Subgroups — #41×D₅₃₆ #31737238

Order 17152.

2406 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×D₂₆₈	C₂	1	2	yes	maximal, complemented
2	#22573184	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×D₅₃₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#22572845	C₂	1	2	yes	maximal, complemented
5	#22572557	C₂	1	3	yes	maximal, complemented
6	#22572023	C₂	1	6	yes	maximal, complemented
7	D₄×D₅₃₆	C₂	1	3	yes	maximal, complemented
8	#22570788	C₂	1	3	yes	maximal, complemented
9	#22570210	C₂	1	6	yes	maximal, complemented
10	C₂×C₄×D₅₃₆	C₂	1	3	yes	maximal, complemented
11	C₈×#41×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×D₆₇	—	2	2	no	—
13	C₂²×D₅₃₆	C₂²	1	3	yes	—
14	#16864068	C₂²	1	2	yes	complemented
15	Q₈×D₂₆₈	C₂²	1	2	yes	complemented
16	#16864058	—	2	2	no	—
17	#16864050	C₂²	1	6	yes	complemented
18	#16864041	—	2	6	no	—
19	D₄×D₂₆₈	C₂²	1	6	yes	complemented
20	#16863946	C₂²	1	6	yes	complemented
21	#16863908	—	2	6	no	—
22	C₂×C₄×D₂₆₈	C₂²	1	6	yes	complemented
23	#16863801	C₂²	1	3	yes	complemented
24	#16863773	C₂²	1	3	yes	complemented
25	#16863743	C₂²	1	6	yes	complemented
26	#16863710	C₂²	1	12	yes	complemented
27	#16863677	C₂²	1	6	yes	complemented
28	#16863677	C₂²	1	6	yes	complemented
29	#16863632	C₂²	1	6	yes	complemented
30	#16863632	C₂²	1	6	yes	complemented
31	#16863592	C₂²	1	6	yes	complemented
32	#16863592	C₂²	1	6	yes	complemented
33	#16863532	C₂²	1	12	yes	complemented
34	C₄×D₅₃₆	C₂²	1	1	yes	characteristic
35	C₄×D₅₃₆	C₂²	1	3	yes	—
36	C₄×D₅₃₆	C₂²	1	3	yes	complemented
37	C₄×D₅₃₆	C₂²	1	3	yes	complemented
38	C₄×D₅₃₆	C₂²	1	6	yes	complemented
39	C₄×D₅₃₆	C₂²	1	6	yes	complemented
40	C₄×D₅₃₆	C₂²	1	6	yes	complemented
41	C₂×#14523977	C₂²	1	3	yes	complemented
42	C₂×#14523965	C₂²	1	6	yes	complemented
43	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
44	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
45	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
46	C₈×D₄×C₆₇	C₂²	1	3	yes	complemented
47	C₈×D₄×C₆₇	C₂²	1	3	yes	complemented
48	#431×C₆₇	C₂²	1	3	yes	complemented
49	C₂×C₄×C₈×C₆₇	C₂²	1	3	yes	complemented
50	C₂×#11761629	—	2	2	no	—
51	C₂×#11761628	—	2	6	no	—
52	C₂×#11761627	—	2	6	no	—
53	#41×D₆₇	—	4	16	no	—
54	C₂×Q₈×D₆₇	—	2	2	no	—
55	C₂²×D₂₆₈	C₂³	1	6	yes	—
