Subgroups — C₆₇×#33107 #30712358

Order 17152.

1104 conjugacy classes of subgroups.

The subgroup lattice has 470 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	2	yes	maximal, complemented
2	C₆₇×#2856	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1982	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1948	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1926	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1874	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1797	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×#121×C₆₇	—	2	1	no	—
11	C₂×#121×C₆₇	—	2	1	no	—
12	C₂×#121×C₆₇	C₂²	1	2	yes	—
13	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₆₇	—	2	1	no	—
16	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
17	#481×C₆₇	—	2	8	no	—
18	C₂×#106×C₆₇	C₂²	1	4	yes	—
19	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
20	C₂×#102×C₆₇	C₂²	1	4	yes	—
21	C₂×#102×C₆₇	C₂²	1	4	yes	—
22	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
23	#390×C₆₇	—	2	1	no	—
24	#390×C₆₇	—	2	1	no	—
25	#390×C₆₇	C₂²	1	2	yes	—
26	#384×C₆₇	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
28	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
29	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
30	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
31	C₂×#95×C₆₇	—	2	1	no	—
32	#340×C₆₇	C₂²	1	4	yes	—
33	#336×C₆₇	C₂²	1	2	yes	—
34	C₂²×D₄×C₆₇	—	2	2	no	—
35	C₂²×D₄×C₆₇	—	2	1	no	—
36	C₂²×D₄×C₆₇	—	2	1	no	—
37	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
38	C₂³×C₄×C₆₇	—	2	1	no	—
39	C₂³×C₄×C₆₇	—	2	1	no	—
40	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
42	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
44	#127×C₆₇	—	2	4	no	—
45	#121×C₆₇	—	2	4	no	—
46	#121×C₆₇	—	2	4	no	—
47	#121×C₆₇	—	4	4	no	—
48	#121×C₆₇	—	4	4	no	—
49	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
50	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
51	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
52	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
53	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
54	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
55	C₂×C₄⋊C₄×C₆₇	D₄	1	4	yes	complemented
56	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
57	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
58	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
59	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
60	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
61	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
62	C₂×C₄²×C₆₇	—	2	4	no	—
63	C₂×C₄²×C₆₇	—	2	2	no	—
64	C₂×C₄²×C₆₇	—	2	2	no	—
65	C₂×C₄²×C₆₇	—	4	4	no	—
66	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
67	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
68	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
70	#106×C₆₇	—	2	8	no	—
71	C₄⋊C₈×C₆₇	—	2	4	no	—
72	#102×C₆₇	—	2	8	no	—
73	#102×C₆₇	—	2	8	no	—
74	C₂²⋊C₈×C₆₇	—	2	4	no	—
75	#95×C₆₇	—	2	2	no	—
76	#95×C₆₇	—	2	2	no	—
77	C₂⁴×C₆₇	—	2	2	no	—
78	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂×D₄×C₆₇	—	2	2	no	—
80	C₂×D₄×C₆₇	—	2	4	no	—
81	C₂×D₄×C₆₇	—	2	2	no	—
