Subgroups — #30825874

Order 17152.

692 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570269	C₂	1	1	yes	maximal, characteristic
2	#41×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#22568920	C₂	1	1	yes	maximal, characteristic, complemented
4	#22559049	C₂	1	2	yes	maximal, complemented
5	#22559035	C₂	1	2	yes	maximal, complemented
6	#22559030	C₂	1	2	yes	maximal, complemented
7	#22559024	C₂	1	2	yes	maximal, complemented
8	#22559000	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558627	C₂	1	1	yes	maximal, characteristic, complemented
10	#22558496	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1489	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524051	—	2	1	no	—
13	#16863946	C₂²	1	1	yes	characteristic
14	#16863720	C₄	1	2	yes	—
15	#16863691	C₄	1	1	yes	characteristic
16	#16863686	C₄	1	1	yes	characteristic
17	#16863596	C₄	1	1	yes	characteristic
18	#16863591	C₄	1	1	yes	characteristic
19	#16863541	C₄	1	2	yes	—
20	Q₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
21	Q₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
22	C₄×#11761615	C₂²	1	2	yes	—
23	C₄×#11761613	C₂²	1	2	yes	—
24	#16863473	—	2	1	no	—
25	#16863473	C₂²	1	2	yes	—
26	D₄×C₆₇⋊C₈	—	2	1	no	—
27	D₄×C₆₇⋊C₈	—	2	1	no	—
28	D₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
29	D₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
30	#16863461	—	2	1	no	—
31	#16863461	C₂²	1	2	yes	—
32	#16863449	—	2	1	no	—
33	#16863449	C₂²	1	1	yes	characteristic
34	#16863449	C₂²	1	1	yes	characteristic
35	#16863364	—	2	1	no	—
36	C₂×C₄×C₆₇⋊C₈	—	2	1	no	—
37	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
38	#16863148	C₂²	1	2	yes	—
39	#16863146	C₂²	1	2	yes	—
40	#16863125	C₂²	1	2	yes	—
41	#16863125	C₂²	1	2	yes	—
42	#16863118	C₂²	1	2	yes	—
43	#16863118	C₂²	1	2	yes	—
44	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
45	#423×C₆₇	C₂²	1	1	yes	characteristic
46	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
47	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
48	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
49	C₂×#11761628	—	2	1	no	—
50	C₂×#11761615	C₂×C₄	1	2	yes	—
51	C₂×#11761613	C₂×C₄	1	2	yes	—
52	#14524051	—	4	4	no	—
53	C₂×#11761611	—	2	2	no	—
54	C₂²×C₆₇⋊C₈	—	2	1	no	—
55	C₂²×C₆₇⋊C₈	—	2	1	no	—
56	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
57	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
58	C₂×C₈×D₆₇	—	2	2	no	—
59	#14524043	C₂×C₄	1	1	yes	characteristic
60	#14524040	C₂×C₄	1	1	yes	characteristic
61	#14524031	—	2	1	no	—
62	#14524017	—	2	1	no	—
63	#14524017	C₂×C₄	1	2	yes	—
64	C₄×D₂₆₈	—	2	1	no	—
65	C₄×D₂₆₈	C₂³	1	2	yes	—
66	#14523976	C₂×C₄	1	2	yes	—
67	#14523974	C₂×C₄	1	2	yes	—
