Subgroups — #30825636

Order 17152.

684 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569249	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#14524068	C₂	1	1	yes	maximal, characteristic
3	#22559121	C₂	1	2	yes	maximal, complemented
4	#22559109	C₂	1	2	yes	maximal, complemented
5	#22559099	C₂	1	2	yes	maximal, complemented
6	#22559092	C₂	1	2	yes	maximal, complemented
7	#22558959	C₂	1	1	yes	maximal, characteristic, complemented
8	#22558591	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#16863123	C₂	1	1	yes	maximal, characteristic, complemented
10	#22558490	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2989	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524068	C₄	1	1	yes	characteristic
13	C₂×C₄×D₂₆₈	C₂²	1	1	yes	characteristic
14	#16863506	C₄	1	2	yes	—
15	#16863494	C₄	1	2	yes	—
16	C₄×#11761615	C₂²	1	2	yes	—
17	C₄×#11761613	C₂²	1	2	yes	—
18	#16863475	C₂²	1	2	yes	—
19	#16863475	C₂²	1	2	yes	—
20	#16863473	—	2	1	no	—
21	#16863473	—	2	1	no	—
22	#16863449	C₂²	1	1	yes	characteristic
23	#16863439	C₄	1	1	yes	characteristic
24	#16863400	C₄	1	1	yes	characteristic
25	#16863397	C₄	1	1	yes	characteristic
26	#16863358	—	2	2	no	—
27	C₄×#11761612	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
29	#16863148	C₂²	1	2	yes	complemented
30	#16863146	C₂²	1	2	yes	complemented
31	Q₈⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic, complemented
32	Q₈⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic, complemented
33	D₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic, complemented
34	D₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic, complemented
35	#16863123	C₂²	1	2	yes	—
36	#16863123	C₂²	1	2	yes	—
37	#16863118	C₂²	1	2	yes	complemented
38	C₈×C₆₇⋊C₈	C₂²	1	2	yes	complemented
39	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
40	#446×C₆₇	C₂²	1	2	yes	complemented
41	#436×C₆₇	C₂²	1	2	yes	complemented
42	#435×C₆₇	—	2	2	no	—
43	#432×C₆₇	—	2	1	no	—
44	#431×C₆₇	C₂²	1	1	yes	characteristic
45	#423×C₆₇	—	2	1	no	—
46	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
47	C₂²×D₂₆₈	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×D₆₇	—	2	1	no	—
49	#14524068	—	2	4	no	—
50	C₂×#11761615	C₂×C₄	1	2	yes	—
51	C₂×#11761613	C₂×C₄	1	2	yes	—
52	C₂×#11761612	C₂×C₄	1	1	yes	characteristic
53	C₂×#11761612	C₂×C₄	1	1	yes	characteristic
54	C₂×#11761611	—	2	2	no	—
55	C₂×#11761611	—	2	2	no	—
56	C₂²×C₆₇⋊C₈	—	2	1	no	—
57	C₄×D₂₆₈	—	2	1	no	—
58	C₄×D₂₆₈	—	2	1	no	—
59	C₄×D₂₆₈	C₂×C₄	1	2	yes	—
60	C₄×D₂₆₈	C₂³	1	2	yes	—
61	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
62	C₂×(C₂×D₆₇)⋊C₄	—	2	1	no	—
63	#14523976	C₂×C₄	1	2	yes	—
64	#14523974	C₂×C₄	1	2	yes	—
