Subgroups — C₆₇×#33128 #30712379

Order 17152.

972 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#465×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2855	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2854	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1968	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1941	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1934	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1933	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1893	C₂	1	1	yes	maximal, characteristic
10	C₆₇×#1892	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1807	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#123×C₆₇	—	2	1	no	—
16	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
19	#465×C₆₇	—	2	4	no	—
20	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#106×C₆₇	C₂²	1	2	yes	—
22	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#103×C₆₇	C₂²	1	2	yes	—
24	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#102×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₆₇	—	2	4	no	—
27	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
29	#391×C₆₇	C₂²	1	2	yes	—
30	#390×C₆₇	C₂²	1	2	yes	—
31	#389×C₆₇	C₂²	1	2	yes	—
32	#389×C₆₇	C₂²	1	2	yes	—
33	#388×C₆₇	—	2	2	no	—
34	#387×C₆₇	—	2	2	no	—
35	#386×C₆₇	—	2	1	no	—
36	#385×C₆₇	—	2	1	no	—
37	#381×C₆₇	—	2	1	no	—
38	#380×C₆₇	C₂²	1	1	yes	characteristic
39	#380×C₆₇	C₂²	1	1	yes	characteristic
40	#340×C₆₇	C₂²	1	2	yes	—
41	#340×C₆₇	C₂²	1	2	yes	—
42	#340×C₆₇	C₂²	1	2	yes	—
43	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
44	C₂²×D₄×C₆₇	—	2	1	no	—
45	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
47	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
49	C₂×SD₁₆×C₆₇	—	2	2	no	—
50	C₂×SD₁₆×C₆₇	—	2	4	no	—
51	C₂²×C₈×C₆₇	—	2	2	no	—
52	C₂²×C₈×C₆₇	—	2	4	no	—
53	C₂²×C₈×C₆₇	—	2	4	no	—
54	C₂²×C₈×C₆₇	D₄	1	4	yes	—
55	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
57	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
58	#123×C₆₇	—	4	4	no	—
59	C₂²⋊Q₈×C₆₇	—	2	2	no	—
60	C₂²⋊Q₈×C₆₇	—	2	2	no	—
61	#121×C₆₇	—	2	2	no	—
62	#121×C₆₇	—	2	2	no	—
63	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
64	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
65	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
66	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
69	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
70	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
71	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
72	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
73	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
