Subgroups — #30862280

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	#22569658	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863516	C₂	1	2	yes	maximal, complemented
3	C₄×#14524049	C₂	1	1	yes	maximal, characteristic, complemented
4	#22561129	C₂	1	1	yes	maximal, characteristic, complemented
5	#22560982	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560854	C₂	1	2	yes	maximal, complemented
7	#22560539	C₂	1	2	yes	maximal, complemented
8	#22560226	C₂	1	1	yes	maximal, characteristic, complemented
9	#22560076	C₂	1	1	yes	maximal, characteristic, complemented
10	#22560051	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2931	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524063	C₂²	1	2	yes	—
13	C₂×#14524049	C₂²	1	1	yes	characteristic
14	C₂×#14524049	C₂²	1	1	yes	characteristic
15	C₄×#11761628	C₂²	1	1	yes	characteristic
16	#16863615	C₄	1	2	yes	complemented
17	#16863608	C₄	1	2	yes	complemented
18	#16863603	C₄	1	1	yes	characteristic, complemented
19	#16863598	C₄	1	1	yes	characteristic, complemented
20	#16863593	C₄	1	1	yes	characteristic, complemented
21	#16863588	C₄	1	1	yes	characteristic, complemented
22	#16863516	—	2	8	no	—
23	C₂×#14523978	C₂²	1	2	yes	—
24	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
25	C₂×#14523970	C₂²	1	2	yes	—
26	C₂×#14523970	C₂²	1	2	yes	—
27	C₂×#14523969	C₂²	1	2	yes	—
28	#16863392	C₂²	1	2	yes	—
29	#16863351	C₂²	1	1	yes	characteristic
30	#16863351	C₂²	1	1	yes	characteristic
31	#16863350	—	2	1	no	—
32	#16863350	—	2	1	no	—
33	C₄×#11761612	C₂²	1	1	yes	characteristic
34	C₄×#11761611	—	2	1	no	—
35	C₂×#14523953	C₂²	1	2	yes	—
36	C₄×C₈×D₆₇	—	2	1	no	—
37	C₄×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
38	#16863160	C₂²	1	2	yes	—
39	#16863157	C₂²	1	2	yes	—
40	#16863150	C₂²	1	2	yes	—
41	C₂×#131×C₆₇	C₂²	1	2	yes	—
42	#532×C₆₇	C₂²	1	1	yes	characteristic
43	#436×C₆₇	—	2	1	no	—
44	#436×C₆₇	—	2	1	no	—
45	C₈×D₄×C₆₇	—	2	1	no	—
46	C₈×D₄×C₆₇	—	2	1	no	—
47	#423×C₆₇	C₂²	1	1	yes	characteristic
48	#408×C₆₇	C₂²	1	2	yes	—
49	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
50	C₂×#11761628	C₂³	1	1	yes	characteristic
51	#14524063	—	2	4	no	—
52	#14524063	—	2	4	no	—
53	C₂×#11761618	C₂×C₄	1	2	yes	—
54	C₂×#11761616	C₂×C₄	1	2	yes	—
55	C₂×#11761615	C₂×C₄	1	2	yes	—
56	C₂×#11761613	C₂×C₄	1	2	yes	—
57	#14524049	—	2	2	no	—
58	#14524049	—	2	2	no	—
59	#14524049	—	2	2	no	—
60	#14524049	D₄	1	4	yes	complemented
61	C₂×#11761612	C₂³	1	1	yes	characteristic
62	C₂×#11761612	C₂³	1	1	yes	characteristic
63	C₂×#11761611	—	2	1	no	—
64	C₂×#11761611	—	2	1	no	—
65	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
66	C₂×C₈×D₆₇	—	2	1	no	—
67	C₂×C₈×D₆₇	—	2	1	no	—
68	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
