Subgroups — C₄×#16863831 #31009147

Order 17152.

738 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863831	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₅₃₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#22562083	C₂	1	2	yes	maximal, complemented
4	#22562071	C₂	1	2	yes	maximal, complemented
5	C₄×#14523986	C₂	1	2	yes	maximal, complemented
6	C₄×#14523984	C₂	1	2	yes	maximal, complemented
7	#22561998	C₂	1	1	yes	maximal, characteristic, complemented
8	#22561924	C₂	1	1	yes	maximal, characteristic, complemented
9	#22561921	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#14523983	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#135×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₅₃₆	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₂₆₈	—	2	1	no	—
14	#16863831	C₄	1	8	yes	direct product factor
15	C₂×#14523986	C₂²	1	2	yes	—
16	C₂×#14523984	C₂²	1	2	yes	—
17	C₂×#14523983	C₂²	1	1	yes	characteristic
18	C₂×#14523983	C₂²	1	1	yes	characteristic
19	C₄×D₅₃₆	—	2	1	no	—
20	C₄×D₅₃₆	—	2	1	no	—
21	C₄×D₅₃₆	C₂²	1	2	yes	—
22	C₄×D₅₃₆	C₂²	1	2	yes	—
23	C₂×#14523977	C₂²	1	1	yes	characteristic
24	C₂×#14523965	—	2	1	no	—
25	#16863223	C₂²	1	2	yes	complemented
26	#16863221	C₂²	1	2	yes	complemented
27	#16863200	C₂²	1	2	yes	complemented
28	#16863197	C₂²	1	2	yes	complemented
29	#16863196	C₂²	1	2	yes	—
30	#16863196	C₂²	1	2	yes	—
31	#16863172	C₂²	1	2	yes	complemented
32	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	complemented
33	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
34	C₄×#41×C₆₇	—	2	1	no	—
35	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
36	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
37	C₄×SD₁₆×C₆₇	—	2	1	no	—
38	C₄×SD₁₆×C₆₇	—	2	1	no	—
39	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
40	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
41	#427×C₆₇	C₂²	1	1	yes	characteristic
42	#416×C₆₇	—	2	1	no	—
43	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
44	C₂²×D₂₆₈	—	2	1	no	—
45	C₂²×C₄×D₆₇	—	4	1	no	—
46	C₂×D₅₃₆	—	2	1	no	—
47	C₂×D₅₃₆	—	2	2	no	—
48	C₂×D₅₃₆	C₂×C₄	1	4	yes	complemented
49	C₂×D₅₃₆	C₂³	1	2	yes	—
50	C₄×D₂₆₈	—	2	2	no	—
51	C₄×D₂₆₈	—	2	2	no	—
52	C₄×D₂₆₈	—	4	1	no	—
53	C₄×D₂₆₈	—	4	1	no	—
54	C₂×C₄⋊Dic₆₇	—	2	1	no	—
55	C₂×(C₂×D₆₇)⋊C₄	—	4	1	no	—
56	#14523986	C₂×C₄	1	8	yes	complemented
57	#14523984	C₂×C₄	1	8	yes	complemented
58	#14523983	—	2	2	no	—
59	#14523983	C₂×C₄	1	4	yes	complemented
60	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
61	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
62	#14523977	—	2	1	no	—
63	#14523977	C₂³	1	2	yes	—
