Subgroups — #30827307

Order 17152.

708 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569160	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×#11761617	C₂	1	1	yes	maximal, characteristic
3	#22568944	C₂	1	1	yes	maximal, characteristic
4	#22559110	C₂	1	2	yes	maximal, complemented
5	#22559086	C₂	1	2	yes	maximal, complemented
6	#22559065	C₂	1	2	yes	maximal, complemented
7	#22559020	C₂	1	2	yes	maximal, complemented
8	#22558637	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558633	C₂	1	1	yes	maximal, characteristic
10	#22558498	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1567	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#11761617	C₄	1	1	yes	characteristic
13	C₂×M₁₆×D₆₇	—	2	1	no	—
14	C₂×C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₂₆₈	C₂²	1	1	yes	characteristic
16	#16863525	—	2	2	no	—
17	#16863523	C₂²	1	2	yes	complemented
18	#16863520	C₂²	1	2	yes	complemented
19	C₄×#11761617	C₄	1	1	yes	characteristic
20	C₄×#11761617	C₄	1	1	yes	characteristic
21	C₄×#11761617	C₄	1	1	yes	characteristic
22	C₄×#11761617	C₄	1	1	yes	characteristic
23	C₄×#11761617	C₂²	1	1	yes	characteristic
24	C₄×#11761617	C₂²	1	1	yes	characteristic
25	C₄×#11761617	C₂²	1	1	yes	characteristic
26	C₄×#11761617	C₂²	1	1	yes	characteristic
27	#16863473	C₂²	1	2	yes	—
28	#16863466	—	2	2	no	—
29	#16863461	C₂²	1	2	yes	—
30	#16863449	C₂²	1	1	yes	characteristic
31	C₂×#14523975	C₄	1	1	yes	characteristic
32	#16863419	—	2	1	no	—
33	C₂×#14523968	C₄	1	1	yes	characteristic
34	C₂×#14523965	C₄	1	1	yes	characteristic
35	#16863359	—	2	1	no	—
36	C₄×#11761612	C₂²	1	1	yes	characteristic
37	M₁₆×Dic₆₇	—	2	1	no	—
38	#16863126	C₂²	1	2	yes	complemented
39	#16863126	C₂²	1	2	yes	complemented
40	#16863123	C₂²	1	2	yes	—
41	#16863123	C₂²	1	2	yes	—
42	#16863118	C₂²	1	2	yes	complemented
43	#16863117	C₂²	1	2	yes	complemented
44	#423×C₆₇	C₂²	1	1	yes	characteristic
45	#423×C₆₇	C₂²	1	1	yes	characteristic
46	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
47	#335×C₆₇	C₂²	1	2	yes	complemented
48	#335×C₆₇	C₂²	1	2	yes	complemented
49	C₂²×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
50	C₂²×D₂₆₈	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₄×D₆₇	—	2	1	no	—
52	C₂×#11761617	—	2	4	no	—
53	C₂×#11761617	C₂×C₄	1	1	yes	characteristic
54	C₂×#11761617	C₂×C₄	1	1	yes	characteristic
55	C₂×#11761617	C₂×C₄	1	1	yes	characteristic
56	C₂×#11761617	C₂×C₄	1	1	yes	characteristic
57	C₂×#11761612	D₄	1	1	yes	characteristic
58	C₂×#11761612	D₄	1	1	yes	characteristic
59	C₂×#11761611	—	2	2	no	—
