Subgroups — #30652363

Order 16256.

1900 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861255	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×D₅₀₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#22550522	C₂	1	4	yes	maximal, complemented
4	#22550517	C₂	1	4	yes	maximal, complemented
5	D₄×D₅₀₈	C₂	1	4	yes	maximal, complemented
6	#22550510	C₂	1	4	yes	maximal, complemented
7	#22550445	C₂	1	1	yes	maximal, characteristic, complemented
8	#22550440	C₂	1	1	yes	maximal, characteristic, complemented
9	#22550437	C₂	1	4	yes	maximal, complemented
10	#22550387	C₂	1	1	yes	maximal, characteristic, complemented
11	#120×D₁₂₇	C₂	1	4	yes	maximal, complemented
12	#22550350	C₂	1	1	yes	maximal, characteristic, complemented
13	#525×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂⁴×D₁₂₇	C₂²	1	2	yes	complemented
15	#16861255	—	2	8	no	—
16	C₂×#14523249	C₂²	1	2	yes	complemented
17	C₂×#14523248	C₂²	1	1	yes	characteristic, complemented
18	C₂×#14523248	C₂²	1	2	yes	complemented
19	C₂²×D₅₀₈	—	2	8	no	—
20	C₂²×D₅₀₈	C₂²	1	1	yes	characteristic
21	C₂²×D₅₀₈	C₂²	1	2	yes	complemented
22	C₂²×D₅₀₈	C₂²	1	1	yes	characteristic, complemented
23	C₂²×D₅₀₈	C₂²	1	4	yes	complemented
24	C₂²×D₅₀₈	C₂²	1	4	yes	complemented
25	C₂²×D₅₀₈	C₂²	1	2	yes	complemented
26	C₂×D₄×D₁₂₇	C₂²	1	4	yes	complemented
27	C₂×D₄×D₁₂₇	C₂²	1	8	yes	complemented
28	C₂×D₄×D₁₂₇	C₂²	1	2	yes	complemented
29	#16861206	C₂²	1	2	yes	complemented
30	#16861205	C₂²	1	2	yes	complemented
31	#16861198	C₂²	1	2	yes	complemented
32	#16861197	C₂²	1	2	yes	complemented
33	#16861194	C₂²	1	8	yes	complemented
34	#16861192	C₂²	1	2	yes	complemented
35	#16861188	C₂²	1	2	yes	complemented
36	#16861184	C₂²	1	8	yes	complemented
37	#16861184	C₂²	1	4	yes	complemented
38	#16861183	C₂²	1	2	yes	complemented
39	#16861183	C₂²	1	4	yes	complemented
40	#16861182	C₂²	1	8	yes	complemented
41	#16861179	C₂²	1	4	yes	complemented
42	#16861179	C₂²	1	8	yes	complemented
43	#16861179	C₂²	1	8	yes	complemented
44	#16861178	C₂²	1	2	yes	complemented
45	#16861177	C₂²	1	4	yes	complemented
