Subgroups — #30750076

Order 17152.

602 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22563109	C₂	1	1	yes	maximal, characteristic
2	#22560973	C₂	1	1	yes	maximal, characteristic, complemented
3	#22560336	C₂	1	1	yes	maximal, characteristic, complemented
4	#22560210	C₂	1	1	yes	maximal, characteristic
5	C₂×#16863123	C₂	1	1	yes	maximal, characteristic
6	#22558048	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1296	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₂₆₈	C₂²	1	1	yes	characteristic
9	C₂×#14524005	—	2	1	no	—
10	C₂×C₄⋊C₄×D₆₇	—	2	1	no	—
11	C₂×#14523975	—	2	1	no	—
12	C₂×#14523965	—	2	1	no	—
13	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
14	#16863292	—	2	1	no	—
15	#16863267	—	2	2	no	—
16	#16863266	C₂²	1	1	yes	characteristic
17	#16863253	—	2	1	no	—
18	C₄×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
19	#16863159	—	2	1	no	—
20	#16863159	—	2	1	no	—
21	#16863158	—	2	2	no	—
22	#16863123	C₄	1	4	yes	complemented
23	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
25	#384×C₆₇	C₂²	1	1	yes	characteristic
26	C₂²×D₂₆₈	D₄	1	1	yes	characteristic
27	C₂²×C₄×D₆₇	—	2	2	no	—
28	C₂²×C₄×D₆₇	—	2	1	no	—
29	C₂²×C₆₇⋊C₈	—	2	1	no	—
30	C₄×D₂₆₈	—	2	1	no	—
31	C₄×D₂₆₈	—	2	1	no	—
32	C₄×D₂₆₈	C₂×C₄	1	2	yes	—
33	C₄×D₂₆₈	Q₈	1	2	yes	—
34	#14524005	—	4	2	no	—
35	#14524005	—	4	2	no	—
36	C₂×C₄⋊Dic₆₇	—	2	1	no	—
37	C₂×C₄⋊Dic₆₇	—	2	1	no	—
38	C₂×C₄⋊Dic₆₇	—	2	1	no	—
39	C₂×C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
40	C₂×C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
41	C₂×C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
42	C₂×Dic₆₇⋊C₄	—	2	2	no	—
43	C₄⋊C₄×D₆₇	—	4	2	no	—
44	C₄⋊C₄×D₆₇	—	4	2	no	—
45	C₂×(C₂×D₆₇)⋊C₄	—	2	2	no	—
46	C₂×(C₂×D₆₇)⋊C₄	—	2	1	no	—
47	C₂×C₄×Dic₆₇	—	2	1	no	—
48	#14523975	—	4	2	no	—
49	#14523965	—	4	2	no	—
50	C₄×C₆₇⋊C₈	—	2	1	no	—
51	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	complemented
52	C₆₇⋊#95	—	2	2	no	—
53	C₆₇⋊#95	—	2	1	no	—
54	C₂²×C₈×C₆₇	—	2	1	no	—
55	C₂²×C₈×C₆₇	—	2	1	no	—
56	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
57	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
58	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
59	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₆₇	—	2	1	no	—
61	C₄⋊C₈×C₆₇	—	2	1	no	—
62	C₄⋊C₈×C₆₇	—	2	1	no	—
63	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	complemented
64	C₂³×D₆₇	—	2	1	no	—
65	C₂×D₂₆₈	—	2	1	no	—
66	C₂×D₂₆₈	—	2	1	no	—
67	C₂×D₂₆₈	—	2	1	no	—
68	C₂×D₂₆₈	C₄⋊C₄	1	2	yes	—
69	C₂×D₂₆₈	SD₁₆	1	2	yes	—
70	C₂×D₂₆₈	SD₁₆	1	2	yes	—
71	C₂²×Dic₆₇	—	2	1	no	—
72	C₂²×Dic₆₇	—	2	1	no	—
73	C₂²×Dic₆₇	—	2	1	no	—
74	C₂²×Dic₆₇	—	4	1	no	—
75	C₂×C₄×D₆₇	—	2	2	no	—
76	C₂×C₄×D₆₇	—	2	2	no	—
