Subgroups — #30827016

Order 17152.

600 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	#22570274	C₂	1	1	yes	maximal, characteristic
2	#22569335	C₂	1	1	yes	maximal, characteristic
3	#22569311	C₂	1	1	yes	maximal, characteristic, complemented
4	#22559103	C₂	1	2	yes	maximal, complemented
5	#22559077	C₂	1	2	yes	maximal, complemented
6	#22559069	C₂	1	2	yes	maximal, complemented
7	#22559006	C₂	1	2	yes	maximal, complemented
8	#22559000	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558592	C₂	1	1	yes	maximal, characteristic
10	#22558492	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1493	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#11761628	C₂²	1	1	yes	characteristic
13	#16863734	C₄	1	2	yes	—
14	#16863678	C₄	1	1	yes	characteristic
15	#16863673	C₄	1	1	yes	characteristic
16	#16863603	C₄	1	1	yes	characteristic
17	#16863598	C₄	1	1	yes	characteristic
18	#16863556	C₄	1	2	yes	—
19	#16863527	—	2	1	no	—
20	#16863527	C₂²	1	2	yes	—
21	#16863523	C₂²	1	2	yes	—
22	Q₈×C₆₇⋊C₈	—	2	1	no	—
23	Q₈×C₆₇⋊C₈	C₂²	1	2	yes	—
24	C₄×#11761618	C₂²	1	2	yes	—
25	C₄×#11761615	C₂²	1	2	yes	—
26	#16863465	—	2	2	no	—
27	#16863461	C₂²	1	2	yes	—
28	#16863457	—	2	2	no	—
29	#16863452	—	2	1	no	—
30	#16863449	C₂²	1	1	yes	characteristic
31	#16863449	C₂²	1	1	yes	characteristic
32	#16863449	C₂²	1	1	yes	characteristic
33	C₄⋊C₈×D₆₇	—	2	1	no	—
34	#16863148	C₂²	1	2	yes	—
35	#16863146	C₂²	1	2	yes	—
36	#16863128	C₂²	1	1	yes	characteristic
37	#16863128	C₂²	1	1	yes	characteristic
38	#16863125	C₂²	1	1	yes	characteristic
39	#16863125	C₂²	1	1	yes	characteristic
40	#16863117	C₂²	1	2	yes	—
41	C₈×C₆₇⋊C₈	C₂²	1	2	yes	—
42	#533×C₆₇	C₂²	1	1	yes	characteristic
43	#423×C₆₇	C₂²	1	1	yes	characteristic
44	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
45	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
46	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
47	C₂×#11761628	C₂×C₄	1	1	yes	characteristic
48	C₂×#11761618	C₂×C₄	1	2	yes	—
49	C₂×#11761615	C₂×C₄	1	2	yes	—
50	C₂×C₈×D₆₇	—	2	2	no	—
51	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
52	C₄×Dic₁₃₄	C₂³	1	1	yes	characteristic
53	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
54	C₄×D₂₆₈	C₂³	1	1	yes	characteristic
55	C₄×C₂²⋊D₆₇	—	2	1	no	—
56	C₄×C₂²⋊D₆₇	—	2	1	no	—
57	#14524002	C₂×C₄	1	1	yes	characteristic
58	#14524000	—	2	1	no	—
59	C₄²×D₆₇	—	2	1	no	—
60	#14523976	C₂×C₄	1	2	yes	—
61	#14523974	C₂×C₄	1	2	yes	—
62	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
63	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
64	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
65	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
66	C₄⋊(C₆₇⋊C₈)	D₄	1	1	yes	characteristic
67	C₄⋊(C₆₇⋊C₈)	D₄	1	1	yes	characteristic
68	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
69	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
