Subgroups — #30825124

Order 17152.

616 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569336	C₂	1	2	yes	maximal
2	C₈×#11761628	C₂	1	1	yes	maximal, characteristic
3	#22558999	C₂	1	2	yes	maximal, complemented
4	#22558995	C₂	1	2	yes	maximal, complemented
5	#22558993	C₂	1	2	yes	maximal, complemented
6	#22558969	C₂	1	1	yes	maximal, characteristic, complemented
7	#22558963	C₂	1	1	yes	maximal, characteristic, complemented
8	#22558954	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#16863147	C₂	1	1	yes	maximal, characteristic, complemented
10	#334×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1568	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#11761628	C₂²	1	1	yes	characteristic
13	#16863524	—	2	2	no	—
14	C₈×Dic₁₃₄	C₄	1	1	yes	characteristic
15	C₈×Dic₁₃₄	C₄	1	1	yes	characteristic
16	#16863469	—	2	2	no	—
17	#16863464	—	2	2	no	—
18	C₈×D₂₆₈	C₄	1	1	yes	characteristic
19	C₈×D₂₆₈	C₄	1	1	yes	characteristic
20	C₈×C₂²⋊D₆₇	C₄	1	1	yes	characteristic
21	C₈×C₂²⋊D₆₇	C₄	1	1	yes	characteristic
22	C₈×C₂²⋊D₆₇	C₄	1	1	yes	characteristic
23	C₈×C₂²⋊D₆₇	C₄	1	1	yes	characteristic
24	(C₂²⋊D₆₇)⋊C₈	—	2	2	no	—
25	#16863452	C₂²	1	2	yes	—
26	#16863449	C₂²	1	1	yes	characteristic
27	#16863448	C₂²	1	2	yes	—
28	#16863447	C₂²	1	1	yes	characteristic
29	#16863447	C₂²	1	1	yes	characteristic
30	#16863418	C₂²	1	2	yes	—
31	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
32	C₄⋊C₈×D₆₇	C₂²	1	2	yes	—
33	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
34	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
35	#16863148	C₂²	1	2	yes	—
36	#16863148	C₂²	1	2	yes	—
37	#16863147	C₂²	1	2	yes	—
38	#16863147	C₂²	1	1	yes	characteristic
39	#16863147	C₂²	1	1	yes	characteristic
40	#16863146	C₂²	1	2	yes	—
41	#16863146	C₂²	1	2	yes	—
42	#432×C₆₇	C₂²	1	2	yes	—
43	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
44	#335×C₆₇	C₂²	1	2	yes	—
45	#334×C₆₇	C₂²	1	1	yes	characteristic
46	#334×C₆₇	C₂²	1	1	yes	characteristic
47	C₂×#11761628	Q₈	1	1	yes	characteristic
48	C₂×#11761611	—	2	2	no	—
49	C₂²×C₆₇⋊C₈	—	2	2	no	—
50	C₂×C₈×D₆₇	—	2	2	no	—
51	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
54	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
55	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
56	C₄×Dic₁₃₄	D₄	1	1	yes	characteristic
57	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
58	C₄×D₂₆₈	D₄	1	1	yes	characteristic
59	C₄×C₂²⋊D₆₇	C₂×C₄	1	1	yes	characteristic
60	C₄×C₂²⋊D₆₇	C₂×C₄	1	1	yes	characteristic
61	C₄×C₂²⋊D₆₇	D₄	1	1	yes	characteristic
62	C₄×C₂²⋊D₆₇	D₄	1	1	yes	characteristic
63	#14524002	Q₈	1	1	yes	characteristic
64	#14524000	Q₈	1	1	yes	characteristic
65	#14524000	Q₈	1	1	yes	characteristic
66	C₄²×D₆₇	C₂³	1	1	yes	characteristic
67	C₄²×D₆₇	C₂³	1	1	yes	characteristic
68	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
69	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
70	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
71	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
72	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
73	Dic₆₇⋊C₈	—	2	2	no	—
74	Dic₆₇⋊C₈	—	2	2	no	—
75	Dic₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
76	Dic₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
77	Dic₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
78	Dic₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
