Subgroups — #30854101

Order 17152.

926 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863457	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863352	C₂	1	1	yes	maximal, characteristic
3	#22560084	C₂	1	1	yes	maximal, characteristic, complemented
4	#22560084	C₂	1	1	yes	maximal, characteristic, complemented
5	#22560078	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄⋊C₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#22559993	C₂	1	1	yes	maximal, characteristic
8	#22559993	C₂	1	1	yes	maximal, characteristic
9	#22559987	C₂	1	1	yes	maximal, characteristic
10	C₂²⋊C₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
11	#22559961	C₂	1	1	yes	maximal, characteristic, complemented
12	#22559955	C₂	1	1	yes	maximal, characteristic, complemented
13	#22559896	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#14523954	C₂	1	1	yes	maximal, characteristic, complemented
15	C₆₇×#2298	C₂	1	1	yes	maximal, characteristic
16	C₂²×#11761612	C₂²	1	1	yes	characteristic
17	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
18	#16863457	C₄	1	8	yes	complemented
19	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
23	#16863352	C₄	1	8	yes	complemented
24	C₂×#14523954	C₂²	1	1	yes	characteristic
25	C₂×#14523954	C₂²	1	1	yes	characteristic
26	C₂×#14523954	C₂²	1	1	yes	characteristic
27	C₂×#14523954	C₂²	1	1	yes	characteristic
28	C₂×#14523954	C₂²	1	1	yes	characteristic
29	C₂×#14523954	C₂²	1	1	yes	characteristic
30	C₂×#14523954	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
35	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
38	#16863151	C₂²	1	1	yes	characteristic
39	#16863151	C₂²	1	1	yes	characteristic
40	#16863151	C₂²	1	1	yes	characteristic
41	#16863151	C₂²	1	1	yes	characteristic
42	#16863151	C₂²	1	1	yes	characteristic
43	#16863151	C₂²	1	1	yes	characteristic
44	#16863151	C₂²	1	1	yes	characteristic
45	#16863151	C₂²	1	1	yes	characteristic
46	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
47	#388×C₆₇	C₄	1	2	yes	—
48	#388×C₆₇	C₄	1	2	yes	—
49	#387×C₆₇	C₄	1	2	yes	—
50	#387×C₆₇	C₄	1	2	yes	—
51	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
52	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
53	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
54	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
55	#376×C₆₇	C₂²	1	1	yes	characteristic
56	C₄³×C₆₇	C₂²	1	1	yes	characteristic
57	C₂×#11761612	—	2	2	no	—
58	C₂×#11761612	—	2	2	no	—
59	C₂×#11761612	C₂×C₄	1	4	yes	—
60	C₂²×C₆₇⋊C₈	—	2	2	no	—
61	C₂²×C₆₇⋊C₈	—	2	2	no	—
62	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
63	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
64	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
65	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
66	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
67	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
68	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
69	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
70	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
71	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
72	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
73	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
74	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
75	#14523954	—	2	2	no	—
76	#14523954	—	2	2	no	—
77	#14523954	—	2	2	no	—
78	#14523954	—	2	2	no	—
79	#14523954	C₂×C₄	1	4	yes	—
80	#14523954	C₂×C₄	1	4	yes	—
81	#14523954	C₂×C₄	1	4	yes	—
82	C₄×C₆₇⋊C₈	—	2	2	no	—
83	C₄×C₆₇⋊C₈	—	2	2	no	—
84	C₄×C₆₇⋊C₈	—	2	2	no	—
85	C₄×C₆₇⋊C₈	—	2	2	no	—
86	C₄×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