70	Dic₆₇⋊C₈	—	2	2	no	—
71	#14523967	C₂×C₄	1	2	yes	—
72	#14523967	C₂×C₄	1	2	yes	—
73	#14523966	C₂×C₄	1	2	yes	—
74	#14523963	C₂×C₄	1	2	yes	—
75	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
76	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
77	C₂²⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
78	(C₂×D₆₇)⋊C₈	—	2	2	no	—
79	#14523954	—	2	2	no	—
80	#14523953	—	2	2	no	—
81	C₄×C₆₇⋊C₈	—	2	2	no	—
82	C₄×C₆₇⋊C₈	—	2	1	no	—
83	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
84	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
85	C₂×M₁₆×C₆₇	—	2	1	no	—
86	C₂²×C₈×C₆₇	—	2	1	no	—
87	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
88	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
89	#125×C₆₇	—	2	1	no	—
90	C₂²⋊Q₈×C₆₇	—	2	1	no	—
91	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
92	C₄×Q₈×C₆₇	—	2	1	no	—
93	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
94	#117×C₆₇	—	2	1	no	—
95	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
96	C₄⋊C₈×C₆₇	—	2	1	no	—
97	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
98	C₄×C₈×C₆₇	—	2	1	no	—
99	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
100	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
101	#11761628	—	2	2	no	—
102	#11761628	—	4	1	no	—
103	C₂×Dic₁₃₄	C₂²×C₄	1	1	yes	characteristic
104	C₂×D₂₆₈	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₂²⋊D₆₇	—	2	1	no	—
106	C₂×C₄×D₆₇	—	2	1	no	—
107	C₂×C₄×D₆₇	—	2	1	no	—
108	C₂×C₄×D₆₇	—	2	1	no	—
109	#11761618	—	4	2	no	—
110	#11761615	—	2	4	no	—
111	C₂×C₆₇⋊C₈	—	2	1	no	—
112	C₂×C₆₇⋊C₈	—	2	1	no	—
113	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
114	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
115	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
116	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
117	C₈×D₆₇	—	8	2	no	—
118	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
119	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
120	Dic₆₇⋊C₄	—	2	1	no	—
121	Dic₆₇⋊C₄	—	2	1	no	—
122	C₂²⋊Dic₆₇	—	2	1	no	—
123	(C₂×D₆₇)⋊C₄	—	2	1	no	—
124	(C₂×D₆₇)⋊C₄	—	2	1	no	—
125	C₄×Dic₆₇	—	2	1	no	—
126	C₄×Dic₆₇	—	2	1	no	—
127	C₄×Dic₆₇	—	2	1	no	—
128	C₂×Q₈×C₆₇	—	2	1	no	—
129	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
130	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
131	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
133	M₁₆×C₆₇	—	8	1	no	—
134	C₂×C₈×C₆₇	—	2	4	no	—
135	C₂×C₈×C₆₇	—	2	1	no	—
136	C₂×C₈×C₆₇	—	2	1	no	—
137	C₂×C₈×C₆₇	—	2	2	no	—
138	C₄⋊C₄×C₆₇	—	2	1	no	—
139	C₄⋊C₄×C₆₇	—	2	1	no	—
140	C₄⋊C₄×C₆₇	—	2	1	no	—
141	C₄⋊C₄×C₆₇	—	2	2	no	—
142	C₄⋊C₄×C₆₇	—	2	2	no	—
143	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
144	C₂²⋊C₄×C₆₇	—	2	2	no	—
145	C₄²×C₆₇	—	2	2	no	—
146	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
147	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