79	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
80	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
81	(C₂×D₆₇)⋊C₈	—	2	2	no	—
82	(C₂×D₆₇)⋊C₈	—	2	2	no	—
83	(C₂×D₆₇)⋊C₈	C₂×C₄	1	1	yes	characteristic
84	(C₂×D₆₇)⋊C₈	C₂×C₄	1	1	yes	characteristic
85	(C₂×D₆₇)⋊C₈	C₂×C₄	1	1	yes	characteristic
86	(C₂×D₆₇)⋊C₈	C₂×C₄	1	1	yes	characteristic
87	#14523953	—	2	2	no	—
88	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
89	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
90	C₈×Dic₆₇	—	2	2	no	—
91	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
92	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
93	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
94	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
95	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
96	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
97	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
98	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
99	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
100	C₂²⋊C₈×C₆₇	—	2	2	no	—
101	#97×C₆₇	—	2	2	no	—
102	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
103	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
104	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
105	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
106	#11761628	—	2	4	no	—
107	C₂×Dic₁₃₄	C₄⋊C₄	1	1	yes	characteristic
108	C₂×D₂₆₈	C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₂²⋊D₆₇	C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₂²⋊D₆₇	C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×D₆₇	C₂×Q₈	1	1	yes	characteristic
116	C₂×C₄×D₆₇	C₂×Q₈	1	1	yes	characteristic
117	#11761611	—	8	2	no	—
118	C₂×C₆₇⋊C₈	—	2	2	no	—
119	C₂×C₆₇⋊C₈	—	2	2	no	—
120	C₂×C₆₇⋊C₈	—	8	2	no	—
121	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
125	C₈×D₆₇	—	2	2	no	—
126	C₈×D₆₇	—	2	2	no	—
127	C₈×D₆₇	—	4	1	no	—
128	C₈×D₆₇	—	4	1	no	—
129	C₈×D₆₇	—	8	2	no	—
130	C₄⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
131	C₄⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
132	Dic₆₇⋊C₄	C₄⋊C₄	1	1	yes	characteristic
133	Dic₆₇⋊C₄	C₄⋊C₄	1	1	yes	characteristic
134	Dic₆₇⋊C₄	C₄⋊C₄	1	1	yes	characteristic
135	Dic₆₇⋊C₄	C₄⋊C₄	1	1	yes	characteristic
136	C₂²⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
137	C₂²⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
138	(C₂×D₆₇)⋊C₄	C₄⋊C₄	1	1	yes	characteristic
139	(C₂×D₆₇)⋊C₄	C₄⋊C₄	1	1	yes	characteristic
140	(C₂×D₆₇)⋊C₄	C₄⋊C₄	1	1	yes	characteristic
141	(C₂×D₆₇)⋊C₄	C₄⋊C₄	1	1	yes	characteristic
142	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
143	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
144	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
145	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
146	C₄×Dic₆₇	C₂×Q₈	1	1	yes	characteristic
147	C₄×Dic₆₇	C₂×Q₈	1	1	yes	characteristic
148	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
149	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
150	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
151	C₂×C₈×C₆₇	—	2	2	no	—
152	C₂×C₈×C₆₇	—	2	2	no	—
153	C₂×C₈×C₆₇	—	2	2	no	—
154	C₂×C₈×C₆₇	—	4	1	no	—
155	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
156	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
157	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
