Subgroups — #30862288

Order 17152.

910 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569666	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863519	C₂	1	2	yes	maximal, complemented
3	#22568922	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#14524049	C₂	1	1	yes	maximal, characteristic, complemented
5	#22561127	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560985	C₂	1	1	yes	maximal, characteristic, complemented
7	#22560858	C₂	1	2	yes	maximal, complemented
8	#22560535	C₂	1	2	yes	maximal, complemented
9	#22560230	C₂	1	1	yes	maximal, characteristic, complemented
10	#22560053	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1770	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524065	C₂²	1	2	yes	—
13	C₂×#14524049	C₂²	1	1	yes	characteristic
14	C₂×#14524049	C₂²	1	2	yes	—
15	C₂×#14524049	C₂²	1	1	yes	characteristic
16	#16863946	C₂²	1	1	yes	characteristic
17	C₄×#11761628	C₂²	1	1	yes	characteristic
18	#16863619	C₄	1	2	yes	complemented
19	#16863612	C₄	1	2	yes	complemented
20	#16863607	C₄	1	1	yes	characteristic, complemented
21	#16863602	C₄	1	1	yes	characteristic, complemented
22	#16863597	C₄	1	1	yes	characteristic, complemented
23	#16863592	C₄	1	1	yes	characteristic, complemented
24	#16863519	—	2	8	no	—
25	#16863475	—	2	1	no	—
26	#16863475	—	2	1	no	—
27	C₈×D₂₆₈	—	2	1	no	—
28	C₈×D₂₆₈	—	2	1	no	—
29	C₂×#14523979	C₂²	1	2	yes	—
30	#16863420	C₂²	1	1	yes	characteristic
31	C₂×#14523971	C₂²	1	2	yes	—
32	C₂×#14523971	C₂²	1	2	yes	—
33	C₂×#14523971	C₂²	1	2	yes	—
34	#16863393	C₂²	1	2	yes	—
35	#16863362	C₂²	1	2	yes	—
36	#16863351	C₂²	1	1	yes	characteristic
37	#16863351	C₂²	1	1	yes	characteristic
38	#16863350	—	2	1	no	—
39	#16863350	—	2	1	no	—
40	C₄×#11761612	C₂²	1	1	yes	characteristic
41	C₄×#11761611	—	2	1	no	—
42	C₄×C₈×D₆₇	—	2	1	no	—
43	#16863161	C₂²	1	2	yes	—
44	#16863156	C₂²	1	2	yes	—
45	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
46	C₂×#97×C₆₇	C₂²	1	2	yes	—
47	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
48	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
49	#336×C₆₇	C₂²	1	2	yes	—
50	C₂×#11761628	C₂³	1	2	yes	—
51	C₂×#11761628	C₂³	1	1	yes	characteristic
52	#14524065	—	2	4	no	—
53	#14524065	—	2	4	no	—
54	C₂×Dic₂₆₈	C₂×C₄	1	2	yes	—
55	C₂×#11761617	C₂×C₄	1	2	yes	—
56	C₂×#11761617	C₂×C₄	1	2	yes	—
57	C₂×D₅₃₆	C₂×C₄	1	2	yes	—
58	#14524049	—	2	2	no	—
59	#14524049	—	2	2	no	—
60	#14524049	—	2	4	no	—
61	#14524049	—	2	2	no	—
62	#14524049	—	2	4	no	—
63	#14524049	D₄	1	4	yes	complemented
64	C₂×#11761612	C₂³	1	2	yes	—
65	C₂×#11761612	C₂³	1	1	yes	characteristic
66	C₂×#11761612	C₂³	1	1	yes	characteristic
67	C₂×#11761611	—	2	1	no	—
68	C₂×#11761611	—	2	1	no	—
69	C₂×#11761611	—	2	2	no	—