60	M₁₆×D₆₇	—	4	4	no	—
61	C₂×C₈×D₆₇	—	2	2	no	—
62	C₄×Dic₁₃₄	—	2	1	no	—
63	C₄×Dic₁₃₄	—	2	1	no	—
64	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₁₃₄	C₂³	1	1	yes	characteristic
67	C₄×Dic₁₃₄	C₂³	1	1	yes	characteristic
68	C₄×D₂₆₈	—	2	1	no	—
69	C₄×D₂₆₈	—	2	1	no	—
70	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
71	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
72	C₄×D₂₆₈	C₂³	1	1	yes	characteristic
73	C₄×D₂₆₈	C₂³	1	1	yes	characteristic
74	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
76	C₂×Dic₆₇⋊C₄	—	2	1	no	—
77	C₂×(C₂×D₆₇)⋊C₄	—	2	1	no	—
78	C₂×C₄×Dic₆₇	—	2	1	no	—
79	#14523975	C₂×C₄	1	1	yes	characteristic
80	#14523975	C₂×C₄	1	1	yes	characteristic
81	#14523975	C₂×C₄	1	1	yes	characteristic
82	#14523975	C₂×C₄	1	1	yes	characteristic
83	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
84	Dic₆₇⋊C₈	—	2	2	no	—
85	Dic₆₇⋊C₈	—	2	2	no	—
86	#14523968	C₂×C₄	1	1	yes	characteristic
87	#14523968	C₂×C₄	1	1	yes	characteristic
88	#14523968	C₂×C₄	1	1	yes	characteristic
89	#14523968	C₂×C₄	1	1	yes	characteristic
90	#14523965	C₂×C₄	1	1	yes	characteristic
91	#14523965	C₂×C₄	1	1	yes	characteristic
92	#14523965	C₂×C₄	1	1	yes	characteristic
93	#14523965	C₂×C₄	1	1	yes	characteristic
94	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
95	(C₂×D₆₇)⋊C₈	—	2	2	no	—
96	(C₂×D₆₇)⋊C₈	—	2	2	no	—
97	#14523954	D₄	1	2	yes	—
98	#14523953	—	2	2	no	—
99	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
100	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
101	C₈×Dic₆₇	—	2	2	no	—
102	C₂×M₁₆×C₆₇	—	2	1	no	—
103	C₂×M₁₆×C₆₇	—	2	1	no	—
104	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
105	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
106	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
107	C₄⋊C₈×C₆₇	—	2	1	no	—
108	C₄⋊C₈×C₆₇	—	2	1	no	—
109	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
110	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
111	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
112	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
113	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
114	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
115	C₂³×D₆₇	—	2	1	no	—
116	C₂×Dic₁₃₄	—	2	2	no	—
117	C₂×Dic₁₃₄	—	2	1	no	—
118	C₂×Dic₁₃₄	—	4	1	no	—
119	C₂×Dic₁₃₄	C₂²×C₄	1	1	yes	characteristic
120	C₂×Dic₁₃₄	C₂²×C₄	1	1	yes	characteristic
121	C₂×D₂₆₈	—	2	2	no	—
122	C₂×D₂₆₈	—	2	1	no	—
123	C₂×D₂₆₈	—	4	1	no	—
124	C₂×D₂₆₈	C₂²×C₄	1	1	yes	characteristic
125	C₂×D₂₆₈	C₂²×C₄	1	1	yes	characteristic
126	C₂²×Dic₆₇	—	2	1	no	—