74	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
75	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
76	C₂×C₄²×C₆₇	—	2	2	no	—
77	C₂×C₄²×C₆₇	—	2	2	no	—
78	C₂×C₄²×C₆₇	D₄	1	2	yes	—
79	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
80	#106×C₆₇	—	2	2	no	—
81	#106×C₆₇	—	2	4	no	—
82	#103×C₆₇	—	2	2	no	—
83	#103×C₆₇	—	2	4	no	—
84	#102×C₆₇	—	2	2	no	—
85	#102×C₆₇	—	2	4	no	—
86	C₄×C₈×C₆₇	—	2	4	no	—
87	C₄×C₈×C₆₇	—	2	4	no	—
88	#95×C₆₇	—	2	1	no	—
89	#95×C₆₇	—	2	1	no	—
90	#95×C₆₇	—	2	2	no	—
91	#95×C₆₇	—	2	2	no	—
92	#95×C₆₇	—	2	2	no	—
93	#95×C₆₇	—	2	2	no	—
94	C₂⁴×C₆₇	—	2	1	no	—
95	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
96	C₂×Q₈×C₆₇	—	2	1	no	—
97	C₂×Q₈×C₆₇	—	2	2	no	—
98	C₂×Q₈×C₆₇	—	2	1	no	—
99	C₂×Q₈×C₆₇	—	2	2	no	—
100	C₂×D₄×C₆₇	—	2	1	no	—
101	C₂×D₄×C₆₇	—	2	2	no	—
102	C₂×D₄×C₆₇	—	2	1	no	—
103	C₂×D₄×C₆₇	—	2	2	no	—
104	C₂×D₄×C₆₇	—	4	2	no	—
105	C₂×D₄×C₆₇	—	4	2	no	—
106	C₂×D₄×C₆₇	—	4	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	2	no	—
112	C₂²×C₄×C₆₇	—	2	2	no	—
113	C₂²×C₄×C₆₇	—	2	2	no	—
114	C₂²×C₄×C₆₇	—	2	1	no	—
115	C₂²×C₄×C₆₇	—	2	1	no	—
116	C₂²×C₄×C₆₇	—	2	2	no	—
117	C₂²×C₄×C₆₇	—	2	1	no	—
118	C₂²×C₄×C₆₇	SD₁₆	1	4	yes	complemented
119	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
121	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
122	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
123	C₂²×C₄×C₆₇	#41	1	2	yes	—
124	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
125	SD₁₆×C₆₇	—	4	4	no	—
126	C₂×C₈×C₆₇	—	2	8	no	—
127	C₂×C₈×C₆₇	—	2	2	no	—
128	C₂×C₈×C₆₇	—	2	2	no	—
129	C₂×C₈×C₆₇	—	2	2	no	—
130	C₂×C₈×C₆₇	—	2	8	no	—
131	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
132	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
133	C₂×C₈×C₆₇	#41	1	4	yes	—
134	C₂×C₈×C₆₇	#41	1	4	yes	—
135	C₄⋊C₄×C₆₇	—	2	1	no	—
136	C₄⋊C₄×C₆₇	—	2	1	no	—
137	C₄⋊C₄×C₆₇	—	2	2	no	—
138	C₄⋊C₄×C₆₇	—	2	2	no	—
139	C₄⋊C₄×C₆₇	—	2	2	no	—
140	C₄⋊C₄×C₆₇	—	2	2	no	—
141	C₄⋊C₄×C₆₇	—	2	1	no	—
142	C₄⋊C₄×C₆₇	—	2	1	no	—
143	C₄⋊C₄×C₆₇	—	4	2	no	—
144	C₄⋊C₄×C₆₇	—	4	4	no	—
145	C₄⋊C₄×C₆₇	—	4	4	no	—
146	C₂²⋊C₄×C₆₇	—	4	2	no	—
147	C₂²⋊C₄×C₆₇	—	4	2	no	—
148	C₂²⋊C₄×C₆₇	—	4	2	no	—
149	C₂²⋊C₄×C₆₇	—	4	2	no	—
150	C₂²⋊C₄×C₆₇	—	4	4	no	—
151	C₂²⋊C₄×C₆₇	—	4	4	no	—
152	C₄²×C₆₇	—	2	4	no	—
153	C₄²×C₆₇	—	2	2	no	—
154	C₄²×C₆₇	—	2	2	no	—
155	C₂³×C₆₇	—	2	2	no	—
156	C₂³×C₆₇	—	2	2	no	—
157	C₂³×C₆₇	—	4	1	no	—
158	C₂³×C₆₇	—	4	2	no	—
159	C₂³×C₆₇	—	4	1	no	—
160	C₂³×C₆₇	—	4	2	no	—
161	C₂³×C₆₇	—	4	1	no	—
162	C₂³×C₆₇	#121	1	2	yes	—
163	C₂³×C₆₇	C₂×SD₁₆	1	2	yes	—
164	C₂³×C₆₇	#135	1	2	yes	—
165	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