77	C₂×C₄×D₆₇	—	4	1	no	—
78	C₂×C₄×D₆₇	—	4	1	no	—
79	C₂×C₄×D₆₇	—	4	1	no	—
80	C₂×C₄×D₆₇	—	4	1	no	—
81	C₂×C₄×D₆₇	—	4	4	no	—
82	C₂×C₄×D₆₇	—	4	4	no	—
83	C₂×C₄×D₆₇	—	8	2	no	—
84	C₂×C₆₇⋊C₈	—	2	2	no	—
85	C₂×C₆₇⋊C₈	—	4	1	no	—
86	C₂×C₆₇⋊C₈	—	4	1	no	—
87	C₄⋊Dic₆₇	—	2	1	no	—
88	C₄⋊Dic₆₇	—	2	1	no	—
89	C₄⋊Dic₆₇	—	2	1	no	—
90	C₄⋊Dic₆₇	—	2	1	no	—
91	C₄⋊Dic₆₇	—	4	2	no	—
92	C₄⋊Dic₆₇	—	4	1	no	—
93	C₄⋊Dic₆₇	—	4	1	no	—
94	C₄⋊Dic₆₇	—	4	1	no	—
95	C₄⋊Dic₆₇	—	4	1	no	—
96	C₄⋊Dic₆₇	C₄⋊C₄	1	2	yes	—
97	C₄⋊Dic₆₇	D₈	1	2	yes	—
98	Dic₆₇⋊C₄	—	4	4	no	—
99	(C₂×D₆₇)⋊C₄	—	2	2	no	—
100	(C₂×D₆₇)⋊C₄	—	2	2	no	—
101	(C₂×D₆₇)⋊C₄	—	4	4	no	—
102	C₄×Dic₆₇	—	4	1	no	—
103	C₄×Dic₆₇	—	4	1	no	—
104	C₂²×C₄×C₆₇	—	2	2	no	—
105	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₆₇	—	2	2	no	—
109	C₂×C₈×C₆₇	—	4	1	no	—
110	C₂×C₈×C₆₇	—	4	1	no	—
111	C₂×C₈×C₆₇	—	4	1	no	—
112	C₂×C₈×C₆₇	—	4	1	no	—
113	C₂×C₈×C₆₇	—	4	1	no	—
114	C₄⋊C₄×C₆₇	—	4	2	no	—
115	C₄⋊C₄×C₆₇	—	4	1	no	—
116	C₄⋊C₄×C₆₇	—	4	1	no	—
117	C₄⋊C₄×C₆₇	—	4	1	no	—
118	C₄⋊C₄×C₆₇	—	4	1	no	—
119	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
120	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
121	C₄²×C₆₇	#41	1	1	yes	characteristic
122	C₄²×C₆₇	#41	1	1	yes	characteristic
123	C₂³×C₆₇	#120	1	1	yes	characteristic
124	C₂²×D₆₇	—	2	2	no	—
125	C₂²×D₆₇	—	2	2	no	—
126	C₂²×D₆₇	—	2	2	no	—
127	C₂²×D₆₇	—	4	1	no	—
128	C₂²×D₆₇	—	4	1	no	—
129	C₂²×D₆₇	—	4	1	no	—
130	C₂²×D₆₇	—	4	1	no	—
131	C₂×C₄×C₆₇	—	2	1	no	—
132	C₂×C₄×C₆₇	—	2	1	no	—
133	C₂×C₄×C₆₇	—	4	2	no	—
134	C₂×C₄×C₆₇	—	4	2	no	—
135	C₂×C₄×C₆₇	—	4	2	no	—
136	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
140	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
141	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
142	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
144	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
145	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
146	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
147	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
149	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
150	C₂×Dic₆₇	—	2	2	no	—
151	C₂×Dic₆₇	—	2	2	no	—
152	C₂×Dic₆₇	—	4	1	no	—
153	C₂×Dic₆₇	—	4	1	no	—
154	C₂×Dic₆₇	—	4	1	no	—
155	C₂×Dic₆₇	—	4	1	no	—
156	C₂×Dic₆₇	—	4	1	no	—
157	C₂×Dic₆₇	—	4	1	no	—
158	C₂×Dic₆₇	—	4	1	no	—
159	C₂×Dic₆₇	—	8	1	no	—
160	C₂×Dic₆₇	—	8	1	no	—
161	C₂×Dic₆₇	—	8	1	no	—
162	D₂₆₈	—	2	2	no	—
163	D₂₆₈	—	2	2	no	—
164	D₂₆₈	—	2	2	no	—
165	D₂₆₈	#106	1	4	yes	complemented
166	C₄×D₆₇	—	4	2	no	—
167	C₄×D₆₇	—	4	2	no	—
168	C₄×D₆₇	—	4	2	no	—
169	C₄×D₆₇	—	4	2	no	—
170	C₄×D₆₇	—	8	8	no	—
171	C₈×C₆₇	—	4	2	no	—
172	C₈×C₆₇	—	8	1	no	—
173	C₆₇⋊C₈	—	4	2	no	—
174	C₂²×C₆₇	#385	1	1	yes	characteristic