158	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
159	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
160	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
161	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
162	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
163	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
164	C₂²×D₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
165	C₂²×D₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
166	C₂×C₄×C₆₇	—	2	2	no	—
167	C₂×C₄×C₆₇	—	2	1	no	—
168	C₂×C₄×C₆₇	—	2	1	no	—
169	C₂×C₄×C₆₇	—	2	2	no	—
170	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
171	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
172	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
173	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
174	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
175	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
176	C₂²⋊D₆₇	—	2	2	no	—
177	C₂²⋊D₆₇	#108	1	4	yes	—
178	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
179	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
180	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
181	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
182	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
183	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
184	D₂₆₈	—	2	2	no	—
185	C₄×D₆₇	—	2	2	no	—
186	C₄×D₆₇	—	2	1	no	—
187	C₄×D₆₇	—	2	1	no	—
188	C₄×D₆₇	—	2	1	no	—
189	C₄×D₆₇	—	2	1	no	—
190	C₄×D₆₇	—	2	2	no	—
191	C₄×D₆₇	—	2	2	no	—
192	C₄×D₆₇	—	2	2	no	—
193	Dic₁₃₄	—	2	2	no	—
194	C₈×C₆₇	—	2	1	no	—
195	C₈×C₆₇	—	2	1	no	—
196	C₈×C₆₇	—	2	1	no	—
197	C₈×C₆₇	—	4	2	no	—
198	C₈×C₆₇	—	4	2	no	—
199	C₈×C₆₇	#131	1	2	yes	—
200	C₆₇⋊C₈	—	2	1	no	—
201	C₆₇⋊C₈	—	2	1	no	—
202	C₆₇⋊C₈	—	2	1	no	—
203	C₆₇⋊C₈	—	2	1	no	—
204	C₆₇⋊C₈	—	4	2	no	—
205	C₆₇⋊C₈	—	4	2	no	—
206	C₂²×C₆₇	—	2	1	no	—
207	C₂²×C₆₇	—	2	1	no	—
208	C₂²×C₆₇	C₂×#108	1	2	yes	—
209	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
210	C₂×D₆₇	—	2	1	no	—
211	C₂×D₆₇	—	2	1	no	—
212	C₂×D₆₇	—	2	1	no	—
213	C₂×D₆₇	—	2	1	no	—
214	C₂×D₆₇	—	2	1	no	—
215	C₂×D₆₇	C₂×#108	1	2	yes	—
216	C₄×C₆₇	—	2	1	no	—
217	C₄×C₆₇	—	2	1	no	—
218	C₄×C₆₇	—	2	1	no	—
219	C₄×C₆₇	—	2	1	no	—
220	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
221	C₄×C₆₇	#568	1	1	yes	characteristic
222	C₄×C₆₇	#577	1	2	yes	—
223	Dic₆₇	—	2	1	no	—
224	Dic₆₇	—	2	1	no	—
225	Dic₆₇	—	2	1	no	—
226	Dic₆₇	—	2	1	no	—
227	Dic₆₇	—	2	1	no	—
228	Dic₆₇	C₂×#108	1	2	yes	—
229	#17834	—	67	1	no	maximal, Sylow
230	C₂×C₆₇	—	2	2	no	—
231	C₂×C₆₇	#2906	1	1	yes	characteristic
232	C₂×C₆₇	#2912	1	1	yes	characteristic
233	C₂×C₆₇	C₂²×#108	1	1	yes	characteristic
234	D₆₇	—	2	2	no	—
235	D₆₇	—	4	1	no	—
236	#2990	—	67	2	no	—
237	C₈×#41	—	67	1	no	—
238	#1574	—	67	2	no	—
239	#1573	—	67	2	no	—
240	#1573	—	67	2	no	—
241	#1568	—	67	1	no	—
242	#1568	D₆₇	1	1	yes	characteristic, complemented
243	#1566	—	67	1	no	—
244	#1564	—	67	1	no	—
245	C₂×#334	—	67	1	no	—
246	C₂×#334	—	67	1	no	—
247	C₆₇	#17834	1	1	yes	characteristic, complemented, Sylow
248	C₄×#41	—	67	1	no	—
249	#446	—	134	2	no	—
250	C₈×Q₈	—	67	1	no	—
251	C₈×Q₈	—	67	1	no	—
252	#435	—	134	2	no	—
253	#435	—	134	2	no	—
254	C₈×D₄	—	67	1	no	—
255	C₈×D₄	—	67	1	no	—
256	C₈×D₄	—	67	1	no	—
257	C₈×D₄	—	67	1	no	—
258	C₈×D₄	—	67	1	no	—
259	C₈×D₄	—	67	1	no	—
