Subgroups — #30827481

Order 17152.

726 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569241	C₂	1	2	yes	maximal, complemented
2	C₄×D₈×D₆₇	C₂	1	1	yes	maximal, characteristic
3	#22559094	C₂	1	1	yes	maximal, characteristic, complemented
4	#22559076	C₂	1	2	yes	maximal, complemented
5	#22559071	C₂	1	2	yes	maximal, complemented
6	C₈×D₅₃₆	C₂	1	1	yes	maximal, characteristic, complemented
7	#22559003	C₂	1	1	yes	maximal, characteristic, complemented
8	#22558971	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558650	C₂	1	2	yes	maximal, complemented
10	C₈×#11761611	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1589	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₈×D₆₇	C₄	1	1	yes	characteristic
13	C₄×D₄×D₆₇	C₂²	1	2	yes	—
14	C₄×D₅₃₆	C₂²	1	1	yes	characteristic
15	D₈×Dic₆₇	C₄	1	1	yes	characteristic
16	#16863478	C₄	1	1	yes	characteristic
17	C₄×#11761613	C₂²	1	2	yes	—
18	#16863476	C₄	1	2	yes	—
19	#16863474	C₂²	1	2	yes	complemented
20	#16863470	—	2	2	no	—
21	#16863465	—	2	2	no	—
22	C₈×D₂₆₈	C₂²	1	2	yes	complemented
23	#16863461	C₂²	1	2	yes	complemented
24	#16863452	C₂²	1	2	yes	—
25	#16863447	C₂²	1	1	yes	characteristic
26	#107×D₆₇	C₄	1	1	yes	characteristic
27	#102×D₆₇	C₄	1	2	yes	—
28	#16863359	—	2	2	no	—
29	C₂²⋊C₈×D₆₇	—	2	2	no	—
30	C₄×#11761611	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
32	#16863149	C₂²	1	1	yes	characteristic, complemented
33	#16863146	C₂²	1	2	yes	complemented
34	#16863125	C₂²	1	2	yes	complemented
35	D₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	complemented
36	#16863123	C₂²	1	2	yes	complemented
37	#16863118	C₂²	1	1	yes	characteristic, complemented
38	#16863117	C₂²	1	1	yes	characteristic, complemented
39	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
40	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
41	#436×C₆₇	C₂²	1	2	yes	complemented
42	#335×C₆₇	C₂²	1	1	yes	characteristic, complemented
43	D₄⋊C₈×C₆₇	C₂²	1	2	yes	complemented
44	C₈²×C₆₇	C₂²	1	1	yes	characteristic, complemented
45	C₂×D₄×D₆₇	C₂×C₄	1	2	yes	—
46	C₂²×C₄×D₆₇	—	2	2	no	—
47	C₂×D₅₃₆	C₂×C₄	1	1	yes	characteristic
48	C₂×#11761613	C₂×C₄	1	2	yes	—
49	D₈×D₆₇	—	2	4	no	—
50	C₂×#11761612	—	2	2	no	—
51	C₂×#11761611	—	2	1	no	—
52	C₂×C₈×D₆₇	—	2	2	no	—
53	C₂×C₈×D₆₇	—	2	2	no	—
54	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
55	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
56	C₄×D₂₆₈	C₂³	1	2	yes	—
57	D₄×Dic₆₇	C₂×C₄	1	2	yes	—
