Subgroups — #30827549

Order 17152.

650 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569341	C₂	1	1	yes	maximal, characteristic, complemented
2	#22569334	C₂	1	1	yes	maximal, characteristic, complemented
3	#22559112	C₂	1	2	yes	maximal, complemented
4	#22559060	C₂	1	2	yes	maximal, complemented
5	#22559039	C₂	1	2	yes	maximal, complemented
6	SD₁₆⋊(C₆₇⋊C₈)	C₂	1	2	yes	maximal, complemented
7	#22558968	C₂	1	1	yes	maximal, characteristic, complemented
8	#22558648	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558648	C₂	1	1	yes	maximal, characteristic, complemented
10	#22558498	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#135×C₆₇	C₂	1	1	yes	maximal, characteristic
12	#16863527	C₂²	1	2	yes	—
13	Q₈×C₆₇⋊C₈	C₂²	1	2	yes	—
14	#16863474	—	2	2	no	—
15	#16863474	C₂²	1	2	yes	—
16	#16863468	—	2	2	no	—
17	D₄×C₆₇⋊C₈	C₂²	1	2	yes	—
18	#16863453	—	2	1	no	—
19	#16863449	—	2	1	no	—
20	#16863449	C₂²	1	1	yes	characteristic
21	#16863420	—	2	1	no	—
22	#16863420	C₂²	1	1	yes	characteristic
23	#16863420	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
25	C₄×#11761612	C₂²	1	1	yes	characteristic
26	#16863149	C₂²	1	2	yes	—
27	#16863149	C₂²	1	2	yes	—
28	Q₈⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
29	Q₈⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
30	D₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
31	D₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
32	#16863118	C₂²	1	2	yes	—
33	#16863117	C₂²	1	2	yes	—
34	C₂×#135×C₆₇	C₄	1	1	yes	characteristic
35	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
36	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
37	C₄×Q₁₆×C₆₇	C₄	1	1	yes	characteristic
38	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
39	C₄×SD₁₆×C₆₇	C₄	1	1	yes	characteristic
40	C₄×SD₁₆×C₆₇	C₄	1	1	yes	characteristic
41	C₄×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
42	C₄×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
43	C₄×D₈×C₆₇	C₄	1	1	yes	characteristic
44	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
45	#427×C₆₇	C₄	1	1	yes	characteristic
46	#416×C₆₇	C₄	1	1	yes	characteristic
47	#416×C₆₇	C₄	1	1	yes	characteristic
48	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
49	C₂×#11761612	—	2	2	no	—
50	C₂×#11761612	—	2	1	no	—
51	C₂×#11761612	—	2	1	no	—
52	C₂×#11761612	—	2	1	no	—
53	C₂²×C₆₇⋊C₈	—	2	2	no	—
54	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
55	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
56	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
57	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
58	C₄⋊(C₆₇⋊C₈)	D₄	1	1	yes	characteristic
59	C₄⋊(C₆₇⋊C₈)	D₄	1	1	yes	characteristic
60	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
61	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
62	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
63	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
64	C₂²⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
65	#14523954	—	2	2	no	—
66	#14523954	—	2	1	no	—
67	C₄×C₆₇⋊C₈	—	2	2	no	—
