Subgroups — #30826835

Order 17152.

662 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570240	C₂	1	1	yes	maximal, characteristic
2	Q₈×#11761612	C₂	1	1	yes	maximal, characteristic, complemented
3	#22568952	C₂	1	1	yes	maximal, characteristic
4	#22559114	C₂	1	2	yes	maximal, complemented
5	#22559084	C₂	1	2	yes	maximal, complemented
6	#22559069	C₂	1	2	yes	maximal, complemented
7	#22559024	C₂	1	2	yes	maximal, complemented
8	#22558983	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558661	C₂	1	1	yes	maximal, characteristic
10	#22558498	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×Q₈⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524049	—	2	1	no	—
13	#16863946	C₂²	1	1	yes	characteristic
14	#16863714	C₄	1	2	yes	—
15	#16863681	C₄	1	1	yes	characteristic
16	#16863681	C₄	1	1	yes	characteristic
17	#16863596	C₄	1	1	yes	characteristic
18	#16863596	C₄	1	1	yes	characteristic
19	#16863535	C₄	1	2	yes	—
20	#16863527	—	2	2	no	—
21	#16863527	C₂²	1	2	yes	—
22	Q₈×C₆₇⋊C₈	C₂²	1	2	yes	—
23	C₄×#11761615	C₂²	1	2	yes	—
24	C₄×#11761615	C₂²	1	2	yes	—
25	#16863473	C₂²	1	2	yes	—
26	#16863466	—	2	2	no	—
27	#16863461	C₂²	1	2	yes	—
28	#16863449	C₂²	1	1	yes	characteristic
29	#16863420	—	2	1	no	—
30	#16863420	C₂²	1	1	yes	characteristic
31	#16863362	—	2	1	no	—
32	C₄×#11761612	—	2	1	no	—
33	C₄×#11761612	C₂²	1	1	yes	characteristic
34	#16863148	C₂²	1	2	yes	—
35	#16863148	C₂²	1	2	yes	—
36	#16863125	C₂²	1	2	yes	—
37	#16863125	C₂²	1	2	yes	—
38	#16863118	C₂²	1	2	yes	—
39	#16863117	C₂²	1	2	yes	—
40	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
42	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
43	Q₈⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
44	Q₈⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
45	Q₈⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
46	Q₈⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
47	C₂×#11761628	—	2	1	no	—
48	C₂×#11761615	C₂×C₄	1	2	yes	—
49	C₂×#11761615	C₂×C₄	1	2	yes	—
50	#14524049	—	4	4	no	—
51	C₂×#11761612	—	2	1	no	—
52	C₂×#11761612	D₄	1	1	yes	characteristic
53	C₂×#11761612	D₄	1	1	yes	characteristic
54	C₂×#11761611	—	2	2	no	—
55	C₂×C₈×D₆₇	—	2	2	no	—
56	#14524043	C₂×C₄	1	1	yes	characteristic
57	#14524040	C₂×C₄	1	1	yes	characteristic
58	#14524031	—	2	1	no	—
59	#14524017	—	2	1	no	—
60	#14524017	C₂×C₄	1	2	yes	—
61	C₄×D₂₆₈	—	2	1	no	—
62	C₄×D₂₆₈	C₂³	1	2	yes	—
63	#14523974	C₂×C₄	1	2	yes	—
64	#14523974	C₂×C₄	1	2	yes	—
65	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
66	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
67	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
68	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
69	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
70	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
71	#14523967	C₂×C₄	1	2	yes	—
72	#14523967	C₂×C₄	1	2	yes	—
73	#14523963	C₂×C₄	1	2	yes	—
74	#14523963	C₂×C₄	1	2	yes	—
75	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
76	(C₂×D₆₇)⋊C₈	—	2	2	no	—
77	(C₂×D₆₇)⋊C₈	—	2	2	no	—
78	#14523954	—	2	2	no	—