46	#16861177	C₂²	1	8	yes	complemented
47	C₄×D₅₀₈	C₂²	1	4	yes	complemented
48	#16861171	C₂²	1	4	yes	complemented
49	C₄×C₂²⋊D₁₂₇	C₂²	1	4	yes	complemented
50	#16861168	C₂²	1	1	yes	characteristic, complemented
51	C₂²⋊C₄×D₁₂₇	C₂²	1	4	yes	complemented
52	C₂²⋊C₄×D₁₂₇	C₂²	1	8	yes	complemented
53	C₂×#41×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
54	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
55	#123×C₁₂₇	C₂²	1	2	yes	complemented
56	C₂²⋊Q₈×C₁₂₇	C₂²	1	2	yes	complemented
57	#121×C₁₂₇	C₂²	1	2	yes	complemented
58	#120×C₁₂₇	C₂²	1	2	yes	complemented
59	C₄×D₄×C₁₂₇	C₂²	1	4	yes	complemented
60	#117×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
61	C₂³×D₁₂₇	—	2	8	no	—
62	C₂³×D₁₂₇	—	4	8	no	—
63	C₂³×D₁₂₇	C₂³	1	8	yes	—
64	C₂³×D₁₂₇	C₂³	1	2	yes	—
65	C₂³×D₁₂₇	C₂³	1	2	yes	—
66	C₂³×D₁₂₇	C₂³	1	4	yes	complemented
67	#14523249	—	2	8	no	—
68	#14523248	—	2	8	no	—
69	#14523248	—	2	4	no	—
70	C₂×Dic₂₅₄	C₂³	1	1	yes	characteristic, complemented
71	C₂×D₅₀₈	—	2	8	no	—
72	C₂×D₅₀₈	—	2	16	no	—
73	C₂×D₅₀₈	—	2	16	no	—
74	C₂×D₅₀₈	—	2	8	no	—
75	C₂×D₅₀₈	—	2	4	no	—
76	C₂×D₅₀₈	D₄	1	8	yes	complemented
77	C₂×D₅₀₈	C₂³	1	2	yes	—
78	C₂×D₅₀₈	C₂³	1	8	yes	—
79	C₂×D₅₀₈	C₂³	1	2	yes	—
80	C₂×D₅₀₈	C₂³	1	2	yes	—
81	C₂×D₅₀₈	C₂³	1	2	yes	—
82	C₂×D₅₀₈	C₂³	1	4	yes	complemented
83	C₂×D₅₀₈	C₂³	1	1	yes	characteristic, complemented
84	C₂×D₅₀₈	C₂³	1	8	yes	complemented
85	C₂×C₂²⋊D₁₂₇	C₂³	1	2	yes	—
86	C₂×C₂²⋊D₁₂₇	C₂³	1	4	yes	—
87	C₂×C₂²⋊D₁₂₇	C₂³	1	4	yes	complemented
88	C₂×C₂²⋊D₁₂₇	C₂³	1	8	yes	complemented
89	D₄×D₁₂₇	—	2	16	no	—
90	D₄×D₁₂₇	—	2	8	no	—
91	D₄×D₁₂₇	—	4	16	no	—
92	C₂×C₄×D₁₂₇	C₂³	1	2	yes	complemented
93	C₂×C₄×D₁₂₇	C₂³	1	4	yes	complemented
94	C₂×C₄×D₁₂₇	C₂³	1	8	yes	complemented
95	C₄⋊Dic₁₂₇	C₂³	1	2	yes	complemented
96	Dic₁₂₇⋊C₄	C₂³	1	4	yes	complemented
97	C₂²⋊Dic₁₂₇	C₂³	1	4	yes	—
98	C₂²⋊Dic₁₂₇	C₂³	1	2	yes	—
99	(C₂×D₁₂₇)⋊C₄	C₂³	1	4	yes	complemented
100	(C₂×D₁₂₇)⋊C₄	C₂³	1	8	yes	complemented
101	(C₂×D₁₂₇)⋊C₄	C₂³	1	8	yes	complemented
102	(C₂×D₁₂₇)⋊C₄	C₂³	1	8	yes	complemented
103	C₄×Dic₁₂₇	C₂³	1	2	yes	complemented
