Subgroups — #47985842

Order 35456.

556 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22579857	C₂	1	1	yes	maximal, characteristic
2	C₂×#22579845	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#22579836	C₂	1	1	yes	maximal, characteristic
4	#31799906	C₂	1	1	yes	maximal, characteristic, complemented
5	#31799900	C₂	1	1	yes	maximal, characteristic, complemented
6	#31799899	C₂	1	1	yes	maximal, characteristic, complemented
7	#31799897	C₂	1	1	yes	maximal, characteristic, complemented
8	#31799896	C₂	1	1	yes	maximal, characteristic
9	#31799896	C₂	1	1	yes	maximal, characteristic
10	#31799895	C₂	1	1	yes	maximal, characteristic
11	#31799885	C₂	1	1	yes	maximal, characteristic, complemented
12	#31799884	C₂	1	1	yes	maximal, characteristic, complemented
13	#31799860	C₂	1	1	yes	maximal, characteristic
14	C₄×#17984542	C₂	1	1	yes	maximal, characteristic, complemented
15	#388×C₂₇₇	C₂	1	1	yes	maximal, characteristic
16	C₂²×Dic₅₅₄	C₂²	1	1	yes	characteristic
17	C₂³×Dic₂₇₇	C₂²	1	1	yes	characteristic
18	#22579857	C₄	1	8	yes	complemented
19	#22579845	—	2	4	no	—
20	#22579836	—	2	4	no	—
21	C₂×C₄⋊Dic₂₇₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₂₇₇	C₂²	1	1	yes	characteristic
23	C₂×#17984542	C₂²	1	1	yes	characteristic
24	C₂×#17984542	C₂²	1	1	yes	characteristic
25	C₂×#17984542	C₂²	1	1	yes	characteristic
26	C₂×#17984542	C₂²	1	1	yes	characteristic
27	C₂×#17984542	C₂²	1	1	yes	characteristic
28	C₂×#17984542	C₂²	1	1	yes	characteristic
29	C₂×#17984542	C₂²	1	1	yes	characteristic
30	C₂×#17984542	C₂²	1	1	yes	characteristic
31	C₂×C₂²⋊Dic₂₇₇	C₂²	1	1	yes	characteristic
32	C₂×C₂²⋊Dic₂₇₇	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊Dic₂₇₇	C₂²	1	1	yes	characteristic
34	C₂×C₄×Dic₂₇₇	C₂²	1	1	yes	characteristic
35	C₂×C₄×Dic₂₇₇	C₂²	1	1	yes	characteristic
36	C₂×C₄×Dic₂₇₇	C₂²	1	1	yes	characteristic
37	C₂×C₄×Dic₂₇₇	C₂²	1	1	yes	characteristic
38	C₂×C₄×Dic₂₇₇	C₂²	1	1	yes	characteristic
39	#22579787	C₂²	1	1	yes	characteristic
40	#22579787	C₂²	1	1	yes	characteristic
41	#22579787	C₂²	1	1	yes	characteristic
42	#22579787	C₂²	1	1	yes	characteristic
43	#22579787	C₂²	1	1	yes	characteristic
44	#22579787	C₂²	1	1	yes	characteristic
45	#22579787	C₂²	1	1	yes	characteristic
46	#22579787	C₂²	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₂₇₇	C₂²	1	1	yes	characteristic
48	C₂×C₂²⋊C₄×C₂₇₇	C₂²	1	1	yes	characteristic
49	C₂×C₂²⋊C₄×C₂₇₇	C₂²	1	1	yes	characteristic