79	#14523954	D₄	1	2	yes	—
80	C₄×C₆₇⋊C₈	—	2	2	no	—
81	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
82	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
83	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
84	C₂²×C₈×C₆₇	—	2	1	no	—
85	C₂²×C₈×C₆₇	—	2	1	no	—
86	C₄×Q₈×C₆₇	—	2	1	no	—
87	C₄×Q₈×C₆₇	—	2	1	no	—
88	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
89	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
90	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
91	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
92	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
93	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
94	C₂×C₄²×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	1	yes	characteristic
98	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
99	C₄×C₈×C₆₇	—	2	1	no	—
100	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
101	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
102	#11761628	—	4	2	no	—
103	#11761628	—	4	2	no	—
104	C₂×Dic₁₃₄	—	2	1	no	—
105	C₂×D₂₆₈	—	2	1	no	—
106	C₂×D₂₆₈	—	2	1	no	—
107	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
108	C₂×C₂²⋊D₆₇	—	2	2	no	—
109	C₂×C₄×D₆₇	—	2	2	no	—
110	#11761615	—	2	4	no	—
111	#11761615	—	4	2	no	—
112	#11761612	—	2	2	no	—
113	#11761612	—	2	2	no	—
114	#11761612	—	4	2	no	—
115	#11761611	—	4	4	no	—
116	C₂×C₆₇⋊C₈	—	2	2	no	—
117	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
118	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
119	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
120	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
121	C₈×D₆₇	—	4	4	no	—
122	C₄⋊Dic₆₇	—	2	1	no	—
123	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
124	(C₂×D₆₇)⋊C₄	—	2	2	no	—
125	(C₂×D₆₇)⋊C₄	—	2	2	no	—
126	C₂×Q₈×C₆₇	—	2	1	no	—
127	C₂×Q₈×C₆₇	—	2	2	no	—
128	C₂×Q₈×C₆₇	—	4	1	no	—
129	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
130	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
131	C₂²×C₄×C₆₇	—	2	1	no	—
132	C₂²×C₄×C₆₇	—	2	1	no	—
133	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
134	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
136	C₂×C₈×C₆₇	—	2	1	no	—
137	C₂×C₈×C₆₇	—	2	1	no	—
138	C₂×C₈×C₆₇	—	2	1	no	—
139	C₂×C₈×C₆₇	—	2	1	no	—
140	C₂×C₈×C₆₇	—	4	2	no	—
141	C₂×C₈×C₆₇	—	4	2	no	—
142	C₄⋊C₄×C₆₇	—	2	1	no	—
143	C₄⋊C₄×C₆₇	—	2	1	no	—
144	C₄⋊C₄×C₆₇	—	2	1	no	—
145	C₄⋊C₄×C₆₇	—	2	1	no	—
146	C₄⋊C₄×C₆₇	—	2	1	no	—
147	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
148	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
149	C₄²×C₆₇	—	2	1	no	—
150	C₄²×C₆₇	—	2	1	no	—
151	C₄²×C₆₇	—	2	1	no	—
152	C₄²×C₆₇	—	2	1	no	—
153	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
154	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
155	C₄²×C₆₇	#41	1	1	yes	characteristic
156	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
157	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
158	C₂²×D₆₇	—	2	2	no	—
159	Q₈×C₆₇	—	2	1	no	—
160	Q₈×C₆₇	—	2	1	no	—
161	Q₈×C₆₇	—	2	1	no	—
162	Q₈×C₆₇	—	4	2	no	—