87	C₄×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
88	C₄×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
89	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
90	#117×C₆₇	C₂×C₄	1	4	yes	—
91	#117×C₆₇	C₂×C₄	1	4	yes	—
92	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
93	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
94	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
95	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
96	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
97	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
98	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
99	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
100	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
101	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
102	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
104	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
105	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
106	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
107	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
108	#95×C₆₇	C₂×C₄	1	2	yes	—
109	#95×C₆₇	C₂×C₄	1	2	yes	—
110	#95×C₆₇	C₂×C₄	1	2	yes	—
111	#95×C₆₇	C₂×C₄	1	2	yes	—
112	#95×C₆₇	C₂×C₄	1	2	yes	—
113	#95×C₆₇	C₂×C₄	1	2	yes	—
114	#11761612	—	2	8	no	—
115	C₂×C₆₇⋊C₈	—	2	1	no	—
116	C₂×C₆₇⋊C₈	—	2	1	no	—
117	C₂×C₆₇⋊C₈	—	2	1	no	—
118	C₂×C₆₇⋊C₈	—	2	1	no	—
119	C₂×C₆₇⋊C₈	—	2	1	no	—
120	C₂×C₆₇⋊C₈	—	2	1	no	—
121	C₂×C₆₇⋊C₈	—	2	1	no	—
122	C₂×C₆₇⋊C₈	—	2	1	no	—
123	C₂×C₆₇⋊C₈	—	2	1	no	—
124	C₂×C₆₇⋊C₈	—	2	1	no	—
125	C₂×C₆₇⋊C₈	—	2	1	no	—
126	C₂×C₆₇⋊C₈	—	2	1	no	—
127	C₂×C₆₇⋊C₈	—	2	8	no	—
128	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
129	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
130	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
131	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
132	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
133	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
134	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
135	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
136	C₂×C₆₇⋊C₈	#41	1	2	yes	—
137	C₂×C₆₇⋊C₈	#41	1	2	yes	—
138	C₂×C₆₇⋊C₈	#41	1	2	yes	—
139	C₂×C₆₇⋊C₈	#41	1	2	yes	—
140	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
141	C₂²×C₄×C₆₇	—	2	1	no	—
142	C₂²×C₄×C₆₇	—	2	1	no	—
143	C₂²×C₄×C₆₇	—	2	1	no	—
144	C₂²×C₄×C₆₇	—	2	1	no	—
145	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
146	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
147	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
148	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
149	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
150	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
151	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
152	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
153	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
154	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
155	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
156	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
157	C₄⋊C₄×C₆₇	—	2	4	no	—
158	C₄⋊C₄×C₆₇	—	4	2	no	—
159	C₄⋊C₄×C₆₇	C₄²	1	4	yes	—
160	C₄⋊C₄×C₆₇	C₄²	1	4	yes	—
161	C₂²⋊C₄×C₆₇	—	2	4	no	—
162	C₂²⋊C₄×C₆₇	—	4	2	no	—
163	C₂²⋊C₄×C₆₇	C₄²	1	4	yes	—
164	C₂²⋊C₄×C₆₇	C₄²	1	4	yes	—
165	C₄²×C₆₇	—	2	4	no	—
166	C₄²×C₆₇	—	2	1	no	—
167	C₄²×C₆₇	—	2	1	no	—
168	C₄²×C₆₇	—	2	1	no	—
169	C₄²×C₆₇	—	2	1	no	—
170	C₄²×C₆₇	—	2	1	no	—
171	C₄²×C₆₇	—	2	1	no	—
172	C₄²×C₆₇	—	2	1	no	—
173	C₄²×C₆₇	—	2	1	no	—
174	C₄²×C₆₇	—	4	2	no	—
175	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
176	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