82	C₂×D₄×C₆₇	—	2	4	no	—
83	C₂×D₄×C₆₇	—	4	2	no	—
84	C₂×D₄×C₆₇	—	4	2	no	—
85	C₂×D₄×C₆₇	—	4	4	no	—
86	C₂×D₄×C₆₇	—	4	4	no	—
87	C₂×D₄×C₆₇	—	4	4	no	—
88	C₂×D₄×C₆₇	—	4	2	no	—
89	C₂×D₄×C₆₇	—	4	2	no	—
90	C₂×D₄×C₆₇	—	4	2	no	—
91	C₂×D₄×C₆₇	—	4	2	no	—
92	C₂²×C₄×C₆₇	—	2	2	no	—
93	C₂²×C₄×C₆₇	—	2	2	no	—
94	C₂²×C₄×C₆₇	—	2	2	no	—
95	C₂²×C₄×C₆₇	—	2	2	no	—
96	C₂²×C₄×C₆₇	—	2	4	no	—
97	C₂²×C₄×C₆₇	—	2	4	no	—
98	C₂²×C₄×C₆₇	—	2	2	no	—
99	C₂²×C₄×C₆₇	—	2	2	no	—
100	C₂²×C₄×C₆₇	—	2	2	no	—
101	C₂²×C₄×C₆₇	—	2	2	no	—
102	C₂²×C₄×C₆₇	—	2	2	no	—
103	C₂²×C₄×C₆₇	—	4	1	no	—
104	C₂²×C₄×C₆₇	—	4	2	no	—
105	C₂²×C₄×C₆₇	—	4	2	no	—
106	C₂²×C₄×C₆₇	—	4	2	no	—
107	C₂²×C₄×C₆₇	—	4	2	no	—
108	C₂²×C₄×C₆₇	SD₁₆	1	4	yes	—
109	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
112	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₆₇	#41	1	2	yes	—
116	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
117	C₂×C₈×C₆₇	—	2	4	no	—
118	C₂×C₈×C₆₇	—	2	4	no	—
119	C₂×C₈×C₆₇	—	2	4	no	—
120	C₄⋊C₄×C₆₇	—	2	4	no	—
121	C₄⋊C₄×C₆₇	—	2	4	no	—
122	C₄⋊C₄×C₆₇	—	2	2	no	—
123	C₄⋊C₄×C₆₇	—	2	2	no	—
124	C₄⋊C₄×C₆₇	—	2	2	no	—
125	C₄⋊C₄×C₆₇	—	2	2	no	—
126	C₄⋊C₄×C₆₇	—	4	2	no	—
127	C₄⋊C₄×C₆₇	—	4	2	no	—
128	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
129	C₄⋊C₄×C₆₇	#41	1	4	yes	—
130	C₂²⋊C₄×C₆₇	—	4	4	no	—
131	C₂²⋊C₄×C₆₇	—	4	4	no	—
132	C₂²⋊C₄×C₆₇	—	4	4	no	—
133	C₄²×C₆₇	—	2	2	no	—
134	C₄²×C₆₇	—	2	2	no	—
135	C₄²×C₆₇	—	2	2	no	—
136	C₄²×C₆₇	—	4	4	no	—
137	C₄²×C₆₇	—	4	4	no	—
138	C₄²×C₆₇	#41	1	4	yes	—
139	C₂³×C₆₇	—	2	1	no	—
140	C₂³×C₆₇	—	2	1	no	—
141	C₂³×C₆₇	—	2	1	no	—
142	C₂³×C₆₇	—	2	1	no	—
143	C₂³×C₆₇	—	4	2	no	—
144	C₂³×C₆₇	—	4	2	no	—
145	C₂³×C₆₇	—	4	2	no	—
146	C₂³×C₆₇	—	4	2	no	—
147	C₂³×C₆₇	—	4	2	no	—
148	C₂³×C₆₇	—	4	2	no	—
149	C₂³×C₆₇	—	4	2	no	—
150	C₂³×C₆₇	#121	1	2	yes	—
151	C₂³×C₆₇	C₂×SD₁₆	1	2	yes	—
152	C₂³×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
153	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
154	D₄×C₆₇	—	4	4	no	—
155	D₄×C₆₇	—	4	4	no	—
156	D₄×C₆₇	—	8	4	no	—
157	D₄×C₆₇	—	8	8	no	—
158	D₄×C₆₇	—	8	4	no	—
159	C₂×C₄×C₆₇	—	2	1	no	—
160	C₂×C₄×C₆₇	—	2	1	no	—
161	C₂×C₄×C₆₇	—	2	1	no	—
162	C₂×C₄×C₆₇	—	2	1	no	—
163	C₂×C₄×C₆₇	—	2	1	no	—
164	C₂×C₄×C₆₇	—	2	1	no	—
165	C₂×C₄×C₆₇	—	2	4	no	—
166	C₂×C₄×C₆₇	—	2	4	no	—
167	C₂×C₄×C₆₇	—	2	8	no	—
168	C₂×C₄×C₆₇	—	2	4	no	—
169	C₂×C₄×C₆₇	—	2	1	no	—
170	C₂×C₄×C₆₇	—	2	4	no	—
171	C₂×C₄×C₆₇	—	2	1	no	—
172	C₂×C₄×C₆₇	—	2	1	no	—
173	C₂×C₄×C₆₇	—	4	4	no	—
174	C₂×C₄×C₆₇	—	4	8	no	—
175	C₂×C₄×C₆₇	—	4	8	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	—	4	2	no	—
182	C₂×C₄×C₆₇	—	4	2	no	—
183	C₂×C₄×C₆₇	#121	1	2	yes	—
184	C₂×C₄×C₆₇	#123	1	2	yes	—
185	C₂×C₄×C₆₇	#123	1	2	yes	—
186	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
187	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
188	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
189	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
190	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
191	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
192	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
193	C₈×C₆₇	—	4	4	no	—
194	C₂²×C₆₇	—	2	2	no	—
195	C₂²×C₆₇	—	2	2	no	—
196	C₂²×C₆₇	—	2	2	no	—
197	C₂²×C₆₇	—	2	2	no	—