58	#14524005	C₂×C₄	1	2	yes	—
59	C₄×C₂²⋊D₆₇	—	2	2	no	—
60	(C₂²⋊D₆₇)⋊C₄	—	2	2	no	—
61	C₄⋊C₄×D₆₇	C₂×C₄	1	2	yes	—
62	C₂²⋊C₄×D₆₇	—	2	2	no	—
63	C₄²×D₆₇	C₂³	1	1	yes	characteristic
64	#14523977	C₂×C₄	1	1	yes	characteristic
65	#14523976	C₂×C₄	1	2	yes	—
66	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
67	Dic₆₇⋊C₈	D₄	1	1	yes	characteristic
68	Dic₆₇⋊C₈	D₄	1	1	yes	characteristic
69	#14523965	C₂×C₄	1	2	yes	—
70	#14523964	C₂×C₄	1	2	yes	—
71	#14523963	C₂×C₄	1	2	yes	—
72	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
73	(C₂×D₆₇)⋊C₈	—	2	2	no	—
74	(C₂×D₆₇)⋊C₈	—	2	2	no	—
75	(C₂×D₆₇)⋊C₈	—	2	2	no	—
76	(C₂×D₆₇)⋊C₈	D₄	1	1	yes	characteristic
77	(C₂×D₆₇)⋊C₈	D₄	1	1	yes	characteristic
78	#14523953	—	2	2	no	—
79	#14523953	—	2	1	no	—
80	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
81	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
82	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
83	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
84	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
85	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic
86	C₂×M₁₆×C₆₇	—	2	2	no	—
87	C₄×D₄×C₆₇	C₂³	1	2	yes	—
88	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
89	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
90	#102×C₆₇	C₂×C₄	1	2	yes	—
91	C₂²⋊C₈×C₆₇	—	2	2	no	—
92	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
95	C₂³×D₆₇	—	2	2	no	—
96	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
97	C₂×C₂²⋊D₆₇	—	2	2	no	—
98	D₄×D₆₇	—	2	4	no	—
99	D₄×D₆₇	—	4	2	no	—
100	C₂²×Dic₆₇	—	2	2	no	—
101	C₂×C₄×D₆₇	—	2	2	no	—
102	C₂×C₄×D₆₇	—	2	2	no	—
103	C₂×C₄×D₆₇	—	2	2	no	—
104	C₂×C₄×D₆₇	—	8	2	no	—
105	C₂×C₄×D₆₇	—	8	2	no	—
106	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×D₆₇	#41	1	1	yes	characteristic
109	D₅₃₆	M₁₆	1	4	yes	complemented
110	#11761613	—	2	4	no	—
111	#11761612	—	8	2	no	—
112	#11761611	—	2	4	no	—
113	C₂×C₆₇⋊C₈	—	2	1	no	—
114	C₂×C₆₇⋊C₈	—	2	2	no	—
115	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
118	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
119	C₈×D₆₇	—	2	2	no	—
120	C₈×D₆₇	—	4	1	no	—
121	C₈×D₆₇	—	8	2	no	—
122	C₈×D₆₇	—	8	2	no	—
123	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
124	Dic₆₇⋊C₄	—	2	2	no	—
125	C₂²⋊Dic₆₇	—	2	2	no	—
126	(C₂×D₆₇)⋊C₄	—	2	2	no	—
127	(C₂×D₆₇)⋊C₄	—	2	2	no	—
128	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
129	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
