Subgroups — #47986041

Order 35456.

548 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₄×D₂₇₇	C₂	1	1	yes	maximal, characteristic
2	#31799899	C₂	1	1	yes	maximal, characteristic, complemented
3	#31799897	C₂	1	1	yes	maximal, characteristic, complemented
4	#31799888	C₂	1	1	yes	maximal, characteristic, complemented
5	#31799887	C₂	1	1	yes	maximal, characteristic, complemented
6	#31799885	C₂	1	1	yes	maximal, characteristic, complemented
7	#31799861	C₂	1	1	yes	maximal, characteristic
8	#31799861	C₂	1	1	yes	maximal, characteristic
9	#31799859	C₂	1	1	yes	maximal, characteristic
10	C₄×#17984542	C₂	1	1	yes	maximal, characteristic, complemented
11	#31799853	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#17984540	C₂	1	1	yes	maximal, characteristic
13	#31799849	C₂	1	1	yes	maximal, characteristic
14	#31799849	C₂	1	1	yes	maximal, characteristic
15	#387×C₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×C₄×D₂₇₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₂₇₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄×D₂₇₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₂₇₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₂₇₇	C₂²	1	1	yes	characteristic
21	C₂×#17984542	C₂²	1	1	yes	characteristic
22	C₂×#17984542	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	C₄⋊C₄×D₂₇₇	C₄	1	8	yes	complemented
28	C₂×#17984540	C₂²	1	1	yes	characteristic
29	C₂×#17984540	C₂²	1	1	yes	characteristic
30	C₂×#17984540	C₂²	1	1	yes	characteristic
31	C₂×#17984540	C₂²	1	1	yes	characteristic
32	C₂×C₄×Dic₂₇₇	C₂²	1	1	yes	characteristic
33	C₂×C₄×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	#22579787	C₂²	1	1	yes	characteristic
38	#22579787	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	C₂×C₄⋊C₄×C₂₇₇	C₂²	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₂₇₇	C₂²	1	1	yes	characteristic
47	C₂×C₄²×C₂₇₇	C₂²	1	1	yes	characteristic
48	#95×C₂₇₇	C₂²	1	1	yes	characteristic
49	#95×C₂₇₇	C₂²	1	1	yes	characteristic
50	#95×C₂₇₇	C₂²	1	1	yes	characteristic
51	#95×C₂₇₇	C₂²	1	1	yes	characteristic
52	C₂³×D₂₇₇	C₂³	1	1	yes	characteristic
53	C₂²×Dic₂₇₇	C₂³	1	1	yes	characteristic
54	C₂²×Dic₂₇₇	C₂³	1	1	yes	characteristic
55	C₂²×Dic₂₇₇	C₂³	1	1	yes	characteristic
56	C₂²×Dic₂₇₇	C₂³	1	1	yes	characteristic
57	C₂²×Dic₂₇₇	C₂³	1	1	yes	characteristic
58	C₂²×Dic₂₇₇	C₂³	1	1	yes	characteristic
59	C₂²×Dic₂₇₇	C₂³	1	1	yes	characteristic
60	C₂×C₄×D₂₇₇	—	2	2	no	—
61	C₂×C₄×D₂₇₇	—	2	2	no	—
62	C₂×C₄×D₂₇₇	—	4	1	no	—
