Subgroups — #30748632

Order 17152.

664 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562771	C₂	1	1	yes	maximal, characteristic
2	#22560360	C₂	1	1	yes	maximal, characteristic, complemented
3	#22560325	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16863139	C₂	1	1	yes	maximal, characteristic
5	C₂×#16863139	C₂	1	1	yes	maximal, characteristic
6	#22558092	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1296	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14524040	C₂²	1	1	yes	characteristic
9	C₂×#14524005	—	2	1	no	—
10	C₂×#14524005	—	2	1	no	—
11	C₂×#14524005	C₄	1	2	yes	—
12	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×Dic₆₇⋊C₈	—	2	1	no	—
15	C₂×Dic₆₇⋊C₈	—	2	1	no	—
16	#16863297	—	2	1	no	—
17	#16863297	—	2	1	no	—
18	#16863297	C₄	1	2	yes	—
19	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
20	#16863150	—	2	1	no	—
21	#16863150	—	2	1	no	—
22	#16863139	—	2	2	no	—
23	#16863139	—	2	2	no	—
24	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
26	#384×C₆₇	C₂²	1	1	yes	characteristic
27	C₂²×D₂₆₈	—	2	1	no	—
28	C₂²×D₂₆₈	—	2	1	no	—
29	C₂²×D₂₆₈	C₂×C₄	1	2	yes	—
30	C₂²×C₄×D₆₇	—	2	2	no	—
31	C₂²×C₆₇⋊C₈	—	2	1	no	—
32	C₂²×C₆₇⋊C₈	—	2	1	no	—
33	#14524040	—	2	1	no	—
34	#14524040	—	2	1	no	—
35	#14524040	D₄	1	2	yes	—
36	#14524040	Q₈	1	2	yes	—
37	#14524005	—	4	2	no	—
38	#14524005	—	4	2	no	—
39	#14524005	—	4	4	no	—
40	#14524005	—	4	2	no	—
41	#14524005	—	4	2	no	—
42	C₂×(C₂×D₆₇)⋊C₄	—	2	2	no	—
43	C₂×(C₂×D₆₇)⋊C₄	—	2	2	no	—
44	C₂×(C₂×D₆₇)⋊C₄	—	2	2	no	—
45	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
48	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
49	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
50	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
51	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
52	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
53	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
54	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
55	Dic₆₇⋊C₈	—	4	2	no	—
56	Dic₆₇⋊C₈	—	4	2	no	—
57	C₂²×C₈×C₆₇	—	2	1	no	—
58	C₂²×C₈×C₆₇	—	2	1	no	—
59	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
60	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
61	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
62	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
63	C₄⋊C₈×C₆₇	—	2	1	no	—
64	C₄⋊C₈×C₆₇	—	2	1	no	—
65	C₄⋊C₈×C₆₇	—	2	1	no	—
66	C₄⋊C₈×C₆₇	—	2	1	no	—
67	C₂³×D₆₇	—	2	2	no	—
68	C₂×D₂₆₈	—	2	2	no	—
69	C₂×D₂₆₈	—	2	2	no	—
70	C₂×D₂₆₈	—	2	2	no	—
71	C₂×D₂₆₈	—	2	2	no	—
72	C₂×D₂₆₈	—	2	2	no	—
73	C₂×D₂₆₈	—	4	2	no	—
74	C₂×D₂₆₈	—	4	2	no	—
75	C₂×D₂₆₈	—	4	1	no	—
76	C₂×D₂₆₈	—	4	1	no	—
77	C₂×D₂₆₈	—	4	1	no	—
78	C₂×D₂₆₈	—	4	1	no	—
79	C₂×D₂₆₈	—	4	1	no	—
80	C₂×D₂₆₈	—	4	1	no	—
81	C₂×D₂₆₈	—	4	1	no	—
