Subgroups — #30825631

Order 17152.

648 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569175	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863526	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#14524076	C₂	1	1	yes	maximal, characteristic
4	#22559119	C₂	1	2	yes	maximal, complemented
5	#22559113	C₂	1	2	yes	maximal, complemented
6	#22559107	C₂	1	2	yes	maximal, complemented
7	#22559101	C₂	1	2	yes	maximal, complemented
8	#22558961	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558605	C₂	1	1	yes	maximal, characteristic, complemented
10	#22558597	C₂	1	1	yes	maximal, characteristic
11	C₈×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524076	C₄	1	1	yes	characteristic
13	C₄×#11761628	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
15	#16863526	—	2	1	no	—
16	#16863526	—	2	1	no	—
17	#16863526	C₂²	1	2	yes	—
18	#16863526	C₂²	1	1	yes	characteristic
19	#16863526	C₂²	1	2	yes	—
20	#16863526	C₂²	1	1	yes	characteristic
21	#16863504	C₄	1	2	yes	—
22	#16863498	C₄	1	2	yes	—
23	C₄×Dic₂₆₈	C₂²	1	2	yes	—
24	C₄×#11761617	C₂²	1	2	yes	—
25	#16863475	C₂²	1	1	yes	characteristic
26	#16863475	C₂²	1	1	yes	characteristic
27	#16863467	—	2	1	no	—
28	#16863467	—	2	1	no	—
29	#16863449	C₂²	1	1	yes	characteristic
30	#16863447	—	2	1	no	—
31	#16863438	C₄	1	1	yes	characteristic
32	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
33	C₂×Dic₆₇⋊C₈	—	2	1	no	—
34	#16863406	C₄	1	1	yes	characteristic
35	#16863398	C₄	1	1	yes	characteristic
36	C₄×#11761611	—	2	1	no	—
37	C₂×#14523953	—	2	1	no	—
38	#16863149	C₂²	1	2	yes	—
39	#16863147	C₂²	1	2	yes	—
40	#16863128	C₂²	1	2	yes	—
41	#16863128	C₂²	1	2	yes	—
42	#16863128	C₂²	1	2	yes	—
43	#16863125	C₂²	1	2	yes	—
44	#431×C₆₇	C₂²	1	1	yes	characteristic
45	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
46	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
47	#322×C₆₇	C₂²	1	2	yes	—
48	C₈²×C₆₇	C₂²	1	2	yes	—
49	C₂×#11761628	C₂×C₄	1	1	yes	characteristic
50	C₂²×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
51	#14524076	—	2	4	no	—
52	C₂×Dic₂₆₈	C₂×C₄	1	2	yes	—
53	C₂×#11761617	C₂×C₄	1	2	yes	—
54	C₂×#11761611	—	2	1	no	—
55	C₂×#11761611	—	2	1	no	—
56	C₂²×C₆₇⋊C₈	—	2	1	no	—
57	C₄×Dic₁₃₄	—	2	1	no	—
58	C₄×Dic₁₃₄	—	2	1	no	—
59	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
60	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₁₃₄	C₂³	1	1	yes	characteristic
62	C₄×Dic₁₃₄	C₂³	1	2	yes	—
63	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
64	C₄×D₂₆₈	C₂³	1	1	yes	characteristic
65	C₄×C₂²⋊D₆₇	—	2	1	no	—
66	C₄×C₂²⋊D₆₇	—	2	1	no	—
67	#14524002	C₂×C₄	1	1	yes	characteristic
68	#14524000	—	2	1	no	—
69	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
70	C₂×Dic₆₇⋊C₄	—	2	1	no	—
71	C₂×C₄×Dic₆₇	—	2	1	no	—
72	C₄²×D₆₇	—	2	1	no	—
73	#14523977	C₂×C₄	1	2	yes	—
74	#14523975	C₂×C₄	1	2	yes	—
75	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
76	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
77	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
78	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
79	Dic₆₇⋊C₈	—	2	2	no	—
80	Dic₆₇⋊C₈	—	2	2	no	—
81	Dic₆₇⋊C₈	—	2	1	no	—
82	Dic₆₇⋊C₈	—	2	1	no	—
