Subgroups — #30752018

Order 17152.

590 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863597	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#22561015	C₂	1	1	yes	maximal, characteristic
4	#22560215	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16863139	C₂	1	1	yes	maximal, characteristic
6	#22558093	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1277	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11761617	—	2	1	no	—
9	C₂×#14524040	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
11	#16863597	—	2	2	no	—
12	#16863597	C₄	1	4	yes	complemented
13	C₈×Dic₁₃₄	—	2	2	no	—
14	C₈×Dic₁₃₄	—	2	2	no	—
15	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×Dic₆₇⋊C₈	—	2	1	no	—
17	C₂×#14523968	—	2	1	no	—
18	C₂×#14523965	—	2	1	no	—
19	C₂×#14523965	—	2	1	no	—
20	C₂×#14523965	—	2	1	no	—
21	C₂×C₈×Dic₆₇	—	2	1	no	—
22	#16863266	C₂²	1	1	yes	characteristic
23	#16863159	—	2	1	no	—
24	#16863139	—	2	2	no	—
25	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
28	C₂²×Dic₁₃₄	D₄	1	1	yes	characteristic
29	C₂²×D₂₆₈	—	2	1	no	—
30	C₂²×D₂₆₈	—	2	1	no	—
31	C₂²×D₂₆₈	—	2	1	no	—
32	C₂×#11761617	—	2	4	no	—
33	C₂×#11761617	—	4	2	no	—
34	C₂×#11761617	—	4	2	no	—
35	C₂²×C₆₇⋊C₈	—	2	1	no	—
36	#14524040	—	2	1	no	—
37	#14524040	—	2	1	no	—
38	#14524040	—	2	1	no	—
39	#14524040	C₂×C₄	1	2	yes	—
40	C₄×Dic₁₃₄	—	2	1	no	—
41	C₄×Dic₁₃₄	—	2	1	no	—
42	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
43	C₄×Dic₁₃₄	D₄	1	2	yes	—
44	C₂×C₄⋊Dic₆₇	—	2	1	no	—
45	C₂×C₄⋊Dic₆₇	—	2	1	no	—
46	C₂×C₄⋊Dic₆₇	—	2	1	no	—
47	C₂×C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
48	C₂×Dic₆₇⋊C₄	—	2	1	no	—
49	C₂×C₄×Dic₆₇	—	2	1	no	—
50	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
51	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
52	Dic₆₇⋊C₈	—	2	4	no	—
53	#14523968	—	4	2	no	—
54	#14523965	—	2	4	no	—
55	#14523965	—	4	2	no	—
56	#14523965	—	4	2	no	—
57	C₈×Dic₆₇	—	2	4	no	—
58	C₂²×C₈×C₆₇	—	2	1	no	—
59	C₂²×C₈×C₆₇	—	2	1	no	—
60	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
61	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
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	C₂×C₄	1	2	yes	—
67	C₄×C₈×C₆₇	—	2	1	no	—
68	C₄×C₈×C₆₇	—	2	1	no	—
69	C₂³×D₆₇	—	4	1	no	—
70	C₂×Dic₁₃₄	—	2	1	no	—
71	C₂×Dic₁₃₄	—	2	1	no	—
72	C₂×Dic₁₃₄	—	2	1	no	—
73	C₂×Dic₁₃₄	C₂²⋊C₄	1	2	yes	—
74	C₂×Dic₁₃₄	D₈	1	2	yes	—
75	C₂×Dic₁₃₄	SD₁₆	1	2	yes	—
76	C₂×D₂₆₈	—	2	2	no	—
77	C₂×D₂₆₈	—	2	2	no	—
78	C₂×D₂₆₈	—	2	2	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	—	4	1	no	—
84	C₂×D₂₆₈	—	4	1	no	—
85	C₂×D₂₆₈	—	4	1	no	—
86	C₂×D₂₆₈	—	4	1	no	—
87	C₂×D₂₆₈	—	4	1	no	—
