Subgroups — C₆₇×#22765 #30702016

Order 17152.

830 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×SD₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2357	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2351	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1936	C₂	1	2	yes	maximal
5	C₆₇×#1920	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1880	C₂	1	2	yes	maximal
7	C₆₇×#1835	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1805	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1804	C₂	1	1	yes	maximal, characteristic
10	C₆₇×#1770	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1763	C₂	1	2	yes	maximal, complemented
12	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
16	C₄×SD₁₆×C₆₇	C₄	1	8	yes	complemented
17	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#103×C₆₇	C₂²	1	2	yes	—
20	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#102×C₆₇	C₂²	1	2	yes	—
22	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#97×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	#390×C₆₇	C₂²	1	2	yes	—
26	#389×C₆₇	—	2	2	no	—
27	#389×C₆₇	C₂²	1	2	yes	—
28	#388×C₆₇	—	2	2	no	—
29	#382×C₆₇	—	2	1	no	—
30	#382×C₆₇	C₂²	1	1	yes	characteristic
31	#382×C₆₇	C₂²	1	1	yes	characteristic
32	#381×C₆₇	—	2	1	no	—
33	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
34	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
35	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
36	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
37	#340×C₆₇	C₂²	1	2	yes	—
38	#340×C₆₇	C₂²	1	2	yes	—
39	#340×C₆₇	C₂²	1	2	yes	—
40	#340×C₆₇	C₂²	1	2	yes	—
41	#336×C₆₇	C₂²	1	2	yes	—
42	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
43	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₆₇	—	2	1	no	—
45	C₂×SD₁₆×C₆₇	—	2	2	no	—
46	C₂×SD₁₆×C₆₇	—	2	2	no	—
47	C₂×SD₁₆×C₆₇	C₂×C₄	1	4	yes	—
48	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
49	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
51	C₄×Q₈×C₆₇	—	2	2	no	—
52	C₄×Q₈×C₆₇	—	2	4	no	—
53	C₄×Q₈×C₆₇	—	4	2	no	—
54	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
55	C₄×D₄×C₆₇	—	2	2	no	—
56	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
57	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
59	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
60	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
61	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
63	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
66	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
67	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
68	C₂×C₄²×C₆₇	—	2	1	no	—
69	C₂×C₄²×C₆₇	—	2	2	no	—
70	C₂×C₄²×C₆₇	—	2	2	no	—
71	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
72	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
73	#106×C₆₇	C₂×C₄	1	4	yes	—
74	C₄⋊C₈×C₆₇	—	2	2	no	—
75	#103×C₆₇	—	2	4	no	—
76	#103×C₆₇	C₂×C₄	1	4	yes	—
77	#102×C₆₇	—	4	2	no	—
78	#102×C₆₇	C₂×C₄	1	4	yes	—
79	#97×C₆₇	—	4	2	no	—
80	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
81	#95×C₆₇	—	2	1	no	—