64	#14523965	—	2	2	no	—
65	#14523965	—	2	2	no	—
66	C₂×#41×C₆₇	—	2	1	no	—
67	#135×C₆₇	—	2	2	no	—
68	#135×C₆₇	C₂×C₄	1	4	yes	complemented
69	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
70	C₂×SD₁₆×C₆₇	—	2	1	no	—
71	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
73	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
74	C₄×Q₈×C₆₇	—	2	1	no	—
75	C₄×Q₈×C₆₇	—	2	1	no	—
76	C₄×D₄×C₆₇	—	2	1	no	—
77	C₄×D₄×C₆₇	—	2	1	no	—
78	C₄×D₄×C₆₇	—	4	1	no	—
79	C₄×D₄×C₆₇	—	4	1	no	—
80	#117×C₆₇	—	2	1	no	—
81	#117×C₆₇	—	4	1	no	—
82	C₂×C₄²×C₆₇	—	4	1	no	—
83	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
84	#107×C₆₇	C₂³	1	1	yes	characteristic
85	#107×C₆₇	C₂³	1	1	yes	characteristic
86	#106×C₆₇	—	2	1	no	—
87	#103×C₆₇	—	2	1	no	—
88	#103×C₆₇	—	2	1	no	—
89	#102×C₆₇	—	2	1	no	—
90	#102×C₆₇	—	2	1	no	—
91	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
92	C₄×C₈×C₆₇	D₄	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	—	4	1	no	—
96	C₂×D₂₆₈	—	2	4	no	—
97	C₂×D₂₆₈	—	2	2	no	—
98	C₂×D₂₆₈	—	4	1	no	—
99	C₂×D₂₆₈	—	4	2	no	—
100	C₂²×Dic₆₇	—	4	1	no	—
101	C₂×C₄×D₆₇	—	4	2	no	—
102	C₂×C₄×D₆₇	—	4	2	no	—
103	C₂×C₄×D₆₇	—	8	4	no	—
104	D₅₃₆	—	2	4	no	—
105	D₅₃₆	—	2	2	no	—
106	D₅₃₆	C₂²×C₄	1	4	yes	—
107	C₄⋊Dic₆₇	—	2	2	no	—
108	C₄⋊Dic₆₇	—	4	1	no	—
109	(C₂×D₆₇)⋊C₄	—	4	2	no	—
110	(C₂×D₆₇)⋊C₄	—	4	2	no	—
111	C₆₇⋊C₁₆	—	2	2	no	—
112	C₆₇⋊C₁₆	C₂²×C₄	1	4	yes	complemented
113	#41×C₆₇	—	2	4	no	—
114	#41×C₆₇	—	4	2	no	—
115	C₂×Q₈×C₆₇	—	2	1	no	—
116	C₂×D₄×C₆₇	—	2	1	no	—
117	C₂×D₄×C₆₇	—	4	1	no	—
118	C₂²×C₄×C₆₇	—	4	1	no	—
119	C₂²×C₄×C₆₇	—	4	1	no	—
120	C₂²×C₄×C₆₇	—	4	1	no	—
121	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
122	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
123	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
124	Q₁₆×C₆₇	—	2	1	no	—
125	Q₁₆×C₆₇	C₂²×C₄	1	2	yes	complemented
126	SD₁₆×C₆₇	—	2	4	no	—
127	D₈×C₆₇	—	2	1	no	—
128	D₈×C₆₇	C₂²×C₄	1	2	yes	complemented
129	C₂×C₈×C₆₇	—	2	2	no	—
130	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
131	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
132	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
133	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
134	C₂×C₈×C₆₇	#41	1	2	yes	—
135	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
136	C₄⋊C₄×C₆₇	—	2	1	no	—
137	C₄⋊C₄×C₆₇	—	2	1	no	—
138	C₄⋊C₄×C₆₇	—	4	1	no	—
139	C₄⋊C₄×C₆₇	—	4	1	no	—
140	C₂²⋊C₄×C₆₇	—	4	1	no	—
141	C₂²⋊C₄×C₆₇	—	4	1	no	—
142	C₂²⋊C₄×C₆₇	—	4	1	no	—
143	C₄²×C₆₇	—	4	1	no	—
144	C₄²×C₆₇	—	4	1	no	—
145	C₄²×C₆₇	—	4	1	no	—
146	C₄²×C₆₇	D₈	1	1	yes	characteristic
147	C₄²×C₆₇	D₈	1	1	yes	characteristic
