Subgroups — #44402776

Order 29056.

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×D₄×Dic₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#20300247	C₂	1	1	yes	maximal, characteristic, complemented
3	#28408935	C₂	1	1	yes	maximal, characteristic, complemented
4	#28408935	C₂	1	2	yes	maximal, complemented
5	#28408931	C₂	1	1	yes	maximal, characteristic, complemented
6	#28408925	C₂	1	2	yes	maximal, complemented
7	#28408912	C₂	1	2	yes	maximal, complemented
8	C₄⋊C₄×Dic₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²⋊C₄×Dic₂₂₇	C₂	1	2	yes	maximal, complemented
10	C₄×C₂²⋊Dic₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#121×C₂₂₇	C₂	1	1	yes	maximal, characteristic
12	C₂³×Dic₂₂₇	C₂²	1	2	yes	—
13	D₄×Dic₂₂₇	—	2	4	no	—
14	#20300247	—	2	4	no	—
15	C₂×C₄⋊Dic₂₂₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₂₂₇	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₂₂₇	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊Dic₂₂₇	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊Dic₂₂₇	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊Dic₂₂₇	C₂²	1	2	yes	—
21	C₂×C₂²⋊Dic₂₂₇	C₂²	1	2	yes	—
22	C₂×C₂²⋊Dic₂₂₇	C₂²	1	2	yes	—
23	C₂×C₂²⋊Dic₂₂₇	C₂²	1	2	yes	—
24	C₂×C₄×Dic₂₂₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×Dic₂₂₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×Dic₂₂₇	C₂²	1	2	yes	—
27	C₂×C₄×Dic₂₂₇	C₂²	1	2	yes	—
28	C₂₂₇⋊#95	C₂²	1	1	yes	characteristic
29	C₂₂₇⋊#95	C₂²	1	1	yes	characteristic
30	C₂₂₇⋊#95	C₂²	1	2	yes	—
31	C₂₂₇⋊#95	C₂²	1	2	yes	—
32	C₂²×D₄×C₂₂₇	C₂²	1	1	yes	characteristic
33	C₂²×D₄×C₂₂₇	C₂²	1	2	yes	—
34	C₂³×C₄×C₂₂₇	C₂²	1	1	yes	characteristic
35	#121×C₂₂₇	C₄	1	8	yes	complemented
36	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₂₂₇	C₂²	1	2	yes	—
38	C₂²×Dic₂₂₇	—	2	4	no	—
39	C₂²×Dic₂₂₇	—	2	4	no	—
40	C₂²×Dic₂₂₇	—	4	2	no	—
41	C₂²×Dic₂₂₇	C₂³	1	1	yes	characteristic
42	C₂²×Dic₂₂₇	C₂³	1	1	yes	characteristic
43	C₂²×Dic₂₂₇	C₂³	1	2	yes	—
44	C₂²×Dic₂₂₇	C₂³	1	2	yes	—
45	C₂²×Dic₂₂₇	C₂³	1	1	yes	characteristic
46	C₂²×Dic₂₂₇	C₂³	1	1	yes	characteristic
47	C₄⋊Dic₂₂₇	—	2	2	no	—
48	C₄⋊Dic₂₂₇	—	4	2	no	—
49	C₄⋊Dic₂₂₇	D₄	1	4	yes	complemented
50	C₂²⋊Dic₂₂₇	—	2	4	no	—
51	C₂²⋊Dic₂₂₇	—	2	4	no	—