166	Q₈×C₆₇	—	4	1	no	—
167	Q₈×C₆₇	—	4	1	no	—
168	Q₈×C₆₇	—	4	2	no	—
169	D₄×C₆₇	—	4	1	no	—
170	D₄×C₆₇	—	4	1	no	—
171	D₄×C₆₇	—	4	2	no	—
172	D₄×C₆₇	—	8	4	no	—
173	C₂×C₄×C₆₇	—	2	8	no	—
174	C₂×C₄×C₆₇	—	2	2	no	—
175	C₂×C₄×C₆₇	—	2	2	no	—
176	C₂×C₄×C₆₇	—	2	2	no	—
177	C₂×C₄×C₆₇	—	2	2	no	—
178	C₂×C₄×C₆₇	—	2	2	no	—
179	C₂×C₄×C₆₇	—	2	8	no	—
180	C₂×C₄×C₆₇	—	2	2	no	—
181	C₂×C₄×C₆₇	—	2	2	no	—
182	C₂×C₄×C₆₇	—	2	4	no	—
183	C₂×C₄×C₆₇	—	4	1	no	—
184	C₂×C₄×C₆₇	—	4	1	no	—
185	C₂×C₄×C₆₇	—	4	1	no	—
186	C₂×C₄×C₆₇	—	4	2	no	—
187	C₂×C₄×C₆₇	—	4	2	no	—
188	C₂×C₄×C₆₇	—	4	2	no	—
189	C₂×C₄×C₆₇	—	4	2	no	—
190	C₂×C₄×C₆₇	—	4	1	no	—
191	C₂×C₄×C₆₇	—	4	2	no	—
192	C₂×C₄×C₆₇	—	4	2	no	—
193	C₂×C₄×C₆₇	—	4	2	no	—
194	C₂×C₄×C₆₇	—	4	2	no	—
195	C₂×C₄×C₆₇	—	4	1	no	—
196	C₂×C₄×C₆₇	#121	1	2	yes	—
197	C₂×C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
198	C₂×C₄×C₆₇	#135	1	4	yes	—
199	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
200	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
201	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
202	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
203	C₈×C₆₇	—	2	2	no	—
204	C₈×C₆₇	—	2	4	no	—
205	C₈×C₆₇	#123	1	4	yes	complemented
206	C₂²×C₆₇	—	2	2	no	—
207	C₂²×C₆₇	—	2	4	no	—
208	C₂²×C₆₇	—	2	4	no	—
209	C₂²×C₆₇	—	2	2	no	—
210	C₂²×C₆₇	—	8	1	no	—
211	C₂²×C₆₇	—	8	2	no	—
212	C₂²×C₆₇	—	8	2	no	—
213	C₂²×C₆₇	—	8	1	no	—
214	C₂²×C₆₇	—	8	1	no	—
215	C₂²×C₆₇	#465	1	4	yes	—
216	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
217	C₂²×C₆₇	#532	1	2	yes	—
218	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
219	C₂²×C₆₇	C₂×#135	1	2	yes	—
220	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
221	C₄×C₆₇	—	2	2	no	—
222	C₄×C₆₇	—	2	4	no	—
223	C₄×C₆₇	—	2	4	no	—
224	C₄×C₆₇	—	8	1	no	—
225	C₄×C₆₇	—	8	2	no	—
226	C₄×C₆₇	—	8	2	no	—
227	C₄×C₆₇	—	8	1	no	—
228	C₄×C₆₇	—	8	1	no	—
229	C₄×C₆₇	C₂×#123	1	1	yes	characteristic
230	C₄×C₆₇	#525	1	1	yes	characteristic
231	C₄×C₆₇	#529	1	2	yes	—
232	#33128	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
233	C₂×C₆₇	—	2	4	no	—
234	C₂×C₆₇	—	16	1	no	—
235	C₂×C₆₇	#2854	1	1	yes	characteristic
236	C₂×C₆₇	C₂×#465	1	1	yes	characteristic
237	C₂×C₆₇	#3050	1	1	yes	characteristic
238	C₂×C₆₇	#3167	1	2	yes	—
239	C₂×C₆₇	#3168	1	2	yes	—
240	C₂×#465	C₂×C₆₇	1	1	yes	characteristic
241	C₂²×C₄×C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
242	#2855	C₂×C₆₇	1	1	yes	characteristic, complemented
243	#2854	C₂×C₆₇	1	1	yes	characteristic
244	#1968	C₂×C₆₇	1	2	yes	complemented
245	#1941	C₂×C₆₇	1	2	yes	complemented
246	#1934	C₂×C₆₇	1	2	yes	complemented
247	#1933	C₂×C₆₇	1	2	yes	complemented
248	#1893	C₂×C₆₇	1	1	yes	characteristic