50	C₂×C₂²⋊C₄×C₂₇₇	C₂²	1	1	yes	characteristic
51	C₂×C₄²×C₂₇₇	C₂²	1	1	yes	characteristic
52	#95×C₂₇₇	C₂²	1	1	yes	characteristic
53	#95×C₂₇₇	C₂²	1	1	yes	characteristic
54	C₂×Dic₅₅₄	—	2	2	no	—
55	C₂×Dic₅₅₄	—	2	2	no	—
56	C₂×Dic₅₅₄	C₂×C₄	1	4	yes	—
57	C₂²×Dic₂₇₇	—	2	2	no	—
58	C₂²×Dic₂₇₇	—	2	2	no	—
59	C₂²×Dic₂₇₇	C₂×C₄	1	4	yes	—
60	C₂²×Dic₂₇₇	C₂³	1	1	yes	characteristic
61	C₂²×Dic₂₇₇	C₂³	1	1	yes	characteristic
62	C₂²×Dic₂₇₇	C₂³	1	1	yes	characteristic
63	C₂²×Dic₂₇₇	C₂³	1	1	yes	characteristic
64	C₂²×Dic₂₇₇	C₂³	1	1	yes	characteristic
65	C₂²×Dic₂₇₇	C₂³	1	1	yes	characteristic
66	C₂²×Dic₂₇₇	C₂³	1	1	yes	characteristic
67	C₂²×Dic₂₇₇	C₂³	1	1	yes	characteristic
68	C₄⋊Dic₂₇₇	—	4	1	no	—
69	C₄⋊Dic₂₇₇	C₂×C₄	1	4	yes	—
70	#17984542	—	2	2	no	—
71	#17984542	—	2	2	no	—
72	#17984542	—	2	2	no	—
73	#17984542	—	2	2	no	—
74	#17984542	—	2	2	no	—
75	#17984542	—	4	1	no	—
76	#17984542	C₂×C₄	1	4	yes	—
77	#17984542	C₂×C₄	1	4	yes	—
78	C₂²⋊Dic₂₇₇	—	2	2	no	—
79	C₂²⋊Dic₂₇₇	—	4	1	no	—
80	C₂²⋊Dic₂₇₇	C₂×C₄	1	4	yes	—
81	C₄×Dic₂₇₇	—	2	2	no	—
82	C₄×Dic₂₇₇	—	2	2	no	—
83	C₄×Dic₂₇₇	—	2	2	no	—
84	C₄×Dic₂₇₇	—	4	1	no	—
85	C₄×Dic₂₇₇	D₄	1	4	yes	—
86	C₂⁴×C₂₇₇	C₂³	1	1	yes	characteristic
87	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
88	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
89	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
90	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
91	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
92	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
93	C₄⋊C₄×C₂₇₇	—	2	2	no	—
94	C₂²⋊C₄×C₂₇₇	—	2	2	no	—
95	C₂²⋊C₄×C₂₇₇	—	4	1	no	—
96	C₂²⋊C₄×C₂₇₇	C₂×C₄	1	4	yes	—
97	C₄²×C₂₇₇	—	4	1	no	—
98	Dic₅₅₄	—	2	8	no	—
99	C₂×Dic₂₇₇	—	2	1	no	—
100	C₂×Dic₂₇₇	—	2	1	no	—
101	C₂×Dic₂₇₇	—	2	1	no	—
102	C₂×Dic₂₇₇	—	2	1	no	—
103	C₂×Dic₂₇₇	—	2	1	no	—
104	C₂×Dic₂₇₇	—	2	1	no	—
105	C₂×Dic₂₇₇	—	2	1	no	—
106	C₂×Dic₂₇₇	—	2	1	no	—
107	C₂×Dic₂₇₇	—	2	8	no	—
108	C₂×Dic₂₇₇	—	2	1	no	—
109	C₂×Dic₂₇₇	—	2	1	no	—
110	C₂×Dic₂₇₇	—	2	1	no	—
111	C₂×Dic₂₇₇	—	2	1	no	—
112	C₂×Dic₂₇₇	—	2	1	no	—
113	C₂×Dic₂₇₇	—	2	1	no	—
114	C₂×Dic₂₇₇	—	2	1	no	—
115	C₂×Dic₂₇₇	—	2	1	no	—
116	C₂×Dic₂₇₇	—	2	1	no	—
117	C₂×Dic₂₇₇	C₂²×C₄	1	2	yes	—
118	C₂×Dic₂₇₇	C₂²×C₄	1	2	yes	—
119	C₂×Dic₂₇₇	C₂²×C₄	1	2	yes	—
120	C₂×Dic₂₇₇	C₂²×C₄	1	2	yes	—
121	C₂×Dic₂₇₇	C₂×D₄	1	2	yes	—
122	C₂×Dic₂₇₇	C₂×D₄	1	2	yes	—
123	C₂×Dic₂₇₇	#41	1	2	yes	—
124	C₂³×C₂₇₇	—	2	1	no	—
125	C₂³×C₂₇₇	—	2	1	no	—
126	C₂³×C₂₇₇	—	2	1	no	—
127	C₂³×C₂₇₇	—	2	1	no	—