130	C₄×Dic₆₇	#41	1	1	yes	characteristic
131	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
132	C₂²×C₄×C₆₇	—	2	2	no	—
133	D₈×C₆₇	—	2	2	no	—
134	M₁₆×C₆₇	—	8	2	no	—
135	C₂×C₈×C₆₇	—	2	1	no	—
136	C₂×C₈×C₆₇	—	2	2	no	—
137	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
138	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
139	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
140	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
141	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
142	C₂²⋊C₄×C₆₇	—	2	2	no	—
143	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
144	C₂³×C₆₇	—	2	2	no	—
145	C₂²×D₆₇	—	2	2	no	—
146	C₂²×D₆₇	—	4	4	no	—
147	C₂²×D₆₇	—	4	4	no	—
148	C₂²×D₆₇	—	8	2	no	—
149	C₂²×D₆₇	C₄×D₄	1	1	yes	characteristic
150	D₄×C₆₇	—	2	2	no	—
151	D₄×C₆₇	—	2	2	no	—
152	C₂×C₄×C₆₇	—	2	2	no	—
153	C₂×C₄×C₆₇	—	4	2	no	—
154	C₂×C₄×C₆₇	—	4	2	no	—
155	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
156	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
158	C₂²⋊D₆₇	—	4	4	no	—
159	C₂×Dic₆₇	—	2	2	no	—
160	C₂×Dic₆₇	—	8	2	no	—
161	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
162	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
163	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
164	D₂₆₈	—	2	2	no	—
165	D₂₆₈	C₂×M₁₆	1	4	yes	—
166	C₄×D₆₇	—	2	2	no	—
167	C₄×D₆₇	—	2	1	no	—
168	C₄×D₆₇	—	2	1	no	—
169	C₄×D₆₇	—	2	1	no	—
170	C₄×D₆₇	—	2	1	no	—
171	C₄×D₆₇	—	4	2	no	—
172	C₄×D₆₇	—	4	2	no	—
173	C₄×D₆₇	—	8	2	no	—
174	C₄×D₆₇	—	8	2	no	—
175	C₈×C₆₇	—	2	2	no	—
176	C₈×C₆₇	—	2	1	no	—
177	C₈×C₆₇	—	2	1	no	—
178	C₈×C₆₇	—	2	1	no	—
179	C₈×C₆₇	—	4	2	no	—
180	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
181	C₆₇⋊C₈	—	2	2	no	—
182	C₆₇⋊C₈	—	2	1	no	—
183	C₆₇⋊C₈	—	2	1	no	—
184	C₆₇⋊C₈	—	2	1	no	—
185	C₆₇⋊C₈	—	2	1	no	—
186	C₆₇⋊C₈	—	4	2	no	—
187	C₂²×C₆₇	—	4	2	no	—
188	C₂²×C₆₇	—	4	2	no	—
189	C₂²×C₆₇	—	4	2	no	—
190	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
191	C₂×D₆₇	—	2	1	no	—
192	C₂×D₆₇	—	2	1	no	—
193	C₂×D₆₇	—	2	4	no	—
194	C₂×D₆₇	—	4	2	no	—
195	C₂×D₆₇	—	4	2	no	—
196	C₂×D₆₇	—	8	8	no	—
197	C₂×D₆₇	#443	1	2	yes	—
198	C₄×C₆₇	—	2	1	no	—
199	C₄×C₆₇	—	4	2	no	—
200	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
201	C₄×C₆₇	#568	1	1	yes	characteristic
202	C₄×C₆₇	#576	1	1	yes	characteristic
203	C₄×C₆₇	#576	1	1	yes	characteristic
204	Dic₆₇	—	2	1	no	—
205	Dic₆₇	—	2	1	no	—
206	Dic₆₇	—	2	1	no	—
207	Dic₆₇	—	4	2	no	—