70	C₂×C₈×D₆₇	—	2	1	no	—
71	C₂×C₈×D₆₇	—	2	1	no	—
72	C₂×C₈×D₆₇	—	2	2	no	—
73	#14524043	C₂×C₄	1	1	yes	characteristic
74	#14524040	C₂×C₄	1	1	yes	characteristic
75	#14524031	C₂×C₄	1	2	yes	—
76	#14524017	—	2	1	no	—
77	#14524017	—	2	1	no	—
78	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
79	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
80	C₄×D₂₆₈	—	2	1	no	—
81	C₄×D₂₆₈	—	2	1	no	—
82	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
83	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
84	C₄×C₂²⋊D₆₇	—	2	1	no	—
85	C₄×C₂²⋊D₆₇	—	2	1	no	—
86	#14524002	C₂³	1	1	yes	characteristic
87	#14524000	—	2	1	no	—
88	C₄²×D₆₇	—	2	1	no	—
89	#14523979	—	2	4	no	—
90	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
91	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
92	#14523971	—	2	4	no	—
93	#14523971	—	2	4	no	—
94	#14523971	—	2	4	no	—
95	#14523968	C₂×C₄	1	2	yes	—
96	#14523968	C₂×C₄	1	2	yes	—
97	#14523965	C₂×C₄	1	2	yes	—
98	#14523965	C₂×C₄	1	2	yes	—
99	(C₂×D₆₇)⋊C₈	—	2	2	no	—
100	(C₂×D₆₇)⋊C₈	—	2	2	no	—
101	#14523954	—	2	1	no	—
102	#14523953	—	2	1	no	—
103	#14523953	—	2	1	no	—
104	C₄×C₆₇⋊C₈	—	2	1	no	—
105	C₈×Dic₆₇	—	2	1	no	—
106	C₈×Dic₆₇	—	2	1	no	—
107	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
108	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
109	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
110	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
111	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
112	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
113	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
114	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
115	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
116	#97×C₆₇	—	2	4	no	—
117	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
118	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
119	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
120	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
121	#95×C₆₇	—	2	1	no	—
122	#11761628	—	2	1	no	—
123	#11761628	—	2	1	no	—
124	#11761628	—	2	2	no	—
125	#11761628	—	2	2	no	—
126	#11761628	—	2	2	no	—
127	#11761628	—	2	2	no	—
128	#11761628	C₂×D₄	1	2	yes	—
129	#11761628	#41	1	2	yes	—
130	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
131	C₂×Dic₁₃₄	C₂²×C₄	1	1	yes	characteristic
132	C₂×D₂₆₈	—	2	1	no	—
133	C₂×D₂₆₈	—	2	1	no	—
134	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
135	C₂×D₂₆₈	C₂²×C₄	1	1	yes	characteristic
136	C₂×C₂²⋊D₆₇	—	2	1	no	—
137	C₂×C₂²⋊D₆₇	—	2	2	no	—
138	C₂×C₄×D₆₇	—	2	1	no	—
139	C₂×C₄×D₆₇	—	2	1	no	—
140	C₂×C₄×D₆₇	—	2	1	no	—
141	C₂×C₄×D₆₇	—	2	2	no	—
142	Dic₂₆₈	—	2	4	no	—
143	#11761617	—	2	4	no	—
144	#11761617	—	2	4	no	—