88	C₂×D₂₆₈	—	4	1	no	—
89	C₂×D₂₆₈	—	4	1	no	—
90	C₂×D₂₆₈	—	8	1	no	—
91	C₂×D₂₆₈	—	8	1	no	—
92	C₂²×Dic₆₇	—	2	1	no	—
93	C₂²×Dic₆₇	—	2	1	no	—
94	C₂²×Dic₆₇	—	4	1	no	—
95	#11761617	—	4	8	no	—
96	C₂×C₆₇⋊C₈	—	2	2	no	—
97	C₂×C₆₇⋊C₈	—	4	1	no	—
98	C₂×C₆₇⋊C₈	—	4	1	no	—
99	C₄⋊Dic₆₇	—	2	1	no	—
100	C₄⋊Dic₆₇	—	4	1	no	—
101	C₄⋊Dic₆₇	—	4	1	no	—
102	C₄⋊Dic₆₇	—	4	1	no	—
103	C₄⋊Dic₆₇	—	4	1	no	—
104	C₄⋊Dic₆₇	—	8	1	no	—
105	C₄⋊Dic₆₇	C₂²⋊C₄	1	2	yes	—
106	Dic₆₇⋊C₄	—	2	2	no	—
107	Dic₆₇⋊C₄	—	2	2	no	—
108	C₄×Dic₆₇	—	2	2	no	—
109	C₄×Dic₆₇	—	2	2	no	—
110	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
113	C₂×C₈×C₆₇	—	2	1	no	—
114	C₂×C₈×C₆₇	—	2	1	no	—
115	C₂×C₈×C₆₇	—	2	2	no	—
116	C₂×C₈×C₆₇	—	2	1	no	—
117	C₂×C₈×C₆₇	—	2	1	no	—
118	C₂×C₈×C₆₇	—	2	1	no	—
119	C₂×C₈×C₆₇	—	4	1	no	—
120	C₂×C₈×C₆₇	—	4	1	no	—
121	C₂×C₈×C₆₇	—	4	1	no	—
122	C₂×C₈×C₆₇	—	4	1	no	—
123	C₂×C₈×C₆₇	—	4	1	no	—
124	C₂×C₈×C₆₇	SD₁₆	1	2	yes	complemented
125	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
126	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₆₇	#41	1	1	yes	characteristic
129	C₂³×C₆₇	#121	1	1	yes	characteristic
130	C₂²×D₆₇	—	4	2	no	—
131	C₂²×D₆₇	—	8	1	no	—
132	C₂²×D₆₇	—	8	1	no	—
133	C₂²×D₆₇	—	8	1	no	—
134	C₂²×D₆₇	—	8	1	no	—
135	C₂²×D₆₇	—	8	1	no	—
136	C₂²×D₆₇	—	8	1	no	—
137	C₂×C₄×C₆₇	—	2	1	no	—
138	C₂×C₄×C₆₇	—	2	1	no	—
139	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
143	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
144	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
147	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
148	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
149	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
150	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
151	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
153	C₂×Dic₆₇	—	2	2	no	—
154	C₂×Dic₆₇	—	2	2	no	—
155	C₂×Dic₆₇	—	2	2	no	—
156	C₂×Dic₆₇	—	2	2	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	—	4	2	no	—
163	D₂₆₈	—	4	2	no	—
164	D₂₆₈	—	4	2	no	—
165	D₂₆₈	—	4	2	no	—
166	D₂₆₈	—	8	2	no	—
167	D₂₆₈	—	8	2	no	—
168	D₂₆₈	—	8	1	no	—
169	D₂₆₈	—	8	1	no	—
170	D₂₆₈	—	8	1	no	—
171	D₂₆₈	—	8	1	no	—
172	Dic₁₃₄	—	2	2	no	—
173	Dic₁₃₄	—	2	2	no	—
174	Dic₁₃₄	—	2	2	no	—
175	Dic₁₃₄	#102	1	4	yes	complemented
176	C₈×C₆₇	—	2	2	no	—
177	C₈×C₆₇	—	4	2	no	—
178	C₈×C₆₇	—	4	1	no	—
179	C₈×C₆₇	—	8	1	no	—
180	C₆₇⋊C₈	—	4	2	no	—
181	C₂²×C₆₇	#386	1	1	yes	characteristic
182	C₂²×C₆₇	#460	1	1	yes	characteristic
183	C₂²×C₆₇	#464	1	1	yes	characteristic
184	C₂²×C₆₇	#465	1	1	yes	characteristic
185	C₂²×C₆₇	#466	1	1	yes	characteristic
186	C₂²×C₆₇	#471	1	1	yes	characteristic