56	C₂×D₄×D₆₇	—	2	6	no	—
57	C₂²×C₄×D₆₇	—	2	6	no	—
58	C₂×D₅₃₆	—	2	12	no	—
59	C₂×D₅₃₆	C₂³	1	3	yes	—
60	C₂×D₅₃₆	C₂³	1	6	yes	—
61	C₂×D₅₃₆	C₂³	1	6	yes	—
62	C₂×D₅₃₆	C₂³	1	1	yes	characteristic
63	#14524043	C₂³	1	6	yes	—
64	#14524040	C₂³	1	6	yes	—
65	#14524031	—	2	6	no	—
66	#14524027	—	2	12	no	—
67	#14524023	—	2	6	no	—
68	#14524023	—	2	6	no	—
69	#14524018	—	2	6	no	—
70	#14524018	—	2	6	no	—
71	#14524017	—	2	6	no	—
72	#14524017	C₂³	1	12	yes	—
73	#14524013	—	2	12	no	—
74	C₄×D₂₆₈	—	2	6	no	—
75	C₄×D₂₆₈	—	2	6	no	—
76	C₄×D₂₆₈	—	2	6	no	—
77	C₄×D₂₆₈	C₂³	1	6	yes	—
78	C₄×D₂₆₈	C₂³	1	6	yes	—
79	C₄×D₂₆₈	C₂³	1	2	yes	—
80	C₄×D₂₆₈	C₂³	1	6	yes	—
81	C₂×C₄⋊Dic₆₇	C₂³	1	6	yes	—
82	C₂×(C₂×D₆₇)⋊C₄	—	2	6	no	—
83	#14523977	C₂³	1	3	yes	—
84	#14523977	C₂³	1	6	yes	—
85	#14523977	C₂³	1	6	yes	—
86	#14523977	C₂³	1	1	yes	characteristic
87	#14523965	C₂³	1	6	yes	—
88	#14523965	C₂³	1	12	yes	—
89	#14523965	C₂³	1	6	yes	—
90	#14523965	C₂³	1	6	yes	—
91	#14523965	C₂³	1	2	yes	—
92	C₂×#41×C₆₇	D₄	1	1	yes	characteristic
93	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
94	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
95	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
96	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
97	C₄×D₄×C₆₇	D₄	1	3	yes	—
98	C₄×D₄×C₆₇	C₂³	1	3	yes	—
99	#117×C₆₇	D₄	1	3	yes	—
100	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
101	C₄⋊C₈×C₆₇	C₂³	1	3	yes	—
102	C₄⋊C₈×C₆₇	C₂³	1	3	yes	—
103	C₂²⋊C₈×C₆₇	C₂³	1	6	yes	—
104	C₄×C₈×C₆₇	C₂³	1	3	yes	—
105	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
106	C₄×C₈×C₆₇	C₂³	1	3	yes	—
107	C₄×C₈×C₆₇	C₂³	1	3	yes	—
108	C₂³×D₆₇	—	2	6	no	—
109	#11761629	—	4	8	no	—
110	#11761628	—	4	24	no	—
111	#11761627	—	4	24	no	—
112	C₂×Dic₁₃₄	—	2	6	no	—
113	Q₈×D₆₇	—	4	8	no	—
114	C₂×D₂₆₈	—	2	12	no	—
115	C₂×D₂₆₈	—	2	6	no	—
116	C₂×D₂₆₈	—	2	6	no	—
117	C₂×D₂₆₈	—	2	6	no	—
118	C₂×D₂₆₈	—	4	6	no	—
119	C₂×D₂₆₈	C₂⁴	1	6	yes	—
120	C₂×D₂₆₈	C₂⁴	1	2	yes	—
121	C₂×C₂²⋊D₆₇	—	2	12	no	—
122	D₄×D₆₇	—	4	24	no	—
123	C₂²×Dic₆₇	—	2	6	no	—
124	C₂×C₄×D₆₇	—	2	2	no	—
125	C₂×C₄×D₆₇	—	2	6	no	—
126	C₂×C₄×D₆₇	—	2	6	no	—
127	C₂×C₄×D₆₇	—	2	6	no	—
128	C₂×C₄×D₆₇	—	4	24	no	—
129	D₅₃₆	—	2	12	no	—
130	D₅₃₆	—	2	6	no	—
131	D₅₃₆	—	4	6	no	—
132	D₅₃₆	#41	1	4	yes	direct product factor