198	C₂²×C₆₇	—	2	2	no	—
199	C₂²×C₆₇	—	2	2	no	—
200	C₂²×C₆₇	—	8	2	no	—
201	C₂²×C₆₇	—	8	2	no	—
202	C₂²×C₆₇	—	8	2	no	—
203	C₂²×C₆₇	—	8	2	no	—
204	C₂²×C₆₇	—	8	2	no	—
205	C₂²×C₆₇	—	8	2	no	—
206	C₂²×C₆₇	—	8	2	no	—
207	C₂²×C₆₇	#460	1	4	yes	—
208	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
209	C₂²×C₆₇	C₂×#123	1	1	yes	characteristic
210	C₂²×C₆₇	#525	1	1	yes	characteristic
211	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
212	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
213	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
214	C₂²×C₆₇	#575	1	1	yes	characteristic
215	C₄×C₆₇	—	2	2	no	—
216	C₄×C₆₇	—	2	2	no	—
217	C₄×C₆₇	—	2	2	no	—
218	C₄×C₆₇	—	4	1	no	—
219	C₄×C₆₇	—	4	4	no	—
220	C₄×C₆₇	—	4	4	no	—
221	C₄×C₆₇	—	8	2	no	—
222	C₄×C₆₇	#481	1	4	yes	complemented
223	C₄×C₆₇	#529	1	2	yes	—
224	C₄×C₆₇	#532	1	2	yes	—
225	#33107	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
226	C₂×C₆₇	—	2	4	no	—
227	C₂×C₆₇	—	16	2	no	—
228	C₂×C₆₇	#2854	1	1	yes	characteristic
229	C₂×C₆₇	C₂×#460	1	1	yes	characteristic
230	C₂×C₆₇	#3036	1	1	yes	characteristic
231	C₂×C₆₇	C₂×#481	1	1	yes	characteristic
232	C₂×C₆₇	#3094	1	1	yes	characteristic
233	C₂×C₆₇	#3103	1	1	yes	characteristic
234	C₂×C₆₇	#3109	1	1	yes	characteristic
235	C₂×#481	C₂×C₆₇	1	2	yes	complemented
236	#2856	C₂×C₆₇	1	1	yes	characteristic
237	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
238	#1982	C₂×C₆₇	1	2	yes	complemented
239	#1948	C₂×C₆₇	1	2	yes	complemented
240	#1926	C₂×C₆₇	1	4	yes	complemented
241	#1874	C₂×C₆₇	1	2	yes	complemented
242	#1797	C₂×C₆₇	1	1	yes	characteristic, complemented
243	C₆₇	#33107	1	1	yes	characteristic, direct product factor, Sylow
244	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
245	C₂×#121	—	2	1	no	—
246	C₂×#121	—	2	1	no	—
247	C₂×#121	C₂²×C₆₇	1	2	yes	—
248	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
249	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
250	C₂²×C₄²	—	2	1	no	—
251	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
252	#481	—	2	8	no	—
253	C₂×#106	C₂²×C₆₇	1	4	yes	—
254	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
255	C₂×#102	C₂²×C₆₇	1	4	yes	—
256	C₂×#102	C₂²×C₆₇	1	4	yes	—
257	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
258	#390	—	2	1	no	—
259	#390	—	2	1	no	—
260	#390	C₂²×C₆₇	1	2	yes	—
261	#384	C₂²×C₆₇	1	1	yes	characteristic
262	C₄×C₄⋊C₄	—	2	1	no	—
263	C₄×C₄⋊C₄	—	2	1	no	—
264	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
265	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
266	C₂×#95	—	2	1	no	—
267	#340	C₂²×C₆₇	1	4	yes	—
268	#336	C₂²×C₆₇	1	2	yes	—
269	C₂²×D₄	—	2	2	no	—
270	C₂²×D₄	—	2	1	no	—
271	C₂²×D₄	—	2	1	no	—
272	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
273	C₂³×C₄	—	2	1	no	—
274	C₂³×C₄	—	2	1	no	—
275	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
276	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
277	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
278	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
279	#127	—	2	4	no	—
280	#121	—	2	4	no	—
281	#121	—	2	4	no	—
282	#121	—	4	4	no	—
283	#121	—	4	4	no	—
284	C₂×C₄⋊C₄	—	2	1	no	—
285	C₂×C₄⋊C₄	—	2	1	no	—
