Subgroups — C₆₇×#20521 #30699772

Order 17152.

866 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#446×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2295	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1771	C₂	1	2	yes	maximal
5	C₆₇×#1770	C₂	1	2	yes	maximal
6	C₆₇×#1770	C₂	1	1	yes	maximal, characteristic
7	C₈×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₆₇×#1768	C₂	1	2	yes	maximal
9	C₄×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
10	C₆₇×#1726	C₂	1	2	yes	maximal
11	C₄×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	#446×C₆₇	C₄	1	8	yes	complemented
14	C₈×Q₈×C₆₇	C₄	1	8	yes	complemented
15	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
19	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#97×C₆₇	C₂²	1	2	yes	—
22	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#97×C₆₇	C₂²	1	2	yes	—
24	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
30	#391×C₆₇	C₄	1	2	yes	—
31	#389×C₆₇	C₄	1	2	yes	—
32	#387×C₆₇	C₄	1	4	yes	—
33	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
35	#376×C₆₇	C₂²	1	2	yes	—
36	C₄³×C₆₇	C₂²	1	1	yes	characteristic
37	#336×C₆₇	C₂²	1	2	yes	—
38	#336×C₆₇	C₂²	1	2	yes	—
39	#336×C₆₇	C₂²	1	2	yes	—
40	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
42	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
43	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
47	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
48	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
49	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
50	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
51	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
52	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
54	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
57	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
59	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
61	C₄⋊C₈×C₆₇	C₂×C₄	1	8	yes	—
62	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
63	C₄⋊C₈×C₆₇	C₂×C₄	1	8	yes	—
64	#97×C₆₇	—	2	2	no	—
65	#97×C₆₇	—	2	2	no	—
66	#97×C₆₇	—	2	4	no	—
67	#97×C₆₇	—	2	2	no	—
68	#97×C₆₇	C₂×C₄	1	8	yes	—
69	#97×C₆₇	Q₈	1	4	yes	complemented
70	C₄×C₈×C₆₇	—	2	4	no	—
71	C₄×C₈×C₆₇	—	2	4	no	—
72	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
73	C₄×C₈×C₆₇	C₂×C₄	1	8	yes	—
74	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
75	#95×C₆₇	C₂×C₄	1	4	yes	—
76	#95×C₆₇	C₂×C₄	1	4	yes	—
77	#95×C₆₇	C₂×C₄	1	2	yes	—
78	#95×C₆₇	C₂×C₄	1	4	yes	—
79	C₂×Q₈×C₆₇	—	2	4	no	—
80	C₂×Q₈×C₆₇	C₄²	1	4	yes	—
81	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
83	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
89	C₂×C₈×C₆₇	—	2	2	no	—
90	C₂×C₈×C₆₇	—	2	2	no	—
91	C₂×C₈×C₆₇	—	2	2	no	—
92	C₂×C₈×C₆₇	—	2	2	no	—
93	C₂×C₈×C₆₇	—	2	1	no	—
94	C₂×C₈×C₆₇	—	2	1	no	—
95	C₂×C₈×C₆₇	—	2	1	no	—
96	C₂×C₈×C₆₇	—	2	1	no	—
97	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
98	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
99	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
101	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
103	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
104	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
105	C₂×C₈×C₆₇	#41	1	2	yes	—
106	C₂×C₈×C₆₇	#41	1	2	yes	—
107	C₄⋊C₄×C₆₇	—	2	4	no	—
108	C₄⋊C₄×C₆₇	—	4	4	no	—
109	C₄⋊C₄×C₆₇	C₄²	1	8	yes	—
110	C₄⋊C₄×C₆₇	C₄²	1	4	yes	—
111	C₄²×C₆₇	—	2	4	no	—
112	C₄²×C₆₇	—	2	2	no	—
113	C₄²×C₆₇	—	2	1	no	—
114	C₄²×C₆₇	—	2	1	no	—
115	C₄²×C₆₇	—	2	2	no	—
116	C₄²×C₆₇	—	4	2	no	—
117	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
118	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
119	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
120	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
121	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
122	C₄²×C₆₇	#41	1	2	yes	—
123	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
124	Q₈×C₆₇	—	2	8	no	—
125	C₂×C₄×C₆₇	—	2	2	no	—
126	C₂×C₄×C₆₇	—	2	4	no	—
127	C₂×C₄×C₆₇	—	2	2	no	—
128	C₂×C₄×C₆₇	—	2	2	no	—
129	C₂×C₄×C₆₇	—	2	4	no	—
130	C₂×C₄×C₆₇	—	2	4	no	—
131	C₂×C₄×C₆₇	—	2	2	no	—
132	C₂×C₄×C₆₇	—	2	2	no	—
133	C₂×C₄×C₆₇	—	2	4	no	—
134	C₂×C₄×C₆₇	—	2	2	no	—
135	C₂×C₄×C₆₇	—	2	4	no	—
136	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
137	C₂×C₄×C₆₇	C₂×C₄²	1	4	yes	—
138	C₂×C₄×C₆₇	C₂×C₄²	1	4	yes	—
139	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