65	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
66	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
67	#14523967	C₂×C₄	1	2	yes	—
68	#14523964	C₂×C₄	1	2	yes	—
69	#14523963	C₂×C₄	1	2	yes	—
70	#14523963	C₂×C₄	1	2	yes	—
71	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
72	(C₂×D₆₇)⋊C₈	—	2	2	no	—
73	(C₂×D₆₇)⋊C₈	—	2	2	no	—
74	#14523954	C₂×C₄	1	2	yes	—
75	C₄×C₆₇⋊C₈	—	2	1	no	—
76	C₄×C₆₇⋊C₈	—	2	1	no	—
77	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
78	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
79	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
80	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
81	C₂×M₁₆×C₆₇	—	2	2	no	—
82	C₂²×C₈×C₆₇	—	2	1	no	—
83	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
84	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
85	C₄×D₄×C₆₇	—	2	1	no	—
86	C₄×D₄×C₆₇	—	2	1	no	—
87	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
88	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
89	#117×C₆₇	—	2	1	no	—
90	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
91	C₂×C₄²×C₆₇	—	2	1	no	—
92	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
93	C₄⋊C₈×C₆₇	—	2	2	no	—
94	C₄⋊C₈×C₆₇	—	2	1	no	—
95	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
96	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
97	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
98	C₂²⋊C₈×C₆₇	—	2	2	no	—
99	C₂²⋊C₈×C₆₇	D₄	1	2	yes	—
100	#97×C₆₇	—	2	2	no	—
101	C₄×C₈×C₆₇	C₂³	1	2	yes	—
102	C₂³×D₆₇	—	2	1	no	—
103	C₂×D₂₆₈	—	2	2	no	—
104	C₂×D₂₆₈	—	2	2	no	—
105	C₂×D₂₆₈	—	2	1	no	—
106	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
107	C₂²×Dic₆₇	—	2	1	no	—
108	C₂×C₄×D₆₇	—	2	2	no	—
109	C₂×C₄×D₆₇	—	2	2	no	—
110	C₂×C₄×D₆₇	—	8	1	no	—
111	C₂×C₄×D₆₇	—	8	1	no	—
112	#11761615	—	2	4	no	—
113	#11761613	—	2	4	no	—
114	#11761612	—	2	2	no	—
115	#11761612	M₁₆	1	4	yes	complemented
116	#11761611	—	8	2	no	—
117	#11761611	—	8	2	no	—
118	C₂×C₆₇⋊C₈	—	2	1	no	—
119	C₂×C₆₇⋊C₈	—	2	1	no	—
120	C₂×C₆₇⋊C₈	—	2	1	no	—
121	C₂×C₆₇⋊C₈	—	2	1	no	—
122	C₂×C₆₇⋊C₈	—	4	2	no	—
123	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
124	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
125	C₄⋊Dic₆₇	—	2	1	no	—
126	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
127	(C₂×D₆₇)⋊C₄	—	2	2	no	—
128	(C₂×D₆₇)⋊C₄	—	2	2	no	—
129	#41×C₆₇	—	2	2	no	—
130	#41×C₆₇	—	2	2	no	—
131	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
132	C₂×D₄×C₆₇	—	2	1	no	—
133	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
134	C₂²×C₄×C₆₇	—	2	1	no	—
135	C₂²×C₄×C₆₇	—	2	1	no	—
136	C₂²×C₄×C₆₇	—	2	1	no	—
137	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