82	#95×C₆₇	—	2	1	no	—
83	#95×C₆₇	—	2	2	no	—
84	#95×C₆₇	—	2	2	no	—
85	#95×C₆₇	—	2	1	no	—
86	#95×C₆₇	—	2	1	no	—
87	#95×C₆₇	D₄	1	1	yes	characteristic
88	#95×C₆₇	D₄	1	1	yes	characteristic
89	C₂⁴×C₆₇	—	2	1	no	—
90	C₂×Q₈×C₆₇	—	2	1	no	—
91	C₂×Q₈×C₆₇	—	2	1	no	—
92	C₂×Q₈×C₆₇	—	2	1	no	—
93	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
94	C₂×Q₈×C₆₇	#41	1	2	yes	—
95	C₂×Q₈×C₆₇	#41	1	2	yes	—
96	C₂×D₄×C₆₇	—	2	1	no	—
97	C₂×D₄×C₆₇	—	2	1	no	—
98	C₂×D₄×C₆₇	—	2	1	no	—
99	C₂×D₄×C₆₇	—	2	1	no	—
100	C₂×D₄×C₆₇	—	2	1	no	—
101	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
102	C₂²×C₄×C₆₇	—	2	1	no	—
103	C₂²×C₄×C₆₇	—	2	1	no	—
104	C₂²×C₄×C₆₇	—	2	1	no	—
105	C₂²×C₄×C₆₇	—	2	4	no	—
106	C₂²×C₄×C₆₇	—	2	2	no	—
107	C₂²×C₄×C₆₇	—	2	2	no	—
108	C₂²×C₄×C₆₇	—	8	2	no	—
109	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
110	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
114	SD₁₆×C₆₇	—	2	8	no	—
115	C₂×C₈×C₆₇	—	2	1	no	—
116	C₂×C₈×C₆₇	—	2	1	no	—
117	C₂×C₈×C₆₇	—	2	1	no	—
118	C₂×C₈×C₆₇	—	2	1	no	—
119	C₂×C₈×C₆₇	—	2	2	no	—
120	C₂×C₈×C₆₇	—	2	2	no	—
121	C₂×C₈×C₆₇	—	2	2	no	—
122	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
123	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
124	C₄⋊C₄×C₆₇	—	2	4	no	—
125	C₄⋊C₄×C₆₇	—	2	1	no	—
126	C₄⋊C₄×C₆₇	—	2	1	no	—
127	C₄⋊C₄×C₆₇	—	2	2	no	—
128	C₄⋊C₄×C₆₇	—	2	2	no	—
129	C₄⋊C₄×C₆₇	—	2	2	no	—
130	C₄⋊C₄×C₆₇	—	2	2	no	—
131	C₄⋊C₄×C₆₇	—	4	2	no	—
132	C₄⋊C₄×C₆₇	—	4	4	no	—
133	C₄⋊C₄×C₆₇	—	4	4	no	—
134	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
135	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
136	C₂²⋊C₄×C₆₇	—	2	4	no	—
137	C₂²⋊C₄×C₆₇	—	4	4	no	—
138	C₂²⋊C₄×C₆₇	—	8	2	no	—
139	C₄²×C₆₇	—	2	4	no	—
140	C₄²×C₆₇	—	2	2	no	—
141	C₄²×C₆₇	—	2	2	no	—
142	C₄²×C₆₇	—	4	4	no	—
143	C₄²×C₆₇	—	8	2	no	—
144	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
145	C₄²×C₆₇	#41	1	2	yes	—
146	C₂³×C₆₇	—	2	2	no	—
147	C₂³×C₆₇	—	4	1	no	—
148	C₂³×C₆₇	—	4	1	no	—
149	C₂³×C₆₇	—	4	2	no	—
150	C₂³×C₆₇	—	4	2	no	—
151	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
152	Q₈×C₆₇	—	2	2	no	—
153	Q₈×C₆₇	—	2	2	no	—
154	Q₈×C₆₇	—	2	2	no	—
155	Q₈×C₆₇	#117	1	4	yes	—
156	D₄×C₆₇	—	2	2	no	—
157	D₄×C₆₇	—	2	2	no	—
158	D₄×C₆₇	—	4	1	no	—
159	D₄×C₆₇	—	4	1	no	—
160	C₂×C₄×C₆₇	—	2	2	no	—
161	C₂×C₄×C₆₇	—	2	2	no	—
162	C₂×C₄×C₆₇	—	2	2	no	—
163	C₂×C₄×C₆₇	—	2	2	no	—
164	C₂×C₄×C₆₇	—	2	2	no	—
165	C₂×C₄×C₆₇	—	2	2	no	—
166	C₂×C₄×C₆₇	—	2	2	no	—
167	C₂×C₄×C₆₇	—	2	2	no	—
168	C₂×C₄×C₆₇	—	2	2	no	—
169	C₂×C₄×C₆₇	—	2	2	no	—
170	C₂×C₄×C₆₇	—	2	2	no	—
171	C₂×C₄×C₆₇	—	2	2	no	—
172	C₂×C₄×C₆₇	—	2	2	no	—
173	C₂×C₄×C₆₇	—	4	2	no	—
174	C₂×C₄×C₆₇	—	4	2	no	—
175	C₂×C₄×C₆₇	—	4	2	no	—
176	C₂×C₄×C₆₇	—	4	2	no	—
177	C₂×C₄×C₆₇	—	4	2	no	—
178	C₂×C₄×C₆₇	—	4	2	no	—
179	C₂×C₄×C₆₇	—	4	2	no	—
180	C₂×C₄×C₆₇	—	4	2	no	—
181	C₂×C₄×C₆₇	—	8	4	no	—
182	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
183	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
184	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
185	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