177	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
178	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
179	C₂³×C₆₇	—	2	1	no	—
180	C₂³×C₆₇	—	2	1	no	—
181	C₂³×C₆₇	—	2	2	no	—
182	C₂³×C₆₇	—	2	2	no	—
183	C₂³×C₆₇	C₂×C₄²	1	2	yes	—
184	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
185	C₂×C₄×C₆₇	—	2	2	no	—
186	C₂×C₄×C₆₇	—	2	1	no	—
187	C₂×C₄×C₆₇	—	2	1	no	—
188	C₂×C₄×C₆₇	—	2	2	no	—
189	C₂×C₄×C₆₇	—	2	2	no	—
190	C₂×C₄×C₆₇	—	2	2	no	—
191	C₂×C₄×C₆₇	—	2	2	no	—
192	C₂×C₄×C₆₇	—	2	2	no	—
193	C₂×C₄×C₆₇	—	2	2	no	—
194	C₂×C₄×C₆₇	—	2	2	no	—
195	C₂×C₄×C₆₇	—	2	2	no	—
196	C₂×C₄×C₆₇	—	2	2	no	—
197	C₂×C₄×C₆₇	—	2	2	no	—
198	C₂×C₄×C₆₇	—	2	2	no	—
199	C₂×C₄×C₆₇	—	2	2	no	—
200	C₂×C₄×C₆₇	—	2	2	no	—
201	C₂×C₄×C₆₇	—	2	2	no	—
202	C₂×C₄×C₆₇	—	2	2	no	—
203	C₂×C₄×C₆₇	—	2	2	no	—
204	C₂×C₄×C₆₇	—	2	2	no	—
205	C₂×C₄×C₆₇	—	2	2	no	—
206	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
207	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
208	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
209	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
210	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
211	C₂×C₄×C₆₇	#131	1	4	yes	—
212	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
213	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
214	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
215	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
216	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
217	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
218	C₆₇⋊C₈	—	2	2	no	—
219	C₆₇⋊C₈	—	2	2	no	—
220	C₆₇⋊C₈	—	2	2	no	—
221	C₆₇⋊C₈	—	2	2	no	—
222	C₆₇⋊C₈	—	2	2	no	—
223	C₆₇⋊C₈	—	2	2	no	—
224	C₆₇⋊C₈	#117	1	4	yes	—
225	C₆₇⋊C₈	#117	1	4	yes	—
226	C₂²×C₆₇	—	2	2	no	—
227	C₂²×C₆₇	—	2	2	no	—
228	C₂²×C₆₇	—	2	2	no	—
229	C₂²×C₆₇	—	4	2	no	—
230	C₂²×C₆₇	—	4	2	no	—
231	C₂²×C₆₇	C₂²×C₄²	1	1	yes	characteristic
232	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
233	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
234	C₂²×C₆₇	C₂×#131	1	2	yes	—
235	C₂²×C₆₇	#568	1	2	yes	—
236	C₄×C₆₇	—	2	2	no	—
237	C₄×C₆₇	—	2	2	no	—
238	C₄×C₆₇	—	2	2	no	—
239	C₄×C₆₇	—	2	2	no	—
240	C₄×C₆₇	—	2	4	no	—
241	C₄×C₆₇	—	2	2	no	—
242	C₄×C₆₇	—	4	2	no	—
243	C₄×C₆₇	—	4	2	no	—
244	C₄×C₆₇	—	4	2	no	—
245	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
246	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
247	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
248	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
249	#20478	—	67	1	no	maximal, Sylow
250	C₂×C₆₇	—	4	2	no	—
251	C₂×C₆₇	C₄×#117	1	1	yes	characteristic
252	C₂×C₆₇	C₄×#131	1	1	yes	characteristic
253	C₂×C₆₇	#2875	1	1	yes	characteristic
254	C₂×C₆₇	#2968	1	2	yes	—
255	C₂×C₆₇	#2969	1	2	yes	—
256	C₂×#433	—	67	1	no	—
257	C₂×#405	—	67	1	no	—
258	#2298	D₆₇	1	1	yes	characteristic
259	#1771	—	67	1	no	—
260	#1771	—	67	1	no	—
261	#1770	—	67	1	no	—
262	C₈×C₄⋊C₄	—	67	1	no	—
263	#1753	—	67	1	no	—
264	#1753	—	67	1	no	—
265	#1752	—	67	1	no	—
266	C₈×C₂²⋊C₄	—	67	1	no	—
267	#1746	—	67	1	no	—
268	#1744	—	67	1	no	—
269	#1726	—	67	1	no	—
270	C₄×#97	—	67	1	no	—
271	C₆₇	#20478	1	1	yes	characteristic, complemented, Sylow
272	C₂²×M₁₆	—	67	1	no	—
273	C₂³×C₈	—	67	1	no	—
274	C₂×#117	C₂×D₆₇	1	1	yes	characteristic
275	#433	—	67	8	no	—
276	C₂×C₄⋊C₈	—	67	1	no	—
277	C₂×C₄⋊C₈	—	67	1	no	—
278	C₂×C₂²⋊C₈	—	67	1	no	—
279	C₂×C₂²⋊C₈	—	67	1	no	—
280	#405	—	67	8	no	—
281	C₂×#97	—	67	1	no	—
282	C₂×#97	—	67	1	no	—
283	C₂×#97	—	67	1	no	—
284	C₂×#97	—	67	1	no	—
285	C₂×#97	—	67	1	no	—
286	C₂×#97	—	67	1	no	—
287	C₂×#97	—	67	1	no	—
288	C₂×C₄×C₈	—	67	1	no	—
289	C₂×C₄×C₈	—	67	1	no	—
290	C₂×C₄×C₈	—	67	1	no	—