208	#18054	—	67	1	no	maximal, Sylow
209	C₂×C₆₇	—	8	2	no	—
210	C₂×C₆₇	D₄×M₁₆	1	1	yes	characteristic
211	C₂×C₆₇	#2948	1	1	yes	characteristic
212	C₂×C₆₇	C₂×#443	1	1	yes	characteristic
213	D₆₇	—	2	2	no	—
214	D₆₇	—	4	4	no	—
215	#2974	—	67	2	no	—
216	C₂×C₄×D₈	—	67	1	no	—
217	#1589	—	67	1	no	—
218	#1589	D₆₇	1	1	yes	characteristic, complemented
219	#1586	—	67	2	no	—
220	#1586	—	67	2	no	—
221	C₈×D₈	—	67	1	no	—
222	C₈×D₈	—	67	1	no	—
223	#1568	—	67	1	no	—
224	#1490	—	67	2	no	—
225	C₈×M₁₆	—	67	1	no	—
226	C₆₇	#18054	1	1	yes	characteristic, complemented, Sylow
227	C₂²×D₈	—	67	1	no	—
228	C₂×C₄×D₄	—	67	2	no	—
229	C₄×D₈	—	67	1	no	—
230	C₄×D₈	—	67	2	no	—
231	C₄×D₈	—	67	1	no	—
232	C₄×D₈	—	67	2	no	—
233	C₄×D₈	—	67	1	no	—
234	C₄×D₈	C₂×D₆₇	1	1	yes	characteristic
235	#436	—	67	2	no	—
236	#436	C₂×D₆₇	1	2	yes	complemented
237	#435	—	134	2	no	—
238	#435	—	134	2	no	—
239	C₈×D₄	—	67	2	no	—
240	C₈×D₄	—	67	2	no	—
241	#432	—	67	2	no	—
242	#431	—	67	1	no	—
243	C₂×#107	—	67	1	no	—
244	C₂×#102	—	67	2	no	—
245	#407	—	134	2	no	—
246	C₂×C₂²⋊C₈	—	134	2	no	—
247	C₄×M₁₆	—	67	1	no	—
248	C₂×C₄×C₈	—	67	1	no	—
249	#335	—	67	1	no	—
250	#335	C₂×D₆₇	1	1	yes	characteristic, complemented
251	#334	—	67	2	no	—
252	D₄⋊C₈	—	67	2	no	—
253	D₄⋊C₈	—	67	2	no	—
254	D₄⋊C₈	—	67	2	no	—
255	D₄⋊C₈	C₂×D₆₇	1	2	yes	complemented
256	#322	—	67	1	no	—
257	#322	—	67	1	no	—
258	C₈²	—	67	1	no	—
259	C₈²	C₂×D₆₇	1	1	yes	characteristic, complemented
260	C₂²×D₄	—	67	2	no	—
261	C₂³×C₄	—	134	2	no	—
262	C₂×D₈	—	67	2	no	—
263	C₂×D₈	—	67	1	no	—
264	C₂×D₈	—	134	4	no	—
265	C₂×D₈	C₄×D₆₇	1	1	yes	characteristic
266	C₂×M₁₆	—	2	2	no	—
267	C₂×M₁₆	—	134	1	no	—
268	C₂×M₁₆	—	134	2	no	—
269	C₂²×C₈	—	67	1	no	—
270	C₂²×C₈	—	67	1	no	—
271	C₂²×C₈	—	134	2	no	—
272	C₂²×C₈	—	134	2	no	—
273	C₄×D₄	—	67	2	no	—
274	C₄×D₄	—	67	2	no	—
275	C₄×D₄	—	67	2	no	—
276	C₄×D₄	—	134	2	no	—
277	C₄×D₄	—	134	2	no	—
278	C₄×D₄	C₂²×D₆₇	1	2	yes	—
279	C₂×C₄⋊C₄	—	67	2	no	—
280	C₂×C₂²⋊C₄	—	134	2	no	—
281	C₂×C₄²	—	67	1	no	—
282	#107	—	67	1	no	—
283	#107	—	67	2	no	—
284	#107	C₄×D₆₇	1	1	yes	characteristic
285	C₄⋊C₈	—	67	1	no	—
286	C₄⋊C₈	—	67	1	no	—
287	C₄⋊C₈	—	67	2	no	—
288	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
289	#102	—	67	2	no	—
290	#102	—	67	2	no	—
291	#102	—	67	2	no	—
292	#102	C₄×D₆₇	1	2	yes	—
293	C₂²⋊C₈	—	2	2	no	—
294	C₂²⋊C₈	—	67	1	no	—
295	C₂²⋊C₈	—	67	1	no	—
296	C₂²⋊C₈	—	134	2	no	—
297	C₂²⋊C₈	—	134	2	no	—
298	C₂²⋊C₈	—	134	2	no	—