63	C₂×C₄×D₂₇₇	—	4	1	no	—
64	C₂×C₄×D₂₇₇	—	4	1	no	—
65	C₂×C₄×D₂₇₇	—	4	1	no	—
66	C₂×C₄×D₂₇₇	C₂×C₄	1	4	yes	—
67	C₂×C₄×D₂₇₇	C₂×C₄	1	4	yes	—
68	C₂×C₄×D₂₇₇	C₂×C₄	1	4	yes	—
69	C₄⋊Dic₂₇₇	—	4	1	no	—
70	C₄⋊Dic₂₇₇	C₂×C₄	1	4	yes	—
71	#17984542	—	2	2	no	—
72	#17984542	—	2	2	no	—
73	#17984542	—	4	1	no	—
74	#17984542	—	4	1	no	—
75	#17984542	C₂×C₄	1	4	yes	—
76	#17984542	C₂×C₄	1	4	yes	—
77	#17984540	—	2	2	no	—
78	#17984540	—	2	2	no	—
79	#17984540	—	2	2	no	—
80	#17984540	Q₈	1	4	yes	—
81	C₄×Dic₂₇₇	—	2	2	no	—
82	C₄×Dic₂₇₇	—	2	2	no	—
83	C₄×Dic₂₇₇	—	4	1	no	—
84	C₄×Dic₂₇₇	—	4	1	no	—
85	C₄×Dic₂₇₇	—	4	1	no	—
86	C₂²×C₄×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	—	4	1	no	—
94	C₄⋊C₄×C₂₇₇	C₂×C₄	1	4	yes	—
95	C₄²×C₂₇₇	—	4	1	no	—
96	C₂²×D₂₇₇	—	2	1	no	—
97	C₂²×D₂₇₇	—	2	1	no	—
98	C₂²×D₂₇₇	—	2	1	no	—
99	C₂²×D₂₇₇	—	2	1	no	—
100	C₂²×D₂₇₇	C₂²×C₄	1	2	yes	—
101	C₂²×D₂₇₇	C₂×Q₈	1	2	yes	—
102	C₂²×D₂₇₇	#41	1	2	yes	—
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	1	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	—	2	1	no	—
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×Q₈	1	2	yes	—
122	C₂×Dic₂₇₇	#41	1	2	yes	—
123	C₂×Dic₂₇₇	#41	1	2	yes	—
124	C₄×D₂₇₇	—	2	8	no	—
125	C₄×D₂₇₇	—	4	4	no	—
126	C₄×D₂₇₇	—	4	4	no	—
127	C₂³×C₂₇₇	C₂⁴	1	1	yes	characteristic
128	C₂×C₄×C₂₇₇	—	2	1	no	—
129	C₂×C₄×C₂₇₇	—	2	1	no	—
130	C₂×C₄×C₂₇₇	—	2	1	no	—
131	C₂×C₄×C₂₇₇	—	2	1	no	—
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	C₂²×C₄	1	2	yes	—
146	C₂×C₄×C₂₇₇	C₂²×C₄	1	2	yes	—
147	C₂×C₄×C₂₇₇	C₂²×C₄	1	2	yes	—
148	C₂×C₄×C₂₇₇	C₂×Q₈	1	2	yes	—
149	C₂²×C₂₇₇	C₂³×C₄	1	1	yes	characteristic
150	C₂²×C₂₇₇	C₂²×Q₈	1	1	yes	characteristic
151	C₂²×C₂₇₇	C₂×#41	1	1	yes	characteristic
152	C₂²×C₂₇₇	C₂×#41	1	1	yes	characteristic
153	C₂²×C₂₇₇	C₂×#41	1	1	yes	characteristic
154	C₂²×C₂₇₇	H(5,𝔽₂)	1	1	yes	characteristic
155	C₂²×C₂₇₇	H(5,𝔽₂)	1	1	yes	characteristic
156	C₂×D₂₇₇	—	2	2	no	—
157	C₂×D₂₇₇	—	2	2	no	—
158	C₂×D₂₇₇	—	2	2	no	—
159	C₂×D₂₇₇	—	2	2	no	—
160	C₂×D₂₇₇	—	2	2	no	—
161	C₂×D₂₇₇	—	2	2	no	—
162	C₂×D₂₇₇	C₄×Q₈	1	4	yes	—
163	C₄×C₂₇₇	—	2	2	no	—
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	—	4	1	no	—
174	Dic₂₇₇	—	4	1	no	—
175	Dic₂₇₇	—	4	1	no	—
176	Dic₂₇₇	#117	1	4	yes	—
177	C₂×C₂₇₇	C₂×#117	1	1	yes	characteristic
178	C₂×C₂₇₇	C₂×C₄×Q₈	1	1	yes	characteristic
179	C₂×C₂₇₇	#518	1	1	yes	characteristic
180	C₂×C₂₇₇	#553	1	1	yes	characteristic
181	C₂×C₂₇₇	#553	1	1	yes	characteristic
182	C₂×C₂₇₇	#554	1	1	yes	characteristic
183	C₂×C₂₇₇	#554	1	1	yes	characteristic