68	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
69	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
70	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
71	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
72	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
73	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
74	#14523967	C₂×C₄	1	2	yes	—
75	#14523964	C₂×C₄	1	2	yes	—
76	#14523963	C₂×C₄	1	2	yes	—
77	#14523963	C₂×C₄	1	2	yes	—
78	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
79	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
80	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
81	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
82	(C₂×D₆₇)⋊C₈	—	2	2	no	—
83	(C₂×D₆₇)⋊C₈	—	2	2	no	—
84	C₄×C₆₇⋊C₈	—	2	1	no	—
85	C₄×C₆₇⋊C₈	—	2	1	no	—
86	C₄×C₆₇⋊C₈	—	2	1	no	—
87	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
88	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
89	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
90	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
91	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
92	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
93	C₂×M₁₆×C₆₇	—	2	1	no	—
94	C₂×M₁₆×C₆₇	—	2	1	no	—
95	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
96	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
97	C₄×D₄×C₆₇	—	2	1	no	—
98	C₄×D₄×C₆₇	—	2	1	no	—
99	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
100	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
101	#117×C₆₇	—	2	1	no	—
102	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
103	C₂×C₄²×C₆₇	—	2	1	no	—
104	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
105	C₄⋊C₈×C₆₇	—	2	1	no	—
106	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
107	#97×C₆₇	—	2	1	no	—
108	C₄×C₈×C₆₇	C₂³	1	2	yes	—
109	#11761628	—	4	2	no	—
110	#11761628	—	4	2	no	—
111	C₂×Dic₁₃₄	—	2	1	no	—
112	C₂×D₂₆₈	—	2	1	no	—
113	C₂×D₂₆₈	—	2	1	no	—
114	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
115	C₂×C₂²⋊D₆₇	—	2	2	no	—
116	C₂×C₄×D₆₇	—	2	2	no	—
117	#11761615	—	2	4	no	—
118	#11761613	—	2	4	no	—
119	#11761611	—	4	4	no	—
120	C₂×C₆₇⋊C₈	—	2	2	no	—
121	C₂×C₆₇⋊C₈	—	2	1	no	—
122	C₂×C₆₇⋊C₈	—	2	1	no	—
123	C₂×C₆₇⋊C₈	—	4	1	no	—
124	C₂×C₆₇⋊C₈	—	4	2	no	—
125	C₂×C₆₇⋊C₈	—	4	2	no	—
126	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
127	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
128	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
129	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
130	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
131	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
132	C₈×D₆₇	—	4	4	no	—
133	C₄⋊Dic₆₇	—	2	1	no	—
134	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
