Subgroups — C₆₇×#33118 #30712369

Order 17152.

930 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#481×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#480×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2619	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2274	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1982	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1949	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1926	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1925	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1875	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1875	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1798	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
18	#481×C₆₇	—	2	4	no	—
19	#480×C₆₇	—	2	4	no	—
20	C₂×#107×C₆₇	C₂²	1	2	yes	—
21	C₂×#106×C₆₇	C₂²	1	2	yes	—
22	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
23	C₂×#102×C₆₇	C₂²	1	2	yes	—
24	C₂×#102×C₆₇	C₂²	1	2	yes	—
25	C₂×#102×C₆₇	C₂²	1	2	yes	—
26	C₂×#102×C₆₇	C₂²	1	2	yes	—
27	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
29	#394×C₆₇	—	2	2	no	—
30	#392×C₆₇	—	2	2	no	—
31	#390×C₆₇	C₂²	1	1	yes	characteristic
32	#390×C₆₇	C₂²	1	1	yes	characteristic
33	#382×C₆₇	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
35	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
36	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
37	C₄×C₂²⋊C₄×C₆₇	—	2	2	no	—
38	#340×C₆₇	C₂²	1	2	yes	—
39	#340×C₆₇	C₂²	1	2	yes	—
40	#336×C₆₇	C₂²	1	2	yes	—
41	C₂²×D₄×C₆₇	—	2	1	no	—
42	C₂²×D₄×C₆₇	—	2	1	no	—
43	C₂²×D₄×C₆₇	—	2	2	no	—
44	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
45	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₆₇	—	2	1	no	—
47	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
49	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
50	#127×C₆₇	—	2	4	no	—
51	#121×C₆₇	—	2	2	no	—
52	#121×C₆₇	—	2	2	no	—
53	#121×C₆₇	—	2	2	no	—
54	#121×C₆₇	—	2	2	no	—
55	#117×C₆₇	—	2	4	no	—
56	#117×C₆₇	—	2	4	no	—
57	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
59	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
60	C₂×C₄⋊C₄×C₆₇	D₄	1	4	yes	complemented
61	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
65	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
66	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
67	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
68	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
69	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