145	D₅₃₆	—	2	4	no	—
146	#11761612	—	2	1	no	—
147	#11761612	—	2	1	no	—
148	#11761612	—	2	2	no	—
149	#11761612	—	2	2	no	—
150	#11761612	C₂×D₄	1	2	yes	—
151	#11761612	#41	1	2	yes	—
152	#11761611	—	2	4	no	—
153	#11761611	—	2	4	no	—
154	#11761611	—	4	4	no	—
155	C₂×C₆₇⋊C₈	—	2	1	no	—
156	C₂×C₆₇⋊C₈	—	2	1	no	—
157	C₂×C₆₇⋊C₈	—	2	2	no	—
158	C₈×D₆₇	—	2	4	no	—
159	C₈×D₆₇	—	2	4	no	—
160	C₈×D₆₇	—	4	4	no	—
161	C₄⋊Dic₆₇	—	2	1	no	—
162	C₄⋊Dic₆₇	—	2	1	no	—
163	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
164	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
165	Dic₆₇⋊C₄	—	2	1	no	—
166	Dic₆₇⋊C₄	—	2	1	no	—
167	C₂²⋊Dic₆₇	—	2	1	no	—
168	(C₂×D₆₇)⋊C₄	—	2	1	no	—
169	(C₂×D₆₇)⋊C₄	—	2	1	no	—
170	(C₂×D₆₇)⋊C₄	—	2	2	no	—
171	(C₂×D₆₇)⋊C₄	—	2	2	no	—
172	C₄×Dic₆₇	—	2	1	no	—
173	C₄×Dic₆₇	—	2	1	no	—
174	C₄×Dic₆₇	—	2	1	no	—
175	C₂²×C₄×C₆₇	—	2	2	no	—
176	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
177	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
178	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
179	C₂×C₈×C₆₇	—	2	2	no	—
180	C₂×C₈×C₆₇	—	2	1	no	—
181	C₂×C₈×C₆₇	—	2	1	no	—
182	C₂×C₈×C₆₇	—	2	2	no	—
183	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
184	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
185	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
186	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
187	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
188	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
189	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
190	C₂×C₈×C₆₇	#41	1	2	yes	—
191	C₄⋊C₄×C₆₇	—	4	2	no	—
192	C₄²×C₆₇	—	2	2	no	—
193	C₄²×C₆₇	—	2	2	no	—
194	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
195	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
196	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
197	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
198	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
199	C₂²×D₆₇	—	2	1	no	—
200	C₂²×D₆₇	—	2	2	no	—
201	C₂×C₄×C₆₇	—	2	2	no	—
202	C₂×C₄×C₆₇	—	4	1	no	—
203	C₂×C₄×C₆₇	—	4	2	no	—
204	C₂×C₄×C₆₇	—	4	2	no	—
205	C₂×C₄×C₆₇	—	4	2	no	—
206	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
207	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
208	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
209	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
210	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
211	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
212	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
213	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
214	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
215	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