135	(C₂×D₆₇)⋊C₄	—	2	2	no	—
136	(C₂×D₆₇)⋊C₄	—	2	2	no	—
137	#41×C₆₇	—	2	2	no	—
138	#41×C₆₇	—	2	2	no	—
139	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
140	C₂×D₄×C₆₇	—	2	1	no	—
141	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
142	C₂²×C₄×C₆₇	—	2	1	no	—
143	C₂²×C₄×C₆₇	—	2	1	no	—
144	C₂²×C₄×C₆₇	—	2	1	no	—
145	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
146	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
148	M₁₆×C₆₇	—	4	2	no	—
149	M₁₆×C₆₇	—	4	2	no	—
150	C₂×C₈×C₆₇	—	2	2	no	—
151	C₂×C₈×C₆₇	—	2	2	no	—
152	C₄⋊C₄×C₆₇	—	2	1	no	—
153	C₄⋊C₄×C₆₇	—	2	1	no	—
154	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
155	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
156	C₂²⋊C₄×C₆₇	—	2	1	no	—
157	C₂²⋊C₄×C₆₇	—	2	1	no	—
158	C₂²⋊C₄×C₆₇	—	2	1	no	—
159	C₄²×C₆₇	—	2	1	no	—
160	C₄²×C₆₇	—	2	1	no	—
161	C₄²×C₆₇	—	2	1	no	—
162	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
163	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
164	C₄²×C₆₇	#41	1	1	yes	characteristic
165	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
166	C₂³×C₆₇	—	2	1	no	—
167	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
168	C₂²×D₆₇	—	2	2	no	—
169	Q₈×C₆₇	—	2	1	no	—
170	Q₈×C₆₇	#131	1	2	yes	complemented
171	D₄×C₆₇	—	2	4	no	—
172	D₄×C₆₇	—	2	1	no	—
173	D₄×C₆₇	#131	1	2	yes	complemented
174	C₂×C₄×C₆₇	—	2	4	no	—
175	C₂×C₄×C₆₇	—	2	1	no	—
176	C₂×C₄×C₆₇	—	2	1	no	—
177	C₂×C₄×C₆₇	—	2	1	no	—
178	C₂×C₄×C₆₇	—	2	1	no	—
179	C₂×C₄×C₆₇	—	2	1	no	—
180	C₂×C₄×C₆₇	—	2	2	no	—
181	C₂×C₄×C₆₇	—	2	1	no	—
182	C₂×C₄×C₆₇	—	2	1	no	—
183	C₂×C₄×C₆₇	—	4	1	no	—
184	C₂×C₄×C₆₇	—	4	1	no	—
185	C₂×C₄×C₆₇	—	4	2	no	—
186	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
187	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
188	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
189	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
190	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
191	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
192	C₂²⋊D₆₇	—	4	4	no	—
193	C₂×Dic₆₇	—	2	2	no	—
194	D₂₆₈	—	2	2	no	—
195	D₂₆₈	—	2	2	no	—
196	D₂₆₈	—	4	2	no	—
197	D₂₆₈	—	4	2	no	—
198	C₄×D₆₇	—	4	2	no	—
199	C₄×D₆₇	—	4	2	no	—
200	Dic₁₃₄	—	4	2	no	—
201	C₈×C₆₇	—	4	2	no	—
202	C₈×C₆₇	—	4	2	no	—
203	C₆₇⋊C₈	—	2	2	no	—
204	C₆₇⋊C₈	—	2	2	no	—
205	C₆₇⋊C₈	—	2	1	no	—
206	C₆₇⋊C₈	—	2	1	no	—
207	C₆₇⋊C₈	—	2	1	no	—
208	C₆₇⋊C₈	—	4	1	no	—
209	C₆₇⋊C₈	—	4	1	no	—
210	C₆₇⋊C₈	AGL(1,ℤ/8)	1	2	yes	—
211	C₂²×C₆₇	—	2	1	no	—
212	C₂²×C₆₇	—	2	2	no	—
213	C₂²×C₆₇	—	2	1	no	—
214	C₂²×C₆₇	—	2	1	no	—
215	C₂²×C₆₇	—	4	1	no	—
216	C₂²×C₆₇	—	4	1	no	—
217	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
218	C₂×D₆₇	—	4	2	no	—
219	C₂×D₆₇	—	4	2	no	—