70	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
71	C₂×C₄²×C₆₇	—	2	1	no	—
72	C₂×C₄²×C₆₇	—	2	2	no	—
73	C₂×C₄²×C₆₇	—	2	2	no	—
74	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
76	#107×C₆₇	—	2	4	no	—
77	#106×C₆₇	—	2	4	no	—
78	C₄⋊C₈×C₆₇	—	2	4	no	—
79	#102×C₆₇	—	2	4	no	—
80	#102×C₆₇	—	2	4	no	—
81	#102×C₆₇	—	2	4	no	—
82	#102×C₆₇	—	2	4	no	—
83	C₂²⋊C₈×C₆₇	—	2	2	no	—
84	C₂²⋊C₈×C₆₇	—	2	2	no	—
85	#95×C₆₇	—	2	2	no	—
86	#95×C₆₇	—	2	2	no	—
87	#95×C₆₇	—	2	2	no	—
88	C₂⁴×C₆₇	—	2	1	no	—
89	C₂⁴×C₆₇	—	2	1	no	—
90	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
91	C₂×D₄×C₆₇	—	2	1	no	—
92	C₂×D₄×C₆₇	—	2	1	no	—
93	C₂×D₄×C₆₇	—	2	2	no	—
94	C₂×D₄×C₆₇	—	2	2	no	—
95	C₂×D₄×C₆₇	—	2	1	no	—
96	C₂×D₄×C₆₇	—	2	1	no	—
97	C₂×D₄×C₆₇	—	2	2	no	—
98	C₂×D₄×C₆₇	—	2	2	no	—
99	C₂×D₄×C₆₇	—	4	2	no	—
100	C₂×D₄×C₆₇	—	4	2	no	—
101	C₂×D₄×C₆₇	—	4	4	no	—
102	C₂×D₄×C₆₇	—	8	2	no	—
103	C₂×D₄×C₆₇	—	8	2	no	—
104	C₂×D₄×C₆₇	—	8	2	no	—
105	C₂×D₄×C₆₇	—	8	2	no	—
106	C₂²×C₄×C₆₇	—	2	2	no	—
107	C₂²×C₄×C₆₇	—	2	2	no	—
108	C₂²×C₄×C₆₇	—	2	2	no	—
109	C₂²×C₄×C₆₇	—	2	2	no	—
110	C₂²×C₄×C₆₇	—	2	4	no	—
111	C₂²×C₄×C₆₇	—	2	1	no	—
112	C₂²×C₄×C₆₇	—	2	1	no	—
113	C₂²×C₄×C₆₇	—	8	2	no	—
114	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
115	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
117	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
118	C₂²×C₄×C₆₇	#41	1	2	yes	—
119	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
120	C₂×C₈×C₆₇	—	2	2	no	—
121	C₂×C₈×C₆₇	—	2	2	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	—	2	2	no	—
126	C₄⋊C₄×C₆₇	—	2	4	no	—
127	C₄⋊C₄×C₆₇	—	2	4	no	—
128	C₄⋊C₄×C₆₇	—	2	2	no	—
129	C₄⋊C₄×C₆₇	—	2	2	no	—
130	C₄⋊C₄×C₆₇	—	2	1	no	—
131	C₄⋊C₄×C₆₇	—	2	1	no	—
132	C₄⋊C₄×C₆₇	—	2	1	no	—
133	C₄⋊C₄×C₆₇	—	2	1	no	—
134	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
135	C₄⋊C₄×C₆₇	#41	1	4	yes	—
136	C₂²⋊C₄×C₆₇	—	2	4	no	—
137	C₂²⋊C₄×C₆₇	—	2	4	no	—
138	C₂²⋊C₄×C₆₇	—	4	2	no	—
139	C₂²⋊C₄×C₆₇	—	4	2	no	—
140	C₂²⋊C₄×C₆₇	—	8	2	no	—
141	C₂²⋊C₄×C₆₇	—	8	2	no	—
142	C₄²×C₆₇	—	2	4	no	—
143	C₄²×C₆₇	—	2	2	no	—
144	C₄²×C₆₇	—	8	2	no	—
145	C₄²×C₆₇	—	8	2	no	—
146	C₄²×C₆₇	#41	1	4	yes	—
147	C₂³×C₆₇	—	2	1	no	—
148	C₂³×C₆₇	—	2	1	no	—
149	C₂³×C₆₇	—	2	2	no	—
150	C₂³×C₆₇	—	2	2	no	—
151	C₂³×C₆₇	—	4	1	no	—
152	C₂³×C₆₇	—	4	1	no	—
153	C₂³×C₆₇	—	4	1	no	—
154	C₂³×C₆₇	—	4	1	no	—
155	C₂³×C₆₇	—	4	1	no	—
156	C₂³×C₆₇	—	4	1	no	—
157	C₂³×C₆₇	—	4	2	no	—
158	C₂³×C₆₇	—	4	2	no	—
159	C₂³×C₆₇	—	4	2	no	—
160	C₂³×C₆₇	—	4	2	no	—
161	C₂³×C₆₇	#121	1	2	yes	—
162	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