69	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
70	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
71	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
72	C₄×C₂²⋊D₆₇	—	2	1	no	—
73	C₄×C₂²⋊D₆₇	—	2	1	no	—
74	#14524002	C₂³	1	1	yes	characteristic
75	#14524000	—	2	1	no	—
76	C₂×C₄⋊Dic₆₇	—	2	1	no	—
77	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
78	C₄²×D₆₇	—	2	1	no	—
79	#14523978	—	2	4	no	—
80	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
81	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
82	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
83	#14523970	—	2	4	no	—
84	#14523970	—	2	4	no	—
85	#14523969	—	2	4	no	—
86	#14523967	C₂×C₄	1	2	yes	—
87	#14523967	C₂×C₄	1	2	yes	—
88	#14523964	C₂×C₄	1	2	yes	—
89	#14523964	C₂×C₄	1	2	yes	—
90	#14523954	—	2	1	no	—
91	#14523953	—	2	4	no	—
92	#14523953	—	2	1	no	—
93	#14523953	—	2	1	no	—
94	C₄×C₆₇⋊C₈	—	2	1	no	—
95	C₈×Dic₆₇	—	2	1	no	—
96	C₈×Dic₆₇	—	2	1	no	—
97	C₆₇⋊#95	—	2	1	no	—
98	C₂×#41×C₆₇	C₂³	1	2	yes	—
99	#131×C₆₇	—	2	4	no	—
100	#131×C₆₇	—	2	4	no	—
101	C₂×M₁₆×C₆₇	—	2	2	no	—
102	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
103	C₂²×C₈×C₆₇	—	2	2	no	—
104	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
105	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
106	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
107	#127×C₆₇	C₂×C₄	1	1	yes	characteristic
108	#124×C₆₇	C₂×C₄	1	2	yes	—
109	#121×C₆₇	—	2	1	no	—
110	#121×C₆₇	—	2	1	no	—
111	C₄×D₄×C₆₇	—	2	1	no	—
112	C₄×D₄×C₆₇	—	2	1	no	—
113	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
114	C₄⋊C₈×C₆₇	—	2	1	no	—
115	C₄⋊C₈×C₆₇	—	2	1	no	—
116	C₂²⋊C₈×C₆₇	—	2	2	no	—
117	C₂²⋊C₈×C₆₇	—	2	2	no	—
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	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
122	#11761628	—	2	1	no	—
123	#11761628	—	2	1	no	—
124	#11761628	C₂×D₄	1	2	yes	—
125	#11761628	#41	1	2	yes	—
126	C₂×Dic₁₃₄	C₂²×C₄	1	1	yes	characteristic
127	C₂×D₂₆₈	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₂²⋊D₆₇	—	2	1	no	—
129	C₂²×Dic₆₇	—	2	2	no	—
130	C₂×C₄×D₆₇	—	2	1	no	—
131	C₂×C₄×D₆₇	—	2	1	no	—
132	C₂×C₄×D₆₇	—	2	1	no	—
133	#11761618	—	2	4	no	—
134	#11761616	—	2	4	no	—
135	#11761615	—	2	4	no	—
136	#11761613	—	2	4	no	—
137	#11761612	—	2	1	no	—
138	#11761612	—	2	1	no	—
139	#11761612	C₂×D₄	1	2	yes	—
140	#11761612	#41	1	2	yes	—
141	#11761611	—	2	4	no	—
142	#11761611	—	2	4	no	—
143	C₂×C₆₇⋊C₈	—	2	1	no	—
144	C₂×C₆₇⋊C₈	—	2	1	no	—
145	C₂×C₆₇⋊C₈	—	2	2	no	—
146	C₂×C₆₇⋊C₈	—	2	2	no	—
147	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
148	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
149	C₈×D₆₇	—	2	4	no	—
150	C₈×D₆₇	—	2	4	no	—
151	C₄⋊Dic₆₇	—	4	2	no	—