220	C₂×D₆₇	—	4	2	no	—
221	C₄×C₆₇	—	2	1	no	—
222	C₄×C₆₇	—	2	2	no	—
223	C₄×C₆₇	—	2	1	no	—
224	C₄×C₆₇	—	2	1	no	—
225	C₄×C₆₇	—	4	1	no	—
226	C₄×C₆₇	—	4	1	no	—
227	C₄×C₆₇	#444	1	2	yes	—
228	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
229	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
230	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
231	C₄×C₆₇	#575	1	1	yes	characteristic
232	Dic₆₇	—	4	2	no	—
233	#17917	—	67	1	no	maximal, Sylow
234	C₂×C₆₇	—	4	1	no	—
235	C₂×C₆₇	—	4	2	no	—
236	C₂×C₆₇	#2929	1	1	yes	characteristic
237	C₂×C₆₇	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
238	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
239	D₆₇	—	8	2	no	—
240	#3106	—	67	1	no	—
241	C₈×#41	—	67	1	no	—
242	#2931	—	67	1	no	—
243	#1583	—	67	2	no	—
244	#1581	—	67	2	no	—
245	#1580	—	67	2	no	—
246	SD₁₆⋊C₈	—	67	2	no	—
247	#1574	—	67	1	no	—
248	#1489	D₆₇	1	1	yes	characteristic, complemented
249	#1484	—	67	1	no	—
250	#1451	—	67	1	no	—
251	C₆₇	#17917	1	1	yes	characteristic, complemented, Sylow
252	C₂×#131	—	134	1	no	—
253	#532	—	67	1	no	—
254	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
255	#482	—	67	2	no	—
256	#477	—	67	1	no	—
257	#476	—	67	1	no	—
258	#460	—	67	1	no	—
259	#459	—	67	1	no	—
260	#449	—	67	2	no	—
261	C₈×Q₈	—	67	1	no	—
262	C₈×Q₈	—	67	1	no	—
263	C₄×SD₁₆	—	67	2	no	—
264	C₄×D₈	—	67	2	no	—
265	#436	—	67	2	no	—
266	#436	—	134	1	no	—
267	C₈×D₄	—	67	1	no	—
268	C₈×D₄	—	67	1	no	—
269	C₈×D₄	—	67	2	no	—
270	C₈×D₄	—	134	1	no	—
271	C₈×D₄	—	134	1	no	—
272	C₈×D₄	—	134	1	no	—
273	#431	—	67	1	no	—
274	#431	—	67	1	no	—
275	#431	—	134	1	no	—
276	#423	C₂×D₆₇	1	1	yes	characteristic
277	#408	—	134	1	no	—
278	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
279	C₂×C₄×C₈	—	67	1	no	—
280	C₂×C₄×C₈	—	134	1	no	—
281	#335	—	67	2	no	—
282	#334	—	67	2	no	—
283	Q₈⋊C₈	C₂×D₆₇	1	2	yes	—
284	D₄⋊C₈	—	67	2	no	—
285	D₄⋊C₈	—	67	2	no	—
286	D₄⋊C₈	C₂×D₆₇	1	2	yes	—
287	#322	—	67	2	no	—
288	#322	—	67	2	no	—
289	C₂×#41	—	134	1	no	—
290	C₂×#41	C₄×D₆₇	1	1	yes	characteristic
291	C₂×SD₁₆	—	67	2	no	—
292	C₂×D₈	—	67	2	no	—
293	#131	—	268	4	no	—
294	C₂×M₁₆	—	2	1	no	—
295	C₂×M₁₆	—	2	1	no	—
296	C₂×M₁₆	—	134	2	no	—
297	C₂²×C₈	—	67	1	no	—
298	C₂²×C₈	—	67	1	no	—
299	C₂²×C₈	—	134	2	no	—
300	C₂²×C₈	—	134	1	no	—
301	C₂²×C₈	—	134	1	no	—
302	C₄⋊Q₈	—	67	1	no	—
303	#127	—	67	1	no	—
304	#124	—	134	1	no	—
305	#121	—	67	2	no	—
306	#121	—	134	1	no	—
307	C₄×Q₈	C₄×D₆₇	1	1	yes	characteristic
308	C₄×Q₈	C₂²×D₆₇	1	1	yes	characteristic
309	C₄×D₄	—	2	1	no	—
310	C₄×D₄	—	2	1	no	—
311	C₄×D₄	—	67	2	no	—
312	C₄×D₄	—	134	1	no	—
313	C₄×D₄	C₄×D₆₇	1	1	yes	characteristic
314	C₄×D₄	C₂²×D₆₇	1	1	yes	characteristic
315	#117	—	2	1	no	—
316	#117	C₄×D₆₇	1	1	yes	characteristic