163	Q₈×C₆₇	#131	1	2	yes	complemented
164	C₂×C₄×C₆₇	—	2	1	no	—
165	C₂×C₄×C₆₇	—	2	1	no	—
166	C₂×C₄×C₆₇	—	2	1	no	—
167	C₂×C₄×C₆₇	—	2	1	no	—
168	C₂×C₄×C₆₇	—	2	1	no	—
169	C₂×C₄×C₆₇	—	2	1	no	—
170	C₂×C₄×C₆₇	—	4	1	no	—
171	C₂×C₄×C₆₇	—	4	2	no	—
172	C₂×C₄×C₆₇	—	8	1	no	—
173	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
174	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
175	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
176	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
177	C₂×C₄×C₆₇	#137	1	2	yes	—
178	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
179	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
180	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
181	C₂²⋊D₆₇	—	4	4	no	—
182	C₂×Dic₆₇	—	2	2	no	—
183	D₂₆₈	—	2	2	no	—
184	D₂₆₈	—	2	2	no	—
185	D₂₆₈	—	4	2	no	—
186	D₂₆₈	—	8	1	no	—
187	C₄×D₆₇	—	4	2	no	—
188	C₄×D₆₇	—	4	2	no	—
189	Dic₁₃₄	—	4	2	no	—
190	C₈×C₆₇	—	2	2	no	—
191	C₈×C₆₇	—	4	1	no	—
192	C₈×C₆₇	—	4	1	no	—
193	C₈×C₆₇	—	4	1	no	—
194	C₆₇⋊C₈	—	2	2	no	—
195	C₆₇⋊C₈	—	2	2	no	—
196	C₆₇⋊C₈	—	2	2	no	—
197	C₆₇⋊C₈	—	2	2	no	—
198	C₆₇⋊C₈	—	4	2	no	—
199	C₂²×C₆₇	—	2	1	no	—
200	C₂²×C₆₇	—	2	1	no	—
201	C₂²×C₆₇	#440	1	2	yes	—
202	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
203	C₂×D₆₇	—	4	2	no	—
204	C₂×D₆₇	—	4	2	no	—
205	C₂×D₆₇	—	4	2	no	—
206	C₄×C₆₇	—	2	1	no	—
207	C₄×C₆₇	—	2	2	no	—
208	C₄×C₆₇	—	2	1	no	—
209	C₄×C₆₇	—	2	1	no	—
210	C₄×C₆₇	—	4	1	no	—
211	C₄×C₆₇	—	4	1	no	—
212	C₄×C₆₇	—	4	1	no	—
213	C₄×C₆₇	#440	1	2	yes	—
214	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
215	C₄×C₆₇	#568	1	1	yes	characteristic
216	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
217	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
218	Dic₆₇	—	4	2	no	—
219	#18004	—	67	1	no	maximal, Sylow
220	C₂×C₆₇	—	2	2	no	—
221	C₂×C₆₇	#2933	1	1	yes	characteristic
222	C₂×C₆₇	C₂×#440	1	1	yes	characteristic
223	C₂×C₆₇	#2956	1	1	yes	characteristic
224	D₆₇	—	8	2	no	—
225	#3103	—	67	1	no	—
226	Q₈×M₁₆	—	67	1	no	—
227	#2935	—	67	1	no	—
228	#1592	—	67	2	no	—
229	#1588	—	67	2	no	—
230	#1585	—	67	2	no	—
231	SD₁₆⋊C₈	—	67	2	no	—
232	#1571	—	67	1	no	—
233	#1492	—	67	1	no	—
234	C₂×Q₈⋊C₈	D₆₇	1	1	yes	characteristic, complemented
235	#1451	—	67	1	no	—
236	C₆₇	#18004	1	1	yes	characteristic, complemented, Sylow
237	C₂×#131	—	134	1	no	—
238	#532	—	67	1	no	—
239	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
240	#481	—	67	2	no	—
241	#475	—	67	1	no	—
242	#475	—	67	1	no	—
243	#460	—	67	1	no	—
244	#460	—	67	1	no	—
245	#448	—	67	2	no	—
246	#446	—	67	2	no	—
247	#446	—	134	2	no	—
248	C₈×Q₈	—	67	2	no	—
249	C₄×SD₁₆	—	67	2	no	—
250	C₄×SD₁₆	—	67	2	no	—
251	#436	—	67	2	no	—
252	#435	—	134	2	no	—
253	C₈×D₄	—	67	2	no	—
254	#431	—	67	1	no	—
255	#423	—	67	1	no	—
256	#423	—	134	1	no	—
257	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
258	#408	—	134	1	no	—
259	C₄×M₁₆	—	67	1	no	—
260	C₄×M₁₆	—	134	1	no	—
261	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
262	#335	—	67	2	no	—
263	#335	—	67	2	no	—
264	Q₈⋊C₈	C₂×D₆₇	1	1	yes	characteristic