140	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
141	C₂×C₄×C₆₇	#131	1	4	yes	—
142	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
146	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
147	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
148	C₈×C₆₇	—	2	4	no	—
149	C₈×C₆₇	—	2	4	no	—
150	C₈×C₆₇	—	2	2	no	—
151	C₈×C₆₇	—	2	2	no	—
152	C₈×C₆₇	C₄×Q₈	1	4	yes	—
153	C₈×C₆₇	C₄×Q₈	1	4	yes	—
154	C₂²×C₆₇	C₂²×C₄²	1	1	yes	characteristic
155	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
156	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
157	C₂²×C₆₇	C₂×#131	1	2	yes	—
158	C₂²×C₆₇	#568	1	2	yes	—
159	C₄×C₆₇	—	2	4	no	—
160	C₄×C₆₇	—	2	2	no	—
161	C₄×C₆₇	—	2	4	no	—
162	C₄×C₆₇	—	2	4	no	—
163	C₄×C₆₇	—	4	2	no	—
164	C₄×C₆₇	—	4	4	no	—
165	C₄×C₆₇	#405	1	4	yes	—
166	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
167	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
168	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
169	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
170	#20521	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
171	C₂×C₆₇	C₄²×Q₈	1	1	yes	characteristic
172	C₂×C₆₇	C₂×#405	1	1	yes	characteristic
173	C₂×C₆₇	#2875	1	1	yes	characteristic
174	C₂×C₆₇	#2962	1	2	yes	—
175	C₂×C₆₇	#2969	1	2	yes	—
176	C₂×#446	C₂×C₆₇	1	1	yes	characteristic
177	C₂×C₈×Q₈	C₂×C₆₇	1	1	yes	characteristic
178	#2295	C₂×C₆₇	1	1	yes	characteristic
179	#1771	C₂×C₆₇	1	2	yes	—
180	#1770	C₂×C₆₇	1	2	yes	—
181	#1770	C₂×C₆₇	1	1	yes	characteristic
182	C₈×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic
183	#1768	C₂×C₆₇	1	2	yes	—
184	C₄×C₄⋊C₈	C₂×C₆₇	1	1	yes	characteristic
185	#1726	C₂×C₆₇	1	2	yes	—
186	C₄×#97	C₂×C₆₇	1	1	yes	characteristic
187	C₆₇	#20521	1	1	yes	characteristic, direct product factor, Sylow
188	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
189	#446	C₄×C₆₇	1	8	yes	complemented
190	C₈×Q₈	C₄×C₆₇	1	8	yes	complemented
191	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
192	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
193	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
194	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
195	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
196	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
197	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
198	C₂×#97	C₂²×C₆₇	1	2	yes	—
199	C₂×#97	C₂²×C₆₇	1	2	yes	—
200	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
201	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
202	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
203	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
204	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
205	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
206	#391	C₄×C₆₇	1	2	yes	—
207	#389	C₄×C₆₇	1	2	yes	—
208	#387	C₄×C₆₇	1	4	yes	—
209	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
210	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
211	#376	C₂²×C₆₇	1	2	yes	—
212	C₄³	C₂²×C₆₇	1	1	yes	characteristic
213	#336	C₂²×C₆₇	1	2	yes	—
214	#336	C₂²×C₆₇	1	2	yes	—
215	#336	C₂²×C₆₇	1	2	yes	—
216	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
217	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
218	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
219	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
220	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
221	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
222	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
223	C₄×Q₈	C₂×C₄×C₆₇	1	4	yes	—
224	C₄×Q₈	C₂×C₄×C₆₇	1	4	yes	—
225	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
226	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
227	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
228	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
229	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
230	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
231	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
232	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
233	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
234	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
235	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
236	C₄⋊C₈	C₂×C₄×C₆₇	1	4	yes	—
237	C₄⋊C₈	C₂×C₄×C₆₇	1	8	yes	—
238	C₄⋊C₈	C₂×C₄×C₆₇	1	4	yes	—