163	D₄×C₆₇	—	4	2	no	—
164	D₄×C₆₇	—	4	2	no	—
165	D₄×C₆₇	—	4	2	no	—
166	D₄×C₆₇	—	4	2	no	—
167	D₄×C₆₇	—	8	8	no	—
168	D₄×C₆₇	—	8	4	no	—
169	D₄×C₆₇	—	8	4	no	—
170	C₂×C₄×C₆₇	—	2	1	no	—
171	C₂×C₄×C₆₇	—	2	1	no	—
172	C₂×C₄×C₆₇	—	2	2	no	—
173	C₂×C₄×C₆₇	—	2	2	no	—
174	C₂×C₄×C₆₇	—	2	4	no	—
175	C₂×C₄×C₆₇	—	2	4	no	—
176	C₂×C₄×C₆₇	—	2	4	no	—
177	C₂×C₄×C₆₇	—	2	2	no	—
178	C₂×C₄×C₆₇	—	2	4	no	—
179	C₂×C₄×C₆₇	—	2	2	no	—
180	C₂×C₄×C₆₇	—	2	2	no	—
181	C₂×C₄×C₆₇	—	4	2	no	—
182	C₂×C₄×C₆₇	—	4	2	no	—
183	C₂×C₄×C₆₇	—	4	2	no	—
184	C₂×C₄×C₆₇	—	4	2	no	—
185	C₂×C₄×C₆₇	—	4	1	no	—
186	C₂×C₄×C₆₇	—	4	1	no	—
187	C₂×C₄×C₆₇	—	4	2	no	—
188	C₂×C₄×C₆₇	—	4	2	no	—
189	C₂×C₄×C₆₇	—	8	4	no	—
190	C₂×C₄×C₆₇	#121	1	2	yes	—
191	C₂×C₄×C₆₇	#123	1	4	yes	—
192	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
193	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
194	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
195	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
196	C₈×C₆₇	—	4	2	no	—
197	C₈×C₆₇	—	4	2	no	—
198	C₂²×C₆₇	—	2	2	no	—
199	C₂²×C₆₇	—	2	2	no	—
200	C₂²×C₆₇	—	2	2	no	—
201	C₂²×C₆₇	—	4	2	no	—
202	C₂²×C₆₇	—	4	2	no	—
203	C₂²×C₆₇	—	8	2	no	—
204	C₂²×C₆₇	—	8	2	no	—
205	C₂²×C₆₇	—	8	1	no	—
206	C₂²×C₆₇	—	8	1	no	—
207	C₂²×C₆₇	—	8	2	no	—
208	C₂²×C₆₇	—	8	2	no	—
209	C₂²×C₆₇	—	8	1	no	—
210	C₂²×C₆₇	—	8	1	no	—
211	C₂²×C₆₇	—	8	1	no	—
212	C₂²×C₆₇	—	8	1	no	—
213	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
214	C₂²×C₆₇	C₂×#123	1	1	yes	characteristic
215	C₂²×C₆₇	#525	1	1	yes	characteristic
216	C₂²×C₆₇	#576	1	2	yes	—
217	C₂²×C₆₇	#577	1	2	yes	—
218	C₄×C₆₇	—	2	4	no	—
219	C₄×C₆₇	—	2	2	no	—
220	C₄×C₆₇	—	4	2	no	—
221	C₄×C₆₇	—	4	4	no	—
222	C₄×C₆₇	—	4	4	no	—
223	C₄×C₆₇	—	8	1	no	—
224	C₄×C₆₇	—	8	1	no	—
225	C₄×C₆₇	#529	1	2	yes	—
226	C₄×C₆₇	#532	1	2	yes	—
227	#33118	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
228	C₂×C₆₇	—	4	2	no	—
229	C₂×C₆₇	—	16	1	no	—
230	C₂×C₆₇	—	16	1	no	—
231	C₂×C₆₇	#2854	1	1	yes	characteristic
232	C₂×C₆₇	#3041	1	1	yes	characteristic
233	C₂×C₆₇	#3043	1	1	yes	characteristic
234	C₂×C₆₇	#3096	1	2	yes	—
235	C₂×C₆₇	#3119	1	2	yes	—
236	C₂×#481	C₂×C₆₇	1	1	yes	characteristic, complemented
237	C₂×#480	C₂×C₆₇	1	1	yes	characteristic, complemented
238	#2619	C₂×C₆₇	1	1	yes	characteristic
239	#2274	C₂×C₆₇	1	1	yes	characteristic, complemented
240	#1982	C₂×C₆₇	1	2	yes	complemented
241	#1949	C₂×C₆₇	1	2	yes	complemented
242	#1926	C₂×C₆₇	1	2	yes	complemented
243	#1925	C₂×C₆₇	1	2	yes	complemented
244	#1875	C₂×C₆₇	1	1	yes	characteristic, complemented