317	C₂×C₄²	—	2	1	no	—
318	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
319	#107	—	67	2	no	—
320	#106	—	67	2	no	—
321	C₄⋊C₈	—	2	1	no	—
322	C₄⋊C₈	—	67	1	no	—
323	C₄⋊C₈	—	67	1	no	—
324	C₄⋊C₈	—	67	1	no	—
325	C₄⋊C₈	—	67	1	no	—
326	C₄⋊C₈	—	134	1	no	—
327	C₄⋊C₈	—	134	1	no	—
328	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
329	#103	—	67	2	no	—
330	#102	—	67	2	no	—
331	#102	—	67	2	no	—
332	#102	—	67	2	no	—
333	C₂²⋊C₈	—	67	2	no	—
334	C₂²⋊C₈	—	134	2	no	—
335	C₂²⋊C₈	—	134	2	no	—
336	C₂²⋊C₈	—	134	1	no	—
337	C₂²⋊C₈	—	134	1	no	—
338	C₂²⋊C₈	—	134	1	no	—
339	#97	—	2	1	no	—
340	C₄×C₈	—	67	1	no	—
341	C₄×C₈	—	67	1	no	—
342	C₄×C₈	—	67	2	no	—
343	C₄×C₈	—	67	1	no	—
344	C₄×C₈	—	67	1	no	—
345	C₄×C₈	—	134	1	no	—
346	C₄×C₈	—	134	1	no	—
347	C₄×C₈	—	134	1	no	—
348	C₄×C₈	C₂²×D₆₇	1	2	yes	—
349	#41	—	2	2	no	—
350	#41	—	2	2	no	—
351	#41	—	268	2	no	—
352	#41	—	268	2	no	—
353	C₂×Q₈	—	134	1	no	—
354	C₂×Q₈	C₂×C₄×D₆₇	1	1	yes	characteristic
355	C₂×D₄	—	2	1	no	—
356	C₂×D₄	—	67	2	no	—
357	C₂×D₄	—	134	2	no	—
358	C₂×D₄	—	134	1	no	—
359	C₂×D₄	—	134	1	no	—
360	C₂×D₄	C₂×C₄×D₆₇	1	1	yes	characteristic
361	C₂²×C₄	—	2	1	no	—
362	C₂²×C₄	—	2	1	no	—
363	C₂²×C₄	—	2	1	no	—
364	C₂²×C₄	—	134	2	no	—
365	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
366	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
367	C₂²×C₄	#11761627	1	1	yes	characteristic
368	SD₁₆	—	134	4	no	—
369	D₈	—	134	4	no	—
370	M₁₆	—	4	2	no	—
371	M₁₆	—	4	2	no	—
372	M₁₆	—	268	4	no	—
373	C₂×C₈	—	2	2	no	—
374	C₂×C₈	—	2	2	no	—
375	C₂×C₈	—	268	4	no	—
376	C₂×C₈	—	268	2	no	—
377	C₂×C₈	—	268	2	no	—
378	C₂×C₈	—	268	1	no	—
379	C₂×C₈	—	67	1	no	—
380	C₂×C₈	—	67	2	no	—
381	C₂×C₈	—	67	2	no	—
382	C₂×C₈	—	67	1	no	—
383	C₂×C₈	—	67	1	no	—
384	C₂×C₈	—	67	1	no	—
385	C₂×C₈	—	134	1	no	—
386	C₂×C₈	—	134	1	no	—
387	C₂×C₈	—	134	2	no	—
388	C₄⋊C₄	—	2	1	no	—
389	C₄⋊C₄	—	2	1	no	—
390	C₄⋊C₄	—	67	2	no	—
391	C₄⋊C₄	—	134	1	no	—
392	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
393	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
394	C₂²⋊C₄	—	2	1	no	—
395	C₂²⋊C₄	—	2	1	no	—
396	C₂²⋊C₄	—	2	1	no	—
397	C₂²⋊C₄	—	134	2	no	—
398	C₂²⋊C₄	—	134	2	no	—
399	C₄²	—	2	1	no	—
400	C₄²	—	2	1	no	—
401	C₄²	—	2	1	no	—
402	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
403	C₄²	D₄×D₆₇	1	1	yes	characteristic
404	C₄²	#11761627	1	1	yes	characteristic
405	C₄²	C₂³×D₆₇	1	1	yes	characteristic
406	C₂³	—	2	1	no	—
407	C₂³	—	134	2	no	—
408	C₂³	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
409	Q₈	—	2	1	no	—
410	Q₈	—	268	2	no	—
411	Q₈	#14524051	1	2	yes	complemented
412	D₄	—	2	4	no	—
413	D₄	—	2	1	no	—
414	D₄	—	268	2	no	—
415	D₄	—	268	4	no	—
416	D₄	—	268	2	no	—
417	D₄	—	134	2	no	—
418	D₄	—	134	2	no	—