133	C₄⋊Dic₆₇	—	2	6	no	—
134	C₄⋊Dic₆₇	—	2	6	no	—
135	C₄⋊Dic₆₇	C₂⁴	1	2	yes	—
136	C₄⋊Dic₆₇	C₂⁴	1	6	yes	—
137	(C₂×D₆₇)⋊C₄	—	2	2	no	—
138	(C₂×D₆₇)⋊C₄	—	2	6	no	—
139	(C₂×D₆₇)⋊C₄	—	2	6	no	—
140	(C₂×D₆₇)⋊C₄	—	2	6	no	—
141	(C₂×D₆₇)⋊C₄	—	2	12	no	—
142	#41×C₆₇	D₈	1	8	yes	complemented
143	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
144	C₂×D₄×C₆₇	C₂×D₄	1	3	yes	—
145	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
146	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
147	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
148	C₂×C₈×C₆₇	—	2	6	no	—
149	C₂×C₈×C₆₇	—	4	3	no	—
150	C₂×C₈×C₆₇	C₂⁴	1	3	yes	—
151	C₂×C₈×C₆₇	C₂⁴	1	3	yes	—
152	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
153	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
154	C₄⋊C₄×C₆₇	C₂×D₄	1	3	yes	—
155	C₄⋊C₄×C₆₇	C₂×D₄	1	3	yes	—
156	C₂²⋊C₄×C₆₇	C₂×D₄	1	6	yes	—
157	C₄²×C₆₇	C₂×D₄	1	3	yes	—
158	C₄²×C₆₇	C₂×D₄	1	3	yes	—
159	C₄²×C₆₇	C₂⁴	1	3	yes	—
160	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
161	C₂³×C₆₇	C₂²×D₄	1	3	yes	—
162	C₂²×D₆₇	—	2	2	no	—
163	C₂²×D₆₇	—	2	6	no	—
164	C₂²×D₆₇	—	4	12	no	—
165	C₂²×D₆₇	—	4	12	no	—
166	C₂²×D₆₇	—	8	6	no	—
167	Q₈×C₆₇	C₂×D₈	1	4	yes	complemented
168	D₄×C₆₇	C₂×D₈	1	12	yes	complemented
169	C₂×C₄×C₆₇	—	2	3	no	—
170	C₂×C₄×C₆₇	—	2	3	no	—
171	C₂×C₄×C₆₇	—	4	3	no	—
172	C₂×C₄×C₆₇	C₂×D₈	1	12	yes	complemented
173	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
174	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
175	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
176	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
177	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
179	C₂²⋊D₆₇	—	4	24	no	—
180	C₂×Dic₆₇	—	2	2	no	—
181	C₂×Dic₆₇	—	2	6	no	—
182	C₂×Dic₆₇	—	4	12	no	—
183	D₂₆₈	—	2	6	no	—
184	D₂₆₈	—	2	2	no	—
185	D₂₆₈	—	2	6	no	—
186	D₂₆₈	—	4	12	no	—
187	D₂₆₈	—	4	12	no	—
188	D₂₆₈	—	8	6	no	—
189	D₂₆₈	C₂×#41	1	4	yes	complemented
190	C₄×D₆₇	—	4	4	no	—
191	C₄×D₆₇	—	4	12	no	—
192	C₄×D₆₇	—	4	12	no	—
193	C₄×D₆₇	—	4	12	no	—
194	Dic₁₃₄	—	4	12	no	—
195	C₈×C₆₇	—	2	3	no	—
196	C₈×C₆₇	—	2	1	no	—
197	C₈×C₆₇	—	2	3	no	—
198	C₈×C₆₇	C₂×#41	1	2	yes	complemented
199	C₂²×C₆₇	—	2	3	no	—
200	C₂²×C₆₇	—	2	3	no	—
201	C₂²×C₆₇	C₂²×D₈	1	6	yes	—