239	C₄⋊C₈	C₂×C₄×C₆₇	1	8	yes	—
240	#97	—	2	2	no	—
241	#97	—	2	2	no	—
242	#97	—	2	4	no	—
243	#97	—	2	2	no	—
244	#97	C₂×C₄×C₆₇	1	8	yes	—
245	#97	Q₈×C₆₇	1	4	yes	complemented
246	C₄×C₈	—	2	4	no	—
247	C₄×C₈	—	2	4	no	—
248	C₄×C₈	C₂×C₄×C₆₇	1	4	yes	—
249	C₄×C₈	C₂×C₄×C₆₇	1	8	yes	—
250	C₄×C₈	C₂×C₄×C₆₇	1	4	yes	—
251	#95	C₂×C₄×C₆₇	1	4	yes	—
252	#95	C₂×C₄×C₆₇	1	4	yes	—
253	#95	C₂×C₄×C₆₇	1	2	yes	—
254	#95	C₂×C₄×C₆₇	1	4	yes	—
255	C₂×Q₈	—	2	4	no	—
256	C₂×Q₈	C₄²×C₆₇	1	4	yes	—
257	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
258	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
259	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
260	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
261	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
262	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
263	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
264	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	—	2	1	no	—
273	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
274	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
275	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
276	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
277	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
278	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
279	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
280	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
281	C₂×C₈	#41×C₆₇	1	2	yes	—
282	C₂×C₈	#41×C₆₇	1	2	yes	—
283	C₄⋊C₄	—	2	4	no	—
284	C₄⋊C₄	—	4	4	no	—
285	C₄⋊C₄	C₄²×C₆₇	1	8	yes	—
286	C₄⋊C₄	C₄²×C₆₇	1	4	yes	—
287	C₄²	—	2	4	no	—
288	C₄²	—	2	2	no	—
289	C₄²	—	2	1	no	—
290	C₄²	—	2	1	no	—
291	C₄²	—	2	2	no	—
292	C₄²	—	4	2	no	—
293	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
294	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
295	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
296	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
297	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
298	C₄²	#41×C₆₇	1	2	yes	—
299	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
300	Q₈	—	2	8	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	4	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	4	no	—
312	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
313	C₂×C₄	C₂×C₄²×C₆₇	1	4	yes	—
314	C₂×C₄	C₂×C₄²×C₆₇	1	4	yes	—
315	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
316	C₂×C₄	C₄×Q₈×C₆₇	1	4	yes	—
317	C₂×C₄	#131×C₆₇	1	4	yes	—
318	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
319	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
320	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
321	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
322	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
323	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
324	C₈	—	2	4	no	—
325	C₈	—	2	4	no	—
326	C₈	—	2	2	no	—
327	C₈	—	2	2	no	—
328	C₈	C₄×Q₈×C₆₇	1	4	yes	—
329	C₈	C₄×Q₈×C₆₇	1	4	yes	—
330	C₂²	C₂²×C₄²×C₆₇	1	1	yes	characteristic
331	C₂²	C₂×C₄×Q₈×C₆₇	1	1	yes	characteristic
332	C₂²	C₄×#41×C₆₇	1	1	yes	characteristic
333	C₂²	C₂×#131×C₆₇	1	2	yes	—
334	C₂²	#568×C₆₇	1	2	yes	—
335	C₄	—	2	4	no	—
336	C₄	—	2	2	no	—
337	C₄	—	2	4	no	—
338	C₄	—	2	4	no	—
339	C₄	—	4	2	no	—
340	C₄	—	4	4	no	—
341	C₄	#405×C₆₇	1	4	yes	—
342	C₄	C₂×C₄×Q₈×C₆₇	1	1	yes	characteristic
343	C₄	C₂×C₄×Q₈×C₆₇	1	1	yes	characteristic
344	C₄	C₄×#41×C₆₇	1	1	yes	characteristic
345	C₄	C₄×#41×C₆₇	1	1	yes	characteristic
346	C₂	C₄²×Q₈×C₆₇	1	1	yes	characteristic
347	C₂	C₂×#405×C₆₇	1	1	yes	characteristic
348	C₂	C₆₇×#2875	1	1	yes	characteristic
349	C₂	C₆₇×#2962	1	2	yes	—
350	C₂	C₆₇×#2969	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂²	C₂²×C₄²×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle, prime radical	C₂³×C₆₇	C₂³×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
13. #446×C₆₇	C₄	semidirect
14. C₈×Q₈×C₆₇	C₄	semidirect
69. #97×C₆₇	Q₈	semidirect
170. #20521	C₆₇	direct (N × H)
187. C₆₇	#20521	direct (N × H)
189. #446	C₄×C₆₇	semidirect
190. C₈×Q₈	C₄×C₆₇	semidirect
245. #97	Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#446×C₆₇	2
C₂×C₈×Q₈×C₆₇	2
C₆₇×#2295	2
C₆₇×#1771	2
C₆₇×#1770	2
C₈×C₄⋊C₄×C₆₇	2
C₆₇×#1768	2
C₄×C₄⋊C₈×C₆₇	2
C₆₇×#1726	2
C₄×#97×C₆₇	2
#20521	67

Subgroups — C67×#20521 #30699772

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20521 #30699772