148	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
149	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
150	C₂³×C₆₇	—	4	1	no	—
151	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
152	C₂²×D₆₇	—	4	2	no	—
153	C₂²×D₆₇	—	4	4	no	—
154	C₂²×D₆₇	—	8	2	no	—
155	C₂²×D₆₇	—	8	2	no	—
156	Q₈×C₆₇	—	2	2	no	—
157	Q₈×C₆₇	—	4	1	no	—
158	D₄×C₆₇	—	2	2	no	—
159	D₄×C₆₇	—	4	4	no	—
160	D₄×C₆₇	—	4	1	no	—
161	C₂×C₄×C₆₇	—	2	2	no	—
162	C₂×C₄×C₆₇	—	4	1	no	—
163	C₂×C₄×C₆₇	—	4	1	no	—
164	C₂×C₄×C₆₇	—	4	4	no	—
165	C₂×C₄×C₆₇	—	4	1	no	—
166	C₂×C₄×C₆₇	—	4	1	no	—
167	C₂×C₄×C₆₇	—	8	2	no	—
168	C₂×C₄×C₆₇	—	8	2	no	—
169	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
170	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
171	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
172	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
173	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
174	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
175	C₂×C₄×C₆₇	#135	1	2	yes	—
176	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
177	C₂×Dic₆₇	—	4	2	no	—
178	C₂×Dic₆₇	—	8	2	no	—
179	D₂₆₈	—	2	4	no	—
180	D₂₆₈	—	4	2	no	—
181	D₂₆₈	—	4	4	no	—
182	C₄×D₆₇	—	8	4	no	—
183	C₄×D₆₇	—	8	4	no	—
184	C₈×C₆₇	—	2	1	no	—
185	C₈×C₆₇	—	2	1	no	—
186	C₈×C₆₇	C₄×D₄	1	2	yes	—
187	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
188	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
189	C₂²×C₆₇	—	2	1	no	—
190	C₂²×C₆₇	—	2	1	no	—
191	C₂²×C₆₇	—	4	2	no	—
192	C₂²×C₆₇	—	8	1	no	—
193	C₂²×C₆₇	—	8	1	no	—
194	C₂²×C₆₇	C₄×D₈	1	2	yes	—
195	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
196	C₂×D₆₇	—	4	4	no	—
197	C₂×D₆₇	—	8	8	no	—
198	C₂×D₆₇	—	8	2	no	—
199	C₂×D₆₇	—	8	2	no	—
200	C₄×C₆₇	—	2	1	no	—
201	C₄×C₆₇	—	2	1	no	—
202	C₄×C₆₇	—	2	1	no	—
203	C₄×C₆₇	—	4	2	no	—
204	C₄×C₆₇	—	8	1	no	—
205	C₄×C₆₇	—	8	1	no	—
206	C₄×C₆₇	C₄×D₈	1	2	yes	—
207	C₄×C₆₇	#509	1	2	yes	complemented
208	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
209	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
210	Dic₆₇	—	8	2	no	—
211	C₄×#509	—	67	1	no	maximal, Sylow
212	C₂×C₆₇	—	2	2	no	—
213	C₂×C₆₇	—	8	2	no	—
214	C₂×C₆₇	C₂×C₄×D₈	1	1	yes	characteristic
215	C₂×C₆₇	C₂×#509	1	1	yes	characteristic
216	C₂×C₆₇	#3414	1	1	yes	characteristic
217	D₆₇	—	8	4	no	—
218	C₂×#509	—	67	1	no	—
219	C₄×#135	D₆₇	1	1	yes	characteristic, complemented
220	C₂×C₄×D₈	—	67	1	no	—
221	#2177	—	67	2	no	—
222	#2175	—	67	2	no	—
223	C₄×SD₃₂	—	67	2	no	—
224	C₄×D₁₆	—	67	2	no	—
225	#2159	—	67	1	no	—
226	#2140	—	67	1	no	—
227	#2139	—	67	1	no	—
228	C₄×M₃₂	—	67	1	no	—
229	C₆₇	C₄×#509	1	1	yes	characteristic, complemented, Sylow