249	#1892	C₂×C₆₇	1	1	yes	characteristic, complemented
250	#1807	C₂×C₆₇	1	1	yes	characteristic, complemented
251	C₆₇	#33128	1	1	yes	characteristic, direct product factor, Sylow
252	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
253	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
254	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
255	C₂×#123	—	2	1	no	—
256	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
257	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
258	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
259	#465	—	2	4	no	—
260	C₂×#106	C₂²×C₆₇	1	2	yes	—
261	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
262	C₂×#103	C₂²×C₆₇	1	2	yes	—
263	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
264	C₂×#102	C₂²×C₆₇	1	2	yes	—
265	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
266	C₂×C₄×C₈	—	2	4	no	—
267	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
268	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
269	#391	C₂²×C₆₇	1	2	yes	—
270	#390	C₂²×C₆₇	1	2	yes	—
271	#389	C₂²×C₆₇	1	2	yes	—
272	#389	C₂²×C₆₇	1	2	yes	—
273	#388	—	2	2	no	—
274	#387	—	2	2	no	—
275	#386	—	2	1	no	—
276	#385	—	2	1	no	—
277	#381	—	2	1	no	—
278	#380	C₂²×C₆₇	1	1	yes	characteristic
279	#380	C₂²×C₆₇	1	1	yes	characteristic
280	#340	C₂²×C₆₇	1	2	yes	—
281	#340	C₂²×C₆₇	1	2	yes	—
282	#340	C₂²×C₆₇	1	2	yes	—
283	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
284	C₂²×D₄	—	2	1	no	—
285	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
286	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
287	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
288	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
289	C₂×SD₁₆	—	2	2	no	—
290	C₂×SD₁₆	—	2	4	no	—
291	C₂²×C₈	—	2	2	no	—
292	C₂²×C₈	—	2	4	no	—
293	C₂²×C₈	—	2	4	no	—
294	C₂²×C₈	D₄×C₆₇	1	4	yes	—
295	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
296	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
297	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
298	#123	—	4	4	no	—
299	C₂²⋊Q₈	—	2	2	no	—
300	C₂²⋊Q₈	—	2	2	no	—
301	#121	—	2	2	no	—
302	#121	—	2	2	no	—
303	C₂×C₄⋊C₄	—	2	1	no	—
304	C₂×C₄⋊C₄	—	2	2	no	—
305	C₂×C₄⋊C₄	—	2	2	no	—
306	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
307	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
308	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
309	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
310	C₂×C₂²⋊C₄	—	2	1	no	—
311	C₂×C₂²⋊C₄	—	2	1	no	—
312	C₂×C₂²⋊C₄	—	2	1	no	—
313	C₂×C₂²⋊C₄	—	2	1	no	—
314	C₂×C₂²⋊C₄	—	2	2	no	—
315	C₂×C₂²⋊C₄	—	2	2	no	—
316	C₂×C₄²	—	2	2	no	—
317	C₂×C₄²	—	2	2	no	—