52	C₂²⋊Dic₂₂₇	—	2	2	no	—
53	C₂²⋊Dic₂₂₇	—	2	4	no	—
54	C₂²⋊Dic₂₂₇	—	2	2	no	—
55	C₂²⋊Dic₂₂₇	—	4	2	no	—
56	C₄×Dic₂₂₇	—	2	4	no	—
57	C₄×Dic₂₂₇	—	2	2	no	—
58	C₄×Dic₂₂₇	—	2	2	no	—
59	C₄×Dic₂₂₇	—	4	2	no	—
60	C₂⁴×C₂₂₇	C₂³	1	2	yes	—
61	C₂⁴×C₂₂₇	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₂₂₇	—	2	2	no	—
63	C₂×D₄×C₂₂₇	—	2	2	no	—
64	C₂×D₄×C₂₂₇	—	4	2	no	—
65	C₂×D₄×C₂₂₇	—	4	2	no	—
66	C₂×D₄×C₂₂₇	C₂×C₄	1	4	yes	—
67	C₂×D₄×C₂₂₇	C₂×C₄	1	8	yes	complemented
68	C₂²×C₄×C₂₂₇	—	2	2	no	—
69	C₂²×C₄×C₂₂₇	—	2	2	no	—
70	C₂²×C₄×C₂₂₇	C₂×C₄	1	4	yes	complemented
71	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
72	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
73	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₂₂₇	C₂×C₄	1	4	yes	complemented
75	C₂²⋊C₄×C₂₂₇	C₂×C₄	1	8	yes	complemented
76	C₂×Dic₂₂₇	—	2	1	no	—
77	C₂×Dic₂₂₇	—	2	1	no	—
78	C₂×Dic₂₂₇	—	2	2	no	—
79	C₂×Dic₂₂₇	—	2	1	no	—
80	C₂×Dic₂₂₇	—	2	2	no	—
81	C₂×Dic₂₂₇	—	2	1	no	—
82	C₂×Dic₂₂₇	—	2	1	no	—
83	C₂×Dic₂₂₇	—	2	2	no	—
84	C₂×Dic₂₂₇	—	2	1	no	—
85	C₂×Dic₂₂₇	—	2	1	no	—
86	C₂×Dic₂₂₇	—	2	2	no	—
87	C₂×Dic₂₂₇	—	2	1	no	—
88	C₂×Dic₂₂₇	—	2	1	no	—
89	C₂×Dic₂₂₇	—	2	2	no	—
90	C₂×Dic₂₂₇	—	2	1	no	—
91	C₂×Dic₂₂₇	—	4	8	no	—
92	C₂×Dic₂₂₇	C₂×D₄	1	4	yes	—
93	C₂×Dic₂₂₇	#41	1	2	yes	—
94	C₂×Dic₂₂₇	#41	1	2	yes	—
95	C₂³×C₂₂₇	—	2	1	no	—
96	C₂³×C₂₂₇	—	2	2	no	—
97	C₂³×C₂₂₇	—	2	1	no	—
98	C₂³×C₂₂₇	—	2	2	no	—
99	C₂³×C₂₂₇	—	2	1	no	—
100	C₂³×C₂₂₇	—	2	2	no	—
101	C₂³×C₂₂₇	—	2	1	no	—
102	C₂³×C₂₂₇	—	2	2	no	—
103	C₂³×C₂₂₇	—	2	1	no	—
104	C₂³×C₂₂₇	—	2	1	no	—
105	C₂³×C₂₂₇	—	2	2	no	—
106	C₂³×C₂₂₇	—	2	2	no	—
107	C₂³×C₂₂₇	C₂²×C₄	1	4	yes	—
108	C₂³×C₂₂₇	C₂²×C₄	1	2	yes	—
109	C₂³×C₂₂₇	C₂⁴	1	1	yes	characteristic
110	D₄×C₂₂₇	—	2	8	no	—
111	D₄×C₂₂₇	—	4	8	no	—
112	C₂×C₄×C₂₂₇	—	2	1	no	—
113	C₂×C₄×C₂₂₇	—	2	1	no	—
114	C₂×C₄×C₂₂₇	—	2	2	no	—
115	C₂×C₄×C₂₂₇	—	2	2	no	—
116	C₂×C₄×C₂₂₇	—	2	8	no	—
117	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
118	C₂×C₄×C₂₂₇	C₂²×C₄	1	4	yes	—
119	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
120	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
121	C₂×C₄×C₂₂₇	#41	1	2	yes	—
122	C₂²×C₂₂₇	—	2	2	no	—
123	C₂²×C₂₂₇	—	2	4	no	—
124	C₂²×C₂₂₇	—	2	2	no	—
125	C₂²×C₂₂₇	—	2	4	no	—