291	C₂×C₄×C₈	—	67	1	no	—
292	C₂×C₄×C₈	—	67	1	no	—
293	C₂×C₄×C₈	—	67	1	no	—
294	C₂×C₄×C₈	—	67	1	no	—
295	#388	Dic₆₇	1	2	yes	—
296	#388	Dic₆₇	1	2	yes	—
297	#387	Dic₆₇	1	2	yes	—
298	#387	Dic₆₇	1	2	yes	—
299	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
300	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
301	C₄×C₂²⋊C₄	C₂×D₆₇	1	1	yes	characteristic
302	C₄×C₂²⋊C₄	C₂×D₆₇	1	1	yes	characteristic
303	#376	C₂×D₆₇	1	1	yes	characteristic
304	C₄³	C₂×D₆₇	1	1	yes	characteristic
305	#336	—	67	1	no	—
306	#336	—	67	1	no	—
307	#336	—	67	1	no	—
308	#336	—	67	1	no	—
309	#336	—	67	1	no	—
310	#336	—	67	1	no	—
311	#336	—	67	1	no	—
312	#336	—	67	1	no	—
313	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
314	C₂×M₁₆	—	67	4	no	—
315	C₂×M₁₆	—	134	2	no	—
316	C₂×M₁₆	—	134	2	no	—
317	C₂²×C₈	—	67	1	no	—
318	C₂²×C₈	—	67	1	no	—
319	C₂²×C₈	—	67	4	no	—
320	C₂²×C₈	—	67	1	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	—	134	2	no	—
327	C₂²×C₈	—	134	2	no	—
328	#117	C₄×D₆₇	1	4	yes	—
329	#117	C₄×D₆₇	1	4	yes	—
330	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
331	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
332	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
333	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
334	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	2	yes	—
335	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	2	yes	—
336	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
337	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
338	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
339	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
340	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
341	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
342	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
343	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
344	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
345	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
346	C₄⋊C₈	—	67	4	no	—
347	C₄⋊C₈	—	67	4	no	—
348	C₂²⋊C₈	—	67	4	no	—
349	C₂²⋊C₈	—	67	4	no	—
350	#97	—	67	4	no	—
351	#97	—	67	4	no	—
352	#97	—	67	4	no	—
353	#97	—	134	2	no	—
354	#97	—	134	2	no	—
355	#97	—	134	2	no	—
356	#97	—	134	2	no	—
357	C₄×C₈	—	67	4	no	—
358	C₄×C₈	—	67	4	no	—
359	C₄×C₈	—	67	4	no	—
360	C₄×C₈	—	134	2	no	—
361	C₄×C₈	—	134	2	no	—
362	C₄×C₈	—	134	2	no	—
363	C₄×C₈	—	134	2	no	—
364	#95	C₂×Dic₆₇	1	2	yes	—
365	#95	C₂×Dic₆₇	1	2	yes	—
366	#95	C₂×Dic₆₇	1	2	yes	—
367	#95	C₂×Dic₆₇	1	2	yes	—
368	#95	C₂×Dic₆₇	1	2	yes	—
369	#95	C₂×Dic₆₇	1	2	yes	—
370	C₂⁴	C₂²×Dic₆₇	1	1	yes	characteristic
371	C₂²×C₄	—	2	1	no	—
372	C₂²×C₄	—	2	1	no	—
373	C₂²×C₄	—	2	1	no	—
374	C₂²×C₄	—	2	1	no	—
375	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
376	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
377	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
378	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
379	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
380	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
381	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
382	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
383	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
384	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
385	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
386	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
387	M₁₆	—	134	8	no	—
388	C₂×C₈	—	67	2	no	—
389	C₂×C₈	—	67	2	no	—
390	C₂×C₈	—	67	2	no	—
391	C₂×C₈	—	67	2	no	—
392	C₂×C₈	—	67	2	no	—
393	C₂×C₈	—	67	2	no	—
394	C₂×C₈	—	67	2	no	—
395	C₂×C₈	—	67	2	no	—
396	C₂×C₈	—	67	2	no	—