216	C₂²⋊D₆₇	—	2	2	no	—
217	C₂²⋊D₆₇	—	2	2	no	—
218	C₂²⋊D₆₇	—	4	2	no	—
219	C₂²⋊D₆₇	—	4	2	no	—
220	C₂×Dic₆₇	—	2	1	no	—
221	C₂×Dic₆₇	—	2	1	no	—
222	C₂×Dic₆₇	—	2	1	no	—
223	C₂×Dic₆₇	—	2	2	no	—
224	D₂₆₈	—	2	2	no	—
225	D₂₆₈	—	2	1	no	—
226	D₂₆₈	—	2	2	no	—
227	D₂₆₈	—	4	2	no	—
228	D₂₆₈	—	4	2	no	—
229	D₂₆₈	C₄×D₄	1	2	yes	—
230	C₄×D₆₇	—	2	2	no	—
231	C₄×D₆₇	—	2	2	no	—
232	C₄×D₆₇	—	4	2	no	—
233	C₄×D₆₇	—	4	2	no	—
234	C₄×D₆₇	—	4	2	no	—
235	C₄×D₆₇	—	4	2	no	—
236	Dic₁₃₄	—	2	1	no	—
237	Dic₁₃₄	—	2	2	no	—
238	Dic₁₃₄	—	2	2	no	—
239	Dic₁₃₄	C₄×D₄	1	2	yes	—
240	C₈×C₆₇	—	2	2	no	—
241	C₈×C₆₇	—	2	2	no	—
242	C₈×C₆₇	—	2	1	no	—
243	C₈×C₆₇	—	2	1	no	—
244	C₈×C₆₇	#121	1	2	yes	—
245	C₈×C₆₇	#121	1	2	yes	—
246	C₆₇⋊C₈	—	2	2	no	—
247	C₆₇⋊C₈	—	2	2	no	—
248	C₆₇⋊C₈	—	4	2	no	—
249	C₂²×C₆₇	—	2	1	no	—
250	C₂²×C₆₇	—	2	1	no	—
251	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
252	C₂²×C₆₇	#529	1	1	yes	characteristic
253	C₂²×C₆₇	#532	1	1	yes	characteristic
254	C₂×D₆₇	—	2	2	no	—
255	C₂×D₆₇	—	4	1	no	—
256	C₂×D₆₇	—	4	1	no	—
257	C₂×D₆₇	—	4	2	no	—
258	C₂×D₆₇	—	4	2	no	—
259	C₂×D₆₇	—	4	2	no	—
260	C₄×C₆₇	—	2	1	no	—
261	C₄×C₆₇	—	2	1	no	—
262	C₄×C₆₇	—	2	1	no	—
263	C₄×C₆₇	—	8	2	no	—
264	C₄×C₆₇	#444	1	2	yes	complemented
265	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
266	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
267	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
268	C₄×C₆₇	#525	1	1	yes	characteristic
269	Dic₆₇	—	2	2	no	—
270	Dic₆₇	—	4	1	no	—
271	Dic₆₇	—	4	1	no	—
272	Dic₆₇	—	4	2	no	—
273	#21518	—	67	1	no	maximal, Sylow
274	C₂×C₆₇	—	2	1	no	—
275	C₂×C₆₇	—	2	1	no	—
276	C₂×C₆₇	C₄×#121	1	1	yes	characteristic
277	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
278	C₂×C₆₇	#2955	1	1	yes	characteristic
279	D₆₇	—	4	2	no	—
280	D₆₇	—	8	2	no	—
281	#3035	—	67	1	no	—
282	C₂×#444	—	67	2	no	—
283	#2931	—	67	1	no	—
284	C₄×#131	—	67	1	no	—
285	#1980	—	67	1	no	—
286	#1950	—	67	1	no	—
287	#1929	—	67	2	no	—
288	#1876	—	67	2	no	—
289	#1806	—	67	1	no	—
290	#1770	D₆₇	1	1	yes	characteristic, complemented
291	#1765	—	67	2	no	—
292	C₆₇	#21518	1	1	yes	characteristic, complemented, Sylow
293	C₂×#135	—	67	2	no	—
294	C₂×#131	—	67	1	no	—
295	C₂×#131	—	67	1	no	—
296	C₂×#131	—	67	2	no	—
297	#532	—	67	1	no	—
298	C₄×#41	—	67	1	no	—
299	#464	—	67	2	no	—
300	#463	—	67	2	no	—
301	#462	—	67	1	no	—
302	#461	—	67	1	no	—
303	#460	—	67	1	no	—
304	#459	—	67	1	no	—
305	#444	—	134	8	no	—
306	#436	—	134	1	no	—
307	#436	—	134	1	no	—
308	C₈×D₄	—	134	1	no	—
309	C₈×D₄	—	134	1	no	—
310	C₂×#108	—	67	2	no	—
311	#423	—	67	1	no	—
312	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
313	C₂×#104	—	67	2	no	—
314	C₂×#104	—	67	2	no	—
315	C₂×#104	—	67	2	no	—