148	C₄²×C₆₇	#41	1	1	yes	characteristic
149	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
150	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
151	C₂²×D₆₇	—	2	1	no	—
152	Q₈×C₆₇	—	2	2	no	—
153	Q₈×C₆₇	—	2	2	no	—
154	Q₈×C₆₇	—	4	2	no	—
155	C₂×C₄×C₆₇	—	2	1	no	—
156	C₂×C₄×C₆₇	—	2	1	no	—
157	C₂×C₄×C₆₇	—	2	1	no	—
158	C₂×C₄×C₆₇	—	2	1	no	—
159	C₂×C₄×C₆₇	—	2	2	no	—
160	C₂×C₄×C₆₇	—	2	2	no	—
161	C₂×C₄×C₆₇	—	2	2	no	—
162	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
163	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
164	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
165	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
166	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
167	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
168	C₂²⋊D₆₇	—	4	2	no	—
169	C₂×Dic₆₇	—	2	1	no	—
170	C₂×Dic₆₇	—	2	1	no	—
171	C₂×Dic₆₇	—	2	1	no	—
172	D₂₆₈	—	2	1	no	—
173	D₂₆₈	#131	1	2	yes	—
174	C₄×D₆₇	—	4	2	no	—
175	C₄×D₆₇	—	4	1	no	—
176	C₄×D₆₇	—	4	1	no	—
177	C₄×D₆₇	—	4	2	no	—
178	Dic₁₃₄	—	2	1	no	—
179	Dic₁₃₄	—	2	1	no	—
180	C₈×C₆₇	—	2	2	no	—
181	C₈×C₆₇	—	2	2	no	—
182	C₈×C₆₇	—	4	2	no	—
183	C₆₇⋊C₈	—	2	2	no	—
184	C₆₇⋊C₈	—	2	2	no	—
185	C₆₇⋊C₈	—	2	2	no	—
186	C₆₇⋊C₈	—	2	2	no	—
187	C₆₇⋊C₈	—	4	1	no	—
188	C₆₇⋊C₈	—	4	1	no	—
189	C₂²×C₆₇	—	2	1	no	—
190	C₂²×C₆₇	—	2	1	no	—
191	C₂²×C₆₇	—	2	1	no	—
192	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
193	C₂×D₆₇	—	2	2	no	—
194	C₂×D₆₇	—	4	1	no	—
195	C₂×D₆₇	—	4	1	no	—
196	C₄×C₆₇	—	2	1	no	—
197	C₄×C₆₇	—	2	1	no	—
198	C₄×C₆₇	—	2	1	no	—
199	C₄×C₆₇	—	4	2	no	—
200	C₄×C₆₇	—	4	2	no	—
201	C₄×C₆₇	#443	1	2	yes	—
202	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
203	C₄×C₆₇	#568	1	1	yes	characteristic
204	C₄×C₆₇	#577	1	1	yes	characteristic
205	C₄×C₆₇	#578	1	1	yes	characteristic
206	Dic₆₇	—	4	1	no	—
207	Dic₆₇	—	4	1	no	—
208	Dic₆₇	—	4	1	no	—
209	#18016	—	67	1	no	maximal, Sylow
210	C₂×C₆₇	—	4	1	no	—
211	C₂×C₆₇	#2933	1	1	yes	characteristic
212	C₂×C₆₇	#2947	1	1	yes	characteristic
213	C₂×C₆₇	C₂×#443	1	1	yes	characteristic
214	D₆₇	—	4	2	no	—
215	#3107	—	67	1	no	—
216	#2990	—	67	1	no	—
217	#2986	—	67	1	no	—
218	#1591	—	67	2	no	—
219	#1587	—	67	2	no	—
220	#1585	—	67	2	no	—
221	C₈×SD₁₆	—	67	2	no	—
222	#1574	—	67	1	no	—
223	#1493	D₆₇	1	1	yes	characteristic, complemented
224	#1476	—	67	1	no	—
225	#1450	—	67	1	no	—
226	C₆₇	#18016	1	1	yes	characteristic, complemented, Sylow
227	#533	C₂×D₆₇	1	1	yes	characteristic
228	C₄×#41	—	67	1	no	—
229	#485	—	67	2	no	—
230	#475	—	67	1	no	—
231	#474	—	67	1	no	—
232	#462	—	67	1	no	—
233	#461	—	67	1	no	—
234	#452	—	67	2	no	—
235	#446	—	67	2	no	—
236	#446	—	67	2	no	—
237	#446	—	134	1	no	—
238	C₈×Q₈	—	67	2	no	—
239	C₈×Q₈	—	134	1	no	—
240	C₄×Q₁₆	—	67	2	no	—
241	C₄×SD₁₆	—	67	2	no	—
242	#435	—	134	2	no	—
243	C₈×D₄	—	67	2	no	—
244	#433	—	134	2	no	—
245	#432	—	134	1	no	—
246	#431	—	67	1	no	—
247	#431	—	67	1	no	—