152	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
153	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
154	Dic₆₇⋊C₄	—	2	1	no	—
155	Dic₆₇⋊C₄	—	2	1	no	—
156	C₂²⋊Dic₆₇	—	2	1	no	—
157	(C₂×D₆₇)⋊C₄	—	2	1	no	—
158	(C₂×D₆₇)⋊C₄	—	2	1	no	—
159	C₄×Dic₆₇	—	2	1	no	—
160	C₄×Dic₆₇	—	2	1	no	—
161	C₄×Dic₆₇	—	2	1	no	—
162	C₄×Dic₆₇	—	2	2	no	—
163	C₄×Dic₆₇	—	2	2	no	—
164	#41×C₆₇	—	2	2	no	—
165	#41×C₆₇	—	2	2	no	—
166	#41×C₆₇	—	2	2	no	—
167	#41×C₆₇	—	2	2	no	—
168	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
169	C₂×D₄×C₆₇	—	2	2	no	—
170	C₂×D₄×C₆₇	—	2	1	no	—
171	C₂×D₄×C₆₇	—	2	1	no	—
172	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
173	C₂²×C₄×C₆₇	—	2	2	no	—
174	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
175	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
176	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
177	M₁₆×C₆₇	—	2	2	no	—
178	M₁₆×C₆₇	—	2	2	no	—
179	M₁₆×C₆₇	—	4	4	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	—	4	4	no	—
184	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
185	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
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	#41	1	2	yes	—
190	C₄⋊C₄×C₆₇	—	2	1	no	—
191	C₄⋊C₄×C₆₇	—	2	1	no	—
192	C₂²⋊C₄×C₆₇	—	2	2	no	—
193	C₂²⋊C₄×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	—	2	2	no	—
199	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
200	C₂²×D₆₇	—	2	1	no	—
201	Q₈×C₆₇	—	2	2	no	—
202	Q₈×C₆₇	—	2	2	no	—
203	D₄×C₆₇	—	2	2	no	—
204	D₄×C₆₇	—	2	2	no	—
205	D₄×C₆₇	—	4	2	no	—
206	D₄×C₆₇	—	4	2	no	—
207	D₄×C₆₇	—	4	2	no	—
208	D₄×C₆₇	—	4	2	no	—
209	C₂×C₄×C₆₇	—	2	2	no	—
210	C₂×C₄×C₆₇	—	2	2	no	—
211	C₂×C₄×C₆₇	—	4	1	no	—
212	C₂×C₄×C₆₇	—	4	2	no	—
213	C₂×C₄×C₆₇	—	4	2	no	—
214	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
215	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
216	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
217	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
218	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
219	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
220	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
221	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
222	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
223	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
224	C₂²⋊D₆₇	—	2	2	no	—
225	C₂²⋊D₆₇	—	2	2	no	—
226	C₂×Dic₆₇	—	2	1	no	—
227	C₂×Dic₆₇	—	2	1	no	—
228	C₂×Dic₆₇	—	2	1	no	—
229	C₂×Dic₆₇	—	4	2	no	—
230	C₂×Dic₆₇	—	4	2	no	—
231	C₂×Dic₆₇	—	4	2	no	—
232	D₂₆₈	—	2	1	no	—
233	D₂₆₈	C₄×D₄	1	2	yes	—
234	C₄×D₆₇	—	2	2	no	—
235	C₄×D₆₇	—	2	2	no	—
236	C₄×D₆₇	—	4	2	no	—
237	C₄×D₆₇	—	4	2	no	—
238	Dic₁₃₄	—	2	1	no	—
239	Dic₁₃₄	C₄×D₄	1	2	yes	—