245	#1875	C₂×C₆₇	1	1	yes	characteristic, complemented
246	#1798	C₂×C₆₇	1	1	yes	characteristic, complemented
247	C₆₇	#33118	1	1	yes	characteristic, direct product factor, Sylow
248	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
249	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
250	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
251	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
252	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
253	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
254	#481	—	2	4	no	—
255	#480	—	2	4	no	—
256	C₂×#107	C₂²×C₆₇	1	2	yes	—
257	C₂×#106	C₂²×C₆₇	1	2	yes	—
258	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
259	C₂×#102	C₂²×C₆₇	1	2	yes	—
260	C₂×#102	C₂²×C₆₇	1	2	yes	—
261	C₂×#102	C₂²×C₆₇	1	2	yes	—
262	C₂×#102	C₂²×C₆₇	1	2	yes	—
263	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
264	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
265	#394	—	2	2	no	—
266	#392	—	2	2	no	—
267	#390	C₂²×C₆₇	1	1	yes	characteristic
268	#390	C₂²×C₆₇	1	1	yes	characteristic
269	#382	C₂²×C₆₇	1	1	yes	characteristic
270	C₄×C₄⋊C₄	—	2	2	no	—
271	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
272	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
273	C₄×C₂²⋊C₄	—	2	2	no	—
274	#340	C₂²×C₆₇	1	2	yes	—
275	#340	C₂²×C₆₇	1	2	yes	—
276	#336	C₂²×C₆₇	1	2	yes	—
277	C₂²×D₄	—	2	1	no	—
278	C₂²×D₄	—	2	1	no	—
279	C₂²×D₄	—	2	2	no	—
280	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
281	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
282	C₂³×C₄	—	2	1	no	—
283	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
284	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
285	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
286	#127	—	2	4	no	—
287	#121	—	2	2	no	—
288	#121	—	2	2	no	—
289	#121	—	2	2	no	—
290	#121	—	2	2	no	—
291	#117	—	2	4	no	—
292	#117	—	2	4	no	—
293	C₂×C₄⋊C₄	—	2	1	no	—
294	C₂×C₄⋊C₄	—	2	2	no	—
295	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
296	C₂×C₄⋊C₄	D₄×C₆₇	1	4	yes	complemented
297	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
298	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
299	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
300	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
301	C₂×C₂²⋊C₄	—	2	1	no	—
302	C₂×C₂²⋊C₄	—	2	1	no	—
303	C₂×C₂²⋊C₄	—	2	1	no	—
304	C₂×C₂²⋊C₄	—	2	2	no	—
305	C₂×C₂²⋊C₄	—	2	2	no	—
306	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
307	C₂×C₄²	—	2	1	no	—
308	C₂×C₄²	—	2	2	no	—
309	C₂×C₄²	—	2	2	no	—