248	#431	—	67	1	no	—
249	#423	C₂×D₆₇	1	1	yes	characteristic
250	C₂×C₄⋊C₈	—	134	1	no	—
251	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
252	#335	—	67	2	no	—
253	#334	—	67	2	no	—
254	Q₈⋊C₈	—	67	1	no	—
255	Q₈⋊C₈	—	67	1	no	—
256	Q₈⋊C₈	C₂×D₆₇	1	2	yes	—
257	Q₈⋊C₈	C₂×D₆₇	1	2	yes	—
258	D₄⋊C₈	—	67	1	no	—
259	D₄⋊C₈	—	67	1	no	—
260	#322	—	67	2	no	—
261	C₈²	—	67	2	no	—
262	C₂×#41	—	67	1	no	—
263	C₂×Q₁₆	—	67	2	no	—
264	C₂×SD₁₆	—	67	2	no	—
265	C₂×M₁₆	—	2	1	no	—
266	C₂²×C₈	—	2	1	no	—
267	C₂²×C₈	—	134	2	no	—
268	C₄⋊Q₈	C₄×D₆₇	1	1	yes	characteristic
269	C₄⋊Q₈	C₄×D₆₇	1	1	yes	characteristic
270	#125	—	2	1	no	—
271	C₂²⋊Q₈	—	2	1	no	—
272	C₂²⋊Q₈	C₄×D₆₇	1	2	yes	—
273	C₄×Q₈	—	2	1	no	—
274	C₄×Q₈	—	67	1	no	—
275	C₄×Q₈	—	67	1	no	—
276	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
277	C₄×D₄	—	67	1	no	—
278	C₄×D₄	—	67	1	no	—
279	C₄×D₄	—	134	1	no	—
280	C₄×D₄	—	134	1	no	—
281	#117	—	2	1	no	—
282	#117	—	67	1	no	—
283	#117	—	134	1	no	—
284	C₂×C₄²	—	134	1	no	—
285	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
286	#107	—	67	2	no	—
287	#106	—	67	2	no	—
288	C₄⋊C₈	—	2	1	no	—
289	C₄⋊C₈	—	67	1	no	—
290	C₄⋊C₈	—	67	1	no	—
291	C₄⋊C₈	—	67	1	no	—
292	C₄⋊C₈	—	67	1	no	—
293	C₄⋊C₈	—	67	1	no	—
294	C₄⋊C₈	—	67	1	no	—
295	C₄⋊C₈	—	134	2	no	—
296	C₄⋊C₈	—	134	2	no	—
297	C₄⋊C₈	—	134	2	no	—
298	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
299	#103	—	67	2	no	—
300	#103	—	67	2	no	—
301	#103	—	67	2	no	—
302	#102	—	67	2	no	—
303	C₂²⋊C₈	—	67	2	no	—
304	C₂²⋊C₈	—	67	2	no	—
305	C₂²⋊C₈	—	134	2	no	—
306	C₂²⋊C₈	—	134	1	no	—
307	#97	—	134	2	no	—
308	#97	—	134	2	no	—
309	C₄×C₈	—	2	1	no	—
310	C₄×C₈	—	67	2	no	—
311	C₄×C₈	—	67	2	no	—
312	C₄×C₈	—	134	2	no	—
313	C₄×C₈	—	134	1	no	—
314	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
315	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
316	#41	—	268	1	no	—
317	#41	—	134	2	no	—
318	C₂×Q₈	—	2	1	no	—
319	C₂×Q₈	—	67	1	no	—
320	C₂×Q₈	C₂×C₄×D₆₇	1	2	yes	—
321	C₂×D₄	—	67	1	no	—
322	C₂×D₄	—	134	1	no	—
323	C₂²×C₄	—	134	1	no	—
324	C₂²×C₄	—	134	1	no	—
325	C₂²×C₄	—	134	1	no	—
326	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
327	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
328	C₂²×C₄	#11761627	1	1	yes	characteristic
329	Q₁₆	—	268	2	no	—
330	SD₁₆	—	134	4	no	—
331	M₁₆	—	8	1	no	—
332	C₂×C₈	—	2	4	no	—
333	C₂×C₈	—	2	1	no	—
334	C₂×C₈	—	2	1	no	—
335	C₂×C₈	—	2	2	no	—
336	C₂×C₈	—	536	2	no	—
337	C₂×C₈	—	67	2	no	—
338	C₂×C₈	—	67	2	no	—
339	C₂×C₈	—	67	2	no	—
340	C₂×C₈	—	67	2	no	—
341	C₂×C₈	—	134	1	no	—
342	C₂×C₈	—	134	1	no	—
343	C₄⋊C₄	—	2	1	no	—
344	C₄⋊C₄	—	2	1	no	—
345	C₄⋊C₄	—	2	1	no	—
346	C₄⋊C₄	—	2	2	no	—
347	C₄⋊C₄	—	2	2	no	—
348	C₄⋊C₄	—	67	1	no	—
349	C₄⋊C₄	—	67	1	no	—
350	C₄⋊C₄	—	134	1	no	—
351	C₄⋊C₄	—	134	1	no	—
352	C₄⋊C₄	C₂×C₄×D₆₇	1	2	yes	—
353	C₂²⋊C₄	—	2	2	no	—
354	C₂²⋊C₄	—	134	1	no	—
355	C₂²⋊C₄	—	134	1	no	—
356	C₂²⋊C₄	—	134	1	no	—