240	C₈×C₆₇	—	2	1	no	—
241	C₈×C₆₇	—	2	1	no	—
242	C₈×C₆₇	—	4	2	no	—
243	C₈×C₆₇	#121	1	2	yes	—
244	C₈×C₆₇	#121	1	2	yes	—
245	C₆₇⋊C₈	—	2	2	no	—
246	C₆₇⋊C₈	—	2	2	no	—
247	C₆₇⋊C₈	—	2	2	no	—
248	C₆₇⋊C₈	—	2	2	no	—
249	C₂²×C₆₇	—	2	1	no	—
250	C₂²×C₆₇	—	2	1	no	—
251	C₂²×C₆₇	—	4	2	no	—
252	C₂²×C₆₇	—	4	2	no	—
253	C₂²×C₆₇	—	4	2	no	—
254	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
255	C₂²×C₆₇	#529	1	1	yes	characteristic
256	C₂²×C₆₇	#532	1	1	yes	characteristic
257	C₂×D₆₇	—	2	2	no	—
258	C₂×D₆₇	—	4	1	no	—
259	C₂×D₆₇	—	4	1	no	—
260	C₄×C₆₇	—	2	1	no	—
261	C₄×C₆₇	—	2	1	no	—
262	C₄×C₆₇	—	2	1	no	—
263	C₄×C₆₇	—	4	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	—	8	2	no	—
273	#21518	—	67	1	no	maximal, Sylow
274	C₂×C₆₇	—	2	1	no	—
275	C₂×C₆₇	—	2	1	no	—
276	C₂×C₆₇	—	8	2	no	—
277	C₂×C₆₇	C₄×#121	1	1	yes	characteristic
278	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
279	C₂×C₆₇	#2955	1	1	yes	characteristic
280	D₆₇	—	4	2	no	—
281	#3035	—	67	1	no	—
282	C₂×#444	—	67	2	no	—
283	#2931	D₆₇	1	1	yes	characteristic, complemented
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	—	67	1	no	—
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	C₂×D₆₇	1	2	yes	—
297	#532	C₂×D₆₇	1	1	yes	characteristic
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	—	2	1	no	—
307	#436	—	2	1	no	—
308	C₈×D₄	—	2	1	no	—
309	C₈×D₄	—	2	1	no	—
310	C₂×#108	—	67	2	no	—
311	#423	C₂×D₆₇	1	1	yes	characteristic
312	C₂×C₄⋊C₈	—	67	1	no	—
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	C₂×D₆₇	1	2	yes	—
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	—	67	2	no	—
325	C₂×C₄×C₈	—	134	1	no	—
326	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
327	C₄×C₄⋊C₄	—	67	1	no	—
328	#341	—	67	2	no	—
329	#339	—	67	2	no	—
330	#336	—	67	2	no	—
331	C₂×#41	—	67	1	no	—
332	C₂×#41	C₂²×D₆₇	1	2	yes	—
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	—	2	4	no	—
340	#131	—	2	4	no	—
341	#131	—	67	4	no	—
342	#131	—	134	2	no	—
343	#131	—	134	2	no	—
344	#131	—	134	2	no	—
345	C₂×M₁₆	—	2	2	no	—
346	C₂×M₁₆	—	67	1	no	—
347	C₂×M₁₆	—	67	1	no	—
348	C₂×M₁₆	—	134	1	no	—
349	C₂×M₁₆	—	134	1	no	—
350	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
351	C₂²×C₈	—	2	2	no	—
352	C₂²×C₈	—	67	2	no	—
353	C₂²×C₈	—	134	1	no	—
354	C₂²×C₈	—	134	1	no	—
355	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
356	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
357	C₄⋊Q₈	C₄×D₆₇	1	1	yes	characteristic
358	#127	C₄×D₆₇	1	1	yes	characteristic
359	#124	C₄×D₆₇	1	2	yes	—
360	#121	—	2	1	no	—
361	#121	—	2	1	no	—
362	C₄×Q₈	—	67	1	no	—
363	C₄×Q₈	—	67	1	no	—
364	C₄×D₄	—	2	1	no	—
365	C₄×D₄	—	2	1	no	—
366	C₄×D₄	—	67	1	no	—
367	C₄×D₄	—	67	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	—	134	1	no	—