68	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
69	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
70	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
71	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
72	#135×C₆₇	—	2	4	no	—
73	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
74	C₂×SD₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
75	C₂×SD₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
76	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic
77	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
78	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
79	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
80	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
81	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
82	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
83	C₄×D₄×C₆₇	—	2	1	no	—
84	C₄×D₄×C₆₇	—	2	1	no	—
85	C₄×D₄×C₆₇	—	2	1	no	—
86	C₄×D₄×C₆₇	—	2	1	no	—
87	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
88	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
89	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
90	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
91	#117×C₆₇	—	2	1	no	—
92	#117×C₆₇	—	2	1	no	—
93	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
94	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
95	C₂×C₄²×C₆₇	—	2	1	no	—
96	C₂×C₄²×C₆₇	—	2	1	no	—
97	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
98	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
99	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
100	#106×C₆₇	C₂×C₄	1	1	yes	characteristic
101	#106×C₆₇	C₂×C₄	1	1	yes	characteristic
102	#103×C₆₇	C₂×C₄	1	1	yes	characteristic
103	#103×C₆₇	C₂×C₄	1	1	yes	characteristic
104	#103×C₆₇	C₂×C₄	1	1	yes	characteristic
105	#103×C₆₇	C₂×C₄	1	1	yes	characteristic
106	#102×C₆₇	C₂×C₄	1	1	yes	characteristic
107	#102×C₆₇	C₂×C₄	1	1	yes	characteristic
108	#102×C₆₇	C₂×C₄	1	1	yes	characteristic
109	#102×C₆₇	C₂×C₄	1	1	yes	characteristic
110	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
111	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
112	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
113	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
114	#11761612	—	4	2	no	—
115	#11761612	—	8	2	no	—
116	#11761612	—	8	1	no	—
117	#11761612	—	8	1	no	—
118	C₂×C₆₇⋊C₈	—	2	2	no	—
119	C₂×C₆₇⋊C₈	—	2	2	no	—
120	C₂×C₆₇⋊C₈	—	2	2	no	—
121	C₂×C₆₇⋊C₈	—	8	2	no	—
122	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
123	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
124	#41×C₆₇	—	2	2	no	—
125	#41×C₆₇	—	2	2	no	—
126	#41×C₆₇	—	4	1	no	—
127	#41×C₆₇	—	4	1	no	—
128	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
129	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
130	C₂×D₄×C₆₇	—	2	1	no	—
131	C₂×D₄×C₆₇	—	2	1	no	—
132	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
133	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
134	C₂²×C₄×C₆₇	—	2	1	no	—
135	C₂²×C₄×C₆₇	—	2	1	no	—
136	C₂²×C₄×C₆₇	—	2	1	no	—