186	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
187	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
188	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
189	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
190	C₈×C₆₇	—	2	2	no	—
191	C₈×C₆₇	—	2	2	no	—
192	C₈×C₆₇	—	4	2	no	—
193	C₂²×C₆₇	—	4	2	no	—
194	C₂²×C₆₇	—	4	2	no	—
195	C₂²×C₆₇	—	4	2	no	—
196	C₂²×C₆₇	—	8	2	no	—
197	C₂²×C₆₇	—	8	2	no	—
198	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
199	C₂²×C₆₇	#546	1	1	yes	characteristic
200	C₂²×C₆₇	#548	1	1	yes	characteristic
201	C₂²×C₆₇	#575	1	2	yes	—
202	C₂²×C₆₇	#577	1	2	yes	—
203	C₄×C₆₇	—	2	4	no	—
204	C₄×C₆₇	—	2	2	no	—
205	C₄×C₆₇	—	2	2	no	—
206	C₄×C₆₇	—	4	2	no	—
207	C₄×C₆₇	—	4	2	no	—
208	C₄×C₆₇	—	4	2	no	—
209	C₄×C₆₇	—	4	2	no	—
210	C₄×C₆₇	—	8	2	no	—
211	C₄×C₆₇	—	8	2	no	—
212	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
213	C₄×C₆₇	#520	1	1	yes	characteristic
214	C₄×C₆₇	#552	1	1	yes	characteristic
215	C₄×C₆₇	#553	1	1	yes	characteristic
216	#22765	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
217	C₂×C₆₇	—	8	2	no	—
218	C₂×C₆₇	#2352	1	1	yes	characteristic
219	C₂×C₆₇	#2943	1	1	yes	characteristic
220	C₂×C₆₇	#2949	1	1	yes	characteristic
221	C₂×C₆₇	#3326	1	2	yes	—
222	C₂×C₆₇	#3371	1	2	yes	—
223	C₂×C₄×SD₁₆	C₂×C₆₇	1	1	yes	characteristic
224	#2357	C₂×C₆₇	1	1	yes	characteristic
225	#2351	C₂×C₆₇	1	1	yes	characteristic, complemented
226	#1936	C₂×C₆₇	1	2	yes	—
227	#1920	C₂×C₆₇	1	2	yes	complemented
228	#1880	C₂×C₆₇	1	2	yes	—
229	#1835	C₂×C₆₇	1	1	yes	characteristic, complemented
230	#1805	C₂×C₆₇	1	1	yes	characteristic, complemented
231	#1804	C₂×C₆₇	1	1	yes	characteristic
232	#1770	C₂×C₆₇	1	1	yes	characteristic, complemented
233	#1763	C₂×C₆₇	1	2	yes	complemented
234	C₆₇	#22765	1	1	yes	characteristic, direct product factor, Sylow
235	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
236	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
237	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
238	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
239	C₄×SD₁₆	C₄×C₆₇	1	8	yes	complemented
240	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
241	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
242	C₂×#103	C₂²×C₆₇	1	2	yes	—
243	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
244	C₂×#102	C₂²×C₆₇	1	2	yes	—
245	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
246	C₂×#97	C₂²×C₆₇	1	2	yes	—
247	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
248	#390	C₂²×C₆₇	1	2	yes	—
249	#389	—	2	2	no	—
250	#389	C₂²×C₆₇	1	2	yes	—
251	#388	—	2	2	no	—
252	#382	—	2	1	no	—
253	#382	C₂²×C₆₇	1	1	yes	characteristic
254	#382	C₂²×C₆₇	1	1	yes	characteristic
255	#381	—	2	1	no	—
256	C₄×C₄⋊C₄	—	2	1	no	—
257	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
258	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
259	C₄×C₂²⋊C₄	—	2	1	no	—
260	#340	C₂²×C₆₇	1	2	yes	—
261	#340	C₂²×C₆₇	1	2	yes	—
262	#340	C₂²×C₆₇	1	2	yes	—
263	#340	C₂²×C₆₇	1	2	yes	—
264	#336	C₂²×C₆₇	1	2	yes	—
265	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