82	C₂×D₂₆₈	—	4	1	no	—
83	C₂×D₂₆₈	C₄⋊C₄	1	4	yes	—
84	C₂²×Dic₆₇	C₂²⋊C₄	1	1	yes	characteristic
85	C₂²×Dic₆₇	C₂²⋊C₄	1	1	yes	characteristic
86	C₂²×Dic₆₇	C₂²⋊C₄	1	1	yes	characteristic
87	C₂²×Dic₆₇	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×D₆₇	—	4	4	no	—
89	C₂×C₄×D₆₇	—	4	4	no	—
90	C₂×C₄×D₆₇	—	4	4	no	—
91	C₂×C₆₇⋊C₈	—	4	1	no	—
92	C₂×C₆₇⋊C₈	—	4	1	no	—
93	C₂×C₆₇⋊C₈	—	4	1	no	—
94	C₂×C₆₇⋊C₈	—	4	1	no	—
95	C₂×C₆₇⋊C₈	—	4	1	no	—
96	C₂×C₆₇⋊C₈	—	4	1	no	—
97	(C₂×D₆₇)⋊C₄	—	4	4	no	—
98	(C₂×D₆₇)⋊C₄	—	4	4	no	—
99	(C₂×D₆₇)⋊C₄	—	4	4	no	—
100	C₄×Dic₆₇	—	2	1	no	—
101	C₄×Dic₆₇	—	2	1	no	—
102	C₄×Dic₆₇	—	2	1	no	—
103	C₄×Dic₆₇	—	2	1	no	—
104	C₄×Dic₆₇	—	2	1	no	—
105	C₄×Dic₆₇	—	2	1	no	—
106	C₄×Dic₆₇	—	2	2	no	—
107	C₄×Dic₆₇	—	4	1	no	—
108	C₄×Dic₆₇	D₈	1	2	yes	—
109	C₄×Dic₆₇	SD₁₆	1	2	yes	—
110	C₂²×C₄×C₆₇	—	2	2	no	—
111	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₆₇	—	4	1	no	—
115	C₂×C₈×C₆₇	—	4	1	no	—
116	C₂×C₈×C₆₇	—	4	1	no	—
117	C₂×C₈×C₆₇	—	4	1	no	—
118	C₂×C₈×C₆₇	—	4	1	no	—
119	C₂×C₈×C₆₇	—	4	1	no	—
120	C₄⋊C₄×C₆₇	—	4	2	no	—
121	C₄⋊C₄×C₆₇	—	4	1	no	—
122	C₄⋊C₄×C₆₇	—	4	1	no	—
123	C₄⋊C₄×C₆₇	—	4	1	no	—
124	C₄⋊C₄×C₆₇	—	4	1	no	—
125	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
127	C₄²×C₆₇	#41	1	1	yes	characteristic
128	C₄²×C₆₇	#41	1	1	yes	characteristic
129	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂²×D₆₇	—	2	4	no	—
131	C₂²×D₆₇	—	4	2	no	—
132	C₂²×D₆₇	—	4	2	no	—
133	C₂²×D₆₇	—	4	2	no	—
134	C₂²×D₆₇	—	4	2	no	—
135	C₂²×D₆₇	—	4	2	no	—
136	C₂²×D₆₇	—	4	2	no	—
137	C₂×C₄×C₆₇	—	2	1	no	—
138	C₂×C₄×C₆₇	—	2	1	no	—
139	C₂×C₄×C₆₇	—	4	2	no	—
140	C₂×C₄×C₆₇	—	4	2	no	—
141	C₂×C₄×C₆₇	—	4	2	no	—
142	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
144	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
145	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
146	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
147	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
148	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
150	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
151	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
152	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
153	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
154	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
155	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
156	C₂×Dic₆₇	—	2	1	no	—
157	C₂×Dic₆₇	—	2	1	no	—
158	C₂×Dic₆₇	—	2	1	no	—
159	C₂×Dic₆₇	—	2	1	no	—
160	C₂×Dic₆₇	—	2	1	no	—
161	C₂×Dic₆₇	—	2	1	no	—
162	C₂×Dic₆₇	—	2	1	no	—
163	C₂×Dic₆₇	—	2	1	no	—
164	C₂×Dic₆₇	—	2	1	no	—
165	C₂×Dic₆₇	#100	1	2	yes	—
166	C₂×Dic₆₇	#102	1	2	yes	—
167	C₂×Dic₆₇	#102	1	2	yes	—
168	D₂₆₈	—	4	4	no	—
169	D₂₆₈	—	4	4	no	—
170	D₂₆₈	—	4	2	no	—
171	D₂₆₈	—	4	2	no	—
172	D₂₆₈	—	4	2	no	—
173	D₂₆₈	—	4	2	no	—