175	C₂²×C₆₇	#447	1	1	yes	characteristic
176	C₂²×C₆₇	#448	1	1	yes	characteristic
177	C₂²×C₆₇	#450	1	1	yes	characteristic
178	C₂²×C₆₇	#452	1	1	yes	characteristic
179	C₂²×C₆₇	#454	1	1	yes	characteristic
180	C₂²×C₆₇	#455	1	1	yes	characteristic
181	C₂×D₆₇	—	2	4	no	—
182	C₂×D₆₇	—	4	2	no	—
183	C₂×D₆₇	—	4	2	no	—
184	C₂×D₆₇	—	4	2	no	—
185	C₂×D₆₇	—	4	2	no	—
186	C₂×D₆₇	—	4	2	no	—
187	C₂×D₆₇	—	4	2	no	—
188	C₄×C₆₇	—	2	1	no	—
189	C₄×C₆₇	—	2	1	no	—
190	C₄×C₆₇	—	8	2	no	—
191	C₄×C₆₇	C₂×#106	1	1	yes	characteristic
192	C₄×C₆₇	#428	1	1	yes	characteristic
193	C₄×C₆₇	C₄×D₈	1	1	yes	characteristic
194	C₄×C₆₇	#440	1	1	yes	characteristic
195	C₄×C₆₇	#474	1	1	yes	characteristic
196	C₄×C₆₇	#478	1	1	yes	characteristic
197	C₄×C₆₇	#481	1	1	yes	characteristic
198	C₄×C₆₇	#483	1	1	yes	characteristic
199	Dic₆₇	—	4	2	no	—
200	Dic₆₇	—	4	2	no	—
201	Dic₆₇	—	8	1	no	—
202	Dic₆₇	—	16	1	no	—
203	#8894	—	67	1	no	maximal, Sylow
204	C₂×C₆₇	#1627	1	1	yes	characteristic
205	C₂×C₆₇	#1635	1	1	yes	characteristic
206	C₂×C₆₇	#1639	1	1	yes	characteristic
207	C₂×C₆₇	#1643	1	1	yes	characteristic
208	C₂×C₆₇	#1853	1	1	yes	characteristic
209	C₂×C₆₇	#1860	1	1	yes	characteristic
210	C₂×C₆₇	#1862	1	1	yes	characteristic
211	D₆₇	—	4	4	no	—
212	D₄×C₄⋊C₄	—	67	1	no	—
213	#1948	—	67	1	no	—
214	#1834	—	67	1	no	—
215	#1802	—	67	1	no	—
216	C₂×D₄⋊C₈	—	67	1	no	—
217	#1296	D₆₇	1	1	yes	characteristic, complemented
218	#1279	—	67	1	no	—
219	C₆₇	#8894	1	1	yes	characteristic, complemented, Sylow
220	C₂×C₄×D₄	—	67	1	no	—
221	C₂×C₄×D₄	—	134	1	no	—
222	C₂²×C₄⋊C₄	—	134	1	no	—
223	C₂×#106	—	134	1	no	—
224	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
225	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
226	C₂×#102	—	134	1	no	—
227	C₂×C₄×C₈	—	67	1	no	—
228	#389	—	134	1	no	—
229	#385	—	134	2	no	—
230	#384	—	67	1	no	—
231	#384	C₂×D₆₇	1	1	yes	characteristic
232	#380	—	134	1	no	—
233	C₄×C₄⋊C₄	—	67	1	no	—
234	#340	—	134	1	no	—
235	#340	—	134	1	no	—
236	#340	—	134	2	no	—
237	D₄⋊C₈	—	67	4	no	—
238	C₂²×D₄	—	67	1	no	—
239	C₂³×C₄	—	134	2	no	—
240	C₂³×C₄	—	134	1	no	—
241	C₂²×C₈	—	2	1	no	—
242	C₂²×C₈	—	2	1	no	—
243	C₂²×C₈	—	134	1	no	—
244	C₄×D₄	—	268	2	no	—
245	C₄×D₄	—	268	2	no	—
246	C₄×D₄	—	67	2	no	—
247	C₄×D₄	—	67	2	no	—
248	C₄×D₄	—	134	1	no	—
249	C₄×D₄	—	134	1	no	—
250	C₂×C₄⋊C₄	—	2	1	no	—
251	C₂×C₄⋊C₄	—	2	1	no	—
252	C₂×C₄⋊C₄	—	268	2	no	—
253	C₂×C₄⋊C₄	—	268	2	no	—
254	C₂×C₄⋊C₄	—	67	1	no	—
255	C₂×C₄⋊C₄	—	67	1	no	—
256	C₂×C₄⋊C₄	—	67	1	no	—
257	C₂×C₄⋊C₄	—	134	2	no	—
258	C₂×C₄⋊C₄	—	134	1	no	—
259	C₂×C₄⋊C₄	—	134	1	no	—
260	C₂×C₄⋊C₄	—	134	1	no	—
261	C₂×C₄⋊C₄	D₂₆₈	1	2	yes	—
262	C₂×C₂²⋊C₄	—	134	2	no	—
263	C₂×C₂²⋊C₄	—	134	1	no	—
264	C₂×C₄²	—	134	1	no	—
