Subgroups — C₆₇×#22760 #30702011

Order 17152.

858 conjugacy classes of subgroups.

The subgroup lattice has 448 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×#2356	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2349	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1935	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1919	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1877	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×#1762	C₂	1	2	yes	maximal
12	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
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	#391×C₆₇	C₂²	1	2	yes	—
26	#389×C₆₇	C₂²	1	2	yes	—
27	#387×C₆₇	—	2	2	no	—
28	#385×C₆₇	—	2	2	no	—
29	#383×C₆₇	—	2	1	no	—
30	#382×C₆₇	C₂²	1	1	yes	characteristic
31	#382×C₆₇	C₂²	1	1	yes	characteristic
32	#380×C₆₇	—	2	1	no	—
33	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
34	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
35	#376×C₆₇	—	2	1	no	—
36	C₂×#95×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	C₂³×C₄×C₆₇	—	2	2	no	—
46	C₂×SD₁₆×C₆₇	—	2	2	no	—
47	C₂×SD₁₆×C₆₇	—	2	2	no	—
48	C₂×SD₁₆×C₆₇	C₂×C₄	1	4	yes	—
49	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
50	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₄×Q₈×C₆₇	—	2	2	no	—
53	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	complemented
54	C₄×D₄×C₆₇	—	2	4	no	—
55	C₄×D₄×C₆₇	—	2	2	no	—
56	C₄×D₄×C₆₇	—	4	2	no	—
57	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
58	C₂×C₄⋊C₄×C₆₇	—	2	1	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×C₄²×C₆₇	—	2	1	no	—
68	C₂×C₄²×C₆₇	—	2	2	no	—
69	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
70	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
71	#106×C₆₇	C₂×C₄	1	4	yes	complemented
72	C₄⋊C₈×C₆₇	—	2	2	no	—
73	#103×C₆₇	—	4	2	no	—
74	#103×C₆₇	C₂×C₄	1	4	yes	complemented
75	#102×C₆₇	—	2	4	no	—
76	#102×C₆₇	C₂×C₄	1	4	yes	—
77	#97×C₆₇	—	4	2	no	—
78	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	complemented
79	#95×C₆₇	—	2	1	no	—
80	#95×C₆₇	—	2	2	no	—
81	#95×C₆₇	—	2	1	no	—
82	#95×C₆₇	—	2	2	no	—
83	#95×C₆₇	—	2	2	no	—
84	#95×C₆₇	—	2	1	no	—
85	#95×C₆₇	—	2	1	no	—
86	#95×C₆₇	—	2	2	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	—	2	1	no	—
94	C₂×Q₈×C₆₇	—	2	1	no	—
95	C₂×Q₈×C₆₇	C₂²×C₄	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	C₂²×C₄	1	2	yes	—
100	C₂×D₄×C₆₇	#41	1	2	yes	—
101	C₂×D₄×C₆₇	#41	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	—	4	4	no	—
109	C₂²×C₄×C₆₇	—	4	4	no	—
110	C₂²×C₄×C₆₇	—	8	2	no	—
111	C₂²×C₄×C₆₇	—	8	2	no	—
112	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
113	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
114	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
117	SD₁₆×C₆₇	—	2	8	no	—
118	C₂×C₈×C₆₇	—	2	1	no	—
119	C₂×C₈×C₆₇	—	2	1	no	—
120	C₂×C₈×C₆₇	—	2	1	no	—
121	C₂×C₈×C₆₇	—	2	1	no	—
122	C₂×C₈×C₆₇	—	2	2	no	—
123	C₂×C₈×C₆₇	—	2	2	no	—
124	C₂×C₈×C₆₇	—	2	2	no	—
125	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
126	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
127	C₄⋊C₄×C₆₇	—	2	4	no	—
128	C₄⋊C₄×C₆₇	—	2	1	no	—
129	C₄⋊C₄×C₆₇	—	2	1	no	—
130	C₄⋊C₄×C₆₇	—	2	2	no	—
131	C₄⋊C₄×C₆₇	—	2	2	no	—
132	C₄⋊C₄×C₆₇	—	2	2	no	—
133	C₄⋊C₄×C₆₇	—	2	2	no	—
134	C₄⋊C₄×C₆₇	—	4	2	no	—
135	C₄⋊C₄×C₆₇	—	8	2	no	—
