Subgroups — #30869052

Order 17152.

806 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568913	C₂	1	2	yes	maximal, complemented
2	#22560859	C₂	1	2	yes	maximal, complemented
3	#22560810	C₂	1	2	yes	maximal, complemented
4	#22560653	C₂	1	2	yes	maximal, complemented
5	#22560538	C₂	1	2	yes	maximal, complemented
6	#22560486	C₂	1	2	yes	maximal, complemented
7	#22560251	C₂	1	1	yes	maximal, characteristic, complemented
8	#22559902	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#3054	C₂	1	1	yes	maximal, characteristic
10	C₂×#14524050	C₂²	1	2	yes	—
11	C₂²×#11761612	C₂²	1	1	yes	characteristic
12	C₂²×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
13	#16863474	—	2	2	no	—
14	#16863474	—	2	2	no	—
15	#16863468	—	2	2	no	—
16	#16863468	—	2	2	no	—
17	C₂×#14523979	C₂²	1	1	yes	characteristic
18	C₂×#14523979	C₂²	1	1	yes	characteristic
19	#16863423	C₂²	1	2	yes	—
20	C₂×#14523970	C₂²	1	2	yes	—
21	C₂×#14523970	C₂²	1	2	yes	—
22	#16863363	C₂²	1	2	yes	—
23	#16863360	C₂²	1	2	yes	—
24	#16863352	C₂²	1	2	yes	—
25	#16863161	C₂²	1	2	yes	—
26	#16863155	C₂²	1	2	yes	—
27	#16863155	C₂²	1	2	yes	—
28	#16863150	C₂²	1	2	yes	—
29	#16863150	C₂²	1	2	yes	—
30	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
31	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
32	#525×C₆₇	C₂²	1	2	yes	—
33	#470×C₆₇	C₄	1	1	yes	characteristic, complemented
34	#470×C₆₇	C₄	1	1	yes	characteristic, complemented
35	#469×C₆₇	C₄	1	1	yes	characteristic, complemented
36	#469×C₆₇	C₄	1	1	yes	characteristic, complemented
37	#468×C₆₇	C₄	1	2	yes	complemented
38	#468×C₆₇	C₄	1	2	yes	complemented
39	#427×C₆₇	C₂²	1	1	yes	characteristic
40	#416×C₆₇	C₂²	1	2	yes	—
41	C₂³×Dic₆₇	—	2	1	no	—
42	#14524050	—	4	2	no	—
43	#14524050	—	4	2	no	—
44	C₂×#11761612	—	2	2	no	—
45	C₂×#11761612	—	4	1	no	—
46	C₂×#11761612	—	4	1	no	—
47	C₂×#11761612	D₄	1	1	yes	characteristic
48	C₂×#11761612	D₄	1	2	yes	—
49	C₂×#11761612	D₄	1	1	yes	characteristic
50	C₂×#11761612	C₂³	1	2	yes	—
51	C₂×#11761612	C₂³	1	2	yes	—
52	C₂²×C₆₇⋊C₈	—	2	2	no	—
53	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
54	C₂×C₄×Dic₆₇	—	2	1	no	—
55	C₂×C₄×Dic₆₇	—	4	1	no	—
56	C₂×C₄×Dic₆₇	—	4	1	no	—
57	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
58	#14523979	—	4	1	no	—
59	#14523979	—	4	1	no	—
60	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
61	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
62	#14523970	—	4	2	no	—
63	#14523970	—	4	2	no	—
64	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
65	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
66	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
67	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
68	#14523954	—	2	2	no	—
69	C₄×C₆₇⋊C₈	—	2	2	no	—
70	C₂×#41×C₆₇	C₂³	1	2	yes	—
71	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
72	#135×C₆₇	—	4	1	no	—
73	#135×C₆₇	—	4	1	no	—
74	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