137	C₂²×C₄×C₆₇	—	2	1	no	—
138	C₂²×C₄×C₆₇	—	2	1	no	—
139	C₂²×C₄×C₆₇	—	2	1	no	—
140	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
141	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
142	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
143	Q₁₆×C₆₇	—	2	2	no	—
144	SD₁₆×C₆₇	—	2	2	no	—
145	SD₁₆×C₆₇	M₁₆	1	4	yes	complemented
146	D₈×C₆₇	—	2	2	no	—
147	C₂×C₈×C₆₇	—	2	2	no	—
148	C₂×C₈×C₆₇	—	4	1	no	—
149	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
150	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
151	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
152	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
153	C₄⋊C₄×C₆₇	—	2	1	no	—
154	C₄⋊C₄×C₆₇	—	2	1	no	—
155	C₄⋊C₄×C₆₇	—	2	1	no	—
156	C₄⋊C₄×C₆₇	—	2	1	no	—
157	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
158	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
159	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
160	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
161	C₂²⋊C₄×C₆₇	—	2	1	no	—
162	C₂²⋊C₄×C₆₇	—	2	1	no	—
163	C₂²⋊C₄×C₆₇	—	2	1	no	—
164	C₂²⋊C₄×C₆₇	—	2	1	no	—
165	C₂²⋊C₄×C₆₇	—	2	1	no	—
166	C₂²⋊C₄×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	C₂²×C₄	1	1	yes	characteristic
174	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
175	C₄²×C₆₇	#41	1	1	yes	characteristic
176	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
177	C₂³×C₆₇	—	2	1	no	—
178	C₂³×C₆₇	—	2	1	no	—
179	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
180	Q₈×C₆₇	—	2	1	no	—
181	Q₈×C₆₇	—	2	1	no	—
182	Q₈×C₆₇	—	2	1	no	—
183	Q₈×C₆₇	C₂×M₁₆	1	2	yes	—
184	D₄×C₆₇	—	2	1	no	—
185	D₄×C₆₇	—	2	1	no	—
186	D₄×C₆₇	—	2	1	no	—
187	D₄×C₆₇	—	4	2	no	—
188	D₄×C₆₇	—	4	2	no	—
189	D₄×C₆₇	C₂×M₁₆	1	2	yes	—
190	C₂×C₄×C₆₇	—	2	1	no	—
191	C₂×C₄×C₆₇	—	2	1	no	—
192	C₂×C₄×C₆₇	—	2	1	no	—
193	C₂×C₄×C₆₇	—	2	1	no	—
194	C₂×C₄×C₆₇	—	2	1	no	—
195	C₂×C₄×C₆₇	—	2	2	no	—
196	C₂×C₄×C₆₇	—	2	1	no	—
197	C₂×C₄×C₆₇	—	2	1	no	—
198	C₂×C₄×C₆₇	—	2	1	no	—
199	C₂×C₄×C₆₇	—	2	1	no	—
200	C₂×C₄×C₆₇	—	2	1	no	—
201	C₂×C₄×C₆₇	—	4	2	no	—
202	C₂×C₄×C₆₇	—	4	2	no	—
203	C₂×C₄×C₆₇	—	4	1	no	—
204	C₂×C₄×C₆₇	—	4	1	no	—
205	C₂×C₄×C₆₇	—	4	1	no	—
206	C₂×C₄×C₆₇	—	4	1	no	—
207	C₂×C₄×C₆₇	—	4	2	no	—
208	C₂×C₄×C₆₇	—	4	2	no	—
209	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
210	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
211	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
212	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
213	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
215	C₈×C₆₇	—	2	1	no	—
216	C₈×C₆₇	—	2	1	no	—
217	C₈×C₆₇	—	2	1	no	—
218	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
219	C₆₇⋊C₈	—	2	2	no	—
220	C₆₇⋊C₈	—	2	2	no	—
221	C₆₇⋊C₈	—	4	2	no	—
222	C₆₇⋊C₈	—	4	2	no	—
223	C₆₇⋊C₈	—	4	2	no	—
224	C₂²×C₆₇	—	2	1	no	—
225	C₂²×C₆₇	—	2	2	no	—
226	C₂²×C₆₇	—	2	1	no	—
227	C₂²×C₆₇	—	2	1	no	—
228	C₂²×C₆₇	—	4	1	no	—
229	C₂²×C₆₇	—	4	1	no	—
230	C₂²×C₆₇	—	4	1	no	—
231	C₂²×C₆₇	—	4	1	no	—
232	C₂²×C₆₇	—	4	1	no	—
233	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
234	C₄×C₆₇	—	2	1	no	—