174	D₂₆₈	—	4	2	no	—
175	D₂₆₈	—	4	2	no	—
176	D₂₆₈	—	4	2	no	—
177	D₂₆₈	—	4	2	no	—
178	C₄×D₆₇	—	8	8	no	—
179	C₈×C₆₇	—	8	1	no	—
180	C₈×C₆₇	—	8	1	no	—
181	C₆₇⋊C₈	—	8	1	no	—
182	C₆₇⋊C₈	—	8	1	no	—
183	C₂²×C₆₇	#380	1	1	yes	characteristic
184	C₂²×C₆₇	C₂×#100	1	1	yes	characteristic
185	C₂²×C₆₇	#413	1	1	yes	characteristic
186	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
187	C₂²×C₆₇	#416	1	1	yes	characteristic
188	C₂²×C₆₇	#417	1	1	yes	characteristic
189	C₂²×C₆₇	#418	1	1	yes	characteristic
190	C₂×D₆₇	—	4	4	no	—
191	C₂×D₆₇	—	4	4	no	—
192	C₂×D₆₇	—	4	4	no	—
193	C₂×D₆₇	—	8	2	no	—
194	C₂×D₆₇	—	8	2	no	—
195	C₂×D₆₇	—	8	2	no	—
196	C₂×D₆₇	—	8	2	no	—
197	C₄×C₆₇	—	2	1	no	—
198	C₄×C₆₇	—	2	1	no	—
199	C₄×C₆₇	—	8	2	no	—
200	C₄×C₆₇	#459	1	1	yes	characteristic
201	C₄×C₆₇	#461	1	1	yes	characteristic
202	C₄×C₆₇	#463	1	1	yes	characteristic
203	C₄×C₆₇	#463	1	1	yes	characteristic
204	C₄×C₆₇	#474	1	1	yes	characteristic
205	C₄×C₆₇	#476	1	1	yes	characteristic
206	C₄×C₆₇	#478	1	1	yes	characteristic
207	C₄×C₆₇	#478	1	1	yes	characteristic
208	Dic₆₇	—	2	2	no	—
209	Dic₆₇	—	2	2	no	—
210	Dic₆₇	—	2	2	no	—
211	Dic₆₇	—	4	1	no	—
212	#8588	—	67	1	no	maximal, Sylow
213	C₂×C₆₇	#1540	1	1	yes	characteristic
214	C₂×C₆₇	#1545	1	1	yes	characteristic
215	C₂×C₆₇	#1551	1	1	yes	characteristic
216	C₂×C₆₇	#1557	1	1	yes	characteristic
217	C₂×C₆₇	#1800	1	1	yes	characteristic
218	C₂×C₆₇	#1802	1	1	yes	characteristic
219	C₂×C₆₇	#1804	1	1	yes	characteristic
220	D₆₇	—	8	4	no	—
221	C₄×#127	—	67	1	no	—
222	#1842	—	67	1	no	—
223	#1831	—	67	1	no	—
224	C₂×#331	—	67	1	no	—
225	C₂×#331	—	67	1	no	—
226	#1296	—	67	1	no	—
227	#1296	D₆₇	1	1	yes	characteristic, complemented
228	C₆₇	#8588	1	1	yes	characteristic, complemented, Sylow
229	C₂×#127	—	67	1	no	—
230	C₂×C₄×D₄	—	67	2	no	—
231	C₂×C₄×D₄	—	134	1	no	—
232	C₂×C₄×D₄	—	134	1	no	—
233	C₂×C₄⋊C₈	—	67	1	no	—
234	C₂×C₄⋊C₈	—	67	1	no	—
235	C₂×C₄⋊C₈	—	134	1	no	—
236	C₂×C₄⋊C₈	—	134	1	no	—
237	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
238	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
239	#390	—	67	2	no	—
240	#390	—	134	1	no	—
241	#390	—	134	1	no	—
242	#384	C₂×D₆₇	1	1	yes	characteristic
243	C₄³	—	67	1	no	—
244	#336	—	134	1	no	—
245	#336	—	134	1	no	—
246	#331	—	134	2	no	—
247	#331	—	134	2	no	—
248	C₂²×D₄	—	67	2	no	—
249	C₂²×D₄	—	134	1	no	—
250	C₂²×D₄	—	134	1	no	—
251	C₂³×C₄	—	134	2	no	—
252	C₂²×C₈	—	2	1	no	—
253	C₂²×C₈	—	2	1	no	—
254	C₂²×C₈	—	134	1	no	—
255	C₂²×C₈	—	134	1	no	—
256	#127	—	67	2	no	—
257	#127	—	67	2	no	—
258	#127	—	134	1	no	—
259	#127	—	134	1	no	—
260	C₄×D₄	—	268	2	no	—
261	C₄×D₄	—	268	2	no	—
262	C₄×D₄	—	268	2	no	—
263	C₄×D₄	—	268	2	no	—
264	C₄×D₄	—	268	4	no	—
265	C₂×C₄⋊C₄	—	2	1	no	—
266	C₂×C₄⋊C₄	—	2	1	no	—
267	C₂×C₄⋊C₄	C₄×D₆₇	1	2	yes	—