310	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
311	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
312	#107	—	2	4	no	—
313	#106	—	2	4	no	—
314	C₄⋊C₈	—	2	4	no	—
315	#102	—	2	4	no	—
316	#102	—	2	4	no	—
317	#102	—	2	4	no	—
318	#102	—	2	4	no	—
319	C₂²⋊C₈	—	2	2	no	—
320	C₂²⋊C₈	—	2	2	no	—
321	#95	—	2	2	no	—
322	#95	—	2	2	no	—
323	#95	—	2	2	no	—
324	C₂⁴	—	2	1	no	—
325	C₂⁴	—	2	1	no	—
326	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
327	C₂×D₄	—	2	1	no	—
328	C₂×D₄	—	2	1	no	—
329	C₂×D₄	—	2	2	no	—
330	C₂×D₄	—	2	2	no	—
331	C₂×D₄	—	2	1	no	—
332	C₂×D₄	—	2	1	no	—
333	C₂×D₄	—	2	2	no	—
334	C₂×D₄	—	2	2	no	—
335	C₂×D₄	—	4	2	no	—
336	C₂×D₄	—	4	2	no	—
337	C₂×D₄	—	4	4	no	—
338	C₂×D₄	—	8	2	no	—
339	C₂×D₄	—	8	2	no	—
340	C₂×D₄	—	8	2	no	—
341	C₂×D₄	—	8	2	no	—
342	C₂²×C₄	—	2	2	no	—
343	C₂²×C₄	—	2	2	no	—
344	C₂²×C₄	—	2	2	no	—
345	C₂²×C₄	—	2	2	no	—
346	C₂²×C₄	—	2	4	no	—
347	C₂²×C₄	—	2	1	no	—
348	C₂²×C₄	—	2	1	no	—
349	C₂²×C₄	—	8	2	no	—
350	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
351	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
352	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
353	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
354	C₂²×C₄	#41×C₆₇	1	2	yes	—
355	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	—	2	2	no	—
358	C₂×C₈	—	2	2	no	—
359	C₂×C₈	—	2	2	no	—
360	C₂×C₈	—	2	2	no	—
361	C₂×C₈	—	2	2	no	—
362	C₄⋊C₄	—	2	4	no	—
363	C₄⋊C₄	—	2	4	no	—
364	C₄⋊C₄	—	2	2	no	—
365	C₄⋊C₄	—	2	2	no	—
366	C₄⋊C₄	—	2	1	no	—
367	C₄⋊C₄	—	2	1	no	—
368	C₄⋊C₄	—	2	1	no	—
369	C₄⋊C₄	—	2	1	no	—
370	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
371	C₄⋊C₄	#41×C₆₇	1	4	yes	—
372	C₂²⋊C₄	—	2	4	no	—
373	C₂²⋊C₄	—	2	4	no	—
374	C₂²⋊C₄	—	4	2	no	—
375	C₂²⋊C₄	—	4	2	no	—
376	C₂²⋊C₄	—	8	2	no	—
377	C₂²⋊C₄	—	8	2	no	—
378	C₄²	—	2	4	no	—
379	C₄²	—	2	2	no	—
380	C₄²	—	8	2	no	—
381	C₄²	—	8	2	no	—
382	C₄²	#41×C₆₇	1	4	yes	—
383	C₂³	—	2	1	no	—
384	C₂³	—	2	1	no	—
385	C₂³	—	2	2	no	—
386	C₂³	—	2	2	no	—
387	C₂³	—	4	1	no	—
388	C₂³	—	4	1	no	—
389	C₂³	—	4	1	no	—
390	C₂³	—	4	1	no	—
391	C₂³	—	4	1	no	—
392	C₂³	—	4	1	no	—
393	C₂³	—	4	2	no	—
394	C₂³	—	4	2	no	—
395	C₂³	—	4	2	no	—
396	C₂³	—	4	2	no	—
397	C₂³	#121×C₆₇	1	2	yes	—
398	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
399	D₄	—	4	2	no	—
400	D₄	—	4	2	no	—
401	D₄	—	4	2	no	—
402	D₄	—	4	2	no	—
403	D₄	—	8	8	no	—
404	D₄	—	8	4	no	—
405	D₄	—	8	4	no	—
406	C₂×C₄	—	2	1	no	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	4	no	—
411	C₂×C₄	—	2	4	no	—