286	C₂×C₄⋊C₄	—	2	2	no	—
287	C₂×C₄⋊C₄	—	2	4	no	—
288	C₂×C₄⋊C₄	—	2	1	no	—
289	C₂×C₄⋊C₄	—	2	1	no	—
290	C₂×C₄⋊C₄	D₄×C₆₇	1	4	yes	complemented
291	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
292	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
293	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
294	C₂×C₂²⋊C₄	—	2	2	no	—
295	C₂×C₂²⋊C₄	—	2	2	no	—
296	C₂×C₂²⋊C₄	—	2	2	no	—
297	C₂×C₄²	—	2	4	no	—
298	C₂×C₄²	—	2	2	no	—
299	C₂×C₄²	—	2	2	no	—
300	C₂×C₄²	—	4	4	no	—
301	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
302	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
303	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
304	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
305	#106	—	2	8	no	—
306	C₄⋊C₈	—	2	4	no	—
307	#102	—	2	8	no	—
308	#102	—	2	8	no	—
309	C₂²⋊C₈	—	2	4	no	—
310	#95	—	2	2	no	—
311	#95	—	2	2	no	—
312	C₂⁴	—	2	2	no	—
313	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
314	C₂×D₄	—	2	2	no	—
315	C₂×D₄	—	2	4	no	—
316	C₂×D₄	—	2	2	no	—
317	C₂×D₄	—	2	4	no	—
318	C₂×D₄	—	4	2	no	—
319	C₂×D₄	—	4	2	no	—
320	C₂×D₄	—	4	4	no	—
321	C₂×D₄	—	4	4	no	—
322	C₂×D₄	—	4	4	no	—
323	C₂×D₄	—	4	2	no	—
324	C₂×D₄	—	4	2	no	—
325	C₂×D₄	—	4	2	no	—
326	C₂×D₄	—	4	2	no	—
327	C₂²×C₄	—	2	2	no	—
328	C₂²×C₄	—	2	2	no	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	—	2	2	no	—
331	C₂²×C₄	—	2	4	no	—
332	C₂²×C₄	—	2	4	no	—
333	C₂²×C₄	—	2	2	no	—
334	C₂²×C₄	—	2	2	no	—
335	C₂²×C₄	—	2	2	no	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	2	2	no	—
338	C₂²×C₄	—	4	1	no	—
339	C₂²×C₄	—	4	2	no	—
340	C₂²×C₄	—	4	2	no	—
341	C₂²×C₄	—	4	2	no	—
342	C₂²×C₄	—	4	2	no	—
343	C₂²×C₄	SD₁₆×C₆₇	1	4	yes	—
344	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
345	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
346	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
347	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
348	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
349	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
350	C₂²×C₄	#41×C₆₇	1	2	yes	—
351	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
352	C₂×C₈	—	2	4	no	—
353	C₂×C₈	—	2	4	no	—
354	C₂×C₈	—	2	4	no	—
355	C₄⋊C₄	—	2	4	no	—
356	C₄⋊C₄	—	2	4	no	—
357	C₄⋊C₄	—	2	2	no	—
358	C₄⋊C₄	—	2	2	no	—
359	C₄⋊C₄	—	2	2	no	—
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	—	4	2	no	—
362	C₄⋊C₄	—	4	2	no	—
363	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
364	C₄⋊C₄	#41×C₆₇	1	4	yes	—
365	C₂²⋊C₄	—	4	4	no	—
366	C₂²⋊C₄	—	4	4	no	—
367	C₂²⋊C₄	—	4	4	no	—
368	C₄²	—	2	2	no	—
369	C₄²	—	2	2	no	—
370	C₄²	—	2	2	no	—
371	C₄²	—	4	4	no	—
372	C₄²	—	4	4	no	—
373	C₄²	#41×C₆₇	1	4	yes	—
374	C₂³	—	2	1	no	—
375	C₂³	—	2	1	no	—
376	C₂³	—	2	1	no	—
377	C₂³	—	2	1	no	—
378	C₂³	—	4	2	no	—
379	C₂³	—	4	2	no	—
380	C₂³	—	4	2	no	—
381	C₂³	—	4	2	no	—
382	C₂³	—	4	2	no	—
383	C₂³	—	4	2	no	—
384	C₂³	—	4	2	no	—
385	C₂³	#121×C₆₇	1	2	yes	—
386	C₂³	C₂×SD₁₆×C₆₇	1	2	yes	—
387	C₂³	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
388	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
389	D₄	—	4	4	no	—
390	D₄	—	4	4	no	—
391	D₄	—	8	4	no	—
392	D₄	—	8	8	no	—
393	D₄	—	8	4	no	—
394	C₂×C₄	—	2	1	no	—
395	C₂×C₄	—	2	1	no	—