104	C₂⁴×C₁₂₇	C₂³	1	1	yes	characteristic
105	#41×C₁₂₇	—	2	4	no	—
106	C₂×Q₈×C₁₂₇	C₂³	1	1	yes	characteristic, complemented
107	C₂×D₄×C₁₂₇	C₂³	1	4	yes	—
108	C₂×D₄×C₁₂₇	C₂³	1	2	yes	—
109	C₂×D₄×C₁₂₇	C₂³	1	1	yes	characteristic, complemented
110	C₂²×C₄×C₁₂₇	—	2	4	no	—
111	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
112	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
113	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
114	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
115	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	complemented
116	C₄⋊C₄×C₁₂₇	C₂³	1	4	yes	complemented
117	C₄⋊C₄×C₁₂₇	C₂³	1	2	yes	complemented
118	C₂²⋊C₄×C₁₂₇	C₂³	1	2	yes	—
119	C₂²⋊C₄×C₁₂₇	C₂³	1	4	yes	—
120	C₂²⋊C₄×C₁₂₇	C₂³	1	4	yes	complemented
121	C₄²×C₁₂₇	C₂³	1	2	yes	complemented
122	C₂²×D₁₂₇	—	2	4	no	—
123	C₂²×D₁₂₇	—	2	16	no	—
124	C₂²×D₁₂₇	—	2	16	no	—
125	C₂²×D₁₂₇	—	2	8	no	—
126	C₂²×D₁₂₇	—	4	4	no	—
127	C₂²×D₁₂₇	—	4	8	no	—
128	C₂²×D₁₂₇	—	4	4	no	—
129	C₂²×D₁₂₇	—	4	8	no	—
130	C₂²×D₁₂₇	—	4	32	no	—
131	C₂²×D₁₂₇	C₂×D₄	1	8	yes	complemented
132	C₂²×D₁₂₇	C₂⁴	1	4	yes	—
133	C₂²×D₁₂₇	C₂⁴	1	2	yes	—
134	C₂²×D₁₂₇	C₂⁴	1	4	yes	—
135	Dic₂₅₄	—	2	2	no	—
136	D₅₀₈	—	2	8	no	—
137	D₅₀₈	—	2	16	no	—
138	D₅₀₈	—	2	4	no	—
139	D₅₀₈	—	2	4	no	—
140	D₅₀₈	—	2	2	no	—
141	D₅₀₈	—	4	8	no	—
142	D₅₀₈	C₂×D₄	1	8	yes	complemented
143	D₅₀₈	C₂×D₄	1	8	yes	complemented
144	C₂²⋊D₁₂₇	—	2	4	no	—
145	C₂²⋊D₁₂₇	—	2	8	no	—
146	C₂²⋊D₁₂₇	—	2	8	no	—
147	C₂²⋊D₁₂₇	—	4	8	no	—
148	C₂×Dic₁₂₇	C₂⁴	1	4	yes	—
149	C₂×Dic₁₂₇	C₂⁴	1	2	yes	—
150	C₄×D₁₂₇	—	2	8	no	—
151	C₄×D₁₂₇	—	2	4	no	—
152	C₄×D₁₂₇	—	4	8	no	—
153	C₂³×C₁₂₇	—	2	2	no	—
154	C₂³×C₁₂₇	—	2	4	no	—
155	C₂³×C₁₂₇	C₂⁴	1	2	yes	—
156	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
157	C₂³×C₁₂₇	C₂⁴	1	2	yes	—
158	Q₈×C₁₂₇	—	2	2	no	—
159	D₄×C₁₂₇	—	2	4	no	—
160	D₄×C₁₂₇	—	2	8	no	—
161	D₄×C₁₂₇	—	2	2	no	—
162	C₂×C₄×C₁₂₇	—	2	2	no	—
163	C₂×C₄×C₁₂₇	—	2	4	no	—
164	C₂×C₄×C₁₂₇	—	2	4	no	—
165	C₂×C₄×C₁₂₇	—	2	4	no	—
166	C₂×C₄×C₁₂₇	C₂×D₄	1	4	yes	complemented
167	C₂×C₄×C₁₂₇	C₂⁴	1	4	yes	—
168	C₂×C₄×C₁₂₇	C₂⁴	1	2	yes	—