235	C₄×C₆₇	—	2	2	no	—
236	C₄×C₆₇	—	2	1	no	—
237	C₄×C₆₇	—	2	1	no	—
238	C₄×C₆₇	—	4	1	no	—
239	C₄×C₆₇	—	4	1	no	—
240	C₄×C₆₇	—	4	1	no	—
241	C₄×C₆₇	—	4	1	no	—
242	C₄×C₆₇	—	4	1	no	—
243	C₄×C₆₇	#444	1	2	yes	—
244	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
245	C₄×C₆₇	#568	1	1	yes	characteristic
246	C₄×C₆₇	#576	1	1	yes	characteristic
247	C₄×C₆₇	#578	1	1	yes	characteristic
248	#18061	—	67	1	no	maximal, Sylow
249	C₂×C₆₇	—	4	1	no	—
250	C₂×C₆₇	—	4	2	no	—
251	C₂×C₆₇	—	8	1	no	—
252	C₂×C₆₇	D₄×M₁₆	1	1	yes	characteristic
253	C₂×C₆₇	#2951	1	1	yes	characteristic
254	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
255	#2991	—	67	1	no	—
256	#2990	—	67	1	no	—
257	C₄×#135	D₆₇	1	1	yes	characteristic
258	#1592	—	67	2	no	—
259	#1584	—	67	2	no	—
260	#1581	—	67	2	no	—
261	SD₁₆⋊C₈	—	67	2	no	—
262	#1567	—	67	1	no	—
263	#1489	—	67	1	no	—
264	#1489	—	67	1	no	—
265	#1451	—	67	1	no	—
266	C₆₇	#18061	1	1	yes	characteristic, complemented, Sylow
267	C₂×#135	Dic₆₇	1	1	yes	characteristic
268	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
269	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
270	#446	—	67	2	no	—
271	C₈×Q₈	—	67	2	no	—
272	C₄×Q₁₆	Dic₆₇	1	1	yes	characteristic
273	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
274	C₄×SD₁₆	Dic₆₇	1	1	yes	characteristic
275	C₄×SD₁₆	Dic₆₇	1	1	yes	characteristic
276	C₄×SD₁₆	C₂×D₆₇	1	1	yes	characteristic
277	C₄×SD₁₆	C₂×D₆₇	1	1	yes	characteristic
278	C₄×D₈	Dic₆₇	1	1	yes	characteristic
279	C₄×D₈	C₂×D₆₇	1	1	yes	characteristic
280	#436	—	67	2	no	—
281	#436	—	134	2	no	—
282	#435	—	134	2	no	—
283	C₈×D₄	—	67	2	no	—
284	#432	—	134	1	no	—
285	#431	—	67	1	no	—
286	#431	—	134	1	no	—
287	#427	Dic₆₇	1	1	yes	characteristic
288	#423	—	67	1	no	—
289	#423	—	67	1	no	—
290	#423	—	134	1	no	—
291	C₂×C₄⋊C₈	—	134	1	no	—
292	#416	Dic₆₇	1	1	yes	characteristic
293	#416	Dic₆₇	1	1	yes	characteristic
294	C₄×M₁₆	—	67	1	no	—
295	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
296	#335	—	67	2	no	—
297	#335	—	67	2	no	—
298	Q₈⋊C₈	—	67	2	no	—
299	Q₈⋊C₈	—	67	2	no	—
300	D₄⋊C₈	—	67	2	no	—
301	D₄⋊C₈	—	67	2	no	—
302	#322	—	67	2	no	—
303	#322	—	67	2	no	—
304	C₂×#41	C₂×Dic₆₇	1	1	yes	characteristic
305	C₂×#41	C₂×Dic₆₇	1	1	yes	characteristic
306	#135	—	2	4	no	—
307	C₂×Q₁₆	C₂×Dic₆₇	1	1	yes	characteristic
308	C₂×SD₁₆	C₂×Dic₆₇	1	1	yes	characteristic
309	C₂×SD₁₆	C₂×Dic₆₇	1	1	yes	characteristic
310	C₂×D₈	C₂×Dic₆₇	1	1	yes	characteristic
311	C₂×M₁₆	—	134	2	no	—
312	C₂×M₁₆	—	134	1	no	—
313	C₂×M₁₆	—	134	1	no	—
314	C₂×M₁₆	—	134	1	no	—
315	C₂²×C₈	—	134	2	no	—
316	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
317	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
318	C₄×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
319	C₄×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
320	C₄×Q₈	C₂²×D₆₇	1	1	yes	characteristic
321	C₄×Q₈	C₂²×D₆₇	1	1	yes	characteristic
322	C₄×D₄	—	2	1	no	—
323	C₄×D₄	—	2	1	no	—
324	C₄×D₄	—	2	1	no	—
325	C₄×D₄	—	2	1	no	—
326	C₄×D₄	C₂×Dic₆₇	1	1	yes	characteristic
327	C₄×D₄	C₂×Dic₆₇	1	1	yes	characteristic