127	C₂²×Dic₆₇	—	2	1	no	—
128	C₂²×Dic₆₇	—	2	1	no	—
129	C₂×C₄×D₆₇	—	2	1	no	—
130	C₂×C₄×D₆₇	—	2	1	no	—
131	C₂×C₄×D₆₇	—	2	1	no	—
132	C₂×C₄×D₆₇	—	2	1	no	—
133	C₂×C₄×D₆₇	—	4	2	no	—
134	C₂×C₄×D₆₇	—	4	2	no	—
135	#11761617	—	2	2	no	—
136	#11761617	—	2	2	no	—
137	#11761617	—	2	2	no	—
138	#11761617	M₁₆	1	4	yes	complemented
139	#11761612	—	2	2	no	—
140	#11761612	—	2	2	no	—
141	#11761611	—	4	4	no	—
142	C₂×C₆₇⋊C₈	—	2	2	no	—
143	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
144	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
145	C₈×D₆₇	—	4	4	no	—
146	C₄⋊Dic₆₇	—	2	1	no	—
147	C₄⋊Dic₆₇	—	2	1	no	—
148	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
149	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
150	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
151	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
152	Dic₆₇⋊C₄	—	2	1	no	—
153	Dic₆₇⋊C₄	—	2	1	no	—
154	Dic₆₇⋊C₄	—	2	1	no	—
155	Dic₆₇⋊C₄	—	2	1	no	—
156	(C₂×D₆₇)⋊C₄	—	2	1	no	—
157	(C₂×D₆₇)⋊C₄	—	2	1	no	—
158	(C₂×D₆₇)⋊C₄	—	2	1	no	—
159	(C₂×D₆₇)⋊C₄	—	2	1	no	—
160	C₄×Dic₆₇	—	2	1	no	—
161	C₄×Dic₆₇	—	2	1	no	—
162	C₄×Dic₆₇	—	2	1	no	—
163	C₄×Dic₆₇	—	2	1	no	—
164	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
165	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
167	M₁₆×C₆₇	—	4	2	no	—
168	M₁₆×C₆₇	—	4	2	no	—
169	C₂×C₈×C₆₇	—	2	2	no	—
170	C₂×C₈×C₆₇	—	2	2	no	—
171	C₂×C₈×C₆₇	—	2	2	no	—
172	C₂×C₈×C₆₇	—	4	1	no	—
173	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
174	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
175	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
176	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
177	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
178	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
179	C₄²×C₆₇	#41	1	1	yes	characteristic
180	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
181	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
182	C₂²×D₆₇	—	2	1	no	—
183	C₂²×D₆₇	—	2	1	no	—
184	C₂²×D₆₇	—	4	2	no	—
185	C₂²×D₆₇	—	4	2	no	—
186	C₂²×D₆₇	—	8	1	no	—
187	C₂×C₄×C₆₇	—	2	1	no	—
188	C₂×C₄×C₆₇	—	2	1	no	—
189	C₂×C₄×C₆₇	—	4	1	no	—
190	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
191	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
192	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
193	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
194	C₂×C₄×C₆₇	#137	1	2	yes	—
195	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
196	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