187	C₂²×C₆₇	#472	1	1	yes	characteristic
188	C₂×D₆₇	—	8	2	no	—
189	C₂×D₆₇	—	8	2	no	—
190	C₂×D₆₇	—	8	2	no	—
191	C₂×D₆₇	—	16	1	no	—
192	C₂×D₆₇	—	16	1	no	—
193	C₂×D₆₇	—	16	1	no	—
194	C₂×D₆₇	—	16	1	no	—
195	C₄×C₆₇	—	2	1	no	—
196	C₄×C₆₇	—	2	1	no	—
197	C₄×C₆₇	C₂×#102	1	1	yes	characteristic
198	C₄×C₆₇	#416	1	1	yes	characteristic
199	C₄×C₆₇	C₄×SD₁₆	1	1	yes	characteristic
200	C₄×C₆₇	#440	1	1	yes	characteristic
201	C₄×C₆₇	#448	1	1	yes	characteristic
202	C₄×C₆₇	#449	1	1	yes	characteristic
203	C₄×C₆₇	#465	1	1	yes	characteristic
204	C₄×C₆₇	#466	1	1	yes	characteristic
205	Dic₆₇	—	2	4	no	—
206	Dic₆₇	—	4	2	no	—
207	Dic₆₇	—	16	1	no	—
208	#9222	—	67	1	no	maximal, Sylow
209	C₂×C₆₇	#1666	1	1	yes	characteristic
210	C₂×C₆₇	#1668	1	1	yes	characteristic
211	C₂×C₆₇	#1678	1	1	yes	characteristic
212	C₂×C₆₇	#1682	1	1	yes	characteristic
213	C₂×C₆₇	#1880	1	1	yes	characteristic
214	C₂×C₆₇	#1893	1	1	yes	characteristic
215	C₂×C₆₇	#1907	1	1	yes	characteristic
216	D₆₇	—	16	2	no	—
217	C₂×#460	—	67	1	no	—
218	C₂×C₈×Q₈	—	67	1	no	—
219	#1957	—	67	1	no	—
220	#1803	—	67	1	no	—
221	C₂×#331	—	67	1	no	—
222	#1296	—	67	1	no	—
223	#1277	D₆₇	1	1	yes	characteristic, complemented
224	C₆₇	#9222	1	1	yes	characteristic, complemented, Sylow
225	C₂²×SD₁₆	—	134	1	no	—
226	C₂×#127	—	67	1	no	—
227	C₂×C₄×Q₈	—	67	1	no	—
228	#460	—	67	4	no	—
229	#460	—	134	2	no	—
230	C₈×Q₈	—	134	2	no	—
231	C₈×Q₈	—	134	2	no	—
232	C₂×C₄⋊C₈	—	67	1	no	—
233	C₂×C₄⋊C₈	—	134	1	no	—
234	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
235	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
236	C₂×#103	—	134	1	no	—
237	C₂×#102	—	134	1	no	—
238	C₂×#102	—	134	1	no	—
239	C₂×#102	—	134	1	no	—
240	C₂×C₄×C₈	—	134	1	no	—
241	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
242	#384	—	67	1	no	—
243	#340	—	134	1	no	—
244	#331	—	134	2	no	—
245	C₂²×Q₈	—	67	1	no	—
246	C₂²×D₄	—	134	1	no	—
247	C₂²×D₄	—	134	1	no	—
248	C₂²×D₄	—	134	1	no	—
249	C₂×SD₁₆	—	268	2	no	—
250	C₂×SD₁₆	—	268	2	no	—
251	C₂×SD₁₆	—	134	4	no	—
252	C₂²×C₈	—	2	1	no	—
253	C₂²×C₈	—	2	1	no	—
254	C₂²×C₈	—	134	1	no	—
255	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
256	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
257	#127	—	67	2	no	—
258	#127	—	134	1	no	—
259	#127	—	134	1	no	—
260	#127	—	134	1	no	—
261	C₄×Q₈	—	67	2	no	—
262	C₄×Q₈	—	67	2	no	—
263	C₄×Q₈	—	134	1	no	—
264	C₄×Q₈	—	134	1	no	—
265	C₂×C₄⋊C₄	—	67	1	no	—
266	C₂×C₄⋊C₄	—	134	1	no	—
267	C₂×C₄⋊C₄	—	134	1	no	—
268	C₂×C₄⋊C₄	—	134	1	no	—
269	C₂×C₄⋊C₄	—	134	1	no	—
270	C₂×C₄²	—	134	1	no	—
271	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
272	C₄⋊C₈	—	2	1	no	—
273	C₄⋊C₈	—	2	1	no	—