184	D₂₇₇	—	2	4	no	—
185	C₂₇₇	#2345	1	1	yes	characteristic, complemented, Sylow
186	#2345	—	277	1	no	maximal, Sylow
187	C₂²×C₄⋊C₄	—	277	1	no	—
188	#389	—	277	1	no	—
189	#389	—	277	1	no	—
190	#387	—	277	1	no	—
191	#387	—	277	1	no	—
192	#387	—	277	1	no	—
193	#387	D₂₇₇	1	1	yes	characteristic, complemented
194	#380	—	277	1	no	—
195	#380	—	277	1	no	—
196	#380	—	277	1	no	—
197	C₄×C₄⋊C₄	—	277	1	no	—
198	C₄×C₄⋊C₄	—	277	1	no	—
199	C₄×C₂²⋊C₄	—	277	1	no	—
200	C₄×C₂²⋊C₄	—	277	1	no	—
201	C₄×C₂²⋊C₄	—	277	1	no	—
202	C₂³×C₄	—	277	1	no	—
203	C₂³×C₄	—	277	1	no	—
204	C₂³×C₄	—	277	1	no	—
205	C₂×C₄⋊C₄	—	277	1	no	—
206	C₂×C₄⋊C₄	—	277	1	no	—
207	C₂×C₄⋊C₄	—	277	1	no	—
208	C₂×C₄⋊C₄	—	277	8	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	C₂×D₂₇₇	1	1	yes	characteristic
215	C₂×C₄⋊C₄	C₂×D₂₇₇	1	1	yes	characteristic
216	C₂×C₂²⋊C₄	—	277	1	no	—
217	C₂×C₂²⋊C₄	—	277	1	no	—
218	C₂×C₂²⋊C₄	—	277	1	no	—
219	C₂×C₂²⋊C₄	—	277	1	no	—
220	C₂×C₄²	—	277	1	no	—
221	C₂×C₄²	—	277	1	no	—
222	C₂×C₄²	—	277	1	no	—
223	C₂×C₄²	—	277	1	no	—
224	C₂×C₄²	—	277	1	no	—
225	C₂×C₄²	C₂×D₂₇₇	1	1	yes	characteristic
226	#95	—	277	1	no	—
227	#95	—	277	1	no	—
228	#95	—	277	1	no	—
229	#95	—	277	1	no	—
230	#95	—	277	1	no	—
231	#95	—	277	1	no	—
232	#95	—	277	1	no	—
233	#95	—	277	1	no	—
234	#95	C₂×D₂₇₇	1	1	yes	characteristic
235	#95	C₂×D₂₇₇	1	1	yes	characteristic
236	#95	C₂×D₂₇₇	1	1	yes	characteristic
237	#95	C₂×D₂₇₇	1	1	yes	characteristic
238	C₂⁴	—	277	1	no	—
239	C₂²×C₄	—	277	1	no	—
240	C₂²×C₄	—	277	1	no	—
241	C₂²×C₄	—	277	1	no	—
242	C₂²×C₄	—	277	1	no	—
243	C₂²×C₄	—	277	4	no	—
244	C₂²×C₄	—	277	4	no	—
245	C₂²×C₄	—	277	4	no	—
246	C₂²×C₄	—	277	1	no	—
247	C₂²×C₄	—	277	1	no	—
248	C₂²×C₄	—	277	1	no	—
249	C₂²×C₄	—	554	2	no	—
250	C₂²×C₄	—	554	2	no	—
251	C₂²×C₄	—	1108	1	no	—
252	C₂²×C₄	—	1108	1	no	—
253	C₂²×C₄	—	1108	1	no	—
254	C₂²×C₄	—	1108	1	no	—
255	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
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	—	4	1	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	—	1108	1	no	—
269	C₄⋊C₄	—	1108	1	no	—
270	C₄⋊C₄	—	1108	1	no	—
271	C₄⋊C₄	C₄×D₂₇₇	1	4	yes	—
272	C₂²⋊C₄	—	277	4	no	—
273	C₂²⋊C₄	—	554	2	no	—
274	C₂²⋊C₄	—	554	2	no	—
275	C₂²⋊C₄	—	554	2	no	—
276	C₄²	—	4	1	no	—
277	C₄²	—	554	2	no	—
278	C₄²	—	554	2	no	—
279	C₄²	—	1108	1	no	—
280	C₄²	—	1108	1	no	—
281	C₄²	—	1108	1	no	—
282	C₂³	—	277	2	no	—
283	C₂³	—	277	2	no	—
284	C₂³	—	277	2	no	—
285	C₂³	—	554	1	no	—
286	C₂³	—	554	1	no	—
287	C₂³	—	554	1	no	—