83	#14523968	C₂×C₄	1	2	yes	—
84	#14523968	C₂×C₄	1	2	yes	—
85	#14523968	C₂×C₄	1	2	yes	—
86	#14523965	C₂×C₄	1	2	yes	—
87	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
88	(C₂×D₆₇)⋊C₈	—	2	1	no	—
89	(C₂×D₆₇)⋊C₈	—	2	1	no	—
90	#14523953	—	2	2	no	—
91	#14523953	—	2	2	no	—
92	#14523953	—	2	1	no	—
93	#14523953	—	2	1	no	—
94	C₄×C₆₇⋊C₈	—	2	1	no	—
95	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
96	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
97	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
98	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
99	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
100	C₄⋊C₈×C₆₇	—	2	1	no	—
101	C₂²⋊C₈×C₆₇	—	2	1	no	—
102	#97×C₆₇	C₂×C₄	1	2	yes	—
103	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
104	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
105	C₄×C₈×C₆₇	C₂³	1	2	yes	—
106	C₄×C₈×C₆₇	C₂³	1	2	yes	—
107	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
108	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
109	#11761628	—	2	2	no	—
110	#11761628	—	2	2	no	—
111	C₂×Dic₁₃₄	—	2	2	no	—
112	C₂×Dic₁₃₄	—	2	2	no	—
113	C₂×Dic₁₃₄	—	2	1	no	—
114	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
115	C₂×Dic₁₃₄	C₂²×C₄	1	1	yes	characteristic
116	C₂×D₂₆₈	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₂²⋊D₆₇	—	2	1	no	—
118	C₂²×Dic₆₇	—	2	1	no	—
119	C₂²×Dic₆₇	—	2	1	no	—
120	C₂×C₄×D₆₇	—	2	1	no	—
121	C₂×C₄×D₆₇	—	2	1	no	—
122	C₂×C₄×D₆₇	—	2	1	no	—
123	Dic₂₆₈	—	2	4	no	—
124	#11761617	—	2	4	no	—
125	#11761611	—	4	2	no	—
126	#11761611	—	4	2	no	—
127	C₂×C₆₇⋊C₈	—	2	1	no	—
128	C₂×C₆₇⋊C₈	—	2	1	no	—
129	C₂×C₆₇⋊C₈	—	2	2	no	—
130	C₂×C₆₇⋊C₈	—	4	2	no	—
131	C₄⋊Dic₆₇	—	2	1	no	—
132	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
133	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
134	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
135	Dic₆₇⋊C₄	—	2	2	no	—
136	Dic₆₇⋊C₄	—	2	1	no	—
137	Dic₆₇⋊C₄	—	2	2	no	—
138	Dic₆₇⋊C₄	—	2	1	no	—
139	C₂²⋊Dic₆₇	—	2	1	no	—
140	(C₂×D₆₇)⋊C₄	—	2	1	no	—
141	(C₂×D₆₇)⋊C₄	—	2	1	no	—
142	C₄×Dic₆₇	—	2	1	no	—
143	C₄×Dic₆₇	—	2	1	no	—
144	C₄×Dic₆₇	—	2	2	no	—
145	C₄×Dic₆₇	—	2	1	no	—
146	C₄×Dic₆₇	—	2	2	no	—
147	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
148	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
149	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
150	M₁₆×C₆₇	—	2	2	no	—
151	M₁₆×C₆₇	M₁₆	1	4	yes	complemented
152	C₂×C₈×C₆₇	—	2	2	no	—
153	C₂×C₈×C₆₇	—	2	2	no	—
154	C₂×C₈×C₆₇	—	4	1	no	—
155	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
156	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
157	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
158	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
159	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
160	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
161	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
162	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
163	C₄²×C₆₇	#41	1	1	yes	characteristic
164	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
165	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
166	C₂²×D₆₇	—	2	1	no	—