419	D₄	#14524051	1	2	yes	complemented
420	C₂×C₄	—	2	4	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	1	no	—
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	1	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	4	2	no	—
430	C₂×C₄	—	4	1	no	—
431	C₂×C₄	—	4	1	no	—
432	C₂×C₄	—	268	2	no	—
433	C₂×C₄	—	268	2	no	—
434	C₂×C₄	—	134	2	no	—
435	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
436	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
437	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
438	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
439	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
440	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
441	C₈	—	4	2	no	—
442	C₈	—	4	2	no	—
443	C₈	—	268	1	no	—
444	C₈	—	268	1	no	—
445	C₈	—	67	2	no	—
446	C₈	—	134	2	no	—
447	C₈	—	134	2	no	—
448	C₈	—	134	1	no	—
449	C₈	—	134	1	no	—
450	C₈	—	134	1	no	—
451	C₂²	—	2	1	no	—
452	C₂²	—	2	1	no	—
453	C₂²	—	2	1	no	—
454	C₂²	—	2	2	no	—
455	C₂²	—	4	1	no	—
456	C₂²	—	4	1	no	—
457	C₂²	—	268	2	no	—
458	C₂²	—	268	2	no	—
459	C₂²	—	268	2	no	—
460	C₂²	C₂×(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
461	C₄	—	2	1	no	—
462	C₄	—	2	1	no	—
463	C₄	—	2	1	no	—
464	C₄	—	2	2	no	—
465	C₄	—	4	1	no	—
466	C₄	—	4	1	no	—
467	C₄	—	268	2	no	—
468	C₄	#16863515	1	2	yes	—
469	C₄	C₂×#14524051	1	1	yes	characteristic
470	C₄	C₂×#14524051	1	1	yes	characteristic
471	C₄	AGL(1,ℤ/8)×D₆₇	1	1	yes	characteristic
472	C₄	#16864195	1	1	yes	characteristic
473	C₂	—	4	1	no	—
474	C₂	—	4	2	no	—
475	C₂	—	536	2	no	—
476	C₂	#22568899	1	1	yes	characteristic
477	C₂	#22569022	1	1	yes	characteristic
478	C₂	C₂×#16863515	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×#1489	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#22570269	C₂
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1489	D₆₇
hypocenter	C₆₇	#17917
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1489

Complements

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

N	H ≅ G/N	type
2. #41×C₆₇⋊C₈	C₂	semidirect
3. #22568920	C₂	semidirect
4. #22559049	C₂	semidirect
5. #22559035	C₂	semidirect
6. #22559030	C₂	semidirect
7. #22559024	C₂	semidirect
8. #22559000	C₂	semidirect
9. #22558627	C₂	semidirect
10. #22558496	C₂	semidirect
11. C₆₇×#1489	C₂	semidirect
170. Q₈×C₆₇	#131	semidirect
173. D₄×C₆₇	#131	semidirect
248. #1489	D₆₇	semidirect
251. C₆₇	#17917	semidirect
411. Q₈	#14524051	semidirect
419. D₄	#14524051	semidirect

Maximal subgroups

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

subgroup	index
#22570269	2
#41×C₆₇⋊C₈	2
#22568920	2
#22559049	2
#22559035	2
#22559030	2
#22559024	2
#22559000	2
#22558627	2
#22558496	2
C₆₇×#1489	2
#17917	67