138	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
140	M₁₆×C₆₇	—	8	2	no	—
141	C₂×C₈×C₆₇	—	2	2	no	—
142	C₂×C₈×C₆₇	—	8	1	no	—
143	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
144	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
145	C₄⋊C₄×C₆₇	—	2	1	no	—
146	C₄⋊C₄×C₆₇	—	2	1	no	—
147	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
148	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
149	C₂²⋊C₄×C₆₇	—	2	1	no	—
150	C₂²⋊C₄×C₆₇	—	2	1	no	—
151	C₂²⋊C₄×C₆₇	—	2	1	no	—
152	C₄²×C₆₇	—	2	1	no	—
153	C₄²×C₆₇	—	2	1	no	—
154	C₄²×C₆₇	—	2	1	no	—
155	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
156	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
157	C₄²×C₆₇	#41	1	1	yes	characteristic
158	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
159	C₂³×C₆₇	—	2	1	no	—
160	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
161	C₂²×D₆₇	—	2	2	no	—
162	C₂²×D₆₇	—	4	2	no	—
163	C₂²×D₆₇	—	8	1	no	—
164	C₂²×D₆₇	—	8	1	no	—
165	Q₈×C₆₇	—	2	1	no	—
166	Q₈×C₆₇	—	2	1	no	—
167	D₄×C₆₇	—	2	1	no	—
168	D₄×C₆₇	—	2	1	no	—
169	D₄×C₆₇	—	4	2	no	—
170	C₂×C₄×C₆₇	—	2	1	no	—
171	C₂×C₄×C₆₇	—	2	1	no	—
172	C₂×C₄×C₆₇	—	2	1	no	—
173	C₂×C₄×C₆₇	—	2	1	no	—
174	C₂×C₄×C₆₇	—	2	1	no	—
175	C₂×C₄×C₆₇	—	4	1	no	—
176	C₂×C₄×C₆₇	—	4	1	no	—
177	C₂×C₄×C₆₇	—	4	1	no	—
178	C₂×C₄×C₆₇	—	4	2	no	—
179	C₂×C₄×C₆₇	—	8	1	no	—
180	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
181	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
182	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
183	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
184	C₂×C₄×C₆₇	#131	1	2	yes	—
185	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
186	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
187	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
188	C₂×Dic₆₇	—	2	2	no	—
189	C₂×Dic₆₇	—	8	1	no	—
190	D₂₆₈	—	2	2	no	—
191	D₂₆₈	—	2	2	no	—
192	D₂₆₈	—	4	2	no	—
193	C₄×D₆₇	—	4	2	no	—
194	C₄×D₆₇	—	4	2	no	—
195	C₄×D₆₇	—	8	2	no	—
196	C₈×C₆₇	—	2	2	no	—
197	C₈×C₆₇	—	2	2	no	—
198	C₈×C₆₇	—	4	2	no	—
199	C₆₇⋊C₈	—	2	2	no	—
200	C₆₇⋊C₈	—	2	2	no	—
201	C₆₇⋊C₈	—	4	1	no	—
202	C₆₇⋊C₈	—	4	1	no	—
203	C₆₇⋊C₈	—	4	1	no	—
204	C₆₇⋊C₈	C₂×M₁₆	1	4	yes	—
205	C₂²×C₆₇	—	2	1	no	—
206	C₂²×C₆₇	—	2	1	no	—
207	C₂²×C₆₇	—	4	1	no	—
208	C₂²×C₆₇	—	4	1	no	—
209	C₂²×C₆₇	—	4	1	no	—
210	C₂²×C₆₇	#436	1	2	yes	—
211	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
212	C₂×D₆₇	—	4	2	no	—
213	C₂×D₆₇	—	4	2	no	—
214	C₂×D₆₇	—	4	2	no	—
215	C₂×D₆₇	—	8	4	no	—
216	C₄×C₆₇	—	2	1	no	—
217	C₄×C₆₇	—	2	1	no	—
218	C₄×C₆₇	—	2	1	no	—
219	C₄×C₆₇	—	4	1	no	—
220	C₄×C₆₇	—	4	1	no	—
221	C₄×C₆₇	—	4	1	no	—
222	C₄×C₆₇	#436	1	2	yes	—
223	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