265	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
266	#106	—	268	2	no	—
267	C₄⋊C₈	—	2	1	no	—
268	C₄⋊C₈	—	2	1	no	—
269	C₄⋊C₈	—	2	1	no	—
270	C₄⋊C₈	C₄×D₆₇	1	2	yes	complemented
271	#102	—	268	2	no	—
272	C₄×C₈	—	67	2	no	—
273	C₄×C₈	—	134	1	no	—
274	#95	—	134	2	no	—
275	#95	—	134	1	no	—
276	C₂⁴	—	134	1	no	—
277	C₂×D₄	—	67	2	no	—
278	C₂×D₄	—	67	2	no	—
279	C₂×D₄	—	67	2	no	—
280	C₂×D₄	—	134	1	no	—
281	C₂×D₄	—	134	1	no	—
282	C₂×D₄	—	134	1	no	—
283	C₂²×C₄	—	2	2	no	—
284	C₂²×C₄	—	268	4	no	—
285	C₂²×C₄	—	268	4	no	—
286	C₂²×C₄	—	268	1	no	—
287	C₂²×C₄	—	268	1	no	—
288	C₂²×C₄	—	268	1	no	—
289	C₂²×C₄	—	268	1	no	—
290	C₂²×C₄	—	268	1	no	—
291	C₂²×C₄	—	536	2	no	—
292	C₂²×C₄	—	134	1	no	—
293	C₂²×C₄	—	134	1	no	—
294	C₂²×C₄	—	134	1	no	—
295	C₂²×C₄	—	134	2	no	—
296	C₂²×C₄	—	134	2	no	—
297	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
298	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
299	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	4	1	no	—
302	C₂×C₈	—	4	1	no	—
303	C₂×C₈	—	4	1	no	—
304	C₂×C₈	—	4	1	no	—
305	C₂×C₈	—	4	1	no	—
306	C₂×C₈	—	268	1	no	—
307	C₂×C₈	—	268	1	no	—
308	C₂×C₈	—	134	2	no	—
309	C₄⋊C₄	—	4	2	no	—
310	C₄⋊C₄	—	4	1	no	—
311	C₄⋊C₄	—	4	1	no	—
312	C₄⋊C₄	—	4	1	no	—
313	C₄⋊C₄	—	4	1	no	—
314	C₄⋊C₄	—	268	2	no	—
315	C₄⋊C₄	—	268	4	no	—
316	C₄⋊C₄	—	268	1	no	—
317	C₄⋊C₄	—	268	1	no	—
318	C₄⋊C₄	—	268	1	no	—
319	C₄⋊C₄	—	268	1	no	—
320	C₄⋊C₄	—	67	2	no	—
321	C₄⋊C₄	—	67	2	no	—
322	C₄⋊C₄	—	134	1	no	—
323	C₄⋊C₄	—	134	1	no	—
324	C₄⋊C₄	—	134	1	no	—
325	C₄⋊C₄	—	134	1	no	—
326	C₂²⋊C₄	—	268	4	no	—
327	C₂²⋊C₄	—	134	2	no	—
328	C₂²⋊C₄	—	134	2	no	—
329	C₄²	—	268	1	no	—
330	C₄²	—	268	1	no	—
331	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
332	C₄²	Q₈×D₆₇	1	1	yes	characteristic
333	C₄²	#11761627	1	1	yes	characteristic
334	C₄²	#11761628	1	1	yes	characteristic
335	C₂³	—	268	1	no	—
336	C₂³	—	268	1	no	—
337	C₂³	—	268	1	no	—
338	C₂³	—	268	1	no	—
339	C₂³	—	134	2	no	—
340	C₂³	—	134	2	no	—
341	C₂³	—	134	2	no	—
342	C₂³	#14524013	1	1	yes	characteristic
343	D₄	—	67	4	no	—
344	D₄	—	134	2	no	—
345	D₄	—	134	2	no	—
346	D₄	—	134	2	no	—
347	C₂×C₄	—	2	1	no	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	4	2	no	—
351	C₂×C₄	—	4	2	no	—
352	C₂×C₄	—	268	1	no	—
353	C₂×C₄	—	268	1	no	—
354	C₂×C₄	—	268	1	no	—
355	C₂×C₄	—	268	1	no	—
356	C₂×C₄	—	268	2	no	—
357	C₂×C₄	—	268	2	no	—
358	C₂×C₄	—	268	2	no	—
359	C₂×C₄	—	268	2	no	—
360	C₂×C₄	—	268	1	no	—
361	C₂×C₄	—	268	1	no	—
362	C₂×C₄	—	268	1	no	—
363	C₂×C₄	—	536	8	no	—
364	C₂×C₄	—	536	1	no	—
365	C₂×C₄	—	536	1	no	—
366	C₂×C₄	—	536	1	no	—
367	C₂×C₄	—	134	2	no	—
368	C₂×C₄	—	134	2	no	—
369	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