316	#416	—	67	2	no	—
317	#408	—	67	2	no	—
318	#405	—	67	1	no	—
319	#405	—	67	1	no	—
320	#405	—	134	1	no	—
321	#405	—	134	1	no	—
322	C₄×M₁₆	—	67	1	no	—
323	C₄×M₁₆	—	134	1	no	—
324	C₂×#97	C₂×D₆₇	1	2	yes	—
325	C₂×C₄×C₈	—	134	1	no	—
326	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
327	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
328	#341	—	67	2	no	—
329	#339	—	67	2	no	—
330	#336	C₂×D₆₇	1	2	yes	—
331	C₂×#41	—	67	1	no	—
332	C₂×#41	—	67	2	no	—
333	#135	—	134	4	no	—
334	#135	—	134	4	no	—
335	C₂×Q₁₆	—	67	2	no	—
336	C₂×SD₁₆	—	67	2	no	—
337	C₂×SD₁₆	—	67	2	no	—
338	C₂×D₈	—	67	2	no	—
339	#131	—	67	4	no	—
340	#131	—	134	2	no	—
341	#131	—	134	2	no	—
342	#131	—	134	4	no	—
343	#131	—	134	4	no	—
344	#131	—	134	2	no	—
345	C₂×M₁₆	—	67	1	no	—
346	C₂×M₁₆	—	67	1	no	—
347	C₂×M₁₆	—	67	2	no	—
348	C₂×M₁₆	—	134	2	no	—
349	C₂×M₁₆	—	134	1	no	—
350	C₂×M₁₆	—	134	1	no	—
351	C₂²×C₈	—	134	2	no	—
352	C₂²×C₈	—	134	1	no	—
353	C₂²×C₈	—	134	1	no	—
354	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
355	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
356	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
357	C₄⋊Q₈	—	67	1	no	—
358	#127	—	67	1	no	—
359	#124	—	67	2	no	—
360	#121	—	134	1	no	—
361	#121	—	134	1	no	—
362	C₄×Q₈	—	67	1	no	—
363	C₄×Q₈	—	67	1	no	—
364	C₄×D₄	—	67	1	no	—
365	C₄×D₄	—	67	1	no	—
366	C₄×D₄	—	134	1	no	—
367	C₄×D₄	—	134	1	no	—
368	C₄×D₄	—	134	1	no	—
369	C₄×D₄	—	134	1	no	—
370	#117	—	67	1	no	—
371	#117	—	134	1	no	—
372	C₂×C₄⋊C₄	—	2	1	no	—
373	C₂×C₄²	—	134	1	no	—
374	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
375	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
376	#108	—	134	4	no	—
377	C₄⋊C₈	—	134	1	no	—
378	C₄⋊C₈	—	134	1	no	—
379	C₄⋊C₈	C₄×D₆₇	1	1	yes	characteristic
380	C₄⋊C₈	C₄×D₆₇	1	1	yes	characteristic
381	C₄⋊C₈	C₄×D₆₇	1	2	yes	—
382	#104	—	134	4	no	—
383	#104	—	134	4	no	—
384	#104	—	134	4	no	—
385	#103	—	67	2	no	—
386	#103	—	67	2	no	—
387	#102	—	67	2	no	—
388	#102	—	67	2	no	—
389	C₂²⋊C₈	—	134	2	no	—
390	C₂²⋊C₈	—	134	2	no	—
391	#97	—	2	4	no	—
392	#97	—	134	1	no	—
393	#97	—	134	1	no	—
394	#97	—	134	1	no	—
395	C₄×C₈	—	134	1	no	—
396	C₄×C₈	—	134	1	no	—
397	C₄×C₈	—	134	1	no	—
398	C₄×C₈	D₂₆₈	1	1	yes	characteristic
399	C₄×C₈	D₂₆₈	1	1	yes	characteristic
400	C₄×C₈	D₂₆₈	1	1	yes	characteristic
401	C₄×C₈	D₂₆₈	1	1	yes	characteristic
402	#95	—	2	1	no	—
403	#41	—	67	2	no	—
404	#41	—	67	2	no	—
405	#41	—	134	2	no	—
406	#41	—	134	2	no	—
407	#41	—	134	2	no	—
408	#41	—	134	2	no	—
409	#41	—	134	1	no	—
410	#41	—	134	1	no	—
411	C₂×Q₈	—	67	1	no	—
412	C₂×Q₈	—	67	2	no	—
413	C₂×D₄	—	67	1	no	—
414	C₂×D₄	—	67	2	no	—
415	C₂×D₄	—	134	2	no	—
416	C₂×D₄	—	134	1	no	—
417	C₂×D₄	—	134	1	no	—