299	C₂²⋊C₈	—	134	2	no	—
300	#97	—	134	1	no	—
301	#97	—	134	2	no	—
302	C₄×C₈	—	67	1	no	—
303	C₄×C₈	—	67	1	no	—
304	C₄×C₈	—	67	1	no	—
305	C₄×C₈	—	67	1	no	—
306	C₄×C₈	—	67	1	no	—
307	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
308	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
309	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
310	C₂⁴	—	134	2	no	—
311	C₂×D₄	—	268	2	no	—
312	C₂×D₄	—	67	2	no	—
313	C₂×D₄	—	134	2	no	—
314	C₂×D₄	—	134	4	no	—
315	C₂×D₄	C₂×C₄×D₆₇	1	2	yes	—
316	C₂²×C₄	—	2	2	no	—
317	C₂²×C₄	—	536	2	no	—
318	C₂²×C₄	—	536	2	no	—
319	C₂²×C₄	—	67	1	no	—
320	C₂²×C₄	—	67	1	no	—
321	C₂²×C₄	—	67	1	no	—
322	C₂²×C₄	—	134	2	no	—
323	C₂²×C₄	—	134	2	no	—
324	C₂²×C₄	—	134	2	no	—
325	C₂²×C₄	—	134	2	no	—
326	D₈	—	2	2	no	—
327	D₈	—	67	4	no	—
328	D₈	—	134	4	no	—
329	M₁₆	—	8	2	no	—
330	M₁₆	—	536	2	no	—
331	M₁₆	—	134	4	no	—
332	C₂×C₈	—	2	1	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	268	1	no	—
335	C₂×C₈	—	536	2	no	—
336	C₂×C₈	—	536	2	no	—
337	C₂×C₈	—	67	1	no	—
338	C₂×C₈	—	67	1	no	—
339	C₂×C₈	—	67	1	no	—
340	C₂×C₈	—	67	1	no	—
341	C₂×C₈	—	134	2	no	—
342	C₂×C₈	—	134	1	no	—
343	C₂×C₈	—	134	2	no	—
344	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
345	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
346	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
347	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
348	C₄⋊C₄	—	67	2	no	—
349	C₄⋊C₄	—	134	2	no	—
350	C₄⋊C₄	C₂×C₄×D₆₇	1	2	yes	—
351	C₂²⋊C₄	—	2	2	no	—
352	C₂²⋊C₄	—	134	2	no	—
353	C₂²⋊C₄	—	134	2	no	—
354	C₂²⋊C₄	—	134	2	no	—
355	C₄²	—	67	1	no	—
356	C₄²	—	67	1	no	—
357	C₄²	—	67	1	no	—
358	C₄²	C₂³×D₆₇	1	1	yes	characteristic
359	C₂³	—	2	2	no	—
360	C₂³	—	268	4	no	—
361	C₂³	—	268	4	no	—
362	C₂³	—	536	2	no	—
363	C₂³	—	67	1	no	—
364	C₂³	—	134	2	no	—
365	D₄	—	2	2	no	—
366	D₄	—	2	2	no	—
367	D₄	—	268	4	no	—
368	D₄	—	67	4	no	—
369	D₄	—	134	2	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	4	2	no	—
372	C₂×C₄	—	4	2	no	—
373	C₂×C₄	—	268	2	no	—
374	C₂×C₄	—	268	2	no	—
375	C₂×C₄	—	536	2	no	—
376	C₂×C₄	—	536	2	no	—
377	C₂×C₄	—	536	2	no	—
378	C₂×C₄	—	67	1	no	—
379	C₂×C₄	—	67	1	no	—
380	C₂×C₄	—	67	1	no	—
381	C₂×C₄	—	134	2	no	—
382	C₂×C₄	—	134	2	no	—
383	C₂×C₄	—	134	1	no	—
384	C₂×C₄	—	134	1	no	—
385	C₂×C₄	—	134	1	no	—
386	C₂×C₄	—	134	1	no	—
387	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
388	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