274	C₄⋊C₈	—	2	1	no	—
275	C₄⋊C₈	—	67	2	no	—
276	C₄⋊C₈	—	134	4	no	—
277	C₄⋊C₈	—	134	1	no	—
278	C₄⋊C₈	C₄×D₆₇	1	2	yes	—
279	#103	—	268	2	no	—
280	#102	—	268	2	no	—
281	#102	—	268	2	no	—
282	#102	—	134	4	no	—
283	C₄×C₈	—	2	1	no	—
284	C₄×C₈	—	2	1	no	—
285	C₄×C₈	—	134	4	no	—
286	C₂⁴	—	268	1	no	—
287	C₂×Q₈	—	67	2	no	—
288	C₂×Q₈	—	67	2	no	—
289	C₂×Q₈	—	67	2	no	—
290	C₂×Q₈	—	134	1	no	—
291	C₂×Q₈	—	134	1	no	—
292	C₂×Q₈	—	134	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	—	268	1	no	—
300	C₂×D₄	—	268	1	no	—
301	C₂×D₄	—	268	1	no	—
302	C₂×D₄	—	268	1	no	—
303	C₂×D₄	—	268	1	no	—
304	C₂×D₄	—	536	1	no	—
305	C₂×D₄	—	536	1	no	—
306	C₂×D₄	—	134	2	no	—
307	C₂×D₄	—	134	2	no	—
308	C₂×D₄	—	134	2	no	—
309	C₂²×C₄	—	268	1	no	—
310	C₂²×C₄	—	134	1	no	—
311	C₂²×C₄	—	134	1	no	—
312	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
313	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
314	C₂²×C₄	#11761629	1	1	yes	characteristic
315	SD₁₆	—	268	8	no	—
316	C₂×C₈	—	2	1	no	—
317	C₂×C₈	—	2	1	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	—	2	1	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	2	1	no	—
322	C₂×C₈	—	4	1	no	—
323	C₂×C₈	—	4	1	no	—
324	C₂×C₈	—	4	1	no	—
325	C₂×C₈	—	4	1	no	—
326	C₂×C₈	—	4	1	no	—
327	C₂×C₈	—	268	1	no	—
328	C₂×C₈	—	268	1	no	—
329	C₂×C₈	—	134	2	no	—
330	C₂×C₈	#11761617	1	2	yes	complemented
331	C₄⋊C₄	—	268	1	no	—
332	C₄⋊C₄	—	268	1	no	—
333	C₄⋊C₄	—	268	1	no	—
334	C₄⋊C₄	—	268	1	no	—
335	C₄⋊C₄	—	536	1	no	—
336	C₄⋊C₄	—	67	2	no	—
337	C₄⋊C₄	—	134	2	no	—
338	C₄⋊C₄	—	134	1	no	—
339	C₄⋊C₄	—	134	2	no	—
340	C₄²	—	134	2	no	—
341	C₄²	—	134	2	no	—
342	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
343	C₄²	D₄×D₆₇	1	1	yes	characteristic
344	C₄²	D₄×D₆₇	1	1	yes	characteristic
345	C₄²	#11761628	1	1	yes	characteristic
346	C₂³	—	268	2	no	—
347	C₂³	—	536	1	no	—
348	C₂³	—	536	1	no	—
349	C₂³	—	536	1	no	—
350	C₂³	—	536	1	no	—
351	C₂³	—	536	1	no	—
352	C₂³	—	536	1	no	—
353	C₂³	#14524018	1	1	yes	characteristic
354	Q₈	—	67	4	no	—
355	Q₈	—	134	2	no	—
356	Q₈	—	134	2	no	—
357	Q₈	—	134	2	no	—
358	D₄	—	268	2	no	—
359	D₄	—	268	2	no	—
360	D₄	—	268	2	no	—
361	D₄	—	268	2	no	—
362	D₄	—	536	2	no	—
363	D₄	—	536	2	no	—
364	D₄	—	536	1	no	—
365	D₄	—	536	1	no	—
366	D₄	—	536	1	no	—
367	D₄	—	536	1	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	268	1	no	—
371	C₂×C₄	—	268	1	no	—
372	C₂×C₄	—	536	1	no	—
373	C₂×C₄	—	536	1	no	—
374	C₂×C₄	—	536	1	no	—
375	C₂×C₄	—	134	2	no	—
376	C₂×C₄	—	134	2	no	—
377	C₂×C₄	—	134	2	no	—
378	C₂×C₄	—	134	2	no	—
379	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
380	C₂×C₄	#14524005	1	1	yes	characteristic