202	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
203	C₂×D₆₇	—	2	4	no	—
204	C₂×D₆₇	—	4	2	no	—
205	C₂×D₆₇	—	4	2	no	—
206	C₂×D₆₇	—	4	6	no	—
207	C₂×D₆₇	—	4	6	no	—
208	C₂×D₆₇	—	4	6	no	—
209	C₂×D₆₇	—	8	24	no	—
210	C₄×C₆₇	—	2	3	no	—
211	C₄×C₆₇	—	2	1	no	—
212	C₄×C₆₇	—	2	3	no	—
213	C₄×C₆₇	C₂²×D₈	1	6	yes	—
214	C₄×C₆₇	C₂²×D₈	1	2	yes	—
215	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
216	C₄×C₆₇	#585	1	1	yes	characteristic
217	Dic₆₇	—	4	6	no	—
218	Dic₆₇	—	4	2	no	—
219	D₈×#41	—	67	1	no	maximal, Sylow
220	C₂×C₆₇	—	2	6	no	—
221	C₂×C₆₇	D₄×#41	1	1	yes	characteristic
222	C₂×C₆₇	C₂³×D₈	1	1	yes	characteristic
223	C₂×C₆₇	#3584	1	1	yes	characteristic
224	D₆₇	—	4	4	no	—
225	D₆₇	—	8	6	no	—
226	D₄×#41	—	67	2	no	—
227	#3391	—	67	1	no	—
228	Q₈×D₈	—	67	1	no	—
229	#3358	—	67	2	no	—
230	#3334	—	67	3	no	—
231	#3294	—	67	6	no	—
232	D₄×D₈	—	67	3	no	—
233	#3169	—	67	3	no	—
234	#3100	—	67	6	no	—
235	C₈×#41	D₆₇	1	1	yes	characteristic, complemented
236	C₂×C₄×D₈	—	67	3	no	—
237	C₆₇	D₈×#41	1	1	yes	characteristic, complemented, Sylow
238	C₂²×#41	—	134	2	no	—
239	C₂²×D₈	—	67	3	no	—
240	#550	—	67	2	no	—
241	D₄×Q₈	—	67	2	no	—
242	#548	—	134	2	no	—
243	#547	—	67	6	no	—
244	#546	—	134	6	no	—
245	D₄²	—	67	6	no	—
246	#532	—	67	6	no	—
247	#525	—	134	6	no	—
248	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
249	C₂×C₄×D₄	—	67	6	no	—
250	#500	—	67	3	no	—
251	#493	—	67	3	no	—
252	#486	—	67	6	no	—
253	#480	—	67	12	no	—
254	#474	—	67	6	no	—
255	#474	—	67	6	no	—
256	#466	—	67	6	no	—
257	#466	—	67	6	no	—
258	#459	—	67	6	no	—
259	#459	—	67	6	no	—
260	#447	—	67	12	no	—
261	C₈×Q₈	C₂×D₆₇	1	1	yes	characteristic, complemented
262	C₈×Q₈	C₂×D₆₇	1	1	yes	characteristic, complemented
263	C₄×D₈	—	67	3	no	—
264	C₄×D₈	—	67	6	no	—
265	C₄×D₈	—	67	6	no	—
266	C₄×D₈	—	67	6	no	—
267	C₄×D₈	—	67	3	no	—
268	C₄×D₈	—	67	3	no	—
269	C₄×D₈	—	67	1	no	—
270	C₈×D₄	C₂×D₆₇	1	3	yes	complemented
271	C₈×D₄	C₂×D₆₇	1	3	yes	complemented
272	#431	C₂×D₆₇	1	3	yes	complemented
273	C₂×#107	—	67	3	no	—
274	C₂×#102	—	67	6	no	—
275	C₂×C₄×C₈	C₂×D₆₇	1	3	yes	complemented
276	C₂×#41	—	268	16	no	—
277	C₂×#41	—	134	2	no	—
278	C₂×#41	—	134	6	no	—
279	C₂×#41	—	134	6	no	—
280	C₂×#41	D₂₆₈	1	1	yes	characteristic
281	C₂²×Q₈	—	134	2	no	—
282	C₂²×D₄	—	67	6	no	—