373	C₂×C₄²	—	67	2	no	—
374	C₂×C₄²	—	134	1	no	—
375	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
376	#108	—	134	4	no	—
377	C₄⋊C₈	—	2	1	no	—
378	C₄⋊C₈	—	2	1	no	—
379	C₄⋊C₈	—	67	1	no	—
380	C₄⋊C₈	—	67	1	no	—
381	C₄⋊C₈	—	67	2	no	—
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	—	2	2	no	—
390	C₂²⋊C₈	—	2	2	no	—
391	#97	—	134	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	C₂²⋊D₆₇	1	1	yes	characteristic
399	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
400	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
401	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
402	#95	—	134	1	no	—
403	#41	—	2	2	no	—
404	#41	—	2	2	no	—
405	#41	—	2	2	no	—
406	#41	—	2	2	no	—
407	#41	—	67	2	no	—
408	#41	—	67	2	no	—
409	#41	—	134	1	no	—
410	#41	—	134	1	no	—
411	C₂×Q₈	—	67	1	no	—
412	C₂×Q₈	C₂×C₄×D₆₇	1	2	yes	—
413	C₂×D₄	—	2	2	no	—
414	C₂×D₄	—	2	1	no	—
415	C₂×D₄	—	2	1	no	—
416	C₂×D₄	—	67	1	no	—
417	C₂×D₄	—	134	1	no	—
418	C₂×D₄	C₂×C₄×D₆₇	1	2	yes	—
419	C₂²×C₄	—	2	2	no	—
420	C₂²×C₄	—	134	2	no	—
421	C₂²×C₄	—	134	1	no	—
422	C₂²×C₄	—	134	1	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	—	2	2	no	—
432	M₁₆	—	2	2	no	—
433	M₁₆	—	4	4	no	—
434	M₁₆	—	67	2	no	—
435	M₁₆	—	67	2	no	—
436	M₁₆	—	134	4	no	—
437	M₁₆	—	134	1	no	—
438	M₁₆	—	134	1	no	—
439	M₁₆	—	134	4	no	—
440	C₂×C₈	—	2	1	no	—
441	C₂×C₈	—	2	1	no	—
442	C₂×C₈	—	2	2	no	—
443	C₂×C₈	—	4	4	no	—
444	C₂×C₈	—	67	2	no	—
445	C₂×C₈	—	67	2	no	—
446	C₂×C₈	—	134	4	no	—
447	C₂×C₈	—	134	2	no	—
448	C₂×C₈	—	134	1	no	—
449	C₂×C₈	—	134	1	no	—
450	C₂×C₈	—	134	2	no	—
451	C₂×C₈	—	134	4	no	—
452	C₂×C₈	D₄×D₆₇	1	2	yes	—
453	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
454	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
455	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
456	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
457	C₂×C₈	#11761627	1	2	yes	—
458	C₄⋊C₄	—	2	1	no	—
459	C₄⋊C₄	—	2	1	no	—
460	C₄⋊C₄	—	268	2	no	—
461	C₄⋊C₄	—	67	1	no	—
462	C₄⋊C₄	—	67	1	no	—
463	C₄⋊C₄	—	134	1	no	—
464	C₄⋊C₄	—	134	1	no	—
465	C₂²⋊C₄	—	2	2	no	—
466	C₂²⋊C₄	—	2	2	no	—
467	C₂²⋊C₄	—	134	1	no	—
468	C₂²⋊C₄	—	134	1	no	—
469	C₂²⋊C₄	—	134	1	no	—
470	C₄²	—	134	2	no	—
471	C₄²	—	134	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×C₂²⋊D₆₇	1	1	yes	characteristic
478	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
479	C₂³	—	2	2	no	—
480	C₂³	—	134	1	no	—
481	C₂³	C₂×#11761628	1	1	yes	characteristic
482	Q₈	—	2	2	no	—
483	Q₈	—	2	2	no	—
484	Q₈	—	67	2	no	—
485	Q₈	—	134	1	no	—
486	D₄	—	2	2	no	—
487	D₄	—	2	2	no	—
488	D₄	—	4	2	no	—
489	D₄	—	4	2	no	—
490	D₄	—	4	2	no	—
491	D₄	—	4	2	no	—
492	D₄	—	67	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	—	2	2	no	—