396	C₂×C₄	—	2	1	no	—
397	C₂×C₄	—	2	1	no	—
398	C₂×C₄	—	2	1	no	—
399	C₂×C₄	—	2	1	no	—
400	C₂×C₄	—	2	8	no	—
401	C₂×C₄	—	2	4	no	—
402	C₂×C₄	—	2	4	no	—
403	C₂×C₄	—	2	4	no	—
404	C₂×C₄	—	2	1	no	—
405	C₂×C₄	—	2	4	no	—
406	C₂×C₄	—	2	1	no	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	4	4	no	—
409	C₂×C₄	—	4	8	no	—
410	C₂×C₄	—	4	8	no	—
411	C₂×C₄	—	4	2	no	—
412	C₂×C₄	—	4	2	no	—
413	C₂×C₄	—	4	2	no	—
414	C₂×C₄	—	4	2	no	—
415	C₂×C₄	—	4	2	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	#121×C₆₇	1	2	yes	—
419	C₂×C₄	#123×C₆₇	1	2	yes	—
420	C₂×C₄	#123×C₆₇	1	2	yes	—
421	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
422	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
423	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
424	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
425	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
426	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
427	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
428	C₈	—	4	4	no	—
429	C₂²	—	2	2	no	—
430	C₂²	—	2	2	no	—
431	C₂²	—	2	2	no	—
432	C₂²	—	2	2	no	—
433	C₂²	—	2	2	no	—
434	C₂²	—	2	2	no	—
435	C₂²	—	8	2	no	—
436	C₂²	—	8	2	no	—
437	C₂²	—	8	2	no	—
438	C₂²	—	8	2	no	—
439	C₂²	—	8	2	no	—
440	C₂²	—	8	2	no	—
441	C₂²	—	8	2	no	—
442	C₂²	#460×C₆₇	1	4	yes	—
443	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
444	C₂²	C₂×#123×C₆₇	1	1	yes	characteristic
445	C₂²	#525×C₆₇	1	1	yes	characteristic
446	C₂²	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
447	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
448	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
449	C₂²	#575×C₆₇	1	1	yes	characteristic
450	C₄	—	2	2	no	—
451	C₄	—	2	2	no	—
452	C₄	—	2	2	no	—
453	C₄	—	4	4	no	—
454	C₄	—	4	1	no	—
455	C₄	—	4	4	no	—
456	C₄	—	8	2	no	—
457	C₄	#481×C₆₇	1	4	yes	complemented
458	C₄	#529×C₆₇	1	2	yes	—
459	C₄	#532×C₆₇	1	2	yes	—
460	C₂	—	2	4	no	—
461	C₂	—	16	2	no	—
462	C₂	C₆₇×#2854	1	1	yes	characteristic
463	C₂	C₂×#460×C₆₇	1	1	yes	characteristic
464	C₂	C₆₇×#3036	1	1	yes	characteristic
465	C₂	C₂×#481×C₆₇	1	1	yes	characteristic
466	C₂	C₆₇×#3094	1	1	yes	characteristic
467	C₂	C₆₇×#3103	1	1	yes	characteristic
468	C₂	C₆₇×#3109	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₆₇×#2856	C₂
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
1. C₂×#481×C₆₇	C₂	semidirect
3. C₂×C₄×C₄⋊C₄×C₆₇	C₂	semidirect
4. C₆₇×#1982	C₂	semidirect
5. C₆₇×#1948	C₂	semidirect
6. C₆₇×#1926	C₂	semidirect
7. C₆₇×#1874	C₂	semidirect
8. C₆₇×#1797	C₂	semidirect
55. C₂×C₄⋊C₄×C₆₇	D₄	semidirect
222. C₄×C₆₇	#481	semidirect
225. #33107	C₆₇	direct (N × H)
235. C₂×#481	C₂×C₆₇	semidirect
237. C₂×C₄×C₄⋊C₄	C₂×C₆₇	semidirect
238. #1982	C₂×C₆₇	semidirect
239. #1948	C₂×C₆₇	semidirect
240. #1926	C₂×C₆₇	semidirect
241. #1874	C₂×C₆₇	semidirect
242. #1797	C₂×C₆₇	semidirect
243. C₆₇	#33107	direct (N × H)
290. C₂×C₄⋊C₄	D₄×C₆₇	semidirect
457. C₄	#481×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
C₆₇×#2856	2
C₂×C₄×C₄⋊C₄×C₆₇	2
C₆₇×#1982	2
C₆₇×#1948	2
C₆₇×#1926	2
C₆₇×#1874	2
C₆₇×#1797	2
#33107	67

Subgroups — C67×#33107 #30712358

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33107 #30712358