370	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
371	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
372	C₂×C₄	#14524021	1	1	yes	characteristic
373	C₂×C₄	#14524023	1	1	yes	characteristic
374	C₂×C₄	#14524025	1	1	yes	characteristic
375	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
376	C₂×C₄	SD₁₆×D₆₇	1	1	yes	characteristic
377	C₂×C₄	SD₁₆×D₆₇	1	1	yes	characteristic
378	C₂×C₄	#14524064	1	1	yes	characteristic
379	C₂×C₄	#14524067	1	1	yes	characteristic
380	C₂×C₄	#14524070	1	1	yes	characteristic
381	C₂×C₄	#14524075	1	1	yes	characteristic
382	C₂×C₄	#14524075	1	1	yes	characteristic
383	C₈	—	4	2	no	—
384	C₈	—	8	1	no	—
385	C₈	—	268	2	no	—
386	C₂²	—	268	2	no	—
387	C₂²	—	268	2	no	—
388	C₂²	—	268	2	no	—
389	C₂²	—	268	2	no	—
390	C₂²	—	268	2	no	—
391	C₂²	—	268	2	no	—
392	C₂²	—	134	4	no	—
393	C₂²	#16863270	1	1	yes	characteristic
394	C₂²	#16863531	1	1	yes	characteristic
395	C₂²	#16863536	1	1	yes	characteristic
396	C₂²	#16863548	1	1	yes	characteristic
397	C₂²	#16863561	1	1	yes	characteristic
398	C₂²	#16863571	1	1	yes	characteristic
399	C₂²	#16863576	1	1	yes	characteristic
400	C₄	—	2	1	no	—
401	C₄	—	2	1	no	—
402	C₄	—	8	2	no	—
403	C₄	—	268	2	no	—
404	C₄	—	268	2	no	—
405	C₄	—	536	1	no	—
406	C₄	—	1072	1	no	—
407	C₄	#106×D₆₇	1	1	yes	characteristic
408	C₄	#16863437	1	1	yes	characteristic
409	C₄	#16863476	1	1	yes	characteristic
410	C₄	#16863499	1	1	yes	characteristic
411	C₄	#16863675	1	1	yes	characteristic
412	C₄	#16863695	1	1	yes	characteristic
413	C₄	#16863713	1	1	yes	characteristic
414	C₄	#16863725	1	1	yes	characteristic
415	C₂	—	268	4	no	—
416	C₂	#22559313	1	1	yes	characteristic
417	C₂	#22559361	1	1	yes	characteristic
418	C₂	#22559389	1	1	yes	characteristic
419	C₂	#22559417	1	1	yes	characteristic
420	C₂	#22560403	1	1	yes	characteristic
421	C₂	#22560445	1	1	yes	characteristic
422	C₂	#22560457	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524013
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1296	C₂
socle	C₂³×C₆₇	#120
prime radical	C₂²×D₂₆₈	D₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1296	D₆₇
hypocenter	C₆₇	#8894
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1296

Complements

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

N	H ≅ G/N	type
2. #22560973	C₂	semidirect
3. #22560336	C₂	semidirect
6. #22558048	C₂	semidirect
7. C₆₇×#1296	C₂	semidirect
22. #16863123	C₄	semidirect
51. C₄×C₆₇⋊C₈	C₂×C₄	semidirect
63. C₄⋊C₈×C₆₇	C₂×C₄	semidirect
165. D₂₆₈	#106	semidirect
217. #1296	D₆₇	semidirect
219. C₆₇	#8894	semidirect
270. C₄⋊C₈	C₄×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22563109	2
#22560973	2
#22560336	2
#22560210	2
C₂×#16863123	2
#22558048	2
C₆₇×#1296	2
#8894	67