197	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
198	C₂×Dic₆₇	—	2	1	no	—
199	C₂×Dic₆₇	—	2	1	no	—
200	C₂×Dic₆₇	—	2	1	no	—
201	C₂×Dic₆₇	—	2	1	no	—
202	C₂×Dic₆₇	—	2	1	no	—
203	C₂×Dic₆₇	—	2	1	no	—
204	C₂×Dic₆₇	—	4	2	no	—
205	C₂×Dic₆₇	—	4	2	no	—
206	C₂×Dic₆₇	—	8	1	no	—
207	D₂₆₈	—	2	1	no	—
208	D₂₆₈	—	2	1	no	—
209	D₂₆₈	—	2	1	no	—
210	D₂₆₈	—	4	2	no	—
211	D₂₆₈	C₂×M₁₆	1	2	yes	—
212	C₄×D₆₇	—	4	1	no	—
213	C₄×D₆₇	—	4	1	no	—
214	C₄×D₆₇	—	4	1	no	—
215	C₄×D₆₇	—	4	1	no	—
216	C₄×D₆₇	—	4	2	no	—
217	C₄×D₆₇	—	4	2	no	—
218	Dic₁₃₄	—	2	1	no	—
219	Dic₁₃₄	—	2	1	no	—
220	Dic₁₃₄	—	2	1	no	—
221	Dic₁₃₄	—	4	2	no	—
222	Dic₁₃₄	C₂×M₁₆	1	2	yes	complemented
223	C₈×C₆₇	—	2	1	no	—
224	C₈×C₆₇	—	2	1	no	—
225	C₈×C₆₇	—	2	1	no	—
226	C₈×C₆₇	—	4	2	no	—
227	C₈×C₆₇	—	4	2	no	—
228	C₈×C₆₇	C₂×M₁₆	1	2	yes	complemented
229	C₆₇⋊C₈	—	2	2	no	—
230	C₆₇⋊C₈	—	2	2	no	—
231	C₆₇⋊C₈	—	4	2	no	—
232	C₂²×C₆₇	—	2	1	no	—
233	C₂²×C₆₇	—	2	1	no	—
234	C₂²×C₆₇	#440	1	2	yes	—
235	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
236	C₂×D₆₇	—	2	2	no	—
237	C₂×D₆₇	—	4	1	no	—
238	C₂×D₆₇	—	4	1	no	—
239	C₂×D₆₇	—	4	1	no	—
240	C₂×D₆₇	—	4	1	no	—
241	C₂×D₆₇	—	4	1	no	—
242	C₂×D₆₇	—	8	4	no	—
243	C₄×C₆₇	—	2	1	no	—
244	C₄×C₆₇	—	2	1	no	—
245	C₄×C₆₇	—	2	1	no	—
246	C₄×C₆₇	#440	1	2	yes	—
247	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
248	C₄×C₆₇	#568	1	1	yes	characteristic
249	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
250	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
251	Dic₆₇	—	2	2	no	—
252	Dic₆₇	—	4	1	no	—
253	Dic₆₇	—	4	1	no	—
254	Dic₆₇	—	4	1	no	—
255	Dic₆₇	—	4	1	no	—
256	Dic₆₇	—	4	1	no	—
257	#18040	—	67	1	no	maximal, Sylow
258	C₂×C₆₇	—	2	2	no	—
259	C₂×C₆₇	D₄×M₁₆	1	1	yes	characteristic
260	C₂×C₆₇	C₂×#440	1	1	yes	characteristic
261	C₂×C₆₇	#2957	1	1	yes	characteristic
262	D₆₇	—	4	2	no	—
263	D₆₇	—	8	1	no	—
264	Q₈×M₁₆	—	67	1	no	—
265	C₂×C₄×SD₁₆	—	67	1	no	—
266	D₄×M₁₆	—	67	1	no	—
267	#1592	—	67	2	no	—
268	#1588	—	67	2	no	—
269	#1585	—	67	2	no	—
270	SD₁₆⋊C₈	—	67	2	no	—
271	#1567	D₆₇	1	1	yes	characteristic, complemented
272	#1487	—	67	1	no	—
273	#1486	—	67	1	no	—
274	#1451	—	67	1	no	—
275	C₆₇	#18040	1	1	yes	characteristic, complemented, Sylow
276	C₂²×SD₁₆	—	67	1	no	—
277	C₂²×M₁₆	—	134	1	no	—
278	C₂×C₄×Q₈	—	67	1	no	—
279	C₂×C₄×D₄	—	67	1	no	—
280	#446	—	67	2	no	—
281	#446	—	134	2	no	—
282	C₈×Q₈	—	67	2	no	—
283	C₄×SD₁₆	—	67	1	no	—
284	C₄×SD₁₆	—	67	1	no	—
285	C₄×SD₁₆	—	67	1	no	—
286	C₄×SD₁₆	—	67	1	no	—
287	C₄×SD₁₆	—	67	1	no	—
288	C₄×SD₁₆	—	67	1	no	—
289	C₄×SD₁₆	—	67	1	no	—
290	C₄×SD₁₆	—	67	1	no	—