268	C₂×C₂²⋊C₄	—	134	2	no	—
269	C₂×C₂²⋊C₄	—	134	2	no	—
270	C₂×C₂²⋊C₄	—	134	2	no	—
271	C₂×C₄²	—	67	1	no	—
272	C₂×C₄²	—	67	1	no	—
273	C₂×C₄²	—	67	1	no	—
274	C₂×C₄²	—	67	1	no	—
275	C₂×C₄²	—	67	1	no	—
276	C₂×C₄²	—	67	1	no	—
277	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
278	C₄⋊C₈	—	2	1	no	—
279	C₄⋊C₈	—	2	1	no	—
280	C₄⋊C₈	—	2	1	no	—
281	C₄⋊C₈	—	2	1	no	—
282	C₄⋊C₈	—	268	2	no	—
283	C₄⋊C₈	—	268	2	no	—
284	C₄⋊C₈	—	134	1	no	—
285	C₄⋊C₈	—	134	1	no	—
286	C₄⋊C₈	—	134	1	no	—
287	C₄⋊C₈	—	134	1	no	—
288	C₂⁴	—	134	2	no	—
289	C₂×D₄	—	268	2	no	—
290	C₂×D₄	—	268	2	no	—
291	C₂×D₄	—	268	1	no	—
292	C₂×D₄	—	268	1	no	—
293	C₂×D₄	—	268	1	no	—
294	C₂×D₄	—	268	1	no	—
295	C₂×D₄	—	268	1	no	—
296	C₂×D₄	—	268	1	no	—
297	C₂×D₄	—	268	1	no	—
298	C₂×D₄	—	268	1	no	—
299	C₂×D₄	—	67	4	no	—
300	C₂×D₄	—	134	2	no	—
301	C₂×D₄	—	134	2	no	—
302	C₂×D₄	—	134	2	no	—
303	C₂×D₄	—	134	2	no	—
304	C₂×D₄	—	134	2	no	—
305	C₂²×C₄	—	2	2	no	—
306	C₂²×C₄	—	268	4	no	—
307	C₂²×C₄	—	268	4	no	—
308	C₂²×C₄	—	268	4	no	—
309	C₂²×C₄	—	67	1	no	—
310	C₂²×C₄	—	67	1	no	—
311	C₂²×C₄	—	67	1	no	—
312	C₂²×C₄	—	67	1	no	—
313	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
314	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
315	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
316	C₂×C₈	—	4	1	no	—
317	C₂×C₈	—	4	1	no	—
318	C₂×C₈	—	4	1	no	—
319	C₂×C₈	—	4	1	no	—
320	C₂×C₈	—	4	1	no	—
321	C₂×C₈	—	4	1	no	—
322	C₂×C₈	—	268	1	no	—
323	C₂×C₈	—	268	1	no	—
324	C₂×C₈	—	268	1	no	—
325	C₂×C₈	—	268	1	no	—
326	C₂×C₈	—	268	1	no	—
327	C₂×C₈	—	268	1	no	—
328	C₄⋊C₄	—	4	2	no	—
329	C₄⋊C₄	—	4	1	no	—
330	C₄⋊C₄	—	4	1	no	—
331	C₄⋊C₄	—	4	1	no	—
332	C₄⋊C₄	—	4	1	no	—
333	C₂²⋊C₄	—	268	4	no	—
334	C₂²⋊C₄	—	268	4	no	—
335	C₂²⋊C₄	—	268	4	no	—
336	C₄²	—	268	1	no	—
337	C₄²	—	67	2	no	—
338	C₄²	—	67	2	no	—
339	C₄²	—	134	2	no	—
340	C₄²	—	134	1	no	—
341	C₄²	—	134	1	no	—
342	C₄²	—	134	1	no	—
343	C₄²	—	134	1	no	—
344	C₄²	—	134	1	no	—
345	C₄²	—	134	1	no	—
346	C₄²	D₄×D₆₇	1	1	yes	characteristic
347	C₄²	Q₈×D₆₇	1	1	yes	characteristic
348	C₄²	#11761628	1	1	yes	characteristic
349	C₄²	#11761628	1	1	yes	characteristic
350	C₂³	—	268	2	no	—
351	C₂³	—	268	2	no	—
352	C₂³	—	268	2	no	—
353	C₂³	—	268	2	no	—
354	C₂³	—	268	2	no	—
355	C₂³	—	268	2	no	—
356	C₂³	—	134	4	no	—
357	C₂³	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
358	D₄	—	268	4	no	—
359	D₄	—	268	4	no	—
360	D₄	—	268	2	no	—
361	D₄	—	268	2	no	—
362	D₄	—	268	2	no	—
363	D₄	—	268	2	no	—
364	D₄	—	268	2	no	—
365	D₄	—	268	2	no	—
366	D₄	—	268	2	no	—
367	D₄	—	268	2	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	4	2	no	—
371	C₂×C₄	—	4	2	no	—
372	C₂×C₄	—	4	2	no	—
373	C₂×C₄	—	536	8	no	—
374	C₂×C₄	—	67	2	no	—
375	C₂×C₄	—	67	2	no	—
376	C₂×C₄	—	67	2	no	—
377	C₂×C₄	—	134	1	no	—
378	C₂×C₄	—	134	1	no	—