230	C₂×#135	C₂×D₆₇	1	1	yes	characteristic
231	C₂²×D₈	—	67	1	no	—
232	C₄×#41	—	2	1	no	—
233	C₂×C₄×D₄	—	134	1	no	—
234	#509	—	67	8	no	—
235	C₂×SD₃₂	—	67	2	no	—
236	C₂×D₁₆	—	67	2	no	—
237	C₂×M₃₂	—	67	1	no	—
238	C₂×M₃₂	—	67	1	no	—
239	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic, complemented
240	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic, complemented
241	C₄×SD₁₆	—	2	1	no	—
242	C₄×SD₁₆	—	2	1	no	—
243	C₄×D₈	—	67	2	no	—
244	C₄×D₈	—	67	2	no	—
245	C₄×D₈	—	134	1	no	—
246	C₄×D₈	—	134	1	no	—
247	C₄×D₈	C₂×D₆₇	1	1	yes	characteristic, complemented
248	C₄×D₈	C₂×D₆₇	1	1	yes	characteristic, complemented
249	#427	C₂×D₆₇	1	1	yes	characteristic
250	C₂×#107	—	67	1	no	—
251	#416	—	2	1	no	—
252	C₂×#102	—	134	1	no	—
253	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
254	#367	—	67	2	no	—
255	#366	—	67	2	no	—
256	#358	—	67	2	no	—
257	#357	—	67	2	no	—
258	#357	—	67	2	no	—
259	#357	—	67	2	no	—
260	#346	—	67	2	no	—
261	C₄×C₁₆	—	67	2	no	—
262	C₂×#41	—	2	1	no	—
263	C₂²×D₄	—	134	1	no	—
264	C₂³×C₄	—	268	1	no	—
265	#135	—	2	2	no	—
266	#135	C₄×D₆₇	1	4	yes	complemented
267	C₂×Q₁₆	C₂²×D₆₇	1	1	yes	characteristic
268	C₂×SD₁₆	—	2	1	no	—
269	C₂×D₈	—	67	4	no	—
270	C₂×D₈	—	67	2	no	—
271	C₂×D₈	—	134	1	no	—
272	C₂×D₈	—	134	2	no	—
273	C₂×D₈	C₂²×D₆₇	1	1	yes	characteristic
274	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
275	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
276	C₄×Q₈	—	2	1	no	—
277	C₄×Q₈	—	2	1	no	—
278	C₄×D₄	—	2	1	no	—
279	C₄×D₄	—	2	1	no	—
280	C₄×D₄	—	4	1	no	—
281	C₄×D₄	—	4	1	no	—
282	C₄×D₄	—	268	1	no	—
283	C₄×D₄	—	268	1	no	—
284	C₄×D₄	—	134	2	no	—
285	C₄×D₄	—	134	2	no	—
286	#117	—	2	1	no	—
287	#117	—	4	1	no	—
288	C₂×C₄⋊C₄	—	134	1	no	—
289	C₂×C₂²⋊C₄	—	268	1	no	—
290	C₂×C₄²	—	4	1	no	—
291	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
292	SD₃₂	—	67	8	no	—
293	D₁₆	—	67	8	no	—
294	M₃₂	—	67	4	no	—
295	M₃₂	—	134	2	no	—
296	C₂×C₁₆	—	67	2	no	—
297	C₂×C₁₆	—	67	2	no	—
298	#107	—	67	2	no	—
299	#107	—	134	1	no	—
300	#107	C₂²×D₆₇	1	1	yes	characteristic
301	#107	C₂²×D₆₇	1	1	yes	characteristic
302	#106	—	2	1	no	—
303	#103	—	2	1	no	—
304	#103	—	2	1	no	—
305	#102	—	2	1	no	—
306	#102	—	2	1	no	—
307	#102	—	134	2	no	—
308	#102	—	134	2	no	—
309	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
310	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
311	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
312	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
313	C₂⁴	—	268	1	no	—
314	#41	—	2	4	no	—
315	#41	—	4	2	no	—
316	C₂×Q₈	—	2	1	no	—
317	C₂×D₄	—	2	1	no	—
318	C₂×D₄	—	4	1	no	—
319	C₂×D₄	—	268	1	no	—