136	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
137	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
138	C₂²⋊C₄×C₆₇	—	2	4	no	—
139	C₂²⋊C₄×C₆₇	—	4	4	no	—
140	C₄²×C₆₇	—	2	4	no	—
141	C₄²×C₆₇	—	2	2	no	—
142	C₄²×C₆₇	—	2	2	no	—
143	C₄²×C₆₇	—	4	4	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	—	2	2	no	—
148	C₂³×C₆₇	—	2	2	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	—	4	1	no	—
155	Q₈×C₆₇	—	4	1	no	—
156	D₄×C₆₇	—	2	2	no	—
157	D₄×C₆₇	—	2	2	no	—
158	D₄×C₆₇	—	2	2	no	—
159	D₄×C₆₇	#117	1	4	yes	—
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	—	4	1	no	—
172	C₂×C₄×C₆₇	—	4	1	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	—	8	8	no	—
183	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
184	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
185	C₂×C₄×C₆₇	C₂³×C₄	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	C₂×#41	1	1	yes	characteristic
189	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
190	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
191	C₈×C₆₇	—	2	2	no	—
192	C₈×C₆₇	—	2	2	no	—
193	C₈×C₆₇	—	4	2	no	—
194	C₂²×C₆₇	—	2	4	no	—
195	C₂²×C₆₇	—	4	4	no	—
196	C₂²×C₆₇	—	4	2	no	—
197	C₂²×C₆₇	—	4	2	no	—
198	C₂²×C₆₇	—	4	4	no	—
199	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
200	C₂²×C₆₇	#546	1	1	yes	characteristic
201	C₂²×C₆₇	#548	1	1	yes	characteristic
202	C₂²×C₆₇	#575	1	2	yes	—
203	C₂²×C₆₇	#577	1	2	yes	—
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	—	4	2	no	—
211	C₄×C₆₇	—	8	2	no	—
212	C₄×C₆₇	—	8	2	no	—
213	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
214	C₄×C₆₇	#520	1	1	yes	characteristic
215	C₄×C₆₇	#552	1	1	yes	characteristic
216	C₄×C₆₇	#553	1	1	yes	characteristic
217	#22760	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
218	C₂×C₆₇	—	4	4	no	—
219	C₂×C₆₇	#2352	1	1	yes	characteristic
220	C₂×C₆₇	#2943	1	1	yes	characteristic
221	C₂×C₆₇	#2949	1	1	yes	characteristic
222	C₂×C₆₇	#3323	1	2	yes	—
223	C₂×C₆₇	#3376	1	2	yes	—
224	C₂×C₄×SD₁₆	C₂×C₆₇	1	1	yes	characteristic
225	#2356	C₂×C₆₇	1	1	yes	characteristic, complemented
226	#2349	C₂×C₆₇	1	1	yes	characteristic
227	#1935	C₂×C₆₇	1	2	yes	complemented
228	#1919	C₂×C₆₇	1	2	yes	complemented
229	#1877	C₂×C₆₇	1	2	yes	—
230	#1835	C₂×C₆₇	1	1	yes	characteristic, complemented
231	#1805	C₂×C₆₇	1	1	yes	characteristic, complemented
232	#1804	C₂×C₆₇	1	1	yes	characteristic
233	#1770	C₂×C₆₇	1	1	yes	characteristic, complemented
234	#1762	C₂×C₆₇	1	2	yes	—
235	C₆₇	#22760	1	1	yes	characteristic, direct product factor, Sylow
236	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
237	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
238	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
239	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
240	C₄×SD₁₆	C₄×C₆₇	1	8	yes	complemented
241	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
242	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
243	C₂×#103	C₂²×C₆₇	1	2	yes	—
244	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
245	C₂×#102	C₂²×C₆₇	1	2	yes	—
246	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
247	C₂×#97	C₂²×C₆₇	1	2	yes	—
248	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
249	#391	C₂²×C₆₇	1	2	yes	—
250	#389	C₂²×C₆₇	1	2	yes	—
251	#387	—	2	2	no	—
252	#385	—	2	2	no	—
253	#383	—	2	1	no	—