260	C₈×D₄	—	134	2	no	—
261	#432	—	67	2	no	—
262	#432	C₂×D₆₇	1	2	yes	—
263	#431	—	67	1	no	—
264	#431	—	67	1	no	—
265	#431	—	67	1	no	—
266	#431	—	67	2	no	—
267	#423	—	67	2	no	—
268	C₂×C₄⋊C₈	—	67	2	no	—
269	C₂×C₄⋊C₈	—	67	2	no	—
270	C₂×C₄×C₈	—	67	1	no	—
271	C₂×C₄×C₈	—	67	1	no	—
272	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
273	#335	—	67	2	no	—
274	#335	—	67	2	no	—
275	#335	C₂×D₆₇	1	2	yes	—
276	#334	—	67	2	no	—
277	#334	—	67	2	no	—
278	#334	—	67	1	no	—
279	#334	—	67	1	no	—
280	#334	—	67	2	no	—
281	#334	C₂×D₆₇	1	1	yes	characteristic
282	#334	C₂×D₆₇	1	1	yes	characteristic
283	C₂×#41	—	67	1	no	—
284	C₂×M₁₆	—	134	2	no	—
285	C₂²×C₈	—	67	1	no	—
286	C₂²×C₈	—	67	1	no	—
287	C₂²×C₈	—	67	1	no	—
288	C₂²×C₈	—	67	1	no	—
289	C₂²×C₈	—	134	2	no	—
290	C₂²×C₈	—	134	2	no	—
291	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
292	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
293	C₄×Q₈	—	67	1	no	—
294	C₄×Q₈	—	67	1	no	—
295	C₄×D₄	—	67	1	no	—
296	C₄×D₄	—	67	1	no	—
297	C₄×D₄	—	67	1	no	—
298	C₄×D₄	—	67	1	no	—
299	C₄×D₄	—	67	1	no	—
300	C₄×D₄	—	67	1	no	—
301	#117	—	67	1	no	—
302	#117	—	67	1	no	—
303	#117	—	67	1	no	—
304	C₂×C₄²	—	67	1	no	—
305	C₂×C₄²	—	67	1	no	—
306	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
307	C₄⋊C₈	—	67	1	no	—
308	C₄⋊C₈	—	67	1	no	—
309	C₄⋊C₈	—	67	1	no	—
310	C₄⋊C₈	—	67	1	no	—
311	C₄⋊C₈	—	67	1	no	—
312	C₄⋊C₈	—	67	1	no	—
313	C₄⋊C₈	—	67	2	no	—
314	C₄⋊C₈	—	67	2	no	—
315	C₄⋊C₈	—	134	2	no	—
316	C₄⋊C₈	—	134	2	no	—
317	C₄⋊C₈	—	134	2	no	—
318	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
319	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
320	C₂²⋊C₈	—	2	2	no	—
321	C₂²⋊C₈	—	67	1	no	—
322	C₂²⋊C₈	—	67	1	no	—
323	C₂²⋊C₈	—	67	1	no	—
324	C₂²⋊C₈	—	67	1	no	—
325	C₂²⋊C₈	—	67	1	no	—
326	C₂²⋊C₈	—	67	1	no	—
327	C₂²⋊C₈	—	134	2	no	—
328	C₂²⋊C₈	—	134	2	no	—
329	#97	—	2	2	no	—
330	#97	—	134	2	no	—
331	C₄×C₈	—	67	1	no	—
332	C₄×C₈	—	67	1	no	—
333	C₄×C₈	—	67	1	no	—
334	C₄×C₈	—	67	1	no	—
335	C₄×C₈	—	67	1	no	—
336	C₄×C₈	—	67	1	no	—
337	C₄×C₈	—	134	2	no	—
338	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
339	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
340	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
341	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
342	#41	—	134	4	no	—
343	C₂×Q₈	—	67	1	no	—
344	C₂×D₄	—	67	1	no	—
345	C₂×D₄	—	67	1	no	—
346	C₂×D₄	—	67	1	no	—
347	C₂²×C₄	—	67	1	no	—
348	C₂²×C₄	—	67	1	no	—
349	C₂²×C₄	—	67	1	no	—
350	C₂²×C₄	—	67	1	no	—
351	C₂²×C₄	—	67	1	no	—
352	C₂²×C₄	—	67	1	no	—
353	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
354	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
355	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
356	M₁₆	—	536	2	no	—
357	C₂×C₈	—	2	2	no	—
358	C₂×C₈	—	2	2	no	—
359	C₂×C₈	—	2	2	no	—
360	C₂×C₈	—	4	1	no	—
361	C₂×C₈	—	268	1	no	—
362	C₂×C₈	—	268	1	no	—
363	C₂×C₈	—	536	2	no	—
364	C₂×C₈	—	536	2	no	—
365	C₂×C₈	—	67	1	no	—
366	C₂×C₈	—	67	1	no	—
367	C₂×C₈	—	67	1	no	—
368	C₂×C₈	—	67	1	no	—
369	C₂×C₈	—	134	2	no	—
370	C₂×C₈	—	134	2	no	—
371	C₂×C₈	—	134	2	no	—
372	C₂×C₈	—	134	2	no	—
373	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
374	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
375	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
376	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
377	C₄⋊C₄	—	67	1	no	—