128	C₂³×C₂₇₇	—	2	1	no	—
129	C₂³×C₂₇₇	—	2	1	no	—
130	C₂³×C₂₇₇	C₂²×C₄	1	2	yes	—
131	C₂³×C₂₇₇	C₂⁴	1	1	yes	characteristic
132	C₂×C₄×C₂₇₇	—	2	1	no	—
133	C₂×C₄×C₂₇₇	—	2	1	no	—
134	C₂×C₄×C₂₇₇	—	2	1	no	—
135	C₂×C₄×C₂₇₇	—	2	1	no	—
136	C₂×C₄×C₂₇₇	—	2	1	no	—
137	C₂×C₄×C₂₇₇	—	2	1	no	—
138	C₂×C₄×C₂₇₇	—	2	1	no	—
139	C₂×C₄×C₂₇₇	—	2	1	no	—
140	C₂×C₄×C₂₇₇	—	2	1	no	—
141	C₂×C₄×C₂₇₇	—	2	1	no	—
142	C₂×C₄×C₂₇₇	—	2	1	no	—
143	C₂×C₄×C₂₇₇	—	2	1	no	—
144	C₂×C₄×C₂₇₇	—	2	1	no	—
145	C₂×C₄×C₂₇₇	—	2	1	no	—
146	C₂×C₄×C₂₇₇	—	2	1	no	—
147	C₂×C₄×C₂₇₇	C₂²×C₄	1	2	yes	—
148	C₂×C₄×C₂₇₇	C₂²×C₄	1	2	yes	—
149	C₂×C₄×C₂₇₇	C₂×D₄	1	2	yes	—
150	C₂²×C₂₇₇	—	2	2	no	—
151	C₂²×C₂₇₇	—	2	2	no	—
152	C₂²×C₂₇₇	—	2	2	no	—
153	C₂²×C₂₇₇	—	4	1	no	—
154	C₂²×C₂₇₇	—	4	1	no	—
155	C₂²×C₂₇₇	—	4	1	no	—
156	C₂²×C₂₇₇	—	4	1	no	—
157	C₂²×C₂₇₇	C₂³×C₄	1	1	yes	characteristic
158	C₂²×C₂₇₇	C₂²×D₄	1	1	yes	characteristic
159	C₂²×C₂₇₇	C₂×#41	1	1	yes	characteristic
160	C₂²×C₂₇₇	H(5,𝔽₂)	1	1	yes	characteristic
161	C₂²×C₂₇₇	2₋¹⁺⁴	1	1	yes	characteristic
162	C₂²×C₂₇₇	2₋¹⁺⁴	1	1	yes	characteristic
163	C₂²×C₂₇₇	2₋¹⁺⁴	1	1	yes	characteristic
164	C₄×C₂₇₇	—	2	2	no	—
165	C₄×C₂₇₇	—	2	2	no	—
166	C₄×C₂₇₇	—	2	2	no	—
167	C₄×C₂₇₇	—	4	1	no	—
168	C₄×C₂₇₇	—	4	1	no	—
169	C₄×C₂₇₇	—	4	1	no	—
170	Dic₂₇₇	—	2	2	no	—
171	Dic₂₇₇	—	2	2	no	—
172	Dic₂₇₇	—	2	2	no	—
173	Dic₂₇₇	—	2	2	no	—
174	Dic₂₇₇	—	4	1	no	—
175	Dic₂₇₇	—	4	1	no	—
176	Dic₂₇₇	—	4	1	no	—
177	Dic₂₇₇	C₄×D₄	1	4	yes	—
178	C₂×C₂₇₇	—	4	2	no	—
179	C₂×C₂₇₇	C₂×C₄×D₄	1	1	yes	characteristic
180	C₂×C₂₇₇	#519	1	1	yes	characteristic
181	C₂×C₂₇₇	#520	1	1	yes	characteristic
182	C₂×C₂₇₇	#536	1	1	yes	characteristic
183	C₂×C₂₇₇	#537	1	1	yes	characteristic
184	C₂×C₂₇₇	#541	1	1	yes	characteristic
185	C₂×C₂₇₇	#542	1	1	yes	characteristic
186	C₂₇₇	#2323	1	1	yes	characteristic, complemented, Sylow
187	#2323	—	277	1	no	maximal, Sylow
188	C₂×C₂²⋊Q₈	—	277	1	no	—
189	C₂×C₄×Q₈	—	277	1	no	—
190	C₂×#117	—	277	1	no	—
191	#391	—	277	1	no	—
192	#389	—	277	1	no	—
193	#389	—	277	1	no	—
194	#389	—	277	1	no	—
195	#388	—	277	1	no	—
196	#388	—	277	1	no	—
197	#388	—	277	1	no	—
198	#388	D₂₇₇	1	1	yes	characteristic
199	#387	—	277	1	no	—
200	#387	—	277	1	no	—
201	#380	—	277	1	no	—
202	C₄×C₄⋊C₄	—	277	1	no	—
203	C₂²×Q₈	—	277	1	no	—
204	C₂³×C₄	—	277	1	no	—
205	C₂²⋊Q₈	—	277	8	no	—
206	C₄×Q₈	—	554	4	no	—
207	#117	—	554	4	no	—
208	C₂×C₄⋊C₄	—	277	1	no	—
209	C₂×C₄⋊C₄	—	277	1	no	—
210	C₂×C₄⋊C₄	—	277	1	no	—
211	C₂×C₄⋊C₄	—	277	1	no	—
212	C₂×C₄⋊C₄	—	277	1	no	—