169	C₂×C₄×C₁₂₇	C₂⁴	1	2	yes	—
170	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
171	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
172	C₂²×C₁₂₇	—	2	8	no	—
173	C₂²×C₁₂₇	—	2	2	no	—
174	C₂²×C₁₂₇	—	2	2	no	—
175	C₂²×C₁₂₇	—	2	2	no	—
176	C₂²×C₁₂₇	—	2	2	no	—
177	C₂²×C₁₂₇	—	2	4	no	—
178	C₂²×C₁₂₇	C₂²×D₄	1	2	yes	—
179	C₂²×C₁₂₇	H(5,𝔽₂)	1	4	yes	complemented
180	C₂²×C₁₂₇	C₂⁵	1	1	yes	characteristic
181	C₂×D₁₂₇	—	2	8	no	—
182	C₂×D₁₂₇	—	2	2	no	—
183	C₂×D₁₂₇	—	2	4	no	—
184	C₂×D₁₂₇	—	2	4	no	—
185	C₂×D₁₂₇	—	2	8	no	—
186	C₂×D₁₂₇	—	4	8	no	—
187	C₂×D₁₂₇	—	4	32	no	—
188	C₂×D₁₂₇	—	4	8	no	—
189	C₂×D₁₂₇	—	8	8	no	—
190	C₂×D₁₂₇	C₂²×D₄	1	8	yes	—
191	C₄×C₁₂₇	—	2	2	no	—
192	C₄×C₁₂₇	—	2	2	no	—
193	C₄×C₁₂₇	—	2	4	no	—
194	C₄×C₁₂₇	C₂²×D₄	1	2	yes	—
195	C₄×C₁₂₇	C₂²×D₄	1	2	yes	—
196	Dic₁₂₇	—	2	2	no	—
197	Dic₁₂₇	—	2	4	no	—
198	C₂×C₁₂₇	—	2	2	no	—
199	C₂×C₁₂₇	—	2	4	no	—
200	C₂×C₁₂₇	—	4	2	no	—
201	C₂×C₁₂₇	C₂³×D₄	1	1	yes	characteristic
202	C₂×C₁₂₇	C₂×H(5,𝔽₂)	1	2	yes	—
203	D₁₂₇	—	2	8	no	—
204	D₁₂₇	—	4	4	no	—
205	D₁₂₇	—	4	2	no	—
206	#3484	—	127	1	no	maximal, Sylow
207	C₁₂₇	#3484	1	1	yes	characteristic, complemented, Sylow
208	C₂×H(5,𝔽₂)	—	127	1	no	—
209	C₂³×D₄	—	127	1	no	—
210	#546	—	127	4	no	—
211	#546	—	127	4	no	—
212	D₄²	—	127	4	no	—
213	D₄²	—	127	4	no	—
214	#535	—	127	1	no	—
215	#535	—	127	1	no	—
216	#534	—	127	4	no	—
217	#525	—	127	1	no	—
218	#525	D₁₂₇	1	1	yes	characteristic, complemented
219	C₂×#120	—	127	4	no	—
220	#518	—	127	1	no	—
221	C₂⁵	—	127	2	no	—
222	H(5,𝔽₂)	—	254	8	no	—
223	C₂×#41	—	127	2	no	—
224	C₂×#41	—	127	1	no	—
225	C₂×#41	—	127	2	no	—
226	C₂×#41	C₂×D₁₂₇	1	1	yes	characteristic, complemented
227	C₂²×D₄	—	127	8	no	—
228	C₂²×D₄	—	127	4	no	—
229	C₂²×D₄	—	127	2	no	—
230	C₂²×D₄	—	127	1	no	—
231	C₂²×D₄	—	127	2	no	—
232	C₂²×D₄	—	127	1	no	—
233	C₂²×D₄	—	127	4	no	—
234	C₂²×D₄	—	127	4	no	—
235	C₂²×D₄	—	127	2	no	—
236	C₂²×D₄	—	254	8	no	—
237	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic, complemented
238	#127	—	127	2	no	—
239	#127	—	127	2	no	—
240	#124	—	127	2	no	—
241	#124	—	127	2	no	—