254	#382	C₂²×C₆₇	1	1	yes	characteristic
255	#382	C₂²×C₆₇	1	1	yes	characteristic
256	#380	—	2	1	no	—
257	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
258	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
259	#376	—	2	1	no	—
260	C₂×#95	—	2	1	no	—
261	#340	C₂²×C₆₇	1	2	yes	—
262	#340	C₂²×C₆₇	1	2	yes	—
263	#340	C₂²×C₆₇	1	2	yes	—
264	#340	C₂²×C₆₇	1	2	yes	—
265	#336	C₂²×C₆₇	1	2	yes	—
266	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
267	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
268	C₂³×C₄	—	2	1	no	—
269	C₂³×C₄	—	2	2	no	—
270	C₂×SD₁₆	—	2	2	no	—
271	C₂×SD₁₆	—	2	2	no	—
272	C₂×SD₁₆	C₂×C₄×C₆₇	1	4	yes	—
273	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
274	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
275	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
276	C₄×Q₈	—	2	2	no	—
277	C₄×Q₈	C₂×C₄×C₆₇	1	4	yes	complemented
278	C₄×D₄	—	2	4	no	—
279	C₄×D₄	—	2	2	no	—
280	C₄×D₄	—	4	2	no	—
281	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
282	C₂×C₄⋊C₄	—	2	1	no	—
283	C₂×C₄⋊C₄	—	2	2	no	—
284	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
285	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
286	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
287	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
288	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
289	C₂×C₂²⋊C₄	—	2	1	no	—
290	C₂×C₂²⋊C₄	—	2	2	no	—
291	C₂×C₄²	—	2	1	no	—
292	C₂×C₄²	—	2	2	no	—
293	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
294	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
295	#106	C₂×C₄×C₆₇	1	4	yes	complemented
296	C₄⋊C₈	—	2	2	no	—
297	#103	—	4	2	no	—
298	#103	C₂×C₄×C₆₇	1	4	yes	complemented
299	#102	—	2	4	no	—
300	#102	C₂×C₄×C₆₇	1	4	yes	—
301	#97	—	4	2	no	—
302	C₄×C₈	C₂×C₄×C₆₇	1	4	yes	complemented
303	#95	—	2	1	no	—
304	#95	—	2	2	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	—	2	2	no	—
311	#95	D₄×C₆₇	1	1	yes	characteristic
312	#95	D₄×C₆₇	1	1	yes	characteristic
313	C₂⁴	—	2	1	no	—
314	C₂×Q₈	—	2	1	no	—
315	C₂×Q₈	—	2	1	no	—
316	C₂×Q₈	—	2	1	no	—
317	C₂×Q₈	—	2	1	no	—
318	C₂×Q₈	—	2	1	no	—
319	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
320	C₂×D₄	—	2	1	no	—
321	C₂×D₄	—	2	1	no	—
322	C₂×D₄	—	2	1	no	—
323	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
324	C₂×D₄	#41×C₆₇	1	2	yes	—
325	C₂×D₄	#41×C₆₇	1	2	yes	—
326	C₂²×C₄	—	2	1	no	—
327	C₂²×C₄	—	2	1	no	—
328	C₂²×C₄	—	2	1	no	—
329	C₂²×C₄	—	2	4	no	—
330	C₂²×C₄	—	2	2	no	—
331	C₂²×C₄	—	2	2	no	—
332	C₂²×C₄	—	4	4	no	—
333	C₂²×C₄	—	4	4	no	—
334	C₂²×C₄	—	8	2	no	—
335	C₂²×C₄	—	8	2	no	—
336	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
337	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
338	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
339	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
340	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
341	SD₁₆	—	2	8	no	—
342	C₂×C₈	—	2	1	no	—
343	C₂×C₈	—	2	1	no	—
344	C₂×C₈	—	2	1	no	—
345	C₂×C₈	—	2	1	no	—
346	C₂×C₈	—	2	2	no	—
347	C₂×C₈	—	2	2	no	—
348	C₂×C₈	—	2	2	no	—
349	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
350	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
351	C₄⋊C₄	—	2	4	no	—
352	C₄⋊C₄	—	2	1	no	—
353	C₄⋊C₄	—	2	1	no	—
354	C₄⋊C₄	—	2	2	no	—
355	C₄⋊C₄	—	2	2	no	—
356	C₄⋊C₄	—	2	2	no	—