213	C₂×C₄⋊C₄	—	277	1	no	—
214	C₂×C₄⋊C₄	—	277	1	no	—
215	C₂×C₄⋊C₄	—	277	1	no	—
216	C₂×C₄⋊C₄	—	277	1	no	—
217	C₂×C₄⋊C₄	—	277	1	no	—
218	C₂×C₄⋊C₄	C₂×D₂₇₇	1	1	yes	characteristic
219	C₂×C₂²⋊C₄	—	277	1	no	—
220	C₂×C₂²⋊C₄	—	277	1	no	—
221	C₂×C₂²⋊C₄	—	277	1	no	—
222	C₂×C₂²⋊C₄	C₂×D₂₇₇	1	1	yes	characteristic
223	C₂×C₂²⋊C₄	C₂×D₂₇₇	1	1	yes	characteristic
224	C₂×C₂²⋊C₄	C₂×D₂₇₇	1	1	yes	characteristic
225	C₂×C₄²	—	277	1	no	—
226	C₂×C₄²	—	277	1	no	—
227	C₂×C₄²	—	277	1	no	—
228	C₂×C₄²	—	277	1	no	—
229	C₂×C₄²	—	277	1	no	—
230	C₂×C₄²	C₂×D₂₇₇	1	1	yes	characteristic
231	#95	—	277	1	no	—
232	#95	—	277	1	no	—
233	#95	—	277	1	no	—
234	#95	—	277	1	no	—
235	#95	—	277	1	no	—
236	#95	—	277	1	no	—
237	#95	—	277	1	no	—
238	#95	—	277	1	no	—
239	#95	C₂×D₂₇₇	1	1	yes	characteristic
240	#95	C₂×D₂₇₇	1	1	yes	characteristic
241	C₂⁴	C₂²×D₂₇₇	1	1	yes	characteristic
242	C₂×Q₈	—	277	4	no	—
243	C₂×Q₈	—	554	2	no	—
244	C₂×Q₈	—	554	2	no	—
245	C₂²×C₄	—	277	1	no	—
246	C₂²×C₄	—	277	4	no	—
247	C₂²×C₄	—	277	1	no	—
248	C₂²×C₄	—	277	1	no	—
249	C₂²×C₄	—	277	1	no	—
250	C₂²×C₄	—	277	1	no	—
251	C₂²×C₄	—	277	1	no	—
252	C₂²×C₄	—	277	1	no	—
253	C₂²×C₄	—	277	1	no	—
254	C₂²×C₄	—	554	2	no	—
255	C₂²×C₄	—	554	2	no	—
256	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
257	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
258	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
259	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
260	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
261	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	277	4	no	—
264	C₄⋊C₄	—	277	4	no	—
265	C₄⋊C₄	—	277	4	no	—
266	C₄⋊C₄	—	554	2	no	—
267	C₄⋊C₄	—	554	2	no	—
268	C₄⋊C₄	—	554	2	no	—
269	C₄⋊C₄	—	554	2	no	—
270	C₄⋊C₄	—	554	2	no	—
271	C₄⋊C₄	—	1108	1	no	—
272	C₄⋊C₄	—	1108	1	no	—
273	C₂²⋊C₄	—	2	2	no	—
274	C₂²⋊C₄	—	4	1	no	—
275	C₂²⋊C₄	—	277	4	no	—
276	C₂²⋊C₄	—	554	2	no	—
277	C₂²⋊C₄	—	1108	1	no	—
278	C₂²⋊C₄	C₄×D₂₇₇	1	4	yes	—
279	C₄²	—	4	1	no	—
280	C₄²	—	277	4	no	—
281	C₄²	—	554	2	no	—
282	C₄²	—	554	2	no	—
283	C₄²	—	554	2	no	—
284	C₄²	—	1108	1	no	—
285	C₂³	—	2	1	no	—
286	C₂³	—	2	1	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	—	2	1	no	—
290	C₂³	—	2	1	no	—
291	C₂³	C₂×C₄×D₂₇₇	1	2	yes	—
292	C₂³	C₂³×D₂₇₇	1	1	yes	characteristic
293	Q₈	—	554	8	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	277	2	no	—
310	C₂×C₄	—	277	2	no	—
311	C₂×C₄	—	277	2	no	—
312	C₂×C₄	—	277	2	no	—
313	C₂×C₄	—	277	2	no	—