242	#123	—	127	2	no	—
243	#123	—	127	8	no	—
244	#123	C₂×D₁₂₇	1	2	yes	complemented
245	C₂²⋊Q₈	—	127	2	no	—
246	C₂²⋊Q₈	C₂×D₁₂₇	1	2	yes	complemented
247	#121	—	127	2	no	—
248	#121	—	127	4	no	—
249	#121	—	127	8	no	—
250	#121	—	127	4	no	—
251	#121	—	127	8	no	—
252	#121	C₂×D₁₂₇	1	2	yes	complemented
253	#120	—	127	2	no	—
254	#120	—	127	4	no	—
255	#120	—	127	8	no	—
256	#120	—	127	4	no	—
257	#120	—	127	8	no	—
258	#120	—	127	8	no	—
259	#120	C₂×D₁₂₇	1	2	yes	complemented
260	C₄×D₄	—	127	4	no	—
261	C₄×D₄	—	127	4	no	—
262	C₄×D₄	—	127	4	no	—
263	C₄×D₄	C₂×D₁₂₇	1	4	yes	complemented
264	#117	—	127	1	no	—
265	#117	C₂×D₁₂₇	1	1	yes	characteristic, complemented
266	C₂×C₂²⋊C₄	—	127	4	no	—
267	C₂×C₂²⋊C₄	—	127	8	no	—
268	C₂⁴	—	127	2	no	—
269	C₂⁴	—	127	2	no	—
270	C₂⁴	—	127	4	no	—
271	C₂⁴	—	127	8	no	—
272	C₂⁴	—	508	8	no	—
273	C₂⁴	—	254	8	no	—
274	C₂⁴	C₂²×D₁₂₇	1	1	yes	characteristic
275	#41	—	2	4	no	—
276	#41	—	254	8	no	—
277	#41	—	254	8	no	—
278	#41	—	254	4	no	—
279	C₂×Q₈	—	127	1	no	—
280	C₂×Q₈	C₂²×D₁₂₇	1	1	yes	characteristic, complemented
281	C₂×D₄	—	127	8	no	—
282	C₂×D₄	—	127	4	no	—
283	C₂×D₄	—	127	2	no	—
284	C₂×D₄	—	127	4	no	—
285	C₂×D₄	—	127	8	no	—
286	C₂×D₄	—	127	2	no	—
287	C₂×D₄	—	127	2	no	—
288	C₂×D₄	—	127	8	no	—
289	C₂×D₄	—	127	8	no	—
290	C₂×D₄	—	127	2	no	—
291	C₂×D₄	—	127	2	no	—
292	C₂×D₄	—	127	4	no	—
293	C₂×D₄	—	127	1	no	—
294	C₂×D₄	—	508	16	no	—
295	C₂×D₄	—	254	16	no	—
296	C₂×D₄	—	254	16	no	—
297	C₂×D₄	—	254	8	no	—
298	C₂×D₄	—	254	8	no	—
299	C₂×D₄	—	254	8	no	—
300	C₂×D₄	—	254	16	no	—
301	C₂×D₄	—	254	4	no	—
302	C₂×D₄	C₂²×D₁₂₇	1	4	yes	—
303	C₂×D₄	C₂²×D₁₂₇	1	2	yes	—
304	C₂×D₄	C₂²×D₁₂₇	1	1	yes	characteristic, complemented
305	C₂²×C₄	—	2	4	no	—
306	C₂²×C₄	—	127	8	no	—
307	C₂²×C₄	—	127	4	no	—
308	C₂²×C₄	—	127	2	no	—
309	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
310	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
311	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
312	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
313	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	complemented