167	C₂×C₄×C₆₇	—	2	1	no	—
168	C₂×C₄×C₆₇	—	2	1	no	—
169	C₂×C₄×C₆₇	—	4	1	no	—
170	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
171	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
172	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
173	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
174	C₂×C₄×C₆₇	#131	1	2	yes	—
175	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
176	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
178	C₂²⋊D₆₇	—	4	2	no	—
179	C₂×Dic₆₇	—	2	1	no	—
180	C₂×Dic₆₇	—	2	1	no	—
181	C₂×Dic₆₇	—	2	2	no	—
182	C₂×Dic₆₇	—	2	1	no	—
183	C₂×Dic₆₇	—	2	2	no	—
184	C₂×Dic₆₇	—	4	2	no	—
185	C₂×Dic₆₇	—	4	2	no	—
186	D₂₆₈	—	2	1	no	—
187	D₂₆₈	—	2	1	no	—
188	C₄×D₆₇	—	4	2	no	—
189	C₄×D₆₇	—	4	1	no	—
190	C₄×D₆₇	—	4	1	no	—
191	C₄×D₆₇	—	4	2	no	—
192	Dic₁₃₄	—	2	2	no	—
193	Dic₁₃₄	—	2	1	no	—
194	Dic₁₃₄	—	2	2	no	—
195	Dic₁₃₄	—	2	1	no	—
196	Dic₁₃₄	—	4	2	no	—
197	C₈×C₆₇	—	2	2	no	—
198	C₈×C₆₇	—	2	1	no	—
199	C₈×C₆₇	—	2	1	no	—
200	C₈×C₆₇	—	2	1	no	—
201	C₈×C₆₇	—	4	2	no	—
202	C₈×C₆₇	C₂×M₁₆	1	4	yes	—
203	C₈×C₆₇	#137	1	2	yes	complemented
204	C₆₇⋊C₈	—	4	1	no	—
205	C₆₇⋊C₈	—	4	2	no	—
206	C₆₇⋊C₈	—	4	1	no	—
207	C₂²×C₆₇	—	2	1	no	—
208	C₂²×C₆₇	—	2	1	no	—
209	C₂²×C₆₇	#436	1	2	yes	—
210	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
211	C₂×D₆₇	—	4	1	no	—
212	C₂×D₆₇	—	4	1	no	—
213	C₂×D₆₇	—	4	1	no	—
214	C₄×C₆₇	—	2	1	no	—
215	C₄×C₆₇	—	2	1	no	—
216	C₄×C₆₇	—	2	1	no	—
217	C₄×C₆₇	#436	1	2	yes	—
218	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
219	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
220	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
221	C₄×C₆₇	#575	1	1	yes	characteristic
222	Dic₆₇	—	4	2	no	—
223	Dic₆₇	—	4	1	no	—
224	Dic₆₇	—	4	1	no	—
225	Dic₆₇	—	4	1	no	—
226	Dic₆₇	—	4	2	no	—
227	#17892	—	67	1	no	maximal, Sylow
228	C₂×C₆₇	—	2	2	no	—
229	C₂×C₆₇	C₂×#436	1	1	yes	characteristic
230	C₂×C₆₇	C₄×#137	1	1	yes	characteristic
231	C₂×C₆₇	#2955	1	1	yes	characteristic
232	D₆₇	—	8	1	no	—
233	#2965	—	67	1	no	—
234	C₂×#446	—	67	1	no	—
235	C₄×#137	—	67	1	no	—
236	#1593	—	67	2	no	—
237	#1592	—	67	2	no	—
238	#1591	—	67	2	no	—
239	#1590	—	67	2	no	—
240	#1565	—	67	1	no	—
241	#1479	—	67	1	no	—
242	#1477	—	67	1	no	—
243	C₈×M₁₆	D₆₇	1	1	yes	characteristic, complemented
244	C₆₇	#17892	1	1	yes	characteristic, complemented, Sylow
245	C₂×#137	—	67	1	no	—
246	C₄×#41	—	67	1	no	—
247	C₂×C₄×Q₈	—	67	1	no	—
248	#446	—	67	2	no	—
249	#446	—	67	1	no	—
250	#446	—	67	2	no	—
251	#446	—	67	1	no	—
252	#446	—	134	1	no	—
253	#446	—	134	1	no	—
254	#441	—	67	2	no	—
255	#440	—	67	2	no	—
256	C₄×Q₁₆	—	67	2	no	—
257	C₄×SD₁₆	—	67	2	no	—
258	#436	—	67	1	no	—
259	#436	—	67	1	no	—
260	#435	—	134	1	no	—
261	#435	—	134	1	no	—
262	#431	—	67	1	no	—
263	#431	—	134	1	no	—
264	#431	C₂×D₆₇	1	1	yes	characteristic
265	#428	—	67	1	no	—
266	C₂×C₄⋊C₈	—	67	1	no	—
267	C₂×C₄⋊C₈	—	134	1	no	—
268	#419	—	67	1	no	—
269	#417	—	67	1	no	—
270	C₄×M₁₆	—	134	1	no	—
271	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
272	C₂×#97	—	134	1	no	—
273	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
274	#335	—	67	2	no	—
275	#334	—	67	2	no	—
276	Q₈⋊C₈	—	67	2	no	—