288	C₂³	—	554	1	no	—
289	C₂³	C₂³×D₂₇₇	1	1	yes	characteristic
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	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	—	277	2	no	—
308	C₂×C₄	—	277	2	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	—	554	8	no	—
314	C₂×C₄	—	554	1	no	—
315	C₂×C₄	—	554	1	no	—
316	C₂×C₄	—	554	1	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	—	1108	4	no	—
330	C₂×C₄	—	1108	4	no	—
331	C₂×C₄	C₂×C₄×D₂₇₇	1	2	yes	—
332	C₂×C₄	C₂×C₄×D₂₇₇	1	2	yes	—
333	C₂×C₄	C₂×C₄×D₂₇₇	1	2	yes	—
334	C₂×C₄	Q₈×D₂₇₇	1	2	yes	—
335	C₂²	—	277	4	no	—
336	C₂²	—	554	2	no	—
337	C₂²	—	554	2	no	—
338	C₂²	—	554	2	no	—
339	C₂²	—	554	2	no	—
340	C₂²	—	554	2	no	—
341	C₂²	—	554	2	no	—
342	C₂²	C₂²×C₄×D₂₇₇	1	1	yes	characteristic
343	C₂²	C₂×Q₈×D₂₇₇	1	1	yes	characteristic
344	C₂²	#41×D₂₇₇	1	1	yes	characteristic
345	C₂²	#41×D₂₇₇	1	1	yes	characteristic
346	C₂²	#41×D₂₇₇	1	1	yes	characteristic
347	C₂²	#22579925	1	1	yes	characteristic
348	C₂²	#22579926	1	1	yes	characteristic
349	C₄	—	2	2	no	—
350	C₄	—	2	2	no	—
351	C₄	—	2	2	no	—
352	C₄	—	2	2	no	—
353	C₄	—	4	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	4	1	no	—
356	C₄	—	277	4	no	—
357	C₄	—	554	2	no	—
358	C₄	—	554	2	no	—
359	C₄	—	554	2	no	—
360	C₄	—	1108	1	no	—
361	C₄	—	1108	1	no	—
362	C₄	—	1108	1	no	—
363	C₂	—	554	4	no	—
364	C₂	#117×D₂₇₇	1	1	yes	characteristic
365	C₂	C₄×Q₈×D₂₇₇	1	1	yes	characteristic
366	C₂	#31800474	1	1	yes	characteristic
367	C₂	#31800697	1	1	yes	characteristic
368	C₂	#31800703	1	1	yes	characteristic
369	C₂	#31800706	1	1	yes	characteristic
370	C₂	#31800708	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₇₇
commutator	C₂²×C₂₇₇	C₂³×C₄
Fitting	#387×C₂₇₇	C₂
socle, cosocle	C₂³×C₂₇₇	C₂⁴
prime radical	C₂³×D₂₇₇	C₂³
hypercenter	#387	D₂₇₇
hypocenter	C₂₇₇	#2345
LCS	G ▷ C₂²×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂³ ◁ #387

Complements

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

N	H ≅ G/N	type
2. #31799899	C₂	semidirect
3. #31799897	C₂	semidirect
4. #31799888	C₂	semidirect
5. #31799887	C₂	semidirect
6. #31799885	C₂	semidirect
10. C₄×#17984542	C₂	semidirect
11. #31799853	C₂	semidirect
15. #387×C₂₇₇	C₂	semidirect
27. C₄⋊C₄×D₂₇₇	C₄	semidirect
185. C₂₇₇	#2345	semidirect
193. #387	D₂₇₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊C₄×D₂₇₇	2
#31799899	2
#31799897	2
#31799888	2
#31799887	2
#31799885	2
#31799861	2
#31799859	2
C₄×#17984542	2
#31799853	2
C₄×#17984540	2
#31799849	2
#387×C₂₇₇	2
#2345	277