314	C₄⋊C₄	—	127	4	no	—
315	C₄⋊C₄	—	127	2	no	—
316	C₄⋊C₄	C₂²×D₁₂₇	1	2	yes	complemented
317	C₄⋊C₄	C₂²×D₁₂₇	1	4	yes	complemented
318	C₂²⋊C₄	—	127	2	no	—
319	C₂²⋊C₄	—	127	4	no	—
320	C₂²⋊C₄	—	127	8	no	—
321	C₂²⋊C₄	—	127	8	no	—
322	C₂²⋊C₄	—	127	8	no	—
323	C₂²⋊C₄	—	127	4	no	—
324	C₂²⋊C₄	C₂²×D₁₂₇	1	2	yes	—
325	C₂²⋊C₄	C₂²×D₁₂₇	1	4	yes	—
326	C₂²⋊C₄	C₂²×D₁₂₇	1	4	yes	complemented
327	C₄²	—	127	2	no	—
328	C₄²	C₂²×D₁₂₇	1	2	yes	complemented
329	C₂³	—	2	2	no	—
330	C₂³	—	2	4	no	—
331	C₂³	—	127	8	no	—
332	C₂³	—	127	4	no	—
333	C₂³	—	127	4	no	—
334	C₂³	—	127	2	no	—
335	C₂³	—	508	8	no	—
336	C₂³	—	508	32	no	—
337	C₂³	—	508	4	no	—
338	C₂³	—	508	8	no	—
339	C₂³	—	508	4	no	—
340	C₂³	—	254	8	no	—
341	C₂³	—	254	16	no	—
342	C₂³	—	254	4	no	—
343	C₂³	—	254	16	no	—
344	C₂³	C₂³×D₁₂₇	1	2	yes	—
345	C₂³	C₂³×D₁₂₇	1	1	yes	characteristic
346	C₂³	C₂³×D₁₂₇	1	2	yes	—
347	Q₈	—	2	2	no	—
348	Q₈	—	254	2	no	—
349	D₄	—	2	4	no	—
350	D₄	—	2	8	no	—
351	D₄	—	2	2	no	—
352	D₄	—	127	8	no	—
353	D₄	—	127	8	no	—
354	D₄	—	508	8	no	—
355	D₄	—	508	8	no	—
356	D₄	—	254	16	no	—
357	D₄	—	254	8	no	—
358	D₄	—	254	8	no	—
359	D₄	—	254	4	no	—
360	D₄	—	254	8	no	—
361	D₄	—	254	4	no	—
362	D₄	—	254	4	no	—
363	D₄	—	254	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	4	no	—
366	C₂×C₄	—	2	4	no	—
367	C₂×C₄	—	2	4	no	—
368	C₂×C₄	—	127	4	no	—
369	C₂×C₄	—	127	2	no	—
370	C₂×C₄	—	508	8	no	—
371	C₂×C₄	—	254	8	no	—
372	C₂×C₄	—	254	4	no	—
373	C₂×C₄	D₄×D₁₂₇	1	4	yes	complemented
374	C₂×C₄	C₂³×D₁₂₇	1	4	yes	—
375	C₂×C₄	C₂³×D₁₂₇	1	2	yes	—
376	C₂×C₄	C₂³×D₁₂₇	1	2	yes	—
377	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
378	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
379	C₂²	—	2	8	no	—
380	C₂²	—	2	2	no	—
381	C₂²	—	2	2	no	—
382	C₂²	—	2	2	no	—
383	C₂²	—	2	2	no	—
384	C₂²	—	2	4	no	—
385	C₂²	—	127	8	no	—
386	C₂²	—	1016	8	no	—
387	C₂²	—	508	8	no	—
388	C₂²	—	508	32	no	—
389	C₂²	—	508	8	no	—
390	C₂²	—	254	2	no	—
391	C₂²	—	254	4	no	—
392	C₂²	—	254	4	no	—
393	C₂²	—	254	8	no	—
394	C₂²	—	254	8	no	—
395	C₂²	C₂×D₄×D₁₂₇	1	2	yes	—