418	C₂×D₄	—	134	1	no	—
419	C₂²×C₄	—	2	2	no	—
420	C₂²×C₄	—	134	1	no	—
421	C₂²×C₄	—	134	1	no	—
422	C₂²×C₄	—	134	2	no	—
423	C₂²×C₄	—	134	1	no	—
424	C₂²×C₄	#11761628	1	1	yes	characteristic
425	C₂²×C₄	#11761628	1	1	yes	characteristic
426	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
427	Q₁₆	—	134	4	no	—
428	SD₁₆	—	134	4	no	—
429	SD₁₆	—	134	4	no	—
430	D₈	—	134	4	no	—
431	M₁₆	—	268	4	no	—
432	M₁₆	—	67	2	no	—
433	M₁₆	—	67	2	no	—
434	M₁₆	—	134	4	no	—
435	M₁₆	—	134	2	no	—
436	M₁₆	—	134	2	no	—
437	M₁₆	—	134	1	no	—
438	M₁₆	—	134	1	no	—
439	M₁₆	—	134	4	no	—
440	C₂×C₈	—	2	2	no	—
441	C₂×C₈	—	2	1	no	—
442	C₂×C₈	—	2	1	no	—
443	C₂×C₈	—	2	2	no	—
444	C₂×C₈	—	268	4	no	—
445	C₂×C₈	—	134	4	no	—
446	C₂×C₈	—	134	2	no	—
447	C₂×C₈	—	134	1	no	—
448	C₂×C₈	—	134	1	no	—
449	C₂×C₈	—	134	4	no	—
450	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
451	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
452	C₂×C₈	D₄×D₆₇	1	2	yes	—
453	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
454	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
455	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
456	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
457	C₂×C₈	#11761629	1	2	yes	—
458	C₄⋊C₄	—	4	2	no	—
459	C₄⋊C₄	—	67	1	no	—
460	C₄⋊C₄	—	67	1	no	—
461	C₄⋊C₄	—	134	1	no	—
462	C₄⋊C₄	—	134	1	no	—
463	C₄⋊C₄	—	134	1	no	—
464	C₄⋊C₄	—	134	1	no	—
465	C₂²⋊C₄	—	134	1	no	—
466	C₂²⋊C₄	—	134	1	no	—
467	C₂²⋊C₄	—	134	1	no	—
468	C₂²⋊C₄	—	134	2	no	—
469	C₂²⋊C₄	—	134	2	no	—
470	C₄²	—	2	2	no	—
471	C₄²	—	2	2	no	—
472	C₄²	—	134	1	no	—
473	C₄²	—	134	1	no	—
474	C₄²	—	134	1	no	—
475	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
476	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
477	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
478	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
479	C₂³	—	134	2	no	—
480	C₂³	—	134	1	no	—
481	C₂³	C₂×#11761628	1	1	yes	characteristic
482	Q₈	—	67	2	no	—
483	Q₈	—	134	2	no	—
484	Q₈	—	134	2	no	—
485	Q₈	—	134	1	no	—
486	D₄	—	268	2	no	—
487	D₄	—	268	2	no	—
488	D₄	—	268	2	no	—
489	D₄	—	268	2	no	—
490	D₄	—	67	2	no	—
491	D₄	—	134	2	no	—
492	D₄	—	134	2	no	—
493	D₄	—	134	2	no	—
494	D₄	—	134	2	no	—
495	D₄	—	134	1	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	4	1	no	—
498	C₂×C₄	—	4	2	no	—
499	C₂×C₄	—	4	2	no	—
500	C₂×C₄	—	4	2	no	—
501	C₂×C₄	—	268	2	no	—
502	C₂×C₄	—	268	2	no	—
503	C₂×C₄	—	268	2	no	—
504	C₂×C₄	—	268	2	no	—
505	C₂×C₄	—	134	1	no	—
506	C₂×C₄	—	134	1	no	—
507	C₂×C₄	—	134	2	no	—
508	C₂×C₄	—	134	1	no	—
509	C₂×C₄	—	134	2	no	—
510	C₂×C₄	—	134	2	no	—
511	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
512	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