277	Q₈⋊C₈	—	67	2	no	—
278	Q₈⋊C₈	—	67	2	no	—
279	D₄⋊C₈	—	67	2	no	—
280	#322	C₂×D₆₇	1	2	yes	—
281	C₈²	C₂×D₆₇	1	2	yes	—
282	C₂×#41	—	67	1	no	—
283	C₂²×Q₈	—	67	1	no	—
284	#137	—	134	4	no	—
285	C₂×Q₁₆	—	67	2	no	—
286	C₂×SD₁₆	—	67	2	no	—
287	C₂×M₁₆	—	134	1	no	—
288	C₂×M₁₆	—	134	1	no	—
289	C₂×M₁₆	C₄×D₆₇	1	1	yes	characteristic
290	C₂×M₁₆	C₄×D₆₇	1	1	yes	characteristic
291	C₂²×C₈	—	134	1	no	—
292	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
293	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
294	C₄×Q₈	—	67	2	no	—
295	C₄×Q₈	—	67	1	no	—
296	C₄×Q₈	—	67	2	no	—
297	C₄×Q₈	—	67	1	no	—
298	C₄×Q₈	—	134	1	no	—
299	C₄×Q₈	—	134	1	no	—
300	C₄×D₄	—	67	1	no	—
301	C₄×D₄	—	67	1	no	—
302	C₄×D₄	—	134	1	no	—
303	C₄×D₄	—	134	1	no	—
304	#117	—	67	1	no	—
305	#117	—	134	1	no	—
306	C₂×C₄⋊C₄	—	67	1	no	—
307	C₂×C₄⋊C₄	—	134	1	no	—
308	C₂×C₄²	—	134	1	no	—
309	C₂×C₄²	—	134	1	no	—
310	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
311	#107	—	67	2	no	—
312	#106	—	67	2	no	—
313	C₄⋊C₈	—	2	1	no	—
314	C₄⋊C₈	—	67	2	no	—
315	C₄⋊C₈	—	67	1	no	—
316	C₄⋊C₈	—	67	1	no	—
317	C₄⋊C₈	—	134	2	no	—
318	C₄⋊C₈	—	134	2	no	—
319	C₄⋊C₈	—	134	1	no	—
320	C₄⋊C₈	—	134	1	no	—
321	C₄⋊C₈	—	134	1	no	—
322	#103	—	67	2	no	—
323	#103	—	67	2	no	—
324	#103	—	67	2	no	—
325	#102	—	67	2	no	—
326	C₂²⋊C₈	—	2	1	no	—
327	C₂²⋊C₈	—	134	1	no	—
328	C₂²⋊C₈	—	134	1	no	—
329	C₂²⋊C₈	—	134	1	no	—
330	#97	—	134	2	no	—
331	#97	—	134	2	no	—
332	#97	—	134	1	no	—
333	#97	—	134	1	no	—
334	#97	C₄×D₆₇	1	2	yes	—
335	C₄×C₈	—	134	1	no	—
336	C₄×C₈	D₂₆₈	1	1	yes	characteristic
337	C₄×C₈	D₂₆₈	1	1	yes	characteristic
338	C₄×C₈	C₂²×D₆₇	1	2	yes	—
339	C₄×C₈	C₂²×D₆₇	1	2	yes	—
340	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
341	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
342	#41	—	134	2	no	—
343	#41	—	134	2	no	—
344	C₂×Q₈	—	67	2	no	—
345	C₂×Q₈	—	67	1	no	—
346	C₂×Q₈	—	134	2	no	—
347	C₂×Q₈	—	134	2	no	—
348	C₂×Q₈	—	134	1	no	—
349	C₂×D₄	—	67	1	no	—
350	C₂×D₄	—	134	1	no	—
351	C₂²×C₄	—	134	1	no	—
352	C₂²×C₄	—	134	1	no	—
353	C₂²×C₄	—	134	1	no	—
354	C₂²×C₄	—	134	1	no	—
355	C₂²×C₄	—	134	1	no	—
356	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
357	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
358	C₂²×C₄	#11761628	1	1	yes	characteristic
359	Q₁₆	—	134	4	no	—
360	SD₁₆	—	134	4	no	—
361	M₁₆	—	2	2	no	—
362	M₁₆	—	268	2	no	—
363	M₁₆	—	268	2	no	—
364	M₁₆	#11761611	1	4	yes	complemented
365	C₂×C₈	—	2	2	no	—
366	C₂×C₈	—	2	2	no	—
367	C₂×C₈	—	4	1	no	—
368	C₂×C₈	—	268	2	no	—
369	C₂×C₈	—	134	1	no	—
370	C₂×C₈	—	134	1	no	—
371	C₂×C₈	—	134	2	no	—
372	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
373	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
374	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
375	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
376	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
377	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
378	C₄⋊C₄	—	67	2	no	—
379	C₄⋊C₄	—	67	1	no	—
380	C₄⋊C₄	—	67	1	no	—
381	C₄⋊C₄	—	134	1	no	—
382	C₄⋊C₄	—	134	2	no	—
383	C₄⋊C₄	—	134	1	no	—
384	C₄⋊C₄	—	134	2	no	—
385	C₄⋊C₄	—	134	1	no	—
386	C₂²⋊C₄	—	134	1	no	—
387	C₂²⋊C₄	—	134	1	no	—