381	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
382	C₂×C₄	#14524013	1	1	yes	characteristic
383	C₂×C₄	#14524016	1	1	yes	characteristic
384	C₂×C₄	#14524028	1	1	yes	characteristic
385	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
386	C₂×C₄	SD₁₆×D₆₇	1	1	yes	characteristic
387	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
388	C₂×C₄	#14524064	1	1	yes	characteristic
389	C₂×C₄	#14524065	1	1	yes	characteristic
390	C₂×C₄	#14524066	1	1	yes	characteristic
391	C₂×C₄	#14524071	1	1	yes	characteristic
392	C₂×C₄	#14524076	1	1	yes	characteristic
393	C₈	—	2	2	no	—
394	C₈	—	4	1	no	—
395	C₈	—	4	2	no	—
396	C₈	—	8	1	no	—
397	C₈	—	268	2	no	—
398	C₂²	—	536	2	no	—
399	C₂²	—	536	2	no	—
400	C₂²	—	536	2	no	—
401	C₂²	—	1072	1	no	—
402	C₂²	—	1072	1	no	—
403	C₂²	—	1072	1	no	—
404	C₂²	—	1072	1	no	—
405	C₂²	#16863274	1	1	yes	characteristic
406	C₂²	#16863597	1	1	yes	characteristic
407	C₂²	#16863619	1	1	yes	characteristic
408	C₂²	#16863626	1	1	yes	characteristic
409	C₂²	#16863633	1	1	yes	characteristic
410	C₂²	#16863660	1	1	yes	characteristic
411	C₂²	#16863667	1	1	yes	characteristic
412	C₄	—	2	1	no	—
413	C₄	—	2	1	no	—
414	C₄	—	268	2	no	—
415	C₄	—	1072	1	no	—
416	C₄	—	134	4	no	—
417	C₄	#102×D₆₇	1	1	yes	characteristic
418	C₄	#16863391	1	1	yes	characteristic
419	C₄	C₄×#11761617	1	1	yes	characteristic
420	C₄	#16863498	1	1	yes	characteristic
421	C₄	#16863537	1	1	yes	characteristic
422	C₄	#16863543	1	1	yes	characteristic
423	C₄	#16863628	1	1	yes	characteristic
424	C₄	#16863631	1	1	yes	characteristic
425	C₂	—	1072	2	no	—
426	C₂	#22559558	1	1	yes	characteristic
427	C₂	#22559570	1	1	yes	characteristic
428	C₂	#22559626	1	1	yes	characteristic
429	C₂	#22559654	1	1	yes	characteristic
430	C₂	#22560563	1	1	yes	characteristic
431	C₂	#22560636	1	1	yes	characteristic
432	C₂	#22560716	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524018
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1277	C₂
socle	C₂³×C₆₇	#121
prime radical	C₂×#14524040	C₂²
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1277	D₆₇
hypocenter	C₆₇	#9222
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1277

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×Dic₁₃₄	C₂	semidirect
4. #22560215	C₂	semidirect
6. #22558093	C₂	semidirect
7. C₆₇×#1277	C₂	semidirect
12. #16863597	C₄	semidirect
124. C₂×C₈×C₆₇	SD₁₆	semidirect
175. Dic₁₃₄	#102	semidirect
223. #1277	D₆₇	semidirect
224. C₆₇	#9222	semidirect
330. C₂×C₈	#11761617	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863597	2
C₂×C₈×Dic₁₃₄	2
#22561015	2
#22560215	2
C₂×#16863139	2
#22558093	2
C₆₇×#1277	2
#9222	67