378	C₄⋊C₄	—	67	1	no	—
379	C₄⋊C₄	—	67	1	no	—
380	C₄⋊C₄	—	67	1	no	—
381	C₄⋊C₄	—	67	1	no	—
382	C₄⋊C₄	—	67	1	no	—
383	C₂²⋊C₄	—	67	1	no	—
384	C₂²⋊C₄	—	67	1	no	—
385	C₂²⋊C₄	—	67	1	no	—
386	C₂²⋊C₄	—	67	1	no	—
387	C₂²⋊C₄	—	67	1	no	—
388	C₂²⋊C₄	—	67	1	no	—
389	C₄²	—	67	1	no	—
390	C₄²	—	67	1	no	—
391	C₄²	—	67	1	no	—
392	C₄²	—	67	1	no	—
393	C₄²	—	67	1	no	—
394	C₄²	—	67	1	no	—
395	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
396	C₄²	D₄×D₆₇	1	1	yes	characteristic
397	C₄²	Q₈×D₆₇	1	1	yes	characteristic
398	C₄²	C₂³×D₆₇	1	1	yes	characteristic
399	C₂³	—	67	1	no	—
400	C₂³	—	67	1	no	—
401	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
402	Q₈	—	134	2	no	—
403	D₄	—	67	4	no	—
404	D₄	—	134	2	no	—
405	D₄	—	134	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	67	1	no	—
411	C₂×C₄	—	67	1	no	—
412	C₂×C₄	—	67	1	no	—
413	C₂×C₄	—	67	1	no	—
414	C₂×C₄	—	67	1	no	—
415	C₂×C₄	—	67	1	no	—
416	C₂×C₄	—	134	2	no	—
417	C₂×C₄	—	134	2	no	—
418	C₂×C₄	—	134	1	no	—
419	C₂×C₄	—	134	1	no	—
420	C₂×C₄	—	134	1	no	—
421	C₂×C₄	—	134	1	no	—
422	C₂×C₄	—	134	2	no	—
423	C₂×C₄	—	134	2	no	—
424	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
425	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
426	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
427	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
428	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
429	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
430	C₈	—	2	1	no	—
431	C₈	—	2	1	no	—
432	C₈	—	2	1	no	—
433	C₈	—	4	2	no	—
434	C₈	—	4	2	no	—
435	C₈	—	268	2	no	—
436	C₈	—	268	2	no	—
437	C₈	—	134	1	no	—
438	C₈	—	134	1	no	—
439	C₈	—	134	1	no	—
440	C₈	—	134	1	no	—
441	C₈	#14524051	1	2	yes	—
442	C₂²	—	2	1	no	—
443	C₂²	—	2	1	no	—
444	C₂²	—	67	2	no	—
445	C₂²	—	134	1	no	—
446	C₂²	—	134	1	no	—
447	C₂²	—	134	1	no	—
448	C₂²	—	134	1	no	—
449	C₂²	—	134	1	no	—
450	C₂²	#108×D₆₇	1	2	yes	—
451	C₂²	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
452	C₄	—	2	1	no	—
453	C₄	—	2	1	no	—
454	C₄	—	2	1	no	—
455	C₄	—	2	1	no	—
456	C₄	—	67	2	no	—
457	C₄	—	134	1	no	—
458	C₄	—	134	1	no	—
459	C₄	—	134	1	no	—
460	C₄	—	134	1	no	—
461	C₄	—	134	1	no	—
462	C₄	C₂×#14524051	1	1	yes	characteristic
463	C₄	#16864162	1	1	yes	characteristic
464	C₄	#16864203	1	2	yes	—
465	C₂	—	2	2	no	—
466	C₂	—	268	1	no	—
467	C₂	—	134	2	no	—
468	C₂	#22568778	1	1	yes	characteristic
469	C₂	#22568801	1	1	yes	characteristic
470	C₂	C₂×#108×D₆₇	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×#1568	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂×#11761628	Q₈
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1568	D₆₇
hypocenter	C₆₇	#17834
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1568

Complements

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

N	H ≅ G/N	type
3. #22558999	C₂	semidirect
4. #22558995	C₂	semidirect
5. #22558993	C₂	semidirect
6. #22558969	C₂	semidirect
7. #22558963	C₂	semidirect
8. #22558954	C₂	semidirect
9. C₂×#16863147	C₂	semidirect
10. #334×D₆₇	C₂	semidirect
11. C₆₇×#1568	C₂	semidirect
242. #1568	D₆₇	semidirect
247. C₆₇	#17834	semidirect

Maximal subgroups

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

subgroup	index
#22569336	2
C₈×#11761628	2
#22558999	2
#22558995	2
#22558993	2
#22558969	2
#22558963	2
#22558954	2
C₂×#16863147	2
#334×D₆₇	2
C₆₇×#1568	2
#17834	67