388	C₂²⋊C₄	—	134	1	no	—
389	C₄²	—	134	1	no	—
390	C₄²	—	134	1	no	—
391	C₄²	—	134	2	no	—
392	C₄²	—	134	1	no	—
393	C₄²	—	134	2	no	—
394	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
395	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
396	C₄²	#11761628	1	1	yes	characteristic
397	C₄²	C₂³×D₆₇	1	1	yes	characteristic
398	C₂³	—	134	1	no	—
399	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
400	Q₈	—	268	2	no	—
401	Q₈	—	134	2	no	—
402	Q₈	—	134	1	no	—
403	Q₈	—	134	2	no	—
404	Q₈	—	134	1	no	—
405	D₄	—	268	2	no	—
406	D₄	—	134	1	no	—
407	D₄	—	134	1	no	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	4	1	no	—
411	C₂×C₄	—	268	2	no	—
412	C₂×C₄	—	268	2	no	—
413	C₂×C₄	—	268	2	no	—
414	C₂×C₄	—	268	1	no	—
415	C₂×C₄	—	268	1	no	—
416	C₂×C₄	—	268	2	no	—
417	C₂×C₄	—	134	1	no	—
418	C₂×C₄	—	134	1	no	—
419	C₂×C₄	—	134	2	no	—
420	C₂×C₄	—	134	1	no	—
421	C₂×C₄	—	134	2	no	—
422	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
423	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
424	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
425	C₂×C₄	C₂×#11761611	1	2	yes	—
426	C₂×C₄	#14524049	1	2	yes	—
427	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
428	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
429	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
430	C₈	—	2	2	no	—
431	C₈	—	2	1	no	—
432	C₈	—	2	1	no	—
433	C₈	—	2	1	no	—
434	C₈	—	4	2	no	—
435	C₈	—	268	1	no	—
436	C₈	—	268	2	no	—
437	C₈	—	268	1	no	—
438	C₈	C₂×#11761611	1	4	yes	—
439	C₈	#14524076	1	2	yes	complemented
440	C₂²	—	2	1	no	—
441	C₂²	—	2	1	no	—
442	C₂²	—	268	1	no	—
443	C₂²	—	268	1	no	—
444	C₂²	—	268	1	no	—
445	C₂²	#16863475	1	2	yes	—
446	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
447	C₄	—	2	1	no	—
448	C₄	—	2	1	no	—
449	C₄	—	2	1	no	—
450	C₄	—	268	1	no	—
451	C₄	—	268	1	no	—
452	C₄	—	268	2	no	—
453	C₄	—	268	1	no	—
454	C₄	—	268	2	no	—
455	C₄	#16863475	1	2	yes	—
456	C₄	C₂²×#11761611	1	1	yes	characteristic
457	C₄	C₂×#14524049	1	1	yes	characteristic
458	C₄	C₂×#14524076	1	1	yes	characteristic
459	C₄	#16864196	1	1	yes	characteristic
460	C₂	—	2	2	no	—
461	C₂	—	536	1	no	—
462	C₂	C₂×#16863475	1	1	yes	characteristic
463	C₂	C₄×#14524076	1	1	yes	characteristic
464	C₂	#22569113	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₈×M₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#11761628	C₂×C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₈×M₁₆	D₆₇
hypocenter	C₆₇	#17892
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#16863526	C₂	semidirect
4. #22559119	C₂	semidirect
5. #22559113	C₂	semidirect
6. #22559107	C₂	semidirect
7. #22559101	C₂	semidirect
8. #22558961	C₂	semidirect
9. #22558605	C₂	semidirect
11. C₈×M₁₆×C₆₇	C₂	semidirect
151. M₁₆×C₆₇	M₁₆	semidirect
203. C₈×C₆₇	#137	semidirect
243. C₈×M₁₆	D₆₇	semidirect
244. C₆₇	#17892	semidirect
364. M₁₆	#11761611	semidirect
439. C₈	#14524076	semidirect

Maximal subgroups

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

subgroup	index
#22569175	2
C₂×#16863526	2
C₄×#14524076	2
#22559119	2
#22559113	2
#22559107	2
#22559101	2
#22558961	2
#22558605	2
#22558597	2
C₈×M₁₆×C₆₇	2
#17892	67