318	C₂×C₄²	D₄×C₆₇	1	2	yes	—
319	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
320	#106	—	2	2	no	—
321	#106	—	2	4	no	—
322	#103	—	2	2	no	—
323	#103	—	2	4	no	—
324	#102	—	2	2	no	—
325	#102	—	2	4	no	—
326	C₄×C₈	—	2	4	no	—
327	C₄×C₈	—	2	4	no	—
328	#95	—	2	1	no	—
329	#95	—	2	1	no	—
330	#95	—	2	2	no	—
331	#95	—	2	2	no	—
332	#95	—	2	2	no	—
333	#95	—	2	2	no	—
334	C₂⁴	—	2	1	no	—
335	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
336	C₂×Q₈	—	2	1	no	—
337	C₂×Q₈	—	2	2	no	—
338	C₂×Q₈	—	2	1	no	—
339	C₂×Q₈	—	2	2	no	—
340	C₂×D₄	—	2	1	no	—
341	C₂×D₄	—	2	2	no	—
342	C₂×D₄	—	2	1	no	—
343	C₂×D₄	—	2	2	no	—
344	C₂×D₄	—	4	2	no	—
345	C₂×D₄	—	4	2	no	—
346	C₂×D₄	—	4	2	no	—
347	C₂²×C₄	—	2	2	no	—
348	C₂²×C₄	—	2	2	no	—
349	C₂²×C₄	—	2	2	no	—
350	C₂²×C₄	—	2	4	no	—
351	C₂²×C₄	—	2	2	no	—
352	C₂²×C₄	—	2	2	no	—
353	C₂²×C₄	—	2	2	no	—
354	C₂²×C₄	—	2	1	no	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	—	2	1	no	—
357	C₂²×C₄	—	2	1	no	—
358	C₂²×C₄	SD₁₆×C₆₇	1	4	yes	complemented
359	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
360	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
361	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
362	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
363	C₂²×C₄	#41×C₆₇	1	2	yes	—
364	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
365	SD₁₆	—	4	4	no	—
366	C₂×C₈	—	2	8	no	—
367	C₂×C₈	—	2	2	no	—
368	C₂×C₈	—	2	2	no	—
369	C₂×C₈	—	2	2	no	—
370	C₂×C₈	—	2	8	no	—
371	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
372	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
373	C₂×C₈	#41×C₆₇	1	4	yes	—
374	C₂×C₈	#41×C₆₇	1	4	yes	—
375	C₄⋊C₄	—	2	1	no	—
376	C₄⋊C₄	—	2	1	no	—
377	C₄⋊C₄	—	2	2	no	—
378	C₄⋊C₄	—	2	2	no	—
379	C₄⋊C₄	—	2	2	no	—
380	C₄⋊C₄	—	2	2	no	—
381	C₄⋊C₄	—	2	1	no	—
382	C₄⋊C₄	—	2	1	no	—
383	C₄⋊C₄	—	4	2	no	—
384	C₄⋊C₄	—	4	4	no	—
385	C₄⋊C₄	—	4	4	no	—
386	C₂²⋊C₄	—	4	2	no	—
387	C₂²⋊C₄	—	4	2	no	—
388	C₂²⋊C₄	—	4	2	no	—
389	C₂²⋊C₄	—	4	2	no	—
390	C₂²⋊C₄	—	4	4	no	—
391	C₂²⋊C₄	—	4	4	no	—
392	C₄²	—	2	4	no	—
393	C₄²	—	2	2	no	—
394	C₄²	—	2	2	no	—
395	C₂³	—	2	2	no	—
396	C₂³	—	2	2	no	—
397	C₂³	—	4	1	no	—
398	C₂³	—	4	2	no	—
399	C₂³	—	4	1	no	—
400	C₂³	—	4	2	no	—
401	C₂³	—	4	1	no	—
402	C₂³	#121×C₆₇	1	2	yes	—
403	C₂³	C₂×SD₁₆×C₆₇	1	2	yes	—
404	C₂³	#135×C₆₇	1	2	yes	—
405	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
406	Q₈	—	4	1	no	—
407	Q₈	—	4	1	no	—
408	Q₈	—	4	2	no	—
409	D₄	—	4	1	no	—
410	D₄	—	4	1	no	—
411	D₄	—	4	2	no	—
412	D₄	—	8	4	no	—
413	C₂×C₄	—	2	8	no	—
414	C₂×C₄	—	2	8	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	4	no	—
423	C₂×C₄	—	4	1	no	—
424	C₂×C₄	—	4	1	no	—