265	Q₈⋊C₈	C₂×D₆₇	1	1	yes	characteristic
266	Q₈⋊C₈	C₂×D₆₇	1	1	yes	characteristic
267	Q₈⋊C₈	C₂×D₆₇	1	1	yes	characteristic
268	D₄⋊C₈	—	67	2	no	—
269	D₄⋊C₈	—	67	2	no	—
270	#322	—	67	2	no	—
271	#322	—	67	2	no	—
272	C₂×#41	—	134	1	no	—
273	C₂²×Q₈	C₄×D₆₇	1	1	yes	characteristic
274	C₂×SD₁₆	—	67	2	no	—
275	C₂×SD₁₆	—	67	2	no	—
276	#131	—	268	4	no	—
277	C₂×M₁₆	—	67	1	no	—
278	C₂×M₁₆	—	67	1	no	—
279	C₂×M₁₆	—	134	2	no	—
280	C₂×M₁₆	—	134	1	no	—
281	C₂²×C₈	—	2	1	no	—
282	C₂²×C₈	—	2	1	no	—
283	C₂²×C₈	—	134	2	no	—
284	C₄⋊Q₈	—	67	1	no	—
285	#127	—	67	1	no	—
286	#124	—	134	1	no	—
287	#121	—	67	2	no	—
288	#121	—	134	1	no	—
289	C₄×Q₈	—	2	1	no	—
290	C₄×Q₈	—	2	1	no	—
291	C₄×Q₈	C₄×D₆₇	1	1	yes	characteristic
292	C₄×Q₈	C₄×D₆₇	1	1	yes	characteristic
293	C₄×Q₈	C₂²×D₆₇	1	1	yes	characteristic
294	C₄×Q₈	C₂²×D₆₇	1	1	yes	characteristic
295	C₄×D₄	—	67	2	no	—
296	C₄×D₄	—	134	1	no	—
297	C₂×C₄⋊C₄	—	2	1	no	—
298	C₂×C₄⋊C₄	C₄×D₆₇	1	1	yes	characteristic
299	C₂×C₄²	—	2	1	no	—
300	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
301	#106	—	67	2	no	—
302	#106	—	67	2	no	—
303	C₄⋊C₈	—	2	1	no	—
304	C₄⋊C₈	—	67	1	no	—
305	C₄⋊C₈	—	67	1	no	—
306	C₄⋊C₈	—	67	2	no	—
307	C₄⋊C₈	—	134	2	no	—
308	C₄⋊C₈	—	134	2	no	—
309	C₄⋊C₈	—	134	1	no	—
310	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
311	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
312	#103	—	67	2	no	—
313	#103	—	67	2	no	—
314	#102	—	67	2	no	—
315	#102	—	67	2	no	—
316	C₂²⋊C₈	—	67	2	no	—
317	C₂²⋊C₈	—	134	2	no	—
318	C₂²⋊C₈	—	134	2	no	—
319	#97	—	67	2	no	—
320	#97	—	134	2	no	—
321	C₄×C₈	—	2	1	no	—
322	C₄×C₈	—	67	2	no	—
323	C₄×C₈	—	67	2	no	—
324	C₄×C₈	—	134	2	no	—
325	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
326	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
327	#41	—	268	2	no	—
328	#41	—	268	2	no	—
329	C₂×Q₈	—	2	1	no	—
330	C₂×Q₈	—	2	2	no	—
331	C₂×Q₈	—	4	1	no	—
332	C₂×Q₈	—	134	1	no	—
333	C₂×Q₈	C₂×C₄×D₆₇	1	1	yes	characteristic
334	C₂×Q₈	C₂×C₄×D₆₇	1	1	yes	characteristic
335	C₂×D₄	—	67	2	no	—
336	C₂×D₄	—	134	2	no	—
337	C₂×D₄	—	134	1	no	—
338	C₂×D₄	—	134	1	no	—
339	C₂²×C₄	—	2	1	no	—
340	C₂²×C₄	—	2	1	no	—
341	C₂²×C₄	—	134	2	no	—
342	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
343	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
344	C₂²×C₄	#11761627	1	1	yes	characteristic
345	SD₁₆	—	268	2	no	—
346	SD₁₆	—	134	4	no	—
347	M₁₆	—	268	4	no	—
348	M₁₆	—	268	2	no	—
349	M₁₆	—	134	2	no	—
350	M₁₆	—	134	2	no	—
351	C₂×C₈	—	2	1	no	—
352	C₂×C₈	—	2	1	no	—
353	C₂×C₈	—	2	1	no	—
354	C₂×C₈	—	2	1	no	—
355	C₂×C₈	—	4	2	no	—
356	C₂×C₈	—	4	2	no	—
357	C₂×C₈	—	268	4	no	—
358	C₂×C₈	—	67	2	no	—
359	C₂×C₈	—	67	2	no	—
360	C₂×C₈	—	67	2	no	—
361	C₂×C₈	—	67	2	no	—
362	C₂×C₈	—	134	2	no	—
363	C₄⋊C₄	—	2	1	no	—
364	C₄⋊C₄	—	2	1	no	—
365	C₄⋊C₄	—	2	1	no	—
366	C₄⋊C₄	—	2	1	no	—
367	C₄⋊C₄	—	2	1	no	—
368	C₄⋊C₄	—	67	2	no	—
369	C₄⋊C₄	—	134	1	no	—
370	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
371	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
372	C₂²⋊C₄	—	134	2	no	—
373	C₂²⋊C₄	—	134	2	no	—
374	C₄²	—	2	1	no	—
375	C₄²	—	2	1	no	—