389	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
390	C₈	—	2	2	no	—
391	C₈	—	2	1	no	—
392	C₈	—	2	1	no	—
393	C₈	—	2	1	no	—
394	C₈	—	4	2	no	—
395	C₈	—	268	2	no	—
396	C₈	—	134	2	no	—
397	C₈	—	134	1	no	—
398	C₈	—	134	1	no	—
399	C₈	—	134	1	no	—
400	C₈	—	134	1	no	—
401	C₈	M₁₆×D₆₇	1	2	yes	—
402	C₂²	—	4	2	no	—
403	C₂²	—	4	2	no	—
404	C₂²	—	4	2	no	—
405	C₂²	—	268	2	no	—
406	C₂²	—	268	2	no	—
407	C₂²	—	536	8	no	—
408	C₂²	—	67	2	no	—
409	C₂²	—	134	4	no	—
410	C₂²	—	134	1	no	—
411	C₂²	—	134	1	no	—
412	C₂²	C₄×D₄×D₆₇	1	1	yes	characteristic
413	C₄	—	2	1	no	—
414	C₄	—	4	2	no	—
415	C₄	—	268	2	no	—
416	C₄	—	134	1	no	—
417	C₄	—	134	1	no	—
418	C₄	—	134	1	no	—
419	C₄	C₂×M₁₆×D₆₇	1	1	yes	characteristic
420	C₄	#16864161	1	1	yes	characteristic
421	C₄	#16864197	1	1	yes	characteristic
422	C₄	#16864198	1	1	yes	characteristic
423	C₂	—	8	2	no	—
424	C₂	—	268	4	no	—
425	C₂	—	134	2	no	—
426	C₂	#22568945	1	1	yes	characteristic
427	C₂	#22569056	1	1	yes	characteristic
428	C₂	#443×D₆₇	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×#1589	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×D₈×D₆₇	C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1589	D₆₇
hypocenter	C₆₇	#18054
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1589

Complements

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

N	H ≅ G/N	type
1. #22569241	C₂	semidirect
3. #22559094	C₂	semidirect
4. #22559076	C₂	semidirect
5. #22559071	C₂	semidirect
6. C₈×D₅₃₆	C₂	semidirect
7. #22559003	C₂	semidirect
8. #22558971	C₂	semidirect
9. #22558650	C₂	semidirect
10. C₈×#11761611	C₂	semidirect
11. C₆₇×#1589	C₂	semidirect
19. #16863474	C₂²	semidirect
22. C₈×D₂₆₈	C₂²	semidirect
23. #16863461	C₂²	semidirect
32. #16863149	C₂²	semidirect
33. #16863146	C₂²	semidirect
34. #16863125	C₂²	semidirect
35. D₄⋊(C₆₇⋊C₈)	C₂²	semidirect
36. #16863123	C₂²	semidirect
37. #16863118	C₂²	semidirect
38. #16863117	C₂²	semidirect
39. C₈×C₆₇⋊C₈	C₂²	semidirect
41. #436×C₆₇	C₂²	semidirect
42. #335×C₆₇	C₂²	semidirect
43. D₄⋊C₈×C₆₇	C₂²	semidirect
44. C₈²×C₆₇	C₂²	semidirect
109. D₅₃₆	M₁₆	semidirect
218. #1589	D₆₇	semidirect
226. C₆₇	#18054	semidirect
236. #436	C₂×D₆₇	semidirect
250. #335	C₂×D₆₇	semidirect
255. D₄⋊C₈	C₂×D₆₇	semidirect
259. C₈²	C₂×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22569241	2
C₄×D₈×D₆₇	2
#22559094	2
#22559076	2
#22559071	2
C₈×D₅₃₆	2
#22559003	2
#22558971	2
#22558650	2
C₈×#11761611	2
C₆₇×#1589	2
#18054	67