425	C₂×C₄	—	4	1	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	2	no	—
428	C₂×C₄	—	4	2	no	—
429	C₂×C₄	—	4	2	no	—
430	C₂×C₄	—	4	1	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	4	2	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	4	2	no	—
435	C₂×C₄	—	4	1	no	—
436	C₂×C₄	#121×C₆₇	1	2	yes	—
437	C₂×C₄	C₂×SD₁₆×C₆₇	1	4	yes	—
438	C₂×C₄	#135×C₆₇	1	4	yes	—
439	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
440	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
441	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
442	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
443	C₈	—	2	2	no	—
444	C₈	—	2	4	no	—
445	C₈	#123×C₆₇	1	4	yes	complemented
446	C₂²	—	2	2	no	—
447	C₂²	—	2	2	no	—
448	C₂²	—	2	4	no	—
449	C₂²	—	2	4	no	—
450	C₂²	—	8	1	no	—
451	C₂²	—	8	1	no	—
452	C₂²	—	8	1	no	—
453	C₂²	—	8	2	no	—
454	C₂²	—	8	2	no	—
455	C₂²	#465×C₆₇	1	4	yes	—
456	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
457	C₂²	#532×C₆₇	1	2	yes	—
458	C₂²	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
459	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
460	C₂²	C₂×#135×C₆₇	1	2	yes	—
461	C₄	—	2	4	no	—
462	C₄	—	2	4	no	—
463	C₄	—	2	2	no	—
464	C₄	—	8	1	no	—
465	C₄	—	8	1	no	—
466	C₄	—	8	1	no	—
467	C₄	—	8	2	no	—
468	C₄	—	8	2	no	—
469	C₄	C₂×#123×C₆₇	1	1	yes	characteristic
470	C₄	#525×C₆₇	1	1	yes	characteristic
471	C₄	#529×C₆₇	1	2	yes	—
472	C₂	—	2	4	no	—
473	C₂	—	16	1	no	—
474	C₂	C₆₇×#2854	1	1	yes	characteristic
475	C₂	C₂×#465×C₆₇	1	1	yes	characteristic
476	C₂	C₆₇×#3050	1	1	yes	characteristic
477	C₂	C₆₇×#3167	1	2	yes	—
478	C₂	C₆₇×#3168	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#121×C₆₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×C₈×C₆₇	C₂	semidirect
3. C₆₇×#2855	C₂	semidirect
5. C₆₇×#1968	C₂	semidirect
6. C₆₇×#1941	C₂	semidirect
7. C₆₇×#1934	C₂	semidirect
8. C₆₇×#1933	C₂	semidirect
10. C₆₇×#1892	C₂	semidirect
11. C₆₇×#1807	C₂	semidirect
118. C₂²×C₄×C₆₇	SD₁₆	semidirect
205. C₈×C₆₇	#123	semidirect
232. #33128	C₆₇	direct (N × H)
241. C₂²×C₄×C₈	C₂×C₆₇	semidirect
242. #2855	C₂×C₆₇	semidirect
244. #1968	C₂×C₆₇	semidirect
245. #1941	C₂×C₆₇	semidirect
246. #1934	C₂×C₆₇	semidirect
247. #1933	C₂×C₆₇	semidirect
249. #1892	C₂×C₆₇	semidirect
250. #1807	C₂×C₆₇	semidirect
251. C₆₇	#33128	direct (N × H)
358. C₂²×C₄	SD₁₆×C₆₇	semidirect
445. C₈	#123×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#465×C₆₇	2
C₂²×C₄×C₈×C₆₇	2
C₆₇×#2855	2
C₆₇×#2854	2
C₆₇×#1968	2
C₆₇×#1941	2
C₆₇×#1934	2
C₆₇×#1933	2
C₆₇×#1893	2
C₆₇×#1892	2
C₆₇×#1807	2
#33128	67

Subgroups — C67×#33128 #30712379

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33128 #30712379