396	C₂²	#16861255	1	4	yes	complemented
397	C₂²	C₂⁴×D₁₂₇	1	1	yes	characteristic
398	C₄	—	2	2	no	—
399	C₄	—	2	2	no	—
400	C₄	—	2	4	no	—
401	C₄	—	254	2	no	—
402	C₄	—	254	4	no	—
403	C₄	C₂×D₄×D₁₂₇	1	2	yes	—
404	C₄	C₂×D₄×D₁₂₇	1	2	yes	—
405	C₂	—	2	2	no	—
406	C₂	—	2	4	no	—
407	C₂	—	4	2	no	—
408	C₂	—	508	2	no	—
409	C₂	—	508	4	no	—
410	C₂	—	254	8	no	—
411	C₂	C₂²×D₄×D₁₂₇	1	1	yes	characteristic
412	C₂	C₂×#16861255	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₂₇
commutator, socle, cosocle	C₂²×C₁₂₇	C₂⁵
Fitting	#525×C₁₂₇	C₂
hypercenter	#525	D₁₂₇
hypocenter	C₁₂₇	#3484
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ #525

Complements

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

N	H ≅ G/N	type
1. C₂×#16861255	C₂	semidirect
2. C₂³×D₅₀₈	C₂	semidirect
3. #22550522	C₂	semidirect
4. #22550517	C₂	semidirect
5. D₄×D₅₀₈	C₂	semidirect
6. #22550510	C₂	semidirect
7. #22550445	C₂	semidirect
8. #22550440	C₂	semidirect
9. #22550437	C₂	semidirect
10. #22550387	C₂	semidirect
11. #120×D₁₂₇	C₂	semidirect
12. #22550350	C₂	semidirect
13. #525×C₁₂₇	C₂	semidirect
14. C₂⁴×D₁₂₇	C₂²	semidirect
16. C₂×#14523249	C₂²	semidirect
17. C₂×#14523248	C₂²	semidirect
18. C₂×#14523248	C₂²	semidirect
21. C₂²×D₅₀₈	C₂²	semidirect
22. C₂²×D₅₀₈	C₂²	semidirect
23. C₂²×D₅₀₈	C₂²	semidirect
24. C₂²×D₅₀₈	C₂²	semidirect
25. C₂²×D₅₀₈	C₂²	semidirect
26. C₂×D₄×D₁₂₇	C₂²	semidirect
27. C₂×D₄×D₁₂₇	C₂²	semidirect
28. C₂×D₄×D₁₂₇	C₂²	semidirect
29. #16861206	C₂²	semidirect
30. #16861205	C₂²	semidirect
31. #16861198	C₂²	semidirect
32. #16861197	C₂²	semidirect
33. #16861194	C₂²	semidirect
34. #16861192	C₂²	semidirect
35. #16861188	C₂²	semidirect
36. #16861184	C₂²	semidirect
37. #16861184	C₂²	semidirect
38. #16861183	C₂²	semidirect
39. #16861183	C₂²	semidirect
40. #16861182	C₂²	semidirect
41. #16861179	C₂²	semidirect
42. #16861179	C₂²	semidirect
43. #16861179	C₂²	semidirect
44. #16861178	C₂²	semidirect
45. #16861177	C₂²	semidirect
46. #16861177	C₂²	semidirect
47. C₄×D₅₀₈	C₂²	semidirect
48. #16861171	C₂²	semidirect
49. C₄×C₂²⋊D₁₂₇	C₂²	semidirect