266	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
267	C₂³×C₄	—	2	1	no	—
268	C₂×SD₁₆	—	2	2	no	—
269	C₂×SD₁₆	—	2	2	no	—
270	C₂×SD₁₆	C₂×C₄×C₆₇	1	4	yes	—
271	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
272	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
273	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
274	C₄×Q₈	—	2	2	no	—
275	C₄×Q₈	—	2	4	no	—
276	C₄×Q₈	—	4	2	no	—
277	C₄×Q₈	C₂×C₄×C₆₇	1	4	yes	—
278	C₄×D₄	—	2	2	no	—
279	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
280	C₂×C₄⋊C₄	—	2	1	no	—
281	C₂×C₄⋊C₄	—	2	2	no	—
282	C₂×C₄⋊C₄	—	2	2	no	—
283	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
284	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
285	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
286	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
287	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
288	C₂×C₂²⋊C₄	—	2	1	no	—
289	C₂×C₂²⋊C₄	—	2	2	no	—
290	C₂×C₂²⋊C₄	—	2	2	no	—
291	C₂×C₄²	—	2	1	no	—
292	C₂×C₄²	—	2	2	no	—
293	C₂×C₄²	—	2	2	no	—
294	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
295	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
296	#106	C₂×C₄×C₆₇	1	4	yes	—
297	C₄⋊C₈	—	2	2	no	—
298	#103	—	2	4	no	—
299	#103	C₂×C₄×C₆₇	1	4	yes	—
300	#102	—	4	2	no	—
301	#102	C₂×C₄×C₆₇	1	4	yes	—
302	#97	—	4	2	no	—
303	C₄×C₈	C₂×C₄×C₆₇	1	4	yes	—
304	#95	—	2	1	no	—
305	#95	—	2	1	no	—
306	#95	—	2	2	no	—
307	#95	—	2	2	no	—
308	#95	—	2	1	no	—
309	#95	—	2	1	no	—
310	#95	D₄×C₆₇	1	1	yes	characteristic
311	#95	D₄×C₆₇	1	1	yes	characteristic
312	C₂⁴	—	2	1	no	—
313	C₂×Q₈	—	2	1	no	—
314	C₂×Q₈	—	2	1	no	—
315	C₂×Q₈	—	2	1	no	—
316	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
317	C₂×Q₈	#41×C₆₇	1	2	yes	—
318	C₂×Q₈	#41×C₆₇	1	2	yes	—
319	C₂×D₄	—	2	1	no	—
320	C₂×D₄	—	2	1	no	—
321	C₂×D₄	—	2	1	no	—
322	C₂×D₄	—	2	1	no	—
323	C₂×D₄	—	2	1	no	—
324	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
325	C₂²×C₄	—	2	1	no	—
326	C₂²×C₄	—	2	1	no	—
327	C₂²×C₄	—	2	1	no	—
328	C₂²×C₄	—	2	4	no	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	—	2	2	no	—
331	C₂²×C₄	—	8	2	no	—
332	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
333	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
334	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
335	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
336	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
337	SD₁₆	—	2	8	no	—
338	C₂×C₈	—	2	1	no	—
339	C₂×C₈	—	2	1	no	—
340	C₂×C₈	—	2	1	no	—
341	C₂×C₈	—	2	1	no	—
342	C₂×C₈	—	2	2	no	—
343	C₂×C₈	—	2	2	no	—
344	C₂×C₈	—	2	2	no	—
345	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
346	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
347	C₄⋊C₄	—	2	4	no	—
348	C₄⋊C₄	—	2	1	no	—
349	C₄⋊C₄	—	2	1	no	—
350	C₄⋊C₄	—	2	2	no	—
351	C₄⋊C₄	—	2	2	no	—
352	C₄⋊C₄	—	2	2	no	—
353	C₄⋊C₄	—	2	2	no	—
354	C₄⋊C₄	—	4	2	no	—
355	C₄⋊C₄	—	4	4	no	—
356	C₄⋊C₄	—	4	4	no	—
357	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
358	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
359	C₂²⋊C₄	—	2	4	no	—
360	C₂²⋊C₄	—	4	4	no	—
