁₀₁	—	2	2	no	—
52	C₂×C₄×D₁₀₁	—	2	4	no	—
53	C₂×C₄×D₁₀₁	—	2	4	no	—
54	C₂×C₄×D₁₀₁	—	4	2	no	—
55	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	complemented
56	C₄⋊Dic₁₀₁	D₄	1	4	yes	complemented
57	#13108051	—	2	2	no	—
58	#13108051	—	2	4	no	—
59	#13108051	—	2	4	no	—
60	#13108051	—	2	4	no	—
61	#13108051	—	2	2	no	—
62	#13108051	—	4	2	no	—
63	#13108051	C₂×C₄	1	8	yes	complemented
64	C₄×Dic₁₀₁	—	2	2	no	—
65	C₄×Dic₁₀₁	—	2	4	no	—
66	C₄×Dic₁₀₁	—	4	2	no	—
67	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
68	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
70	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
71	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₁₀₁	—	2	2	no	—
73	C₄⋊C₄×C₁₀₁	—	4	2	no	—
74	C₄⋊C₄×C₁₀₁	C₂×C₄	1	4	yes	complemented
75	C₄²×C₁₀₁	—	2	2	no	—
76	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
77	C₂²×D₁₀₁	—	2	1	no	—
78	C₂²×D₁₀₁	—	2	2	no	—
79	C₂²×D₁₀₁	—	2	1	no	—
80	C₂²×D₁₀₁	—	2	2	no	—
81	C₂²×D₁₀₁	—	2	1	no	—
82	C₂²×D₁₀₁	—	2	1	no	—
83	C₂²×D₁₀₁	—	2	2	no	—
84	C₂²×D₁₀₁	—	2	2	no	—
85	C₂²×D₁₀₁	—	2	2	no	—
86	C₂²×D₁₀₁	—	2	2	no	—
87	C₂²×D₁₀₁	—	2	1	no	—
88	C₂²×D₁₀₁	—	2	1	no	—
89	C₂²×D₁₀₁	C₂²×C₄	1	4	yes	—
90	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₁₀₁	—	2	1	no	—
92	C₂×C₄×C₁₀₁	—	2	2	no	—
93	C₂×C₄×C₁₀₁	—	2	2	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	1	no	—
99	C₂×C₄×C₁₀₁	—	2	1	no	—
100	C₂×C₄×C₁₀₁	—	2	2	no	—
101	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
102	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
103	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
104	C₂×C₄×C₁₀₁	#41	1	2	yes	—
105	C₂×C₄×C₁₀₁	#41	1	2	yes	—
106	C₂×Dic₁₀₁	—	2	1	no	—
107	C₂×Dic₁₀₁	—	2	1	no	—
108	C₂×Dic₁₀₁	—	2	1	no	—
109	C₂×Dic₁₀₁	—	2	2	no	—
110	C₂×Dic₁₀₁	—	2	1	no	—
111	C₂×Dic₁₀₁	—	2	2	no	—
112	C₂×Dic₁₀₁	—	2	1	no	—
113	C₂×Dic₁₀₁	—	2	1	no	—
114	C₂×Dic₁₀₁	—	2	1	no	—
115	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
116	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	—
117	C₂×Dic₁₀₁	#41	1	2	yes	—
118	D₄₀₄	—	2	8	no	—
119	D₄₀₄	—	4	8	no	—
120	C₄×D₁₀₁	—	2	8	no	—
121	C₄×D₁₀₁	—	4	8	no	—
122	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
123	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
124	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
125	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
126	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
127	C₂²×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
128	C₂²×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
129	C₂×D₁₀₁	—	2	2	no	—
130	C₂×D₁₀₁	—	2	4	no	—
131	C₂×D₁₀₁	—	2	2	no	—
132	C₂×D₁₀₁	—	2	4	no	—
133	C₂×D₁₀₁	—	2	2	no	—
134	C₂×D₁₀₁	—	2	4	no	—
135	C₂×D₁₀₁	—	4	2	no	—
136	C₂×D₁₀₁	—	4	1	no	—
137	C₂×D₁₀₁	—	4	1	no	—
138	C₂×D₁₀₁	—	4	2	no	—
139	C₂×D₁₀₁	—	4	1	no	—
140	C₂×D₁₀₁	—	4	2	no	—
141	C₂×D₁₀₁	—	4	1	no	—
142	C₂×D₁₀₁	—	4	2	no	—
143	C₄×C₁₀₁	—	2	4	no	—
144	C₄×C₁₀₁	—	2	2	no	—
145	C₄×C₁₀₁	—	4	1	no	—