376	C₄²	—	2	1	no	—
377	C₄²	—	2	1	no	—
378	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
379	C₄²	D₄×D₆₇	1	1	yes	characteristic
380	C₄²	#11761627	1	1	yes	characteristic
381	C₄²	C₂³×D₆₇	1	1	yes	characteristic
382	C₂³	—	134	2	no	—
383	C₂³	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
384	Q₈	—	2	1	no	—
385	Q₈	—	2	1	no	—
386	Q₈	—	2	1	no	—
387	Q₈	—	4	2	no	—
388	Q₈	—	268	2	no	—
389	Q₈	#14524049	1	2	yes	complemented
390	D₄	—	268	4	no	—
391	D₄	—	268	2	no	—
392	D₄	—	536	1	no	—
393	D₄	—	134	2	no	—
394	D₄	—	134	2	no	—
395	C₂×C₄	—	2	1	no	—
396	C₂×C₄	—	2	1	no	—
397	C₂×C₄	—	2	1	no	—
398	C₂×C₄	—	2	1	no	—
399	C₂×C₄	—	2	1	no	—
400	C₂×C₄	—	2	1	no	—
401	C₂×C₄	—	4	1	no	—
402	C₂×C₄	—	4	2	no	—
403	C₂×C₄	—	8	1	no	—
404	C₂×C₄	—	268	2	no	—
405	C₂×C₄	—	268	2	no	—
406	C₂×C₄	—	134	2	no	—
407	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
408	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
409	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
410	C₂×C₄	#14524070	1	2	yes	—
411	C₂×C₄	#14524075	1	2	yes	—
412	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
413	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
414	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
415	C₈	—	2	2	no	—
416	C₈	—	4	1	no	—
417	C₈	—	4	1	no	—
418	C₈	—	4	1	no	—
419	C₈	—	268	2	no	—
420	C₈	—	134	2	no	—
421	C₈	—	134	2	no	—
422	C₈	—	134	2	no	—
423	C₈	—	134	2	no	—
424	C₂²	—	2	1	no	—
425	C₂²	—	2	1	no	—
426	C₂²	—	268	2	no	—
427	C₂²	—	268	2	no	—
428	C₂²	—	268	2	no	—
429	C₂²	#16863499	1	2	yes	—
430	C₂²	C₂×(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
431	C₄	—	2	1	no	—
432	C₄	—	2	1	no	—
433	C₄	—	2	1	no	—
434	C₄	—	2	2	no	—
435	C₄	—	4	1	no	—
436	C₄	—	4	1	no	—
437	C₄	—	4	1	no	—
438	C₄	—	268	2	no	—
439	C₄	#16863499	1	2	yes	—
440	C₄	C₂×#14524049	1	1	yes	characteristic
441	C₄	#16864162	1	1	yes	characteristic
442	C₄	C₂×#14524070	1	1	yes	characteristic
443	C₄	C₂×#14524075	1	1	yes	characteristic
444	C₂	—	2	2	no	—
445	C₂	—	536	2	no	—
446	C₂	#22568930	1	1	yes	characteristic
447	C₂	C₂×#16863499	1	1	yes	characteristic
448	C₂	#22569115	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×Q₈⋊C₈×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#16863946	C₂²
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₂×Q₈⋊C₈	D₆₇
hypocenter	C₆₇	#18004
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×Q₈⋊C₈

Complements

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

N	H ≅ G/N	type
2. Q₈×#11761612	C₂	semidirect
4. #22559114	C₂	semidirect
5. #22559084	C₂	semidirect
6. #22559069	C₂	semidirect
7. #22559024	C₂	semidirect
8. #22558983	C₂	semidirect
10. #22558498	C₂	semidirect
11. C₂×Q₈⋊C₈×C₆₇	C₂	semidirect
163. Q₈×C₆₇	#131	semidirect
234. C₂×Q₈⋊C₈	D₆₇	semidirect
236. C₆₇	#18004	semidirect
389. Q₈	#14524049	semidirect

Maximal subgroups

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

subgroup	index
#22570240	2
Q₈×#11761612	2
#22568952	2
#22559114	2
#22559084	2
#22559069	2
#22559024	2
#22558983	2
#22558661	2
#22558498	2
C₂×Q₈⋊C₈×C₆₇	2
#18004	67