126	C₂²×C₂₂₇	—	2	2	no	—
127	C₂²×C₂₂₇	—	2	4	no	—
128	C₂²×C₂₂₇	—	4	1	no	—
129	C₂²×C₂₂₇	—	4	2	no	—
130	C₂²×C₂₂₇	—	4	1	no	—
131	C₂²×C₂₂₇	—	4	2	no	—
132	C₂²×C₂₂₇	—	4	1	no	—
133	C₂²×C₂₂₇	—	4	2	no	—
134	C₂²×C₂₂₇	—	4	1	no	—
135	C₂²×C₂₂₇	—	4	2	no	—
136	C₂²×C₂₂₇	C₂³×C₄	1	1	yes	characteristic
137	C₂²×C₂₂₇	C₂²×D₄	1	1	yes	characteristic
138	C₂²×C₂₂₇	C₂×#41	1	1	yes	characteristic
139	C₂²×C₂₂₇	C₂×#41	1	1	yes	characteristic
140	C₂²×C₂₂₇	C₂×#41	1	1	yes	characteristic
141	C₂²×C₂₂₇	H(5,𝔽₂)	1	1	yes	characteristic
142	C₂²×C₂₂₇	H(5,𝔽₂)	1	1	yes	characteristic
143	C₄×C₂₂₇	—	2	2	no	—
144	C₄×C₂₂₇	—	2	4	no	—
145	C₄×C₂₂₇	C₄×D₄	1	4	yes	—
146	Dic₂₂₇	—	2	2	no	—
147	Dic₂₂₇	—	2	4	no	—
148	Dic₂₂₇	—	2	2	no	—
149	Dic₂₂₇	—	4	1	no	—
150	Dic₂₂₇	—	4	2	no	—
151	Dic₂₂₇	—	4	1	no	—
152	C₂×C₂₂₇	—	4	2	no	—
153	C₂×C₂₂₇	—	4	4	no	—
154	C₂×C₂₂₇	C₂×C₄×D₄	1	1	yes	characteristic
155	C₂×C₂₂₇	C₄×#41	1	1	yes	characteristic
156	C₂×C₂₂₇	#518	1	1	yes	characteristic
157	C₂×C₂₂₇	#546	1	1	yes	characteristic
158	C₂×C₂₂₇	#550	1	1	yes	characteristic
159	C₂×C₂₂₇	#553	1	1	yes	characteristic
160	C₂×C₂₂₇	#555	1	1	yes	characteristic
161	C₂₂₇	#2363	1	1	yes	characteristic, complemented, Sylow
162	#2363	—	227	1	no	maximal, Sylow
163	C₂×#121	D₂₂₇	1	1	yes	characteristic
164	C₂×C₄×D₄	—	227	1	no	—
165	C₂×#117	—	227	1	no	—
166	#390	—	227	2	no	—
167	#390	—	227	1	no	—
168	#389	—	227	1	no	—
169	#388	—	227	2	no	—
170	#386	—	227	2	no	—
171	C₄×C₄⋊C₄	—	227	1	no	—
172	C₄×C₂²⋊C₄	—	227	1	no	—
173	C₄×C₂²⋊C₄	—	227	2	no	—
174	C₂²×D₄	C₂×D₂₂₇	1	1	yes	characteristic
175	C₂²×D₄	C₂×D₂₂₇	1	2	yes	—
176	C₂³×C₄	—	227	2	no	—
177	C₂³×C₄	C₂×D₂₂₇	1	1	yes	characteristic
178	#121	Dic₂₂₇	1	8	yes	complemented
179	C₄×D₄	—	454	4	no	—
180	#117	—	454	4	no	—
181	C₂×C₄⋊C₄	—	227	2	no	—
182	C₂×C₄⋊C₄	—	227	1	no	—
183	C₂×C₄⋊C₄	—	227	1	no	—
184	C₂×C₄⋊C₄	C₂×D₂₂₇	1	1	yes	characteristic
185	C₂×C₂²⋊C₄	—	227	2	no	—
186	C₂×C₂²⋊C₄	—	227	2	no	—
187	C₂×C₂²⋊C₄	—	227	2	no	—
188	C₂×C₂²⋊C₄	—	227	1	no	—
189	C₂×C₂²⋊C₄	—	227	1	no	—
190	C₂×C₂²⋊C₄	—	227	2	no	—
191	C₂×C₂²⋊C₄	C₂×D₂₂₇	1	2	yes	—
192	C₂×C₄²	—	227	2	no	—
193	C₂×C₄²	—	227	2	no	—
194	C₂×C₄²	—	227	1	no	—
195	C₂×C₄²	—	227	1	no	—
196	#95	—	227	1	no	—
197	#95	—	227	2	no	—
198	#95	—	227	1	no	—