146	C₄×C₁₀₁	—	4	2	no	—
147	C₄×C₁₀₁	C₄×D₄	1	4	yes	—
148	Dic₁₀₁	—	2	2	no	—
149	Dic₁₀₁	—	2	4	no	—
150	Dic₁₀₁	—	2	2	no	—
151	Dic₁₀₁	—	4	1	no	—
152	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
153	C₂×C₁₀₁	C₄×#41	1	1	yes	characteristic
154	C₂×C₁₀₁	#518	1	1	yes	characteristic
155	C₂×C₁₀₁	#546	1	1	yes	characteristic
156	C₂×C₁₀₁	#550	1	1	yes	characteristic
157	C₂×C₁₀₁	#553	1	1	yes	characteristic
158	C₂×C₁₀₁	#555	1	1	yes	characteristic
159	D₁₀₁	—	4	2	no	—
160	D₁₀₁	—	4	4	no	—
161	#2363	—	101	1	no	maximal, Sylow
162	C₁₀₁	#2363	1	1	yes	characteristic, complemented, Sylow
163	C₂×#121	—	101	1	no	—
164	C₂×C₄×D₄	—	101	1	no	—
165	C₂×#117	—	101	1	no	—
166	#390	—	101	2	no	—
167	#390	—	101	1	no	—
168	#389	D₁₀₁	1	1	yes	characteristic, complemented
169	#388	—	101	2	no	—
170	#386	—	101	2	no	—
171	C₄×C₄⋊C₄	—	101	1	no	—
172	C₄×C₂²⋊C₄	—	101	1	no	—
173	C₄×C₂²⋊C₄	—	101	2	no	—
174	C₂²×D₄	—	101	2	no	—
175	C₂²×D₄	—	101	1	no	—
176	C₂³×C₄	—	101	1	no	—
177	C₂³×C₄	—	101	2	no	—
178	#121	—	101	8	no	—
179	C₄×D₄	—	202	4	no	—
180	#117	—	202	4	no	—
181	C₂×C₄⋊C₄	—	101	1	no	—
182	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
183	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
184	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
185	C₂×C₂²⋊C₄	—	101	2	no	—
186	C₂×C₂²⋊C₄	—	101	2	no	—
187	C₂×C₂²⋊C₄	—	101	2	no	—
188	C₂×C₂²⋊C₄	—	101	2	no	—
189	C₂×C₂²⋊C₄	—	101	1	no	—
190	C₂×C₂²⋊C₄	—	101	1	no	—
191	C₂×C₂²⋊C₄	—	101	2	no	—
192	C₂×C₄²	—	101	2	no	—
193	C₂×C₄²	—	101	2	no	—
194	C₂×C₄²	—	101	1	no	—
195	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
196	#95	—	101	1	no	—
197	#95	—	101	1	no	—
198	#95	—	101	2	no	—
199	#95	C₂×D₁₀₁	1	2	yes	—
200	C₂⁴	—	101	2	no	—
201	C₂⁴	—	101	1	no	—
202	C₂×D₄	—	101	8	no	—
203	C₂×D₄	—	101	4	no	—
204	C₂×D₄	—	404	2	no	—
205	C₂×D₄	—	404	2	no	—
206	C₂×D₄	—	202	2	no	—
207	C₂×D₄	—	202	2	no	—
208	C₂²×C₄	—	101	1	no	—
209	C₂²×C₄	—	101	1	no	—
210	C₂²×C₄	—	101	2	no	—
211	C₂²×C₄	—	101	4	no	—
212	C₂²×C₄	—	101	1	no	—
213	C₂²×C₄	—	404	2	no	—
214	C₂²×C₄	—	202	4	no	—
215	C₂²×C₄	—	202	4	no	—
216	C₂²×C₄	—	202	2	no	—
217	C₂²×C₄	—	202	2	no	—
218	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
219	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
220	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
221	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
222	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₄⋊C₄	—	101	4	no	—
226	C₄⋊C₄	C₄×D₁₀₁	1	4	yes	complemented
227	C₂²⋊C₄	—	101	8	no	—
228	C₂²⋊C₄	—	404	2	no	—
229	C₂²⋊C₄	—	202	2	no	—
230	C₂²⋊C₄	—	202	4	no	—
231	C₂²⋊C₄	—	202	4	no	—
232	C₂²⋊C₄	—	202	4	no	—
233	C₂²⋊C₄	—	202	2	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	404	2	no	—
236	C₄²	—	202	4	no	—
237	C₄²	—	202	2	no	—
238	C₂³	—	101	4	no	—
239	C₂³	—	101	2	no	—
240	C₂³	—	202	1	no	—
241	C₂³	—	202	2	no	—
242	C₂³	—	202	1	no	—
243	C₂³	—	202	2	no	—
244	C₂³	—	202	1	no	—
245	C₂³	—	202	2	no	—
246	C₂³	—	202	1	no	—
247	C₂³	—	202	2	no	—