412	C₂×C₄	—	2	4	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	4	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	4	2	no	—
421	C₂×C₄	—	4	1	no	—
422	C₂×C₄	—	4	1	no	—
423	C₂×C₄	—	4	2	no	—
424	C₂×C₄	—	4	2	no	—
425	C₂×C₄	—	8	4	no	—
426	C₂×C₄	#121×C₆₇	1	2	yes	—
427	C₂×C₄	#123×C₆₇	1	4	yes	—
428	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
429	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
430	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
431	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
432	C₈	—	4	2	no	—
433	C₈	—	4	2	no	—
434	C₂²	—	2	2	no	—
435	C₂²	—	2	2	no	—
436	C₂²	—	2	2	no	—
437	C₂²	—	4	2	no	—
438	C₂²	—	4	2	no	—
439	C₂²	—	8	1	no	—
440	C₂²	—	8	1	no	—
441	C₂²	—	8	1	no	—
442	C₂²	—	8	1	no	—
443	C₂²	—	8	2	no	—
444	C₂²	—	8	2	no	—
445	C₂²	—	8	1	no	—
446	C₂²	—	8	1	no	—
447	C₂²	—	8	2	no	—
448	C₂²	—	8	2	no	—
449	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
450	C₂²	C₂×#123×C₆₇	1	1	yes	characteristic
451	C₂²	#525×C₆₇	1	1	yes	characteristic
452	C₂²	#576×C₆₇	1	2	yes	—
453	C₂²	#577×C₆₇	1	2	yes	—
454	C₄	—	2	4	no	—
455	C₄	—	2	2	no	—
456	C₄	—	4	4	no	—
457	C₄	—	4	2	no	—
458	C₄	—	4	4	no	—
459	C₄	—	8	1	no	—
460	C₄	—	8	1	no	—
461	C₄	#529×C₆₇	1	2	yes	—
462	C₄	#532×C₆₇	1	2	yes	—
463	C₂	—	4	2	no	—
464	C₂	—	16	1	no	—
465	C₂	—	16	1	no	—
466	C₂	C₆₇×#2854	1	1	yes	characteristic
467	C₂	C₆₇×#3041	1	1	yes	characteristic
468	C₂	C₆₇×#3043	1	1	yes	characteristic
469	C₂	C₆₇×#3096	1	2	yes	—
470	C₂	C₆₇×#3119	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₆₇×#2619	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#481×C₆₇	C₂	semidirect
2. C₂×#480×C₆₇	C₂	semidirect
4. C₆₇×#2274	C₂	semidirect
5. C₆₇×#1982	C₂	semidirect
6. C₆₇×#1949	C₂	semidirect
7. C₆₇×#1926	C₂	semidirect
8. C₆₇×#1925	C₂	semidirect
9. C₆₇×#1875	C₂	semidirect
10. C₆₇×#1875	C₂	semidirect
11. C₆₇×#1798	C₂	semidirect
60. C₂×C₄⋊C₄×C₆₇	D₄	semidirect
227. #33118	C₆₇	direct (N × H)
236. C₂×#481	C₂×C₆₇	semidirect
237. C₂×#480	C₂×C₆₇	semidirect
239. #2274	C₂×C₆₇	semidirect
240. #1982	C₂×C₆₇	semidirect
241. #1949	C₂×C₆₇	semidirect
242. #1926	C₂×C₆₇	semidirect
243. #1925	C₂×C₆₇	semidirect
244. #1875	C₂×C₆₇	semidirect
245. #1875	C₂×C₆₇	semidirect
246. #1798	C₂×C₆₇	semidirect
247. C₆₇	#33118	direct (N × H)
296. C₂×C₄⋊C₄	D₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#481×C₆₇	2
C₂×#480×C₆₇	2
C₆₇×#2619	2
C₆₇×#2274	2
C₆₇×#1982	2
C₆₇×#1949	2
C₆₇×#1926	2
C₆₇×#1925	2
C₆₇×#1875	2
C₆₇×#1798	2
#33118	67

Subgroups — C67×#33118 #30712369

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33118 #30712369