199	#95	—	227	2	no	—
200	C₂⁴	C₂²×D₂₂₇	1	1	yes	characteristic
201	C₂⁴	C₂²×D₂₂₇	1	2	yes	—
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	—	4	2	no	—
205	C₂×D₄	—	4	2	no	—
206	C₂×D₄	C₂×Dic₂₂₇	1	4	yes	—
207	C₂×D₄	C₂×Dic₂₂₇	1	8	yes	complemented
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	908	2	no	—
211	C₂²×C₄	—	454	4	no	—
212	C₂²×C₄	—	454	4	no	—
213	C₂²×C₄	—	227	1	no	—
214	C₂²×C₄	—	227	1	no	—
215	C₂²×C₄	—	227	2	no	—
216	C₂²×C₄	—	227	1	no	—
217	C₂²×C₄	—	227	1	no	—
218	C₂²×C₄	—	227	2	no	—
219	C₂²×C₄	C₂×Dic₂₂₇	1	4	yes	complemented
220	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
221	C₂²×C₄	C₂²×D₂₂₇	1	2	yes	—
222	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
223	C₄⋊C₄	—	908	2	no	—
224	C₄⋊C₄	—	454	2	no	—
225	C₄⋊C₄	—	227	4	no	—
226	C₄⋊C₄	C₂×Dic₂₂₇	1	4	yes	complemented
227	C₂²⋊C₄	—	908	2	no	—
228	C₂²⋊C₄	—	454	2	no	—
229	C₂²⋊C₄	—	454	4	no	—
230	C₂²⋊C₄	—	454	2	no	—
231	C₂²⋊C₄	—	454	4	no	—
232	C₂²⋊C₄	—	454	4	no	—
233	C₂²⋊C₄	C₂×Dic₂₂₇	1	8	yes	complemented
234	C₄²	—	908	2	no	—
235	C₄²	—	454	4	no	—
236	C₄²	—	454	2	no	—
237	C₄²	—	454	2	no	—
238	C₂³	—	2	1	no	—
239	C₂³	—	2	2	no	—
240	C₂³	—	2	1	no	—
241	C₂³	—	2	1	no	—
242	C₂³	—	2	2	no	—
243	C₂³	—	2	2	no	—
244	C₂³	—	2	1	no	—
245	C₂³	—	2	2	no	—
246	C₂³	—	2	2	no	—
247	C₂³	—	2	2	no	—
248	C₂³	—	2	1	no	—
249	C₂³	—	2	1	no	—
250	C₂³	C₂²×Dic₂₂₇	1	4	yes	—
251	C₂³	C₂²×Dic₂₂₇	1	2	yes	—
252	C₂³	C₂³×D₂₂₇	1	1	yes	characteristic
253	D₄	—	2	8	no	—
254	D₄	—	4	8	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	8	no	—
260	C₂×C₄	—	908	8	no	—
261	C₂×C₄	—	454	1	no	—
262	C₂×C₄	—	454	2	no	—
263	C₂×C₄	—	454	1	no	—
264	C₂×C₄	—	454	2	no	—
265	C₂×C₄	—	454	1	no	—
266	C₂×C₄	—	454	1	no	—
267	C₂×C₄	—	454	1	no	—
268	C₂×C₄	—	454	2	no	—
269	C₂×C₄	—	454	1	no	—
270	C₂×C₄	—	454	1	no	—
271	C₂×C₄	—	454	2	no	—
272	C₂×C₄	—	454	1	no	—
273	C₂×C₄	—	454	1	no	—
274	C₂×C₄	—	454	2	no	—
275	C₂×C₄	—	454	1	no	—
276	C₂×C₄	—	227	4	no	—
277	C₂×C₄	—	227	2	no	—
278	C₂×C₄	—	227	2	no	—
279	C₂×C₄	C₂²×Dic₂₂₇	1	2	yes	—
280	C₂×C₄	C₂²×Dic₂₂₇	1	4	yes	—
281	C₂×C₄	C₂²×Dic₂₂₇	1	2	yes	—
282	C₂×C₄	D₄×D₂₂₇	1	2	yes	—
283	C₂×C₄	#16813883	1	2	yes	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	4	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	4	no	—