357	C₄⋊C₄	—	2	2	no	—
358	C₄⋊C₄	—	4	2	no	—
359	C₄⋊C₄	—	8	2	no	—
360	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
361	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
362	C₂²⋊C₄	—	2	4	no	—
363	C₂²⋊C₄	—	4	4	no	—
364	C₄²	—	2	4	no	—
365	C₄²	—	2	2	no	—
366	C₄²	—	2	2	no	—
367	C₄²	—	4	4	no	—
368	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
369	C₄²	#41×C₆₇	1	2	yes	—
370	C₂³	—	2	2	no	—
371	C₂³	—	2	2	no	—
372	C₂³	—	2	2	no	—
373	C₂³	—	4	2	no	—
374	C₂³	—	4	2	no	—
375	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
376	Q₈	—	2	2	no	—
377	Q₈	—	2	2	no	—
378	Q₈	—	4	1	no	—
379	Q₈	—	4	1	no	—
380	D₄	—	2	2	no	—
381	D₄	—	2	2	no	—
382	D₄	—	2	2	no	—
383	D₄	#117×C₆₇	1	4	yes	—
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	—	4	1	no	—
396	C₂×C₄	—	4	1	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	—	4	2	no	—
405	C₂×C₄	—	8	4	no	—
406	C₂×C₄	—	8	8	no	—
407	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
408	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
409	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
410	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
411	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
412	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
413	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
414	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
415	C₈	—	2	2	no	—
416	C₈	—	2	2	no	—
417	C₈	—	4	2	no	—
418	C₂²	—	2	4	no	—
419	C₂²	—	4	4	no	—
420	C₂²	—	4	4	no	—
421	C₂²	—	4	2	no	—
422	C₂²	—	4	2	no	—
423	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
424	C₂²	#546×C₆₇	1	1	yes	characteristic
425	C₂²	#548×C₆₇	1	1	yes	characteristic
426	C₂²	#575×C₆₇	1	2	yes	—
427	C₂²	#577×C₆₇	1	2	yes	—
428	C₄	—	2	2	no	—
429	C₄	—	2	2	no	—
430	C₄	—	4	2	no	—
431	C₄	—	4	2	no	—
432	C₄	—	4	2	no	—
433	C₄	—	4	2	no	—
434	C₄	—	4	2	no	—
435	C₄	—	8	2	no	—
436	C₄	—	8	2	no	—
437	C₄	C₂×#117×C₆₇	1	1	yes	characteristic
438	C₄	#520×C₆₇	1	1	yes	characteristic
439	C₄	#552×C₆₇	1	1	yes	characteristic
440	C₄	#553×C₆₇	1	1	yes	characteristic
441	C₂	—	4	4	no	—
442	C₂	C₆₇×#2352	1	1	yes	characteristic
443	C₂	C₆₇×#2943	1	1	yes	characteristic
444	C₂	C₆₇×#2949	1	1	yes	characteristic
445	C₂	C₆₇×#3323	1	2	yes	—
446	C₂	C₆₇×#3376	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
2. C₆₇×#2356	C₂	semidirect
4. C₆₇×#1935	C₂	semidirect
5. C₆₇×#1919	C₂	semidirect
7. C₆₇×#1835	C₂	semidirect
8. C₆₇×#1805	C₂	semidirect
10. C₆₇×#1770	C₂	semidirect
16. C₄×SD₁₆×C₆₇	C₄	semidirect
53. C₄×Q₈×C₆₇	C₂×C₄	semidirect
71. #106×C₆₇	C₂×C₄	semidirect
74. #103×C₆₇	C₂×C₄	semidirect
78. C₄×C₈×C₆₇	C₂×C₄	semidirect
217. #22760	C₆₇	direct (N × H)
225. #2356	C₂×C₆₇	semidirect
227. #1935	C₂×C₆₇	semidirect
228. #1919	C₂×C₆₇	semidirect
230. #1835	C₂×C₆₇	semidirect
231. #1805	C₂×C₆₇	semidirect
233. #1770	C₂×C₆₇	semidirect
235. C₆₇	#22760	direct (N × H)
240. C₄×SD₁₆	C₄×C₆₇	semidirect
277. C₄×Q₈	C₂×C₄×C₆₇	semidirect
295. #106	C₂×C₄×C₆₇	semidirect
298. #103	C₂×C₄×C₆₇	semidirect
302. C₄×C₈	C₂×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×#2356	2
C₆₇×#2349	2
C₆₇×#1935	2
C₆₇×#1919	2
C₆₇×#1877	2
C₆₇×#1835	2
C₆₇×#1805	2
C₆₇×#1804	2
C₆₇×#1770	2
C₆₇×#1762	2
#22760	67

Subgroups — C67×#22760 #30702011

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22760 #30702011