314	C₂×C₄	—	277	2	no	—
315	C₂×C₄	—	277	2	no	—
316	C₂×C₄	—	554	8	no	—
317	C₂×C₄	—	554	1	no	—
318	C₂×C₄	—	554	1	no	—
319	C₂×C₄	—	554	1	no	—
320	C₂×C₄	—	554	1	no	—
321	C₂×C₄	—	554	1	no	—
322	C₂×C₄	—	554	1	no	—
323	C₂×C₄	—	554	1	no	—
324	C₂×C₄	—	554	1	no	—
325	C₂×C₄	—	554	1	no	—
326	C₂×C₄	—	554	1	no	—
327	C₂×C₄	—	554	1	no	—
328	C₂×C₄	—	554	1	no	—
329	C₂×C₄	—	554	1	no	—
330	C₂×C₄	—	554	1	no	—
331	C₂×C₄	—	554	1	no	—
332	C₂×C₄	—	554	1	no	—
333	C₂×C₄	—	554	1	no	—
334	C₂×C₄	C₂×C₄×D₂₇₇	1	2	yes	—
335	C₂×C₄	C₂×C₄×D₂₇₇	1	2	yes	—
336	C₂×C₄	D₄×D₂₇₇	1	2	yes	—
337	C₂²	—	2	2	no	—
338	C₂²	—	2	2	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	4	1	no	—
342	C₂²	—	4	1	no	—
343	C₂²	—	4	1	no	—
344	C₂²	C₂²×C₄×D₂₇₇	1	1	yes	characteristic
345	C₂²	C₂×D₄×D₂₇₇	1	1	yes	characteristic
346	C₂²	#41×D₂₇₇	1	1	yes	characteristic
347	C₂²	#22579925	1	1	yes	characteristic
348	C₂²	#22579927	1	1	yes	characteristic
349	C₂²	#22579927	1	1	yes	characteristic
350	C₂²	#22579927	1	1	yes	characteristic
351	C₄	—	2	2	no	—
352	C₄	—	2	2	no	—
353	C₄	—	2	2	no	—
354	C₄	—	4	1	no	—
355	C₄	—	4	1	no	—
356	C₄	—	4	1	no	—
357	C₄	—	277	4	no	—
358	C₄	—	554	2	no	—
359	C₄	—	554	2	no	—
360	C₄	—	554	2	no	—
361	C₄	—	554	2	no	—
362	C₄	—	1108	1	no	—
363	C₄	—	1108	1	no	—
364	C₄	—	1108	1	no	—
365	C₂	—	4	2	no	—
366	C₂	C₄×D₄×D₂₇₇	1	1	yes	characteristic
367	C₂	#31800479	1	1	yes	characteristic
368	C₂	#31800481	1	1	yes	characteristic
369	C₂	#31800572	1	1	yes	characteristic
370	C₂	#31800579	1	1	yes	characteristic
371	C₂	#31800609	1	1	yes	characteristic
372	C₂	#31800620	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₇₇
commutator	C₂²×C₂₇₇	C₂³×C₄
Fitting	#388×C₂₇₇	C₂
socle, cosocle	C₂³×C₂₇₇	C₂⁴
prime radical	C₂⁴×C₂₇₇	C₂³
hypercenter	#388	D₂₇₇
hypocenter	C₂₇₇	#2323
LCS	G ▷ C₂²×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂³ ◁ #388

Complements

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

N	H ≅ G/N	type
2. C₂×#22579845	C₂	semidirect
4. #31799906	C₂	semidirect
5. #31799900	C₂	semidirect
6. #31799899	C₂	semidirect
7. #31799897	C₂	semidirect
11. #31799885	C₂	semidirect
12. #31799884	C₂	semidirect
14. C₄×#17984542	C₂	semidirect
18. #22579857	C₄	semidirect
186. C₂₇₇	#2323	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22579857	2
C₂×#22579845	2
C₂×#22579836	2
#31799906	2
#31799900	2
#31799899	2
#31799897	2
#31799896	2
#31799895	2
#31799885	2
#31799884	2
#31799860	2
C₄×#17984542	2
#388×C₂₇₇	2
#2323	277