75	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	complemented
76	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
77	C₂×M₁₆×C₆₇	—	4	1	no	—
78	C₂×M₁₆×C₆₇	—	4	1	no	—
79	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
80	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
81	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
82	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
83	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
84	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
85	#123×C₆₇	—	2	2	no	—
86	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
87	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
88	#121×C₆₇	C₂×C₄	1	2	yes	—
89	#121×C₆₇	C₂×C₄	1	2	yes	—
90	#120×C₆₇	—	2	2	no	—
91	C₄×D₄×C₆₇	—	2	2	no	—
92	C₄×D₄×C₆₇	—	2	2	no	—
93	#117×C₆₇	C₂³	1	2	yes	—
94	#107×C₆₇	C₂×C₄	1	2	yes	complemented
95	#106×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
96	#106×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
97	#103×C₆₇	C₂×C₄	1	2	yes	complemented
98	#103×C₆₇	C₂×C₄	1	2	yes	complemented
99	#102×C₆₇	C₂×C₄	1	2	yes	complemented
100	#102×C₆₇	C₂×C₄	1	2	yes	complemented
101	C₂²×Dic₆₇	—	2	2	no	—
102	C₂²×Dic₆₇	—	4	4	no	—
103	C₂²×Dic₆₇	—	4	2	no	—
104	#11761612	—	2	2	no	—
105	#11761612	—	2	2	no	—
106	#11761612	—	2	2	no	—
107	#11761612	—	2	2	no	—
108	#11761612	—	4	1	no	—
109	#11761612	—	4	2	no	—
110	#11761612	—	4	1	no	—
111	#11761612	—	4	4	no	—
112	C₂×C₆₇⋊C₈	—	2	2	no	—
113	C₂×C₆₇⋊C₈	—	2	2	no	—
114	C₂×C₆₇⋊C₈	—	2	2	no	—
115	C₂×C₆₇⋊C₈	—	2	2	no	—
116	C₂×C₆₇⋊C₈	—	4	4	no	—
117	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
118	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
119	C₄×Dic₆₇	—	2	2	no	—
120	C₄×Dic₆₇	—	2	2	no	—
121	C₄×Dic₆₇	—	4	2	no	—
122	C₂⁴×C₆₇	#41	1	1	yes	characteristic
123	#41×C₆₇	—	2	2	no	—
124	#41×C₆₇	—	2	2	no	—
125	#41×C₆₇	—	2	2	no	—
126	#41×C₆₇	—	2	2	no	—
127	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
128	C₂×D₄×C₆₇	—	2	2	no	—
129	C₂×D₄×C₆₇	—	2	2	no	—
130	C₂×D₄×C₆₇	—	2	2	no	—
131	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
132	C₂²×C₄×C₆₇	—	2	1	no	—
133	C₂²×C₄×C₆₇	—	2	1	no	—
134	C₂²×C₄×C₆₇	—	2	1	no	—
135	C₂²×C₄×C₆₇	—	2	1	no	—
136	C₂²×C₄×C₆₇	—	2	2	no	—
137	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
138	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
139	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
142	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
143	Q₁₆×C₆₇	—	4	1	no	—
144	SD₁₆×C₆₇	—	4	2	no	—
145	D₈×C₆₇	—	4	1	no	—
146	M₁₆×C₆₇	—	4	1	no	—
147	M₁₆×C₆₇	—	4	1	no	—
148	M₁₆×C₆₇	—	4	1	no	—
149	M₁₆×C₆₇	—	4	1	no	—
150	C₂×C₈×C₆₇	—	2	1	no	—
151	C₂×C₈×C₆₇	—	2	1	no	—
152	C₂×C₈×C₆₇	—	2	1	no	—
153	C₂×C₈×C₆₇	—	2	1	no	—
154	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
155	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
156	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