397	C₂×C₈	—	67	2	no	—
398	C₂×C₈	—	67	2	no	—
399	C₂×C₈	—	67	2	no	—
400	C₂×C₈	—	134	8	no	—
401	C₂×C₈	—	134	1	no	—
402	C₂×C₈	—	134	1	no	—
403	C₂×C₈	—	134	1	no	—
404	C₂×C₈	—	134	1	no	—
405	C₂×C₈	—	134	1	no	—
406	C₂×C₈	—	134	1	no	—
407	C₂×C₈	—	134	1	no	—
408	C₂×C₈	—	134	1	no	—
409	C₂×C₈	—	134	1	no	—
410	C₂×C₈	—	134	1	no	—
411	C₂×C₈	—	134	1	no	—
412	C₂×C₈	—	134	1	no	—
413	C₄⋊C₄	—	2	4	no	—
414	C₄⋊C₄	—	4	2	no	—
415	C₄⋊C₄	C₄×Dic₆₇	1	4	yes	—
416	C₄⋊C₄	C₄×Dic₆₇	1	4	yes	—
417	C₂²⋊C₄	—	2	4	no	—
418	C₂²⋊C₄	—	4	2	no	—
419	C₂²⋊C₄	C₄×Dic₆₇	1	4	yes	—
420	C₂²⋊C₄	C₄×Dic₆₇	1	4	yes	—
421	C₄²	—	2	4	no	—
422	C₄²	—	2	1	no	—
423	C₄²	—	2	1	no	—
424	C₄²	—	2	1	no	—
425	C₄²	—	2	1	no	—
426	C₄²	—	2	1	no	—
427	C₄²	—	2	1	no	—
428	C₄²	—	2	1	no	—
429	C₄²	—	2	1	no	—
430	C₄²	—	4	2	no	—
431	C₄²	C₂×C₄×D₆₇	1	2	yes	—
432	C₄²	C₂×C₄×D₆₇	1	2	yes	—
433	C₄²	C₂×C₄×D₆₇	1	2	yes	—
434	C₄²	C₂×C₄×D₆₇	1	2	yes	—
435	C₂³	—	2	1	no	—
436	C₂³	—	2	1	no	—
437	C₂³	—	2	2	no	—
438	C₂³	—	2	2	no	—
439	C₂³	C₂×C₄×Dic₆₇	1	2	yes	—
440	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	1	no	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
463	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
464	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
465	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
466	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
467	C₂×C₄	#14524050	1	4	yes	—
468	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
469	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
470	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
471	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
472	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
473	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
474	C₈	—	67	4	no	—
475	C₈	—	67	4	no	—
476	C₈	—	134	2	no	—
477	C₈	—	134	2	no	—
478	C₈	—	134	2	no	—
479	C₈	—	134	2	no	—
480	C₈	—	134	2	no	—
481	C₈	—	134	2	no	—
482	C₂²	—	2	2	no	—
483	C₂²	—	2	2	no	—
484	C₂²	—	2	2	no	—
485	C₂²	—	4	2	no	—
486	C₂²	—	4	2	no	—
487	C₂²	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
488	C₂²	#41×Dic₆₇	1	1	yes	characteristic
489	C₂²	#41×Dic₆₇	1	1	yes	characteristic
490	C₂²	C₂×#14524050	1	2	yes	—
491	C₂²	#16864163	1	2	yes	—
492	C₄	—	2	2	no	—
493	C₄	—	2	2	no	—
494	C₄	—	2	2	no	—
495	C₄	—	2	2	no	—
496	C₄	—	2	4	no	—
497	C₄	—	2	2	no	—
498	C₄	—	4	2	no	—
499	C₄	—	4	2	no	—
500	C₄	—	4	2	no	—
501	C₄	#117×D₆₇	1	1	yes	characteristic
502	C₄	#117×D₆₇	1	1	yes	characteristic
503	C₄	#16863864	1	1	yes	characteristic
504	C₄	#16863864	1	1	yes	characteristic
505	C₂	—	4	2	no	—
506	C₂	#117×Dic₆₇	1	1	yes	characteristic
507	C₂	C₄×#14524050	1	1	yes	characteristic
508	C₂	#22568637	1	1	yes	characteristic
509	C₂	#22569205	1	2	yes	—
510	C₂	#22569214	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator	C₂²×C₆₇	C₂²×C₄²
Fitting	C₆₇×#2298	C₂
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂⁴×C₆₇	C₂²×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#2298	D₆₇
hypocenter	C₆₇	#20478
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ #2298

Complements

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

N	H ≅ G/N	type
3. #22560084	C₂	semidirect
4. #22560084	C₂	semidirect
5. #22560078	C₂	semidirect
6. C₄⋊C₄×C₆₇⋊C₈	C₂	semidirect
11. #22559961	C₂	semidirect
12. #22559955	C₂	semidirect
13. #22559896	C₂	semidirect
14. C₄×#14523954	C₂	semidirect
18. #16863457	C₄	semidirect
23. #16863352	C₄	semidirect
271. C₆₇	#20478	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863457	2
C₂×#16863352	2
#22560084	2
#22560078	2
C₄⋊C₄×C₆₇⋊C₈	2
#22559993	2
#22559987	2
C₂²⋊C₄×C₆₇⋊C₈	2
#22559961	2
#22559955	2
#22559896	2
C₄×#14523954	2
C₆₇×#2298	2
#20478	67