224	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
225	C₄×C₆₇	#576	1	1	yes	characteristic
226	C₄×C₆₇	#577	1	1	yes	characteristic
227	Dic₆₇	—	4	2	no	—
228	#17893	—	67	1	no	maximal, Sylow
229	C₂×C₆₇	—	2	2	no	—
230	C₂×C₆₇	—	8	1	no	—
231	C₂×C₆₇	C₂×#436	1	1	yes	characteristic
232	C₂×C₆₇	#2946	1	1	yes	characteristic
233	C₂×C₆₇	#2956	1	1	yes	characteristic
234	D₆₇	—	8	2	no	—
235	#2989	D₆₇	1	1	yes	characteristic, complemented
236	#2975	—	67	1	no	—
237	C₄×AGL(1,ℤ/8)	—	67	1	no	—
238	#1594	—	67	2	no	—
239	#1592	—	67	2	no	—
240	#1590	—	67	2	no	—
241	#1589	—	67	2	no	—
242	#1565	—	67	1	no	—
243	#1476	—	67	1	no	—
244	C₂×D₄⋊C₈	—	67	1	no	—
245	C₈×M₁₆	—	67	1	no	—
246	C₆₇	#17893	1	1	yes	characteristic, complemented, Sylow
247	C₂×AGL(1,ℤ/8)	—	67	1	no	—
248	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
249	C₂×C₄×D₄	—	67	1	no	—
250	#446	C₂×D₆₇	1	2	yes	complemented
251	#442	—	67	2	no	—
252	#440	—	67	2	no	—
253	C₄×SD₁₆	—	67	2	no	—
254	C₄×D₈	—	67	2	no	—
255	#436	—	67	2	no	—
256	#436	—	67	2	no	—
257	#436	—	134	1	no	—
258	#436	—	134	1	no	—
259	#436	C₂×D₆₇	1	2	yes	complemented
260	#435	—	2	2	no	—
261	#432	—	2	1	no	—
262	#431	—	67	1	no	—
263	#431	C₂×D₆₇	1	1	yes	characteristic
264	#428	—	67	1	no	—
265	#423	—	2	1	no	—
266	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
267	#418	—	67	1	no	—
268	#417	—	67	1	no	—
269	#407	—	134	2	no	—
270	C₄×M₁₆	—	67	1	no	—
271	C₂×C₄×C₈	—	67	1	no	—
272	#335	—	67	2	no	—
273	#334	—	67	2	no	—
274	Q₈⋊C₈	—	67	1	no	—
275	Q₈⋊C₈	—	67	1	no	—
276	D₄⋊C₈	—	67	1	no	—
277	D₄⋊C₈	—	67	1	no	—
278	D₄⋊C₈	—	67	2	no	—
279	D₄⋊C₈	—	67	2	no	—
280	#322	—	67	2	no	—
281	C₈²	—	67	2	no	—
282	C₂×#41	C₄×D₆₇	1	1	yes	characteristic
283	C₂²×D₄	—	67	1	no	—
284	C₂³×C₄	—	134	1	no	—
285	AGL(1,ℤ/8)	—	134	4	no	—
286	C₂×SD₁₆	—	67	2	no	—
287	C₂×D₈	—	67	2	no	—
288	C₂×M₁₆	—	2	2	no	—
289	C₂×M₁₆	—	67	1	no	—
290	C₂×M₁₆	—	67	1	no	—
291	C₂×M₁₆	—	134	2	no	—
292	C₂×M₁₆	—	134	2	no	—
293	C₂²×C₈	—	2	1	no	—
294	C₂²×C₈	—	134	1	no	—
295	C₄×Q₈	C₄×D₆₇	1	1	yes	characteristic
296	C₄×Q₈	C₂²×D₆₇	1	1	yes	characteristic
297	C₄×D₄	—	2	1	no	—
298	C₄×D₄	—	2	1	no	—
299	C₄×D₄	—	67	2	no	—
300	C₄×D₄	—	67	2	no	—
301	C₄×D₄	—	134	1	no	—
302	C₄×D₄	—	134	1	no	—
303	C₄×D₄	C₄×D₆₇	1	1	yes	characteristic
304	C₄×D₄	C₂²×D₆₇	1	1	yes	characteristic
305	#117	—	2	1	no	—
306	#117	C₄×D₆₇	1	1	yes	characteristic
307	C₂×C₄⋊C₄	—	67	1	no	—
308	C₂×C₂²⋊C₄	—	134	1	no	—
309	C₂×C₄²	—	2	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	2	no	—
314	C₄⋊C₈	—	2	1	no	—
315	C₄⋊C₈	—	67	1	no	—
316	C₄⋊C₈	—	67	1	no	—
317	C₄⋊C₈	C₂²⋊D₆₇	1	1	yes	characteristic
318	C₄⋊C₈	C₂²⋊D₆₇	1	1	yes	characteristic
319	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