157	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
158	C₄⋊C₄×C₆₇	—	2	2	no	—
159	C₄⋊C₄×C₆₇	—	2	2	no	—
160	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
161	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
162	C₂²⋊C₄×C₆₇	—	2	2	no	—
163	C₂²⋊C₄×C₆₇	—	2	2	no	—
164	C₂²⋊C₄×C₆₇	—	2	2	no	—
165	C₂²⋊C₄×C₆₇	—	2	2	no	—
166	C₂²⋊C₄×C₆₇	—	2	2	no	—
167	C₄²×C₆₇	—	2	2	no	—
168	C₂³×C₆₇	—	2	1	no	—
169	C₂³×C₆₇	—	2	1	no	—
170	C₂³×C₆₇	—	2	2	no	—
171	C₂³×C₆₇	—	4	2	no	—
172	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
173	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
174	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
175	Q₈×C₆₇	—	2	2	no	—
176	Q₈×C₆₇	—	2	2	no	—
177	D₄×C₆₇	—	2	2	no	—
178	D₄×C₆₇	—	2	2	no	—
179	D₄×C₆₇	—	4	2	no	—
180	D₄×C₆₇	—	4	2	no	—
181	D₄×C₆₇	—	8	2	no	—
182	D₄×C₆₇	—	8	2	no	—
183	C₂×C₄×C₆₇	—	2	1	no	—
184	C₂×C₄×C₆₇	—	2	1	no	—
185	C₂×C₄×C₆₇	—	2	1	no	—
186	C₂×C₄×C₆₇	—	2	1	no	—
187	C₂×C₄×C₆₇	—	2	1	no	—
188	C₂×C₄×C₆₇	—	2	1	no	—
189	C₂×C₄×C₆₇	—	2	1	no	—
190	C₂×C₄×C₆₇	—	2	1	no	—
191	C₂×C₄×C₆₇	—	2	1	no	—
192	C₂×C₄×C₆₇	—	2	1	no	—
193	C₂×C₄×C₆₇	—	2	2	no	—
194	C₂×C₄×C₆₇	—	2	2	no	—
195	C₂×C₄×C₆₇	—	2	2	no	—
196	C₂×C₄×C₆₇	—	4	2	no	—
197	C₂×C₄×C₆₇	—	4	2	no	—
198	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
199	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
200	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
201	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
202	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
203	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
204	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
205	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
206	C₂×Dic₆₇	—	2	4	no	—
207	C₂×Dic₆₇	—	4	2	no	—
208	C₂×Dic₆₇	—	4	8	no	—
209	C₂×Dic₆₇	—	4	2	no	—
210	C₈×C₆₇	—	2	1	no	—
211	C₈×C₆₇	—	2	1	no	—
212	C₈×C₆₇	—	2	1	no	—
213	C₈×C₆₇	—	2	1	no	—
214	C₆₇⋊C₈	—	2	2	no	—
215	C₆₇⋊C₈	—	2	2	no	—
216	C₆₇⋊C₈	—	4	2	no	—
217	C₆₇⋊C₈	—	4	2	no	—
218	C₂²×C₆₇	—	2	2	no	—
219	C₂²×C₆₇	—	2	1	no	—
220	C₂²×C₆₇	—	2	1	no	—
221	C₂²×C₆₇	—	2	2	no	—
222	C₂²×C₆₇	—	2	2	no	—
223	C₂²×C₆₇	—	4	2	no	—
224	C₂²×C₆₇	—	4	2	no	—
225	C₂²×C₆₇	—	4	2	no	—
226	C₂²×C₆₇	—	4	2	no	—
227	C₂²×C₆₇	—	4	2	no	—
228	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
229	C₂²×C₆₇	#538	1	1	yes	characteristic
230	C₂²×C₆₇	#539	1	1	yes	characteristic
231	C₄×C₆₇	—	2	1	no	—
232	C₄×C₆₇	—	2	1	no	—
233	C₄×C₆₇	—	4	2	no	—
234	C₄×C₆₇	—	4	2	no	—
235	C₄×C₆₇	—	4	2	no	—
236	C₄×C₆₇	#518	1	1	yes	characteristic
237	C₄×C₆₇	#520	1	1	yes	characteristic
238	C₄×C₆₇	#534	1	1	yes	characteristic
239	C₄×C₆₇	#537	1	1	yes	characteristic
240	Dic₆₇	—	4	4	no	—
241	#22207	—	67	1	no	maximal, Sylow
242	C₂×C₆₇	—	2	1	no	—
243	C₂×C₆₇	—	2	1	no	—
244	C₂×C₆₇	—	4	1	no	—
245	C₂×C₆₇	—	4	1	no	—
246	C₂×C₆₇	—	8	2	no	—
247	C₂×C₆₇	#2325	1	1	yes	characteristic
248	C₂×C₆₇	#2956	1	2	yes	—
249	#3054	D₆₇	1	1	yes	characteristic
250	#2930	—	67	2	no	—