328	C₄×D₄	C₂²×D₆₇	1	1	yes	characteristic
329	C₄×D₄	C₂²×D₆₇	1	1	yes	characteristic
330	#117	—	2	1	no	—
331	#117	—	2	1	no	—
332	#117	C₂×Dic₆₇	1	1	yes	characteristic
333	#117	C₂×Dic₆₇	1	1	yes	characteristic
334	C₂×C₄²	—	2	1	no	—
335	C₂×C₄²	—	2	1	no	—
336	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
337	#107	C₂×Dic₆₇	1	1	yes	characteristic
338	#107	C₂×Dic₆₇	1	1	yes	characteristic
339	#106	C₂×Dic₆₇	1	1	yes	characteristic
340	#106	C₂×Dic₆₇	1	1	yes	characteristic
341	C₄⋊C₈	—	67	1	no	—
342	C₄⋊C₈	—	67	1	no	—
343	C₄⋊C₈	—	67	2	no	—
344	C₄⋊C₈	—	67	2	no	—
345	C₄⋊C₈	—	134	2	no	—
346	C₄⋊C₈	—	134	2	no	—
347	C₄⋊C₈	—	134	1	no	—
348	C₄⋊C₈	—	134	1	no	—
349	#103	C₂×Dic₆₇	1	1	yes	characteristic
350	#103	C₂×Dic₆₇	1	1	yes	characteristic
351	#103	C₂×Dic₆₇	1	1	yes	characteristic
352	#103	C₂×Dic₆₇	1	1	yes	characteristic
353	#102	C₂×Dic₆₇	1	1	yes	characteristic
354	#102	C₂×Dic₆₇	1	1	yes	characteristic
355	#102	C₂×Dic₆₇	1	1	yes	characteristic
356	#102	C₂×Dic₆₇	1	1	yes	characteristic
357	C₂²⋊C₈	—	67	2	no	—
358	C₂²⋊C₈	—	134	2	no	—
359	C₂²⋊C₈	—	134	2	no	—
360	#97	—	134	2	no	—
361	#97	—	134	1	no	—
362	C₄×C₈	—	67	2	no	—
363	C₄×C₈	—	67	2	no	—
364	C₄×C₈	—	134	2	no	—
365	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
366	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
367	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
368	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
369	#41	—	2	2	no	—
370	#41	—	2	2	no	—
371	#41	—	4	1	no	—
372	#41	—	4	1	no	—
373	C₂×Q₈	C₂²×Dic₆₇	1	1	yes	characteristic
374	C₂×Q₈	C₂²×Dic₆₇	1	1	yes	characteristic
375	C₂×D₄	—	2	1	no	—
376	C₂×D₄	—	2	1	no	—
377	C₂×D₄	C₂²×Dic₆₇	1	1	yes	characteristic
378	C₂×D₄	C₂²×Dic₆₇	1	1	yes	characteristic
379	C₂²×C₄	—	2	1	no	—
380	C₂²×C₄	—	2	1	no	—
381	C₂²×C₄	—	2	1	no	—
382	C₂²×C₄	—	2	1	no	—
383	C₂²×C₄	—	2	1	no	—
384	C₂²×C₄	—	2	1	no	—
385	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
386	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
387	C₂²×C₄	#11761627	1	1	yes	characteristic
388	Q₁₆	—	2	2	no	—
389	SD₁₆	—	2	2	no	—
390	SD₁₆	#11761612	1	4	yes	complemented
391	D₈	—	2	2	no	—
392	M₁₆	—	268	2	no	—
393	M₁₆	—	536	1	no	—
394	M₁₆	—	536	1	no	—
395	M₁₆	—	536	2	no	—
396	C₂×C₈	—	2	2	no	—
397	C₂×C₈	—	4	1	no	—
398	C₂×C₈	—	536	2	no	—
399	C₂×C₈	—	67	2	no	—
400	C₂×C₈	—	67	2	no	—
401	C₂×C₈	—	134	2	no	—
402	C₂×C₈	—	134	2	no	—
403	C₂×C₈	—	134	2	no	—
404	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
405	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
406	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
407	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
408	C₄⋊C₄	—	2	1	no	—
409	C₄⋊C₄	—	2	1	no	—
410	C₄⋊C₄	—	2	1	no	—
411	C₄⋊C₄	—	2	1	no	—
412	C₄⋊C₄	C₂²×Dic₆₇	1	1	yes	characteristic
413	C₄⋊C₄	C₂²×Dic₆₇	1	1	yes	characteristic
414	C₄⋊C₄	C₂²×Dic₆₇	1	1	yes	characteristic
415	C₄⋊C₄	C₂²×Dic₆₇	1	1	yes	characteristic
416	C₂²⋊C₄	—	2	1	no	—
417	C₂²⋊C₄	—	2	1	no	—
418	C₂²⋊C₄	—	2	1	no	—
419	C₂²⋊C₄	—	2	1	no	—
420	C₂²⋊C₄	—	2	1	no	—