513	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
514	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
515	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
516	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
517	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
518	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
519	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
520	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
521	C₈	—	2	2	no	—
522	C₈	—	2	2	no	—
523	C₈	—	2	1	no	—
524	C₈	—	2	1	no	—
525	C₈	—	268	2	no	—
526	C₈	—	134	2	no	—
527	C₈	—	134	2	no	—
528	C₈	#14524018	1	2	yes	—
529	C₈	#14524018	1	2	yes	—
530	C₂²	—	2	1	no	—
531	C₂²	—	2	1	no	—
532	C₂²	—	268	1	no	—
533	C₂²	—	268	1	no	—
534	C₂²	—	268	2	no	—
535	C₂²	—	268	2	no	—
536	C₂²	—	268	2	no	—
537	C₂²	—	134	2	no	—
538	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
539	C₂²	#16863933	1	1	yes	characteristic
540	C₂²	#16863943	1	1	yes	characteristic
541	C₄	—	2	1	no	—
542	C₄	—	2	1	no	—
543	C₄	—	2	1	no	—
544	C₄	—	8	2	no	—
545	C₄	—	268	1	no	—
546	C₄	—	268	1	no	—
547	C₄	—	268	2	no	—
548	C₄	—	134	2	no	—
549	C₄	#16863519	1	2	yes	complemented
550	C₄	C₄×D₄×D₆₇	1	1	yes	characteristic
551	C₄	C₄×#11761629	1	1	yes	characteristic
552	C₄	C₂×#14524018	1	1	yes	characteristic
553	C₄	#16863908	1	1	yes	characteristic
554	C₂	—	2	1	no	—
555	C₂	—	2	1	no	—
556	C₂	—	268	2	no	—
557	C₂	—	536	2	no	—
558	C₂	C₄×#14524018	1	1	yes	characteristic
559	C₂	C₂×#16863519	1	1	yes	characteristic
560	C₂	#22569113	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1770	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#22569666	C₂
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1770	D₆₇
hypocenter	C₆₇	#21518
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1770

Complements

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

N	H ≅ G/N	type
2. C₂×#16863519	C₂	semidirect
3. #22568922	C₂	semidirect
4. C₄×#14524049	C₂	semidirect
5. #22561127	C₂	semidirect
6. #22560985	C₂	semidirect
7. #22560858	C₂	semidirect
8. #22560535	C₂	semidirect
9. #22560230	C₂	semidirect
10. #22560053	C₂	semidirect
11. C₆₇×#1770	C₂	semidirect
18. #16863619	C₄	semidirect
19. #16863612	C₄	semidirect
20. #16863607	C₄	semidirect
21. #16863602	C₄	semidirect
22. #16863597	C₄	semidirect
23. #16863592	C₄	semidirect
63. #14524049	D₄	semidirect
264. C₄×C₆₇	#444	semidirect
290. #1770	D₆₇	semidirect
292. C₆₇	#21518	semidirect
549. C₄	#16863519	semidirect

Maximal subgroups

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

subgroup	index
#22569666	2
C₂×#16863519	2
#22568922	2
C₄×#14524049	2
#22561127	2
#22560985	2
#22560858	2
#22560535	2
#22560230	2
#22560053	2
C₆₇×#1770	2
#21518	67