50. #16861168	C₂²	semidirect
51. C₂²⋊C₄×D₁₂₇	C₂²	semidirect
52. C₂²⋊C₄×D₁₂₇	C₂²	semidirect
53. C₂×#41×C₁₂₇	C₂²	semidirect
54. C₂³×C₄×C₁₂₇	C₂²	semidirect
55. #123×C₁₂₇	C₂²	semidirect
56. C₂²⋊Q₈×C₁₂₇	C₂²	semidirect
57. #121×C₁₂₇	C₂²	semidirect
58. #120×C₁₂₇	C₂²	semidirect
59. C₄×D₄×C₁₂₇	C₂²	semidirect
60. #117×C₁₂₇	C₂²	semidirect
66. C₂³×D₁₂₇	C₂³	semidirect
70. C₂×Dic₂₅₄	C₂³	semidirect
76. C₂×D₅₀₈	D₄	semidirect
82. C₂×D₅₀₈	C₂³	semidirect
83. C₂×D₅₀₈	C₂³	semidirect
84. C₂×D₅₀₈	C₂³	semidirect
87. C₂×C₂²⋊D₁₂₇	C₂³	semidirect
88. C₂×C₂²⋊D₁₂₇	C₂³	semidirect
92. C₂×C₄×D₁₂₇	C₂³	semidirect
93. C₂×C₄×D₁₂₇	C₂³	semidirect
94. C₂×C₄×D₁₂₇	C₂³	semidirect
95. C₄⋊Dic₁₂₇	C₂³	semidirect
96. Dic₁₂₇⋊C₄	C₂³	semidirect
99. (C₂×D₁₂₇)⋊C₄	C₂³	semidirect
100. (C₂×D₁₂₇)⋊C₄	C₂³	semidirect
101. (C₂×D₁₂₇)⋊C₄	C₂³	semidirect
102. (C₂×D₁₂₇)⋊C₄	C₂³	semidirect
103. C₄×Dic₁₂₇	C₂³	semidirect
106. C₂×Q₈×C₁₂₇	C₂³	semidirect
109. C₂×D₄×C₁₂₇	C₂³	semidirect
115. C₂²×C₄×C₁₂₇	C₂³	semidirect
116. C₄⋊C₄×C₁₂₇	C₂³	semidirect
117. C₄⋊C₄×C₁₂₇	C₂³	semidirect
120. C₂²⋊C₄×C₁₂₇	C₂³	semidirect
121. C₄²×C₁₂₇	C₂³	semidirect
131. C₂²×D₁₂₇	C₂×D₄	semidirect
142. D₅₀₈	C₂×D₄	semidirect
143. D₅₀₈	C₂×D₄	semidirect
166. C₂×C₄×C₁₂₇	C₂×D₄	semidirect
179. C₂²×C₁₂₇	H(5,𝔽₂)	semidirect
207. C₁₂₇	#3484	semidirect
218. #525	D₁₂₇	semidirect
226. C₂×#41	C₂×D₁₂₇	semidirect
237. C₂³×C₄	C₂×D₁₂₇	semidirect
244. #123	C₂×D₁₂₇	semidirect
246. C₂²⋊Q₈	C₂×D₁₂₇	semidirect
252. #121	C₂×D₁₂₇	semidirect
259. #120	C₂×D₁₂₇	semidirect
263. C₄×D₄	C₂×D₁₂₇	semidirect
265. #117	C₂×D₁₂₇	semidirect
280. C₂×Q₈	C₂²×D₁₂₇	semidirect
304. C₂×D₄	C₂²×D₁₂₇	semidirect
313. C₂²×C₄	C₂²×D₁₂₇	semidirect
316. C₄⋊C₄	C₂²×D₁₂₇	semidirect
317. C₄⋊C₄	C₂²×D₁₂₇	semidirect
326. C₂²⋊C₄	C₂²×D₁₂₇	semidirect
328. C₄²	C₂²×D₁₂₇	semidirect
373. C₂×C₄	D₄×D₁₂₇	semidirect
396. C₂²	#16861255	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16861255	2
C₂³×D₅₀₈	2
#22550522	2
#22550517	2
D₄×D₅₀₈	2
#22550510	2
#22550445	2
#22550440	2
#22550437	2
#22550387	2
#120×D₁₂₇	2
#22550350	2
#525×C₁₂₇	2
#3484	127