320	#103	—	67	2	no	—
321	#102	—	67	2	no	—
322	#102	—	67	2	no	—
323	#102	—	67	2	no	—
324	C₂²⋊C₈	—	2	2	no	—
325	C₂²⋊C₈	—	134	1	no	—
326	C₂²⋊C₈	—	134	2	no	—
327	C₂²⋊C₈	—	134	2	no	—
328	C₂²⋊C₈	C₂²⋊D₆₇	1	2	yes	—
329	#97	—	2	2	no	—
330	#97	—	67	2	no	—
331	C₄×C₈	—	67	2	no	—
332	C₄×C₈	—	67	1	no	—
333	C₄×C₈	—	67	1	no	—
334	C₄×C₈	—	134	1	no	—
335	C₄×C₈	—	134	1	no	—
336	C₄×C₈	C₂²×D₆₇	1	2	yes	—
337	C₂⁴	—	134	1	no	—
338	#41	—	2	2	no	—
339	#41	—	2	2	no	—
340	C₂×Q₈	C₂×C₄×D₆₇	1	1	yes	characteristic
341	C₂×D₄	—	2	1	no	—
342	C₂×D₄	—	67	2	no	—
343	C₂×D₄	—	134	1	no	—
344	C₂×D₄	—	134	2	no	—
345	C₂×D₄	—	134	2	no	—
346	C₂×D₄	C₂×C₄×D₆₇	1	1	yes	characteristic
347	C₂²×C₄	—	2	1	no	—
348	C₂²×C₄	—	2	1	no	—
349	C₂²×C₄	—	2	1	no	—
350	C₂²×C₄	—	536	1	no	—
351	C₂²×C₄	—	536	1	no	—
352	C₂²×C₄	—	134	2	no	—
353	C₂²×C₄	—	134	1	no	—
354	C₂²×C₄	—	134	2	no	—
355	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
356	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
357	C₂²×C₄	#11761628	1	1	yes	characteristic
358	SD₁₆	—	134	4	no	—
359	D₈	—	134	4	no	—
360	M₁₆	—	8	2	no	—
361	M₁₆	—	536	2	no	—
362	M₁₆	—	536	2	no	—
363	M₁₆	—	67	4	no	—
364	M₁₆	—	134	2	no	—
365	C₂×C₈	—	2	2	no	—
366	C₂×C₈	—	8	1	no	—
367	C₂×C₈	—	268	2	no	—
368	C₂×C₈	—	67	2	no	—
369	C₂×C₈	—	67	2	no	—
370	C₂×C₈	—	134	1	no	—
371	C₂×C₈	—	134	1	no	—
372	C₂×C₈	—	134	1	no	—
373	C₂×C₈	—	134	1	no	—
374	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
375	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
376	C₄⋊C₄	—	2	1	no	—
377	C₄⋊C₄	—	2	1	no	—
378	C₄⋊C₄	—	67	2	no	—
379	C₄⋊C₄	—	134	1	no	—
380	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
381	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
382	C₂²⋊C₄	—	2	1	no	—
383	C₂²⋊C₄	—	2	1	no	—
384	C₂²⋊C₄	—	2	1	no	—
385	C₂²⋊C₄	—	134	2	no	—
386	C₂²⋊C₄	—	134	2	no	—
387	C₄²	—	2	1	no	—
388	C₄²	—	2	1	no	—
389	C₄²	—	2	1	no	—
390	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
391	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
392	C₄²	#11761628	1	1	yes	characteristic
393	C₄²	C₂³×D₆₇	1	1	yes	characteristic
394	C₂³	—	2	1	no	—
395	C₂³	—	268	2	no	—
396	C₂³	—	536	1	no	—
397	C₂³	—	536	1	no	—
398	C₂³	—	134	2	no	—
399	C₂³	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
400	Q₈	—	2	1	no	—
401	Q₈	—	2	1	no	—
402	D₄	—	2	1	no	—
403	D₄	—	2	1	no	—
404	D₄	—	4	2	no	—
405	D₄	—	268	2	no	—
406	D₄	—	134	2	no	—
407	D₄	—	134	2	no	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	4	1	no	—
414	C₂×C₄	—	4	1	no	—
415	C₂×C₄	—	4	1	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	8	1	no	—
418	C₂×C₄	—	268	2	no	—
419	C₂×C₄	—	268	2	no	—
420	C₂×C₄	—	536	2	no	—
421	C₂×C₄	—	536	1	no	—