320	C₂×D₄	—	268	2	no	—
321	C₂×D₄	—	134	4	no	—
322	C₂×D₄	—	134	2	no	—
323	C₂²×C₄	—	4	1	no	—
324	C₂²×C₄	—	4	1	no	—
325	C₂²×C₄	—	4	1	no	—
326	C₂²×C₄	—	268	1	no	—
327	C₂²×C₄	—	268	2	no	—
328	C₂²×C₄	—	268	2	no	—
329	C₂²×C₄	—	536	4	no	—
330	C₂²×C₄	#11761613	1	1	yes	characteristic
331	C₂²×C₄	#11761613	1	1	yes	characteristic
332	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
333	Q₁₆	—	2	1	no	—
334	Q₁₆	C₂×C₄×D₆₇	1	2	yes	complemented
335	SD₁₆	—	2	4	no	—
336	D₈	—	2	1	no	—
337	D₈	—	67	4	no	—
338	D₈	—	134	4	no	—
339	D₈	—	134	2	no	—
340	D₈	C₂×C₄×D₆₇	1	2	yes	complemented
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
343	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
344	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
345	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
346	C₂×C₈	#11761628	1	2	yes	—
347	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
348	C₄⋊C₄	—	2	1	no	—
349	C₄⋊C₄	—	2	1	no	—
350	C₄⋊C₄	—	4	1	no	—
351	C₄⋊C₄	—	4	1	no	—
352	C₄⋊C₄	—	268	1	no	—
353	C₄⋊C₄	—	134	2	no	—
354	C₂²⋊C₄	—	4	1	no	—
355	C₂²⋊C₄	—	4	1	no	—
356	C₂²⋊C₄	—	4	1	no	—
357	C₂²⋊C₄	—	268	2	no	—
358	C₂²⋊C₄	—	268	2	no	—
359	C₄²	—	4	1	no	—
360	C₄²	—	4	1	no	—
361	C₄²	—	4	1	no	—
362	C₄²	#11761613	1	1	yes	characteristic
363	C₄²	#11761613	1	1	yes	characteristic
364	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
365	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
366	C₁₆	—	67	4	no	—
367	C₁₆	—	134	2	no	—
368	C₂³	—	4	1	no	—
369	C₂³	—	268	2	no	—
370	C₂³	—	268	4	no	—
371	C₂³	—	536	2	no	—
372	C₂³	—	536	2	no	—
373	C₂³	C₂×#11761613	1	1	yes	characteristic
374	Q₈	—	2	2	no	—
375	Q₈	—	4	1	no	—
376	D₄	—	2	2	no	—
377	D₄	—	4	4	no	—
378	D₄	—	4	1	no	—
379	D₄	—	268	2	no	—
380	D₄	—	268	4	no	—
381	D₄	—	134	4	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	4	1	no	—
384	C₂×C₄	—	4	1	no	—
385	C₂×C₄	—	4	4	no	—
386	C₂×C₄	—	4	1	no	—
387	C₂×C₄	—	4	1	no	—
388	C₂×C₄	—	8	2	no	—
389	C₂×C₄	—	8	2	no	—
390	C₂×C₄	—	268	2	no	—
391	C₂×C₄	—	536	4	no	—
392	C₂×C₄	—	536	2	no	—
393	C₂×C₄	—	536	4	no	—
394	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
395	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
396	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
397	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
398	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
399	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
400	C₂×C₄	#14524063	1	2	yes	—
401	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
402	C₈	—	2	1	no	—
403	C₈	—	2	1	no	—
404	C₈	C₄×C₂²⋊D₆₇	1	2	yes	—
405	C₈	C₂²×C₄×D₆₇	1	1	yes	characteristic