283	C₂²×D₄	—	134	6	no	—
284	C₂³×C₄	—	134	6	no	—
285	C₂×D₈	—	67	6	no	—
286	C₂×D₈	—	67	1	no	—
287	C₂×D₈	—	67	6	no	—
288	C₂×D₈	—	67	3	no	—
289	C₂×D₈	—	134	12	no	—
290	C₂²×C₈	C₂²×D₆₇	1	3	yes	—
291	C₂²×C₈	C₂²×D₆₇	1	3	yes	—
292	C₄⋊Q₈	—	67	6	no	—
293	#127	—	67	6	no	—
294	#124	—	134	6	no	—
295	#123	—	134	12	no	—
296	C₂²⋊Q₈	—	134	6	no	—
297	C₂²⋊Q₈	—	134	6	no	—
298	#121	—	67	12	no	—
299	#121	—	134	6	no	—
300	#121	—	134	6	no	—
301	#121	—	134	6	no	—
302	#120	—	134	12	no	—
303	C₄×Q₈	D₂₆₈	1	1	yes	characteristic
304	C₄×Q₈	C₂²×D₆₇	1	1	yes	characteristic
305	C₄×D₄	—	67	2	no	—
306	C₄×D₄	—	67	6	no	—
307	C₄×D₄	—	67	6	no	—
308	C₄×D₄	—	67	6	no	—
309	C₄×D₄	—	134	6	no	—
310	C₄×D₄	—	134	6	no	—
311	C₄×D₄	—	134	6	no	—
312	C₄×D₄	D₂₆₈	1	3	yes	—
313	C₄×D₄	C₂²×D₆₇	1	3	yes	—
314	#117	D₂₆₈	1	3	yes	—
315	C₂×C₄⋊C₄	—	67	6	no	—
316	C₂×C₂²⋊C₄	—	134	6	no	—
317	C₂×C₄²	C₂²×D₆₇	1	3	yes	—
318	#107	—	67	1	no	—
319	#107	—	67	3	no	—
320	#107	—	67	6	no	—
321	#107	—	67	6	no	—
322	C₄⋊C₈	C₂²×D₆₇	1	3	yes	—
323	C₄⋊C₈	C₂²×D₆₇	1	3	yes	—
324	#102	—	67	2	no	—
325	#102	—	67	6	no	—
326	#102	—	67	6	no	—
327	#102	—	67	6	no	—
328	#102	—	67	12	no	—
329	C₂²⋊C₈	C₂²×D₆₇	1	6	yes	—
330	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
331	C₄×C₈	C₂²×D₆₇	1	3	yes	—
332	C₄×C₈	C₂²×D₆₇	1	3	yes	—
333	C₄×C₈	C₂²×D₆₇	1	3	yes	—
334	C₂⁴	—	134	6	no	—
335	#41	—	268	8	no	—
336	#41	—	268	24	no	—
337	#41	—	268	24	no	—
338	#41	D₅₃₆	1	8	yes	direct product factor
339	C₂×Q₈	—	268	8	no	—
340	C₂×Q₈	—	134	6	no	—
341	C₂×Q₈	C₂×D₂₆₈	1	1	yes	characteristic
342	C₂×D₄	—	268	6	no	—
343	C₂×D₄	—	268	24	no	—
344	C₂×D₄	—	67	2	no	—
345	C₂×D₄	—	67	6	no	—
346	C₂×D₄	—	134	12	no	—
347	C₂×D₄	—	134	12	no	—
348	C₂×D₄	—	134	6	no	—
349	C₂×D₄	—	134	6	no	—
350	C₂×D₄	—	134	6	no	—
351	C₂×D₄	C₂×D₂₆₈	1	3	yes	—
352	C₂²×C₄	—	268	24	no	—
353	C₂²×C₄	—	134	6	no	—
354	C₂²×C₄	—	134	6	no	—
355	C₂²×C₄	—	134	6	no	—
356	C₂²×C₄	—	134	2	no	—
357	C₂²×C₄	—	134	6	no	—
358	C₂²×C₄	C₂×D₂₆₈	1	3	yes	—
359	C₂²×C₄	C₂×D₂₆₈	1	3	yes	—
360	C₂²×C₄	C₂³×D₆₇	1	3	yes	—
361	D₈	—	268	6	no	—
362	D₈	—	67	4	no	—
363	D₈	—	134	12	no	—
364	D₈	—	134	6	no	—
365	C₂×C₈	—	2	6	no	—
366	C₂×C₈	—	4	3	no	—
367	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
368	C₂×C₈	C₂³×D₆₇	1	3	yes	—
369	C₂×C₈	C₂³×D₆₇	1	3	yes	—