422	C₂×C₄	—	134	2	no	—
423	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
424	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
425	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
426	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
427	C₂×C₄	#14524051	1	2	yes	—
428	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
429	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
430	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
431	C₈	—	2	2	no	—
432	C₈	—	2	2	no	—
433	C₈	—	4	2	no	—
434	C₈	—	268	1	no	—
435	C₈	—	268	1	no	—
436	C₈	—	268	1	no	—
437	C₈	—	67	4	no	—
438	C₈	—	134	2	no	—
439	C₈	—	134	2	no	—
440	C₂²	—	2	1	no	—
441	C₂²	—	2	1	no	—
442	C₂²	—	4	1	no	—
443	C₂²	—	4	1	no	—
444	C₂²	—	4	1	no	—
445	C₂²	—	268	2	no	—
446	C₂²	—	268	2	no	—
447	C₂²	—	268	2	no	—
448	C₂²	—	536	4	no	—
449	C₂²	#16863472	1	2	yes	—
450	C₂²	C₂×C₄×C₂²⋊D₆₇	1	1	yes	characteristic
451	C₄	—	2	1	no	—
452	C₄	—	2	1	no	—
453	C₄	—	2	1	no	—
454	C₄	—	4	1	no	—
455	C₄	—	4	1	no	—
456	C₄	—	4	1	no	—
457	C₄	—	268	2	no	—
458	C₄	#16863472	1	2	yes	—
459	C₄	C₂×M₁₆×D₆₇	1	1	yes	characteristic
460	C₄	C₂×#14524051	1	1	yes	characteristic
461	C₄	#16864199	1	1	yes	characteristic
462	C₄	#16864204	1	1	yes	characteristic
463	C₂	—	2	2	no	—
464	C₂	—	8	1	no	—
465	C₂	—	536	2	no	—
466	C₂	C₂×#16863472	1	1	yes	characteristic
467	C₂	#22569037	1	1	yes	characteristic
468	C₂	#22569116	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#2989	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#14524068	C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#2989	D₆₇
hypocenter	C₆₇	#17893
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #2989

Complements

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

N	H ≅ G/N	type
1. #22569249	C₂	semidirect
3. #22559121	C₂	semidirect
4. #22559109	C₂	semidirect
5. #22559099	C₂	semidirect
6. #22559092	C₂	semidirect
7. #22558959	C₂	semidirect
8. #22558591	C₂	semidirect
9. C₂×#16863123	C₂	semidirect
10. #22558490	C₂	semidirect
11. C₆₇×#2989	C₂	semidirect
29. #16863148	C₂²	semidirect
30. #16863146	C₂²	semidirect
31. Q₈⋊(C₆₇⋊C₈)	C₂²	semidirect
32. Q₈⋊(C₆₇⋊C₈)	C₂²	semidirect
33. D₄⋊(C₆₇⋊C₈)	C₂²	semidirect
34. D₄⋊(C₆₇⋊C₈)	C₂²	semidirect
37. #16863118	C₂²	semidirect
38. C₈×C₆₇⋊C₈	C₂²	semidirect
40. #446×C₆₇	C₂²	semidirect
41. #436×C₆₇	C₂²	semidirect
115. #11761612	M₁₆	semidirect
235. #2989	D₆₇	semidirect
246. C₆₇	#17893	semidirect
250. #446	C₂×D₆₇	semidirect
259. #436	C₂×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22569249	2
C₄×#14524068	2
#22559121	2
#22559109	2
#22559099	2
#22559092	2
#22558959	2
#22558591	2
C₂×#16863123	2
#22558490	2
C₆₇×#2989	2
#17893	67