361	C₂²⋊C₄	—	8	2	no	—
362	C₄²	—	2	4	no	—
363	C₄²	—	2	2	no	—
364	C₄²	—	2	2	no	—
365	C₄²	—	4	4	no	—
366	C₄²	—	8	2	no	—
367	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
368	C₄²	#41×C₆₇	1	2	yes	—
369	C₂³	—	2	2	no	—
370	C₂³	—	4	1	no	—
371	C₂³	—	4	1	no	—
372	C₂³	—	4	2	no	—
373	C₂³	—	4	2	no	—
374	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
375	Q₈	—	2	2	no	—
376	Q₈	—	2	2	no	—
377	Q₈	—	2	2	no	—
378	Q₈	#117×C₆₇	1	4	yes	—
379	D₄	—	2	2	no	—
380	D₄	—	2	2	no	—
381	D₄	—	4	1	no	—
382	D₄	—	4	1	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	2	no	—
395	C₂×C₄	—	2	2	no	—
396	C₂×C₄	—	4	2	no	—
397	C₂×C₄	—	4	2	no	—
398	C₂×C₄	—	4	2	no	—
399	C₂×C₄	—	4	2	no	—
400	C₂×C₄	—	4	2	no	—
401	C₂×C₄	—	4	2	no	—
402	C₂×C₄	—	4	2	no	—
403	C₂×C₄	—	4	2	no	—
404	C₂×C₄	—	8	4	no	—
405	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
406	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
407	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
408	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
409	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
410	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
411	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
412	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
413	C₈	—	2	2	no	—
414	C₈	—	2	2	no	—
415	C₈	—	4	2	no	—
416	C₂²	—	4	2	no	—
417	C₂²	—	4	2	no	—
418	C₂²	—	4	2	no	—
419	C₂²	—	8	2	no	—
420	C₂²	—	8	2	no	—
421	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
422	C₂²	#546×C₆₇	1	1	yes	characteristic
423	C₂²	#548×C₆₇	1	1	yes	characteristic
424	C₂²	#575×C₆₇	1	2	yes	—
425	C₂²	#577×C₆₇	1	2	yes	—
426	C₄	—	2	4	no	—
427	C₄	—	2	2	no	—
428	C₄	—	2	2	no	—
429	C₄	—	4	2	no	—
430	C₄	—	4	2	no	—
431	C₄	—	4	2	no	—
432	C₄	—	4	2	no	—
433	C₄	—	8	2	no	—
434	C₄	—	8	2	no	—
435	C₄	C₂×#117×C₆₇	1	1	yes	characteristic
436	C₄	#520×C₆₇	1	1	yes	characteristic
437	C₄	#552×C₆₇	1	1	yes	characteristic
438	C₄	#553×C₆₇	1	1	yes	characteristic
439	C₂	—	8	2	no	—
440	C₂	C₆₇×#2352	1	1	yes	characteristic
441	C₂	C₆₇×#2943	1	1	yes	characteristic
442	C₂	C₆₇×#2949	1	1	yes	characteristic
443	C₂	C₆₇×#3326	1	2	yes	—
444	C₂	C₆₇×#3371	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×D₄×C₆₇	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄×C₄⋊C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#2351	C₂	semidirect
5. C₆₇×#1920	C₂	semidirect
7. C₆₇×#1835	C₂	semidirect
8. C₆₇×#1805	C₂	semidirect
10. C₆₇×#1770	C₂	semidirect
11. C₆₇×#1763	C₂	semidirect
16. C₄×SD₁₆×C₆₇	C₄	semidirect
216. #22765	C₆₇	direct (N × H)
225. #2351	C₂×C₆₇	semidirect
227. #1920	C₂×C₆₇	semidirect
229. #1835	C₂×C₆₇	semidirect
230. #1805	C₂×C₆₇	semidirect
232. #1770	C₂×C₆₇	semidirect
233. #1763	C₂×C₆₇	semidirect
234. C₆₇	#22765	direct (N × H)
239. C₄×SD₁₆	C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×SD₁₆×C₆₇	2
C₆₇×#2357	2
C₆₇×#2351	2
C₆₇×#1936	2
C₆₇×#1920	2
C₆₇×#1880	2
C₆₇×#1835	2
C₆₇×#1805	2
C₆₇×#1804	2
C₆₇×#1770	2
C₆₇×#1763	2
#22765	67

Subgroups — C67×#22765 #30702016

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22765 #30702016