291	#436	—	67	2	no	—
292	#435	—	134	2	no	—
293	C₈×D₄	—	67	2	no	—
294	#431	—	67	1	no	—
295	C₂×#106	—	67	1	no	—
296	#423	—	134	1	no	—
297	#423	C₂×D₆₇	1	1	yes	characteristic
298	#423	C₂×D₆₇	1	1	yes	characteristic
299	C₂×#103	—	67	1	no	—
300	C₂×#102	—	67	1	no	—
301	#407	—	134	1	no	—
302	C₄×M₁₆	—	67	1	no	—
303	C₄×M₁₆	—	134	1	no	—
304	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
305	#335	C₂×D₆₇	1	2	yes	complemented
306	#335	C₂×D₆₇	1	2	yes	complemented
307	Q₈⋊C₈	—	67	2	no	—
308	Q₈⋊C₈	—	67	2	no	—
309	D₄⋊C₈	—	67	2	no	—
310	D₄⋊C₈	—	67	2	no	—
311	#322	—	67	2	no	—
312	#322	—	67	2	no	—
313	C₂²×Q₈	—	67	1	no	—
314	C₂²×D₄	—	67	1	no	—
315	C₂³×C₄	—	134	1	no	—
316	C₂×SD₁₆	—	67	1	no	—
317	C₂×SD₁₆	—	67	1	no	—
318	C₂×SD₁₆	—	67	1	no	—
319	C₂×SD₁₆	—	67	1	no	—
320	C₂×SD₁₆	—	134	4	no	—
321	C₂×M₁₆	—	2	1	no	—
322	C₂×M₁₆	—	2	1	no	—
323	C₂×M₁₆	—	268	4	no	—
324	C₂×M₁₆	—	67	1	no	—
325	C₂×M₁₆	—	67	1	no	—
326	C₂×M₁₆	—	134	2	no	—
327	C₂²×C₈	—	134	2	no	—
328	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
329	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
330	C₄×Q₈	—	67	1	no	—
331	C₄×Q₈	—	67	1	no	—
332	C₄×Q₈	—	67	1	no	—
333	C₄×Q₈	—	67	1	no	—
334	C₄×Q₈	—	134	1	no	—
335	C₄×Q₈	—	134	1	no	—
336	C₄×D₄	—	67	1	no	—
337	C₄×D₄	—	67	1	no	—
338	C₄×D₄	—	67	1	no	—
339	C₄×D₄	—	67	1	no	—
340	C₄×D₄	—	134	1	no	—
341	C₄×D₄	—	134	1	no	—
342	C₂×C₄⋊C₄	—	67	1	no	—
343	C₂×C₄⋊C₄	—	67	1	no	—
344	C₂×C₄⋊C₄	—	134	1	no	—
345	C₂×C₂²⋊C₄	—	134	1	no	—
346	C₂×C₄²	—	134	1	no	—
347	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
348	#106	—	67	1	no	—
349	#106	—	67	1	no	—
350	#106	—	67	1	no	—
351	#106	—	67	1	no	—
352	C₄⋊C₈	—	2	1	no	—
353	C₄⋊C₈	—	2	1	no	—
354	C₄⋊C₈	—	134	2	no	—
355	C₄⋊C₈	—	134	2	no	—
356	C₄⋊C₈	—	134	1	no	—
357	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
358	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
359	#103	—	67	1	no	—
360	#103	—	67	1	no	—
361	#103	—	67	1	no	—
362	#103	—	67	1	no	—
363	#102	—	67	1	no	—
364	#102	—	67	1	no	—
365	#102	—	67	1	no	—
366	#102	—	67	1	no	—
367	C₂²⋊C₈	—	134	2	no	—
368	C₂²⋊C₈	—	134	2	no	—
369	C₂²⋊C₈	—	134	1	no	—
370	#97	—	67	2	no	—
371	#97	—	134	2	no	—
372	C₄×C₈	—	67	2	no	—
373	C₄×C₈	—	67	2	no	—
374	C₄×C₈	—	134	2	no	—
375	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
376	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
377	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
378	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
379	C₂⁴	—	134	1	no	—
380	C₂×Q₈	—	268	1	no	—
381	C₂×Q₈	—	67	1	no	—
382	C₂×Q₈	—	67	1	no	—