251	#1929	—	67	2	no	—
252	#1922	—	67	2	no	—
253	#1896	—	67	2	no	—
254	#1876	—	67	2	no	—
255	#1866	—	67	2	no	—
256	#1811	—	67	1	no	—
257	#1728	—	67	1	no	—
258	C₆₇	#22207	1	1	yes	characteristic, complemented, Sylow
259	C₂×#135	C₂×D₆₇	1	1	yes	characteristic
260	C₂×#131	—	67	2	no	—
261	C₂²×M₁₆	—	67	1	no	—
262	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
263	#525	C₂×D₆₇	1	2	yes	—
264	C₂²×C₄²	—	67	1	no	—
265	#470	Dic₆₇	1	1	yes	characteristic, complemented
266	#470	Dic₆₇	1	1	yes	characteristic, complemented
267	#469	Dic₆₇	1	1	yes	characteristic, complemented
268	#469	Dic₆₇	1	1	yes	characteristic, complemented
269	#468	Dic₆₇	1	2	yes	complemented
270	#468	Dic₆₇	1	2	yes	complemented
271	#436	—	134	2	no	—
272	#436	—	134	2	no	—
273	#435	—	134	2	no	—
274	#435	—	134	2	no	—
275	C₂×#108	—	67	1	no	—
276	C₂×#108	—	67	1	no	—
277	#427	C₂×D₆₇	1	1	yes	characteristic
278	#424	—	67	2	no	—
279	C₂×#104	—	67	2	no	—
280	C₂×#104	—	67	2	no	—
281	#416	C₂×D₆₇	1	2	yes	—
282	#408	—	67	2	no	—
283	#407	—	67	2	no	—
284	#405	—	67	2	no	—
285	#341	—	67	2	no	—
286	#339	—	67	2	no	—
287	#339	—	67	2	no	—
288	#336	—	67	2	no	—
289	#336	—	67	2	no	—
290	C₂×#41	C₂²×D₆₇	1	2	yes	—
291	C₂³×C₄	—	134	1	no	—
292	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
293	#135	—	4	1	no	—
294	#135	—	4	1	no	—
295	C₂×Q₁₆	C₂×Dic₆₇	1	1	yes	characteristic, complemented
296	C₂×SD₁₆	C₂×Dic₆₇	1	2	yes	complemented
297	C₂×D₈	C₂×Dic₆₇	1	1	yes	characteristic, complemented
298	#131	—	268	2	no	—
299	#131	—	268	2	no	—
300	C₂×M₁₆	—	4	1	no	—
301	C₂×M₁₆	—	4	1	no	—
302	C₂×M₁₆	—	268	1	no	—
303	C₂×M₁₆	—	268	1	no	—
304	C₂×M₁₆	—	67	1	no	—
305	C₂×M₁₆	—	67	2	no	—
306	C₂×M₁₆	—	67	1	no	—
307	C₂×M₁₆	—	67	2	no	—
308	C₂×M₁₆	—	67	2	no	—
309	C₂×M₁₆	—	134	2	no	—
310	C₂×M₁₆	C₂×Dic₆₇	1	1	yes	characteristic
311	C₂×M₁₆	C₂×Dic₆₇	1	1	yes	characteristic
312	C₂×M₁₆	C₂×Dic₆₇	1	1	yes	characteristic
313	C₂×M₁₆	C₂×Dic₆₇	1	1	yes	characteristic
314	C₂²×C₈	—	67	2	no	—
315	C₂²×C₈	—	134	2	no	—
316	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
317	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
318	#123	—	2	2	no	—
319	C₂²⋊Q₈	C₂×Dic₆₇	1	2	yes	—
320	C₂²⋊Q₈	C₂×Dic₆₇	1	2	yes	—
321	#121	C₂×Dic₆₇	1	2	yes	—
322	#121	C₂×Dic₆₇	1	2	yes	—
323	#120	—	2	2	no	—
324	C₄×D₄	—	2	2	no	—
325	C₄×D₄	—	2	2	no	—
326	#117	C₂²×D₆₇	1	2	yes	—
327	C₂×C₄²	—	268	1	no	—
328	C₂×C₄²	—	268	1	no	—
329	C₂×C₄²	—	67	2	no	—
330	C₂×C₄²	—	134	1	no	—
331	#108	—	268	1	no	—
332	#108	—	268	1	no	—
333	#107	C₂×Dic₆₇	1	2	yes	complemented
334	#106	C₂×Dic₆₇	1	1	yes	characteristic, complemented
335	#106	C₂×Dic₆₇	1	1	yes	characteristic, complemented
336	C₄⋊C₈	—	134	2	no	—
337	C₄⋊C₈	—	134	2	no	—
338	#104	—	268	2	no	—
339	#104	—	268	2	no	—
340	#103	C₂×Dic₆₇	1	2	yes	complemented
341	#103	C₂×Dic₆₇	1	2	yes	complemented
342	#102	C₂×Dic₆₇	1	2	yes	complemented
343	#102	C₂×Dic₆₇	1	2	yes	complemented
344	C₂²⋊C₈	—	134	2	no	—