290	C₂²	—	4	2	no	—
291	C₂²	—	4	1	no	—
292	C₂²	—	4	1	no	—
293	C₂²	—	4	2	no	—
294	C₂²	—	4	1	no	—
295	C₂²	—	4	2	no	—
296	C₂²	—	4	1	no	—
297	C₂²	—	4	2	no	—
298	C₂²	C₂³×Dic₂₂₇	1	1	yes	characteristic
299	C₂²	C₂×D₄×D₂₂₇	1	1	yes	characteristic
300	C₂²	#41×D₂₂₇	1	1	yes	characteristic
301	C₂²	#41×D₂₂₇	1	1	yes	characteristic
302	C₂²	C₂×#16813883	1	1	yes	characteristic
303	C₂²	#20300333	1	1	yes	characteristic
304	C₂²	#20300333	1	1	yes	characteristic
305	C₄	—	2	2	no	—
306	C₄	—	2	4	no	—
307	C₄	—	908	1	no	—
308	C₄	—	908	2	no	—
309	C₄	—	908	1	no	—
310	C₄	—	454	2	no	—
311	C₄	—	454	4	no	—
312	C₄	—	454	2	no	—
313	C₄	D₄×Dic₂₂₇	1	4	yes	—
314	C₂	—	4	2	no	—
315	C₂	—	4	4	no	—
316	C₂	C₂×D₄×Dic₂₂₇	1	1	yes	characteristic
317	C₂	#41×Dic₂₂₇	1	1	yes	characteristic
318	C₂	#28409507	1	1	yes	characteristic
319	C₂	#28409679	1	1	yes	characteristic
320	C₂	#28409700	1	1	yes	characteristic
321	C₂	#28409729	1	1	yes	characteristic
322	C₂	#28409746	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₂₇
commutator	C₂²×C₂₂₇	C₂³×C₄
Fitting, prime radical	C₂×#121×C₂₂₇	C₂
socle, cosocle	C₂³×C₂₂₇	C₂⁴
hypercenter	C₂×#121	D₂₂₇
hypocenter	C₂₂₇	#2363
LCS	G ▷ C₂²×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×#121

Complements

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

N	H ≅ G/N	type
1. C₂×D₄×Dic₂₂₇	C₂	semidirect
2. C₂×#20300247	C₂	semidirect
3. #28408935	C₂	semidirect
4. #28408935	C₂	semidirect
5. #28408931	C₂	semidirect
6. #28408925	C₂	semidirect
7. #28408912	C₂	semidirect
8. C₄⋊C₄×Dic₂₂₇	C₂	semidirect
9. C₂²⋊C₄×Dic₂₂₇	C₂	semidirect
10. C₄×C₂²⋊Dic₂₂₇	C₂	semidirect
35. #121×C₂₂₇	C₄	semidirect
49. C₄⋊Dic₂₂₇	D₄	semidirect
67. C₂×D₄×C₂₂₇	C₂×C₄	semidirect
70. C₂²×C₄×C₂₂₇	C₂×C₄	semidirect
74. C₄⋊C₄×C₂₂₇	C₂×C₄	semidirect
75. C₂²⋊C₄×C₂₂₇	C₂×C₄	semidirect
161. C₂₂₇	#2363	semidirect
178. #121	Dic₂₂₇	semidirect
207. C₂×D₄	C₂×Dic₂₂₇	semidirect
219. C₂²×C₄	C₂×Dic₂₂₇	semidirect
226. C₄⋊C₄	C₂×Dic₂₂₇	semidirect
233. C₂²⋊C₄	C₂×Dic₂₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×Dic₂₂₇	2
C₂×#20300247	2
#28408935	2
#28408931	2
#28408925	2
#28408912	2
C₄⋊C₄×Dic₂₂₇	2
C₂²⋊C₄×Dic₂₂₇	2
C₄×C₂²⋊Dic₂₂₇	2
C₂×#121×C₂₂₇	2
#2363	227