370	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
371	C₄⋊C₄	—	67	2	no	—
372	C₄⋊C₄	—	67	6	no	—
373	C₄⋊C₄	—	134	6	no	—
374	C₄⋊C₄	—	134	6	no	—
375	C₄⋊C₄	C₂×D₂₆₈	1	3	yes	—
376	C₄⋊C₄	C₂×D₂₆₈	1	3	yes	—
377	C₂²⋊C₄	—	134	2	no	—
378	C₂²⋊C₄	—	134	6	no	—
379	C₂²⋊C₄	—	134	6	no	—
380	C₂²⋊C₄	—	134	6	no	—
381	C₂²⋊C₄	—	134	12	no	—
382	C₂²⋊C₄	C₂×D₂₆₈	1	6	yes	—
383	C₄²	C₂×D₂₆₈	1	3	yes	—
384	C₄²	C₂×D₂₆₈	1	3	yes	—
385	C₄²	C₂³×D₆₇	1	3	yes	—
386	C₄²	C₂³×D₆₇	1	1	yes	characteristic
387	C₂³	—	268	12	no	—
388	C₂³	—	268	12	no	—
389	C₂³	—	536	6	no	—
390	C₂³	—	134	6	no	—
391	C₂³	—	134	2	no	—
392	C₂³	C₂²×D₂₆₈	1	3	yes	—
393	Q₈	—	268	12	no	—
394	Q₈	C₂×D₅₃₆	1	4	yes	complemented
395	D₄	—	268	12	no	—
396	D₄	—	268	24	no	—
397	D₄	—	268	12	no	—
398	D₄	—	536	6	no	—
399	D₄	—	67	4	no	—
400	D₄	—	134	6	no	—
401	D₄	—	134	6	no	—
402	D₄	—	134	2	no	—
403	D₄	C₂×D₅₃₆	1	12	yes	complemented
404	C₂×C₄	—	2	3	no	—
405	C₂×C₄	—	2	3	no	—
406	C₂×C₄	—	4	3	no	—
407	C₂×C₄	—	268	12	no	—
408	C₂×C₄	—	268	4	no	—
409	C₂×C₄	—	268	12	no	—
410	C₂×C₄	—	268	12	no	—
411	C₂×C₄	—	268	12	no	—
412	C₂×C₄	—	134	2	no	—
413	C₂×C₄	—	134	6	no	—
414	C₂×C₄	C₂×D₅₃₆	1	12	yes	complemented
415	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
416	C₂×C₄	C₂²×D₂₆₈	1	3	yes	—
417	C₂×C₄	C₂²×D₂₆₈	1	3	yes	—
418	C₂×C₄	C₂²×D₂₆₈	1	3	yes	—
419	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
420	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
421	C₈	—	2	3	no	—
422	C₈	—	2	1	no	—
423	C₈	—	2	3	no	—
424	C₈	#41×D₆₇	1	2	yes	complemented
425	C₂²	—	2	3	no	—
426	C₂²	—	2	3	no	—
427	C₂²	—	268	2	no	—
428	C₂²	—	268	2	no	—
429	C₂²	—	268	6	no	—
430	C₂²	—	268	6	no	—
431	C₂²	—	268	6	no	—
432	C₂²	—	536	24	no	—
433	C₂²	—	134	4	no	—
434	C₂²	C₂²×D₅₃₆	1	6	yes	—
435	C₂²	C₂³×D₂₆₈	1	1	yes	characteristic
436	C₄	—	2	3	no	—
437	C₄	—	2	1	no	—
438	C₄	—	2	3	no	—
439	C₄	—	268	2	no	—
440	C₄	—	268	6	no	—
441	C₄	C₂²×D₅₃₆	1	2	yes	—
442	C₄	C₂²×D₅₃₆	1	6	yes	—
443	C₄	C₂×#41×D₆₇	1	1	yes	characteristic
444	C₄	#16864232	1	1	yes	characteristic
445	C₂	—	2	6	no	—
446	C₂	—	268	4	no	—
447	C₂	—	536	6	no	—
448	C₂	#41×D₂₆₈	1	1	yes	characteristic
449	C₂	C₂³×D₅₃₆	1	1	yes	characteristic
450	C₂	#22574983	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator, cosocle	C₂×C₄×C₆₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₈×#41×C₆₇	C₂