345	C₂²⋊C₈	—	134	2	no	—
346	C₂²⋊C₈	—	134	2	no	—
347	C₂²⋊C₈	—	134	2	no	—
348	#97	—	134	2	no	—
349	C₄×C₈	—	134	2	no	—
350	C₂⁴	#11761627	1	1	yes	characteristic
351	#41	—	2	2	no	—
352	#41	—	2	2	no	—
353	#41	—	2	2	no	—
354	#41	—	2	2	no	—
355	C₂×Q₈	C₂²×Dic₆₇	1	2	yes	—
356	C₂×D₄	—	2	2	no	—
357	C₂×D₄	—	2	2	no	—
358	C₂×D₄	—	2	2	no	—
359	C₂×D₄	C₂²×Dic₆₇	1	2	yes	—
360	C₂²×C₄	—	2	1	no	—
361	C₂²×C₄	—	2	1	no	—
362	C₂²×C₄	—	2	1	no	—
363	C₂²×C₄	—	2	1	no	—
364	C₂²×C₄	—	2	2	no	—
365	C₂²×C₄	—	268	2	no	—
366	C₂²×C₄	—	268	4	no	—
367	C₂²×C₄	—	134	2	no	—
368	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
369	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
370	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
371	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
372	C₂²×C₄	#11761627	1	1	yes	characteristic
373	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
374	Q₁₆	—	4	1	no	—
375	SD₁₆	—	4	2	no	—
376	D₈	—	4	1	no	—
377	M₁₆	—	4	1	no	—
378	M₁₆	—	4	1	no	—
379	M₁₆	—	4	1	no	—
380	M₁₆	—	4	1	no	—
381	M₁₆	—	268	4	no	—
382	M₁₆	—	268	1	no	—
383	M₁₆	—	268	2	no	—
384	M₁₆	—	268	1	no	—
385	M₁₆	—	134	2	no	—
386	M₁₆	—	134	2	no	—
387	M₁₆	—	134	2	no	—
388	M₁₆	—	134	2	no	—
389	C₂×C₈	—	2	1	no	—
390	C₂×C₈	—	2	1	no	—
391	C₂×C₈	—	2	1	no	—
392	C₂×C₈	—	2	1	no	—
393	C₂×C₈	—	268	4	no	—
394	C₂×C₈	—	67	2	no	—
395	C₂×C₈	—	67	2	no	—
396	C₂×C₈	—	134	2	no	—
397	C₂×C₈	—	134	2	no	—
398	C₂×C₈	—	134	2	no	—
399	C₂×C₈	—	134	2	no	—
400	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
401	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
402	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
403	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
404	C₄⋊C₄	—	2	2	no	—
405	C₄⋊C₄	—	2	2	no	—
406	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
407	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
408	C₂²⋊C₄	—	2	2	no	—
409	C₂²⋊C₄	—	2	2	no	—
410	C₂²⋊C₄	—	2	2	no	—
411	C₂²⋊C₄	—	2	2	no	—
412	C₂²⋊C₄	—	2	2	no	—
413	C₄²	—	2	2	no	—
414	C₄²	—	268	2	no	—
415	C₄²	—	134	2	no	—
416	C₄²	—	134	2	no	—
417	C₂³	—	2	1	no	—
418	C₂³	—	2	1	no	—
419	C₂³	—	2	2	no	—
420	C₂³	—	4	2	no	—
421	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
422	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
423	C₂³	C₂×#11761627	1	1	yes	characteristic
424	Q₈	—	2	2	no	—
425	Q₈	—	2	2	no	—
426	D₄	—	2	2	no	—
427	D₄	—	2	2	no	—
428	D₄	—	4	2	no	—
429	D₄	—	4	2	no	—
430	D₄	—	8	2	no	—
431	D₄	—	8	2	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	1	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	2	1	no	—
437	C₂×C₄	—	2	1	no	—
438	C₂×C₄	—	2	1	no	—
439	C₂×C₄	—	2	1	no	—
440	C₂×C₄	—	2	1	no	—
441	C₂×C₄	—	2	1	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	4	2	no	—
447	C₂×C₄	—	268	2	no	—
448	C₂×C₄	—	268	8	no	—
449	C₂×C₄	—	268	2	no	—
450	C₂×C₄	—	134	4	no	—
451	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
452	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