498	C₂×C₄	—	4	1	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	—	268	2	no	—
506	C₂×C₄	—	134	1	no	—
507	C₂×C₄	—	134	1	no	—
508	C₂×C₄	—	134	1	no	—
509	C₂×C₄	—	134	2	no	—
510	C₂×C₄	—	134	2	no	—
511	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
512	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
513	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
514	C₂×C₄	C₄×C₂²⋊D₆₇	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×C₂²⋊D₆₇	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×#11761627	1	1	yes	characteristic
521	C₈	—	2	1	no	—
522	C₈	—	2	1	no	—
523	C₈	—	4	2	no	—
524	C₈	—	134	2	no	—
525	C₈	—	134	2	no	—
526	C₈	—	134	2	no	—
527	C₈	—	134	2	no	—
528	C₈	#14524014	1	2	yes	—
529	C₈	#14524014	1	2	yes	—
530	C₂²	—	2	1	no	—
531	C₂²	—	2	1	no	—
532	C₂²	—	4	2	no	—
533	C₂²	—	4	2	no	—
534	C₂²	—	4	2	no	—
535	C₂²	—	268	1	no	—
536	C₂²	—	268	1	no	—
537	C₂²	—	134	2	no	—
538	C₂²	C₂×C₄×C₂²⋊D₆₇	1	1	yes	characteristic
539	C₂²	#16863928	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	—	4	2	no	—
545	C₄	—	268	1	no	—
546	C₄	—	268	1	no	—
547	C₄	—	536	2	no	—
548	C₄	—	134	2	no	—
549	C₄	#16863516	1	2	yes	complemented
550	C₄	C₄×D₄×D₆₇	1	1	yes	characteristic
551	C₄	C₄×#11761627	1	1	yes	characteristic
552	C₄	C₂×#14524014	1	1	yes	characteristic
553	C₄	#16863903	1	1	yes	characteristic
554	C₂	—	2	1	no	—
555	C₂	—	2	1	no	—
556	C₂	—	8	2	no	—
557	C₂	—	268	2	no	—
558	C₂	C₄×#14524014	1	1	yes	characteristic
559	C₂	C₂×#16863516	1	1	yes	characteristic
560	C₂	#22569110	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#2931	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#22569658	C₂
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#2931	D₆₇
hypocenter	C₆₇	#21518
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2931

Complements

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

N	H ≅ G/N	type
2. C₂×#16863516	C₂	semidirect
3. C₄×#14524049	C₂	semidirect
4. #22561129	C₂	semidirect
5. #22560982	C₂	semidirect
6. #22560854	C₂	semidirect
7. #22560539	C₂	semidirect
8. #22560226	C₂	semidirect
9. #22560076	C₂	semidirect
10. #22560051	C₂	semidirect
11. C₆₇×#2931	C₂	semidirect
16. #16863615	C₄	semidirect
17. #16863608	C₄	semidirect
18. #16863603	C₄	semidirect
19. #16863598	C₄	semidirect
20. #16863593	C₄	semidirect
21. #16863588	C₄	semidirect
60. #14524049	D₄	semidirect
264. C₄×C₆₇	#444	semidirect
283. #2931	D₆₇	semidirect
292. C₆₇	#21518	semidirect
549. C₄	#16863516	semidirect

Maximal subgroups

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

subgroup	index
#22569658	2
C₂×#16863516	2
C₄×#14524049	2
#22561129	2
#22560982	2
#22560854	2
#22560539	2
#22560226	2
#22560076	2
#22560051	2
C₆₇×#2931	2
#21518	67