421	C₂²⋊C₄	—	2	1	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	C₂²×Dic₆₇	1	1	yes	characteristic
429	C₄²	D₄×D₆₇	1	1	yes	characteristic
430	C₄²	#11761627	1	1	yes	characteristic
431	C₄²	C₂³×D₆₇	1	1	yes	characteristic
432	C₂³	—	2	1	no	—
433	C₂³	—	2	1	no	—
434	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
435	Q₈	—	2	1	no	—
436	Q₈	—	2	1	no	—
437	Q₈	—	2	1	no	—
438	Q₈	C₂×#11761612	1	2	yes	—
439	D₄	—	2	1	no	—
440	D₄	—	2	1	no	—
441	D₄	—	2	1	no	—
442	D₄	—	4	2	no	—
443	D₄	—	4	2	no	—
444	D₄	C₂×#11761612	1	2	yes	—
445	C₂×C₄	—	2	1	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	1	no	—
448	C₂×C₄	—	2	1	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	2	1	no	—
454	C₂×C₄	—	2	1	no	—
455	C₂×C₄	—	2	1	no	—
456	C₂×C₄	—	4	2	no	—
457	C₂×C₄	—	4	2	no	—
458	C₂×C₄	—	4	2	no	—
459	C₂×C₄	—	4	2	no	—
460	C₂×C₄	—	4	1	no	—
461	C₂×C₄	—	4	1	no	—
462	C₂×C₄	—	4	1	no	—
463	C₂×C₄	—	4	1	no	—
464	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
465	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
466	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
467	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
468	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
469	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
470	C₈	—	2	1	no	—
471	C₈	—	2	1	no	—
472	C₈	—	2	1	no	—
473	C₈	—	268	2	no	—
474	C₈	—	268	2	no	—
475	C₈	—	268	2	no	—
476	C₈	—	134	2	no	—
477	C₈	—	134	2	no	—
478	C₈	C₂×#11761612	1	2	yes	—
479	C₂²	—	2	1	no	—
480	C₂²	—	2	1	no	—
481	C₂²	—	2	1	no	—
482	C₂²	—	2	2	no	—
483	C₂²	—	4	1	no	—
484	C₂²	—	4	1	no	—
485	C₂²	—	4	1	no	—
486	C₂²	—	4	1	no	—
487	C₂²	—	4	1	no	—
488	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
489	C₄	—	2	1	no	—
490	C₄	—	2	1	no	—
491	C₄	—	2	1	no	—
492	C₄	—	2	2	no	—
493	C₄	—	4	1	no	—
494	C₄	—	4	1	no	—
495	C₄	—	4	1	no	—
496	C₄	—	4	1	no	—
497	C₄	—	4	1	no	—
498	C₄	#16863518	1	2	yes	—
499	C₄	C₂²×#11761612	1	1	yes	characteristic
500	C₄	#16864163	1	1	yes	characteristic
501	C₄	#16864198	1	1	yes	characteristic
502	C₄	#16864210	1	1	yes	characteristic
503	C₂	—	4	1	no	—
504	C₂	—	4	2	no	—
505	C₂	—	8	1	no	—
506	C₂	D₄×#11761612	1	1	yes	characteristic
507	C₂	#22569085	1	1	yes	characteristic
508	C₂	C₂×#16863518	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₄×#135×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#135×C₆₇	C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₄×#135	D₆₇
hypocenter	C₆₇	#18061
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
1. #22569341	C₂	semidirect
2. #22569334	C₂	semidirect
3. #22559112	C₂	semidirect
4. #22559060	C₂	semidirect
5. #22559039	C₂	semidirect
6. SD₁₆⋊(C₆₇⋊C₈)	C₂	semidirect
7. #22558968	C₂	semidirect
8. #22558648	C₂	semidirect
9. #22558648	C₂	semidirect
10. #22558498	C₂	semidirect
145. SD₁₆×C₆₇	M₁₆	semidirect
266. C₆₇	#18061	semidirect
390. SD₁₆	#11761612	semidirect

Maximal subgroups

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

subgroup	index
#22569341	2
#22569334	2
#22559112	2
#22559060	2
#22559039	2
SD₁₆⋊(C₆₇⋊C₈)	2
#22558968	2
#22558648	2
#22558498	2
C₄×#135×C₆₇	2
#18061	67