453	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
454	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
455	C₂×C₄	#14524088	1	1	yes	characteristic
456	C₂×C₄	#14524088	1	1	yes	characteristic
457	C₂×C₄	#14524089	1	1	yes	characteristic
458	C₂×C₄	#14524090	1	1	yes	characteristic
459	C₈	—	2	1	no	—
460	C₈	—	2	1	no	—
461	C₈	—	2	1	no	—
462	C₈	—	2	1	no	—
463	C₈	—	268	2	no	—
464	C₈	—	268	2	no	—
465	C₈	—	134	2	no	—
466	C₈	—	134	2	no	—
467	C₂²	—	2	2	no	—
468	C₂²	—	2	1	no	—
469	C₂²	—	2	1	no	—
470	C₂²	—	2	2	no	—
471	C₂²	—	2	2	no	—
472	C₂²	—	4	2	no	—
473	C₂²	—	4	2	no	—
474	C₂²	—	4	2	no	—
475	C₂²	—	4	2	no	—
476	C₂²	—	4	2	no	—
477	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
478	C₂²	#16863983	1	1	yes	characteristic
479	C₂²	#16863991	1	1	yes	characteristic
480	C₄	—	2	1	no	—
481	C₄	—	2	1	no	—
482	C₄	—	4	2	no	—
483	C₄	—	4	2	no	—
484	C₄	—	4	2	no	—
485	C₄	—	268	4	no	—
486	C₄	#16863871	1	1	yes	characteristic
487	C₄	#16863879	1	1	yes	characteristic
488	C₄	#16863954	1	1	yes	characteristic
489	C₄	#16863973	1	1	yes	characteristic
490	C₂	—	2	1	no	—
491	C₂	—	2	1	no	—
492	C₂	—	4	1	no	—
493	C₂	—	4	1	no	—
494	C₂	—	8	2	no	—
495	C₂	#22562914	1	1	yes	characteristic
496	C₂	#22569119	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting, prime radical	C₆₇×#3054	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#3054	D₆₇
hypocenter	C₆₇	#22207
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #3054

Complements

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

N	H ≅ G/N	type
1. #22568913	C₂	semidirect
2. #22560859	C₂	semidirect
3. #22560810	C₂	semidirect
4. #22560653	C₂	semidirect
5. #22560538	C₂	semidirect
6. #22560486	C₂	semidirect
7. #22560251	C₂	semidirect
8. #22559902	C₂	semidirect
33. #470×C₆₇	C₄	semidirect
34. #470×C₆₇	C₄	semidirect
35. #469×C₆₇	C₄	semidirect
36. #469×C₆₇	C₄	semidirect
37. #468×C₆₇	C₄	semidirect
38. #468×C₆₇	C₄	semidirect
74. C₂×Q₁₆×C₆₇	C₂×C₄	semidirect
75. C₂×SD₁₆×C₆₇	C₂×C₄	semidirect
76. C₂×D₈×C₆₇	C₂×C₄	semidirect
94. #107×C₆₇	C₂×C₄	semidirect
95. #106×C₆₇	C₂×C₄	semidirect
96. #106×C₆₇	C₂×C₄	semidirect
97. #103×C₆₇	C₂×C₄	semidirect
98. #103×C₆₇	C₂×C₄	semidirect
99. #102×C₆₇	C₂×C₄	semidirect
100. #102×C₆₇	C₂×C₄	semidirect
258. C₆₇	#22207	semidirect
265. #470	Dic₆₇	semidirect
266. #470	Dic₆₇	semidirect
267. #469	Dic₆₇	semidirect
268. #469	Dic₆₇	semidirect
269. #468	Dic₆₇	semidirect
270. #468	Dic₆₇	semidirect
295. C₂×Q₁₆	C₂×Dic₆₇	semidirect
296. C₂×SD₁₆	C₂×Dic₆₇	semidirect
297. C₂×D₈	C₂×Dic₆₇	semidirect
333. #107	C₂×Dic₆₇	semidirect
334. #106	C₂×Dic₆₇	semidirect
335. #106	C₂×Dic₆₇	semidirect
340. #103	C₂×Dic₆₇	semidirect
341. #103	C₂×Dic₆₇	semidirect
342. #102	C₂×Dic₆₇	semidirect
343. #102	C₂×Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22568913	2
#22560859	2
#22560810	2
#22560653	2
#22560538	2
#22560486	2
#22560251	2
#22559902	2
C₆₇×#3054	2
#22207	67