383	C₂×Q₈	—	134	2	no	—
384	C₂×Q₈	—	134	1	no	—
385	C₂×D₄	—	268	1	no	—
386	C₂×D₄	—	67	1	no	—
387	C₂×D₄	—	67	1	no	—
388	C₂×D₄	—	134	2	no	—
389	C₂×D₄	—	134	1	no	—
390	C₂²×C₄	—	268	2	no	—
391	C₂²×C₄	—	268	2	no	—
392	C₂²×C₄	—	134	1	no	—
393	C₂²×C₄	—	134	1	no	—
394	C₂²×C₄	—	134	1	no	—
395	C₂²×C₄	—	134	1	no	—
396	C₂²×C₄	—	134	1	no	—
397	C₂²×C₄	—	134	1	no	—
398	C₂²×C₄	—	134	1	no	—
399	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
400	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
401	C₂²×C₄	#11761629	1	1	yes	characteristic
402	SD₁₆	—	67	4	no	—
403	SD₁₆	—	134	2	no	—
404	SD₁₆	—	134	2	no	—
405	SD₁₆	—	134	2	no	—
406	M₁₆	—	4	2	no	—
407	M₁₆	—	4	2	no	—
408	M₁₆	—	268	4	no	—
409	M₁₆	—	134	2	no	—
410	M₁₆	—	134	2	no	—
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	—	2	2	no	—
413	C₂×C₈	—	2	2	no	—
414	C₂×C₈	—	4	1	no	—
415	C₂×C₈	—	268	4	no	—
416	C₂×C₈	—	67	2	no	—
417	C₂×C₈	—	67	2	no	—
418	C₂×C₈	—	134	2	no	—
419	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
420	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
421	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
422	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
423	C₄⋊C₄	—	67	1	no	—
424	C₄⋊C₄	—	67	1	no	—
425	C₄⋊C₄	—	67	1	no	—
426	C₄⋊C₄	—	67	1	no	—
427	C₄⋊C₄	—	134	1	no	—
428	C₄⋊C₄	—	134	1	no	—
429	C₄⋊C₄	—	134	1	no	—
430	C₄⋊C₄	—	134	1	no	—
431	C₄⋊C₄	—	134	1	no	—
432	C₄⋊C₄	—	134	1	no	—
433	C₂²⋊C₄	—	134	1	no	—
434	C₂²⋊C₄	—	134	1	no	—
435	C₂²⋊C₄	—	134	1	no	—
436	C₂²⋊C₄	—	134	1	no	—
437	C₄²	—	134	1	no	—
438	C₄²	—	134	1	no	—
439	C₄²	—	134	1	no	—
440	C₄²	—	134	1	no	—
441	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
442	C₄²	D₄×D₆₇	1	1	yes	characteristic
443	C₄²	#11761629	1	1	yes	characteristic
444	C₄²	C₂³×D₆₇	1	1	yes	characteristic
445	C₂³	—	268	2	no	—
446	C₂³	—	268	2	no	—
447	C₂³	—	536	1	no	—
448	C₂³	—	134	1	no	—
449	C₂³	—	134	1	no	—
450	C₂³	#14524005	1	1	yes	characteristic
451	Q₈	—	268	2	no	—
452	Q₈	—	67	2	no	—
453	Q₈	—	134	1	no	—
454	Q₈	—	134	1	no	—
455	Q₈	—	134	1	no	—
456	D₄	—	268	2	no	—
457	D₄	—	67	2	no	—
458	D₄	—	134	1	no	—
459	D₄	—	134	1	no	—
460	D₄	—	134	1	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	1	no	—
463	C₂×C₄	—	4	1	no	—
464	C₂×C₄	—	268	2	no	—
465	C₂×C₄	—	268	2	no	—
466	C₂×C₄	—	268	2	no	—
467	C₂×C₄	—	268	2	no	—
468	C₂×C₄	—	268	1	no	—
469	C₂×C₄	—	268	1	no	—
470	C₂×C₄	—	268	1	no	—
471	C₂×C₄	—	268	1	no	—
472	C₂×C₄	—	536	1	no	—
473	C₂×C₄	—	134	1	no	—
474	C₂×C₄	—	134	1	no	—
475	C₂×C₄	—	134	1	no	—
476	C₂×C₄	—	134	1	no	—
477	C₂×C₄	—	134	1	no	—