406	C₈	C₂×#11761628	1	1	yes	characteristic
407	C₂²	—	2	1	no	—
408	C₂²	—	2	1	no	—
409	C₂²	—	4	2	no	—
410	C₂²	—	8	1	no	—
411	C₂²	—	8	1	no	—
412	C₂²	—	268	4	no	—
413	C₂²	—	536	8	no	—
414	C₂²	—	536	2	no	—
415	C₂²	—	536	2	no	—
416	C₂²	C₄×#11761613	1	2	yes	—
417	C₂²	C₂²×#11761613	1	1	yes	characteristic
418	C₄	—	2	1	no	—
419	C₄	—	2	1	no	—
420	C₄	—	2	1	no	—
421	C₄	—	4	2	no	—
422	C₄	—	8	1	no	—
423	C₄	—	8	1	no	—
424	C₄	—	536	2	no	—
425	C₄	C₄×#11761613	1	2	yes	—
426	C₄	#16863831	1	2	yes	direct product factor
427	C₄	C₂×C₄×C₂²⋊D₆₇	1	1	yes	characteristic
428	C₄	C₂×#14524063	1	1	yes	characteristic
429	C₂	—	2	2	no	—
430	C₂	—	8	2	no	—
431	C₂	—	536	4	no	—
432	C₂	C₂×C₄×#11761613	1	1	yes	characteristic
433	C₂	C₂×#16863831	1	1	yes	characteristic
434	C₂	#22573343	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11761613
commutator	C₈×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₄×#135×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂×#16863831	C₂
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₄×#135	D₆₇
hypocenter	C₆₇	C₄×#509
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₅₃₆	C₂	semidirect
3. #22562083	C₂	semidirect
4. #22562071	C₂	semidirect
5. C₄×#14523986	C₂	semidirect
6. C₄×#14523984	C₂	semidirect
7. #22561998	C₂	semidirect
8. #22561924	C₂	semidirect
9. #22561921	C₂	semidirect
10. C₄×#14523983	C₂	semidirect
11. C₄×#135×C₆₇	C₂	semidirect
14. #16863831	C₄	direct (N × H)
25. #16863223	C₂²	semidirect
26. #16863221	C₂²	semidirect
27. #16863200	C₂²	semidirect
28. #16863197	C₂²	semidirect
31. #16863172	C₂²	semidirect
32. C₄×C₆₇⋊C₁₆	C₂²	semidirect
35. C₄×Q₁₆×C₆₇	C₂²	semidirect
36. C₄×Q₁₆×C₆₇	C₂²	semidirect
39. C₄×D₈×C₆₇	C₂²	semidirect
40. C₄×D₈×C₆₇	C₂²	semidirect
48. C₂×D₅₃₆	C₂×C₄	semidirect
56. #14523986	C₂×C₄	semidirect
57. #14523984	C₂×C₄	semidirect
59. #14523983	C₂×C₄	semidirect
68. #135×C₆₇	C₂×C₄	semidirect
112. C₆₇⋊C₁₆	C₂²×C₄	semidirect
125. Q₁₆×C₆₇	C₂²×C₄	semidirect
128. D₈×C₆₇	C₂²×C₄	semidirect
207. C₄×C₆₇	#509	semidirect
219. C₄×#135	D₆₇	semidirect
229. C₆₇	C₄×#509	semidirect
239. C₄×Q₁₆	C₂×D₆₇	semidirect
240. C₄×Q₁₆	C₂×D₆₇	semidirect
247. C₄×D₈	C₂×D₆₇	semidirect
248. C₄×D₈	C₂×D₆₇	semidirect
266. #135	C₄×D₆₇	semidirect
334. Q₁₆	C₂×C₄×D₆₇	semidirect
340. D₈	C₂×C₄×D₆₇	semidirect
426. C₄	#16863831	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16863831	2
C₂×C₄×D₅₃₆	2
#22562083	2
#22562071	2
C₄×#14523986	2
C₄×#14523984	2
#22561998	2
#22561924	2
#22561921	2
C₄×#14523983	2
C₄×#135×C₆₇	2
C₄×#509	67

Subgroups — C4×#16863831 #31009147

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16863831 #31009147