357	C₄²	—	2	2	no	—
358	C₄²	—	134	1	no	—
359	C₄²	—	134	1	no	—
360	C₄²	—	134	1	no	—
361	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
362	C₄²	D₄×D₆₇	1	1	yes	characteristic
363	C₄²	#11761627	1	1	yes	characteristic
364	C₄²	C₂³×D₆₇	1	1	yes	characteristic
365	C₂³	—	134	1	no	—
366	C₂³	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
367	Q₈	—	2	2	no	—
368	Q₈	—	2	2	no	—
369	Q₈	—	4	2	no	—
370	Q₈	—	134	1	no	—
371	Q₈	—	134	1	no	—
372	D₄	—	268	2	no	—
373	D₄	—	67	2	no	—
374	D₄	—	134	1	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	1	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	268	2	no	—
383	C₂×C₄	—	268	2	no	—
384	C₂×C₄	—	268	1	no	—
385	C₂×C₄	—	268	1	no	—
386	C₂×C₄	—	134	1	no	—
387	C₂×C₄	—	134	1	no	—
388	C₂×C₄	—	134	1	no	—
389	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
390	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
391	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
392	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
393	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
394	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
395	C₈	—	2	2	no	—
396	C₈	—	2	2	no	—
397	C₈	—	4	2	no	—
398	C₈	—	268	1	no	—
399	C₈	—	268	1	no	—
400	C₈	—	134	2	no	—
401	C₈	—	134	2	no	—
402	C₈	—	134	2	no	—
403	C₈	—	134	2	no	—
404	C₂²	—	2	1	no	—
405	C₂²	—	2	1	no	—
406	C₂²	—	2	1	no	—
407	C₂²	—	268	1	no	—
408	C₂²	—	268	1	no	—
409	C₂²	—	134	2	no	—
410	C₂²	C₂×(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
411	C₄	—	2	1	no	—
412	C₄	—	2	1	no	—
413	C₄	—	2	1	no	—
414	C₄	—	4	2	no	—
415	C₄	—	4	2	no	—
416	C₄	—	268	1	no	—
417	C₄	—	268	1	no	—
418	C₄	—	268	1	no	—
419	C₄	#16863510	1	2	yes	—
420	C₄	#131×D₆₇	1	1	yes	characteristic
421	C₄	#16864161	1	1	yes	characteristic
422	C₄	#16864202	1	1	yes	characteristic
423	C₄	#16864209	1	1	yes	characteristic
424	C₂	—	4	1	no	—
425	C₂	—	268	2	no	—
426	C₂	#22568939	1	1	yes	characteristic
427	C₂	#22569046	1	1	yes	characteristic
428	C₂	C₂×#16863510	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×#1493	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#11761628	C₂×C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1493	D₆₇
hypocenter	C₆₇	#18016
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1493

Complements

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

N	H ≅ G/N	type
3. #22569311	C₂	semidirect
4. #22559103	C₂	semidirect
5. #22559077	C₂	semidirect
6. #22559069	C₂	semidirect
7. #22559006	C₂	semidirect
8. #22559000	C₂	semidirect
10. #22558492	C₂	semidirect
11. C₆₇×#1493	C₂	semidirect
223. #1493	D₆₇	semidirect
226. C₆₇	#18016	semidirect

Maximal subgroups

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

subgroup	index
#22570274	2
#22569335	2
#22569311	2
#22559103	2
#22559077	2
#22559069	2
#22559006	2
#22559000	2
#22558592	2
#22558492	2
C₆₇×#1493	2
#18016	67