478	C₂×C₄	—	134	1	no	—
479	C₂×C₄	#14524005	1	1	yes	characteristic
480	C₂×C₄	#14524005	1	1	yes	characteristic
481	C₂×C₄	#14524005	1	1	yes	characteristic
482	C₂×C₄	#14524067	1	2	yes	—
483	C₂×C₄	#14524075	1	2	yes	—
484	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
485	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
486	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
487	C₈	—	2	1	no	—
488	C₈	—	2	1	no	—
489	C₈	—	2	1	no	—
490	C₈	—	4	2	no	—
491	C₈	—	4	2	no	—
492	C₈	—	268	2	no	—
493	C₈	—	134	2	no	—
494	C₈	—	134	2	no	—
495	C₈	M₁₆×D₆₇	1	2	yes	complemented
496	C₂²	—	2	1	no	—
497	C₂²	—	2	1	no	—
498	C₂²	—	268	1	no	—
499	C₂²	—	268	1	no	—
500	C₂²	—	268	1	no	—
501	C₂²	—	268	1	no	—
502	C₂²	—	268	1	no	—
503	C₂²	—	536	4	no	—
504	C₂²	—	134	2	no	—
505	C₂²	#16863495	1	2	yes	—
506	C₂²	C₂×#14524005	1	1	yes	characteristic
507	C₄	—	2	1	no	—
508	C₄	—	2	1	no	—
509	C₄	—	2	1	no	—
510	C₄	—	268	1	no	—
511	C₄	—	268	1	no	—
512	C₄	—	268	1	no	—
513	C₄	—	268	1	no	—
514	C₄	—	268	1	no	—
515	C₄	—	134	2	no	—
516	C₄	#16863495	1	2	yes	—
517	C₄	C₂×M₁₆×D₆₇	1	1	yes	characteristic
518	C₄	#16864162	1	1	yes	characteristic
519	C₄	C₂×#14524067	1	1	yes	characteristic
520	C₄	C₂×#14524075	1	1	yes	characteristic
521	C₂	—	2	2	no	—
522	C₂	—	268	2	no	—
523	C₂	—	536	1	no	—
524	C₂	#22568944	1	1	yes	characteristic
525	C₂	C₂×#16863495	1	1	yes	characteristic
526	C₂	#22569125	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1567	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂²×D₂₆₈	C₂×C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1567	D₆₇
hypocenter	C₆₇	#18040
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
1. #22569160	C₂	semidirect
4. #22559110	C₂	semidirect
5. #22559086	C₂	semidirect
6. #22559065	C₂	semidirect
7. #22559020	C₂	semidirect
8. #22558637	C₂	semidirect
10. #22558498	C₂	semidirect
11. C₆₇×#1567	C₂	semidirect
17. #16863523	C₂²	semidirect
18. #16863520	C₂²	semidirect
38. #16863126	C₂²	semidirect
39. #16863126	C₂²	semidirect
42. #16863118	C₂²	semidirect
43. #16863117	C₂²	semidirect
47. #335×C₆₇	C₂²	semidirect
48. #335×C₆₇	C₂²	semidirect
138. #11761617	M₁₆	semidirect
222. Dic₁₃₄	C₂×M₁₆	semidirect
228. C₈×C₆₇	C₂×M₁₆	semidirect
271. #1567	D₆₇	semidirect
275. C₆₇	#18040	semidirect
305. #335	C₂×D₆₇	semidirect
306. #335	C₂×D₆₇	semidirect
495. C₈	M₁₆×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22569160	2
C₂×C₄×#11761617	2
#22568944	2
#22559110	2
#22559086	2
#22559065	2
#22559020	2
#22558637	2
#22558633	2
#22558498	2
C₆₇×#1567	2
#18040	67