socle	C₂²×C₆₇	C₂³×D₄
hypercenter	C₈×#41	D₆₇
hypocenter	C₆₇	D₈×#41
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
1. #41×D₂₆₈	C₂	semidirect
2. #22573184	C₂	semidirect
3. Q₈×D₅₃₆	C₂	semidirect
4. #22572845	C₂	semidirect
5. #22572557	C₂	semidirect
6. #22572023	C₂	semidirect
7. D₄×D₅₃₆	C₂	semidirect
8. #22570788	C₂	semidirect
9. #22570210	C₂	semidirect
10. C₂×C₄×D₅₃₆	C₂	semidirect
11. C₈×#41×C₆₇	C₂	semidirect
14. #16864068	C₂²	semidirect
15. Q₈×D₂₆₈	C₂²	semidirect
17. #16864050	C₂²	semidirect
19. D₄×D₂₆₈	C₂²	semidirect
20. #16863946	C₂²	semidirect
22. C₂×C₄×D₂₆₈	C₂²	semidirect
23. #16863801	C₂²	semidirect
24. #16863773	C₂²	semidirect
25. #16863743	C₂²	semidirect
26. #16863710	C₂²	semidirect
27. #16863677	C₂²	semidirect
28. #16863677	C₂²	semidirect
29. #16863632	C₂²	semidirect
30. #16863632	C₂²	semidirect
31. #16863592	C₂²	semidirect
32. #16863592	C₂²	semidirect
33. #16863532	C₂²	semidirect
36. C₄×D₅₃₆	C₂²	semidirect
37. C₄×D₅₃₆	C₂²	semidirect
38. C₄×D₅₃₆	C₂²	semidirect
39. C₄×D₅₃₆	C₂²	semidirect
40. C₄×D₅₃₆	C₂²	semidirect
41. C₂×#14523977	C₂²	semidirect
42. C₂×#14523965	C₂²	semidirect
44. C₈×Q₈×C₆₇	C₂²	semidirect
45. C₈×Q₈×C₆₇	C₂²	semidirect
46. C₈×D₄×C₆₇	C₂²	semidirect
47. C₈×D₄×C₆₇	C₂²	semidirect
48. #431×C₆₇	C₂²	semidirect
49. C₂×C₄×C₈×C₆₇	C₂²	semidirect
132. D₅₃₆	#41	direct (N × H)
142. #41×C₆₇	D₈	semidirect
167. Q₈×C₆₇	C₂×D₈	semidirect
168. D₄×C₆₇	C₂×D₈	semidirect
172. C₂×C₄×C₆₇	C₂×D₈	semidirect
189. D₂₆₈	C₂×#41	semidirect
198. C₈×C₆₇	C₂×#41	semidirect
235. C₈×#41	D₆₇	semidirect
237. C₆₇	D₈×#41	semidirect
261. C₈×Q₈	C₂×D₆₇	semidirect
262. C₈×Q₈	C₂×D₆₇	semidirect
270. C₈×D₄	C₂×D₆₇	semidirect
271. C₈×D₄	C₂×D₆₇	semidirect
272. #431	C₂×D₆₇	semidirect
275. C₂×C₄×C₈	C₂×D₆₇	semidirect
338. #41	D₅₃₆	direct (N × H)
394. Q₈	C₂×D₅₃₆	semidirect
403. D₄	C₂×D₅₃₆	semidirect
414. C₂×C₄	C₂×D₅₃₆	semidirect
424. C₈	#41×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
#41×D₂₆₈	2
#22573184	2
Q₈×D₅₃₆	2
#22572845	2
#22572557	2
#22572023	2
D₄×D₅₃₆	2
#22570788	2
#22570210	2
C₂×C₄×D₅₃₆	2
C₈×#41×C₆₇	2
D₈×#41	67

Subgroups — #41×D536 #31737238

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #41×D₅₃₆ #31737238