379	C₂×C₄	—	134	1	no	—
380	C₂×C₄	—	134	1	no	—
381	C₂×C₄	—	134	1	no	—
382	C₂×C₄	—	134	1	no	—
383	C₂×C₄	—	134	1	no	—
384	C₂×C₄	—	134	1	no	—
385	C₂×C₄	—	134	1	no	—
386	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
387	C₂×C₄	#14524000	1	1	yes	characteristic
388	C₂×C₄	#14524016	1	1	yes	characteristic
389	C₂×C₄	#14524016	1	1	yes	characteristic
390	C₂×C₄	#14524021	1	1	yes	characteristic
391	C₂×C₄	#14524021	1	1	yes	characteristic
392	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
393	C₂×C₄	SD₁₆×D₆₇	1	1	yes	characteristic
394	C₂×C₄	#14524064	1	1	yes	characteristic
395	C₂×C₄	#14524066	1	1	yes	characteristic
396	C₂×C₄	#14524067	1	1	yes	characteristic
397	C₂×C₄	#14524070	1	1	yes	characteristic
398	C₂×C₄	#14524070	1	1	yes	characteristic
399	C₂×C₄	#14524075	1	1	yes	characteristic
400	C₈	—	8	1	no	—
401	C₈	—	8	1	no	—
402	C₈	—	536	1	no	—
403	C₈	—	536	1	no	—
404	C₂²	—	268	4	no	—
405	C₂²	—	268	4	no	—
406	C₂²	—	268	4	no	—
407	C₂²	—	536	2	no	—
408	C₂²	—	536	2	no	—
409	C₂²	—	536	2	no	—
410	C₂²	—	536	2	no	—
411	C₂²	#16863255	1	1	yes	characteristic
412	C₂²	#100×D₆₇	1	1	yes	characteristic
413	C₂²	#16863379	1	1	yes	characteristic
414	C₂²	#102×D₆₇	1	1	yes	characteristic
415	C₂²	#16863391	1	1	yes	characteristic
416	C₂²	#16863396	1	1	yes	characteristic
417	C₂²	#16863399	1	1	yes	characteristic
418	C₄	—	2	1	no	—
419	C₄	—	2	1	no	—
420	C₄	—	8	2	no	—
421	C₄	—	268	1	no	—
422	C₄	—	134	2	no	—
423	C₄	—	134	2	no	—
424	C₄	—	134	2	no	—
425	C₄	#16863590	1	1	yes	characteristic
426	C₄	#16863600	1	1	yes	characteristic
427	C₄	#16863610	1	1	yes	characteristic
428	C₄	#16863614	1	1	yes	characteristic
429	C₄	#16863675	1	1	yes	characteristic
430	C₄	#16863685	1	1	yes	characteristic
431	C₄	#16863695	1	1	yes	characteristic
432	C₄	#16863699	1	1	yes	characteristic
433	C₂	—	536	4	no	—
434	C₂	#22558857	1	1	yes	characteristic
435	C₂	#22558875	1	1	yes	characteristic
436	C₂	#22558903	1	1	yes	characteristic
437	C₂	#22558930	1	1	yes	characteristic
438	C₂	#22560200	1	1	yes	characteristic
439	C₂	#22560208	1	1	yes	characteristic
440	C₂	#22560218	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊C₄×D₆₇
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1296	C₂
socle	C₂³×C₆₇	C₂×C₂²⋊C₄
prime radical	C₂×#14524040	C₂²
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1296	D₆₇
hypocenter	C₆₇	#8588
LCS	G ▷ C₂²×C₄×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. #22560360	C₂	semidirect
3. #22560325	C₂	semidirect
6. #22558092	C₂	semidirect
7. C₆₇×#1296	C₂	semidirect
227. #1296	D₆₇	semidirect
228. C₆₇	#8588	semidirect

Maximal subgroups

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

subgroup	index
#22562771	2
#22560360	2
#22560325	2
C₂×#16863139	2
#22558092	2
C₆₇×#1296	2
#8588	67