248	C₂³	—	202	2	no	—
249	C₂³	—	202	1	no	—
250	C₂³	—	202	2	no	—
251	C₂³	—	202	1	no	—
252	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
253	D₄	—	404	8	no	—
254	D₄	—	202	8	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	101	2	no	—
266	C₂×C₄	—	101	4	no	—
267	C₂×C₄	—	101	2	no	—
268	C₂×C₄	—	404	8	no	—
269	C₂×C₄	—	202	1	no	—
270	C₂×C₄	—	202	1	no	—
271	C₂×C₄	—	202	1	no	—
272	C₂×C₄	—	202	1	no	—
273	C₂×C₄	—	202	1	no	—
274	C₂×C₄	—	202	2	no	—
275	C₂×C₄	—	202	1	no	—
276	C₂×C₄	—	202	1	no	—
277	C₂×C₄	—	202	2	no	—
278	C₂×C₄	—	202	8	no	—
279	C₂×C₄	C₂×C₄×D₁₀₁	1	4	yes	—
280	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
281	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
282	C₂×C₄	#13108075	1	2	yes	—
283	C₂×C₄	#13108075	1	2	yes	—
284	C₂²	—	404	2	no	—
285	C₂²	—	404	1	no	—
286	C₂²	—	404	1	no	—
287	C₂²	—	404	2	no	—
288	C₂²	—	404	1	no	—
289	C₂²	—	404	2	no	—
290	C₂²	—	404	1	no	—
291	C₂²	—	404	2	no	—
292	C₂²	—	202	2	no	—
293	C₂²	—	202	4	no	—
294	C₂²	—	202	4	no	—
295	C₂²	—	202	2	no	—
296	C₂²	—	202	2	no	—
297	C₂²	—	202	4	no	—
298	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
299	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
300	C₂²	#41×D₁₀₁	1	1	yes	characteristic
301	C₂²	C₂×#13108075	1	1	yes	characteristic
302	C₂²	C₂×#13108075	1	1	yes	characteristic
303	C₂²	#15692300	1	1	yes	characteristic
304	C₂²	#15692301	1	1	yes	characteristic
305	C₄	—	2	2	no	—
306	C₄	—	2	4	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	2	no	—
309	C₄	—	404	1	no	—
310	C₄	—	202	4	no	—
311	C₄	—	202	2	no	—
312	C₄	—	202	2	no	—
313	C₄	#15692217	1	4	yes	—
314	C₂	—	404	4	no	—
315	C₂	—	404	2	no	—
316	C₂	C₂×#15692217	1	1	yes	characteristic
317	C₂	C₄×#13108075	1	1	yes	characteristic
318	C₂	#20265649	1	1	yes	characteristic
319	C₂	#20265818	1	1	yes	characteristic
320	C₂	#20265845	1	1	yes	characteristic
321	C₂	#20265880	1	1	yes	characteristic
322	C₂	#20265893	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Fitting	#389×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂×#15692230	C₂
hypercenter	#389	D₁₀₁
hypocenter	C₁₀₁	#2363
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ #389

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₄₀₄	C₂	semidirect
3. C₂×#15692213	C₂	semidirect
4. #20265078	C₂	semidirect
5. #20265078	C₂	semidirect
6. #20265067	C₂	semidirect
7. #20265055	C₂	semidirect
8. C₄⋊C₄×Dic₁₀₁	C₂	semidirect
9. C₄×#13108051	C₂	semidirect
10. #20265024	C₂	semidirect
11. #389×C₁₀₁	C₂	semidirect
16. #15692230	C₄	semidirect
45. C₂×D₄₀₄	C₂×C₄	semidirect
55. C₂×C₄×D₁₀₁	C₂×C₄	semidirect
56. C₄⋊Dic₁₀₁	D₄	semidirect
63. #13108051	C₂×C₄	semidirect
74. C₄⋊C₄×C₁₀₁	C₂×C₄	semidirect
162. C₁₀₁	#2363	semidirect
168. #389	D₁₀₁	semidirect
226. C₄⋊C₄	C₄×D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15692230	2
C₂×C₄×D₄₀₄	2
C₂×#15692213	2
#20265078	2
#20265067	2
#20265055	2
C₄⋊C₄×Dic₁₀₁	2
C₄×#13108051	2
#20265024	2
#389×C₁₀₁	2
#2363	101