Subgroups — C₆₇×#10062 #30689313

Order 17152.

806 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#486×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#461×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2541	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1988	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#331×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1279	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×SD₁₆×C₆₇	—	2	1	no	—
9	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×#127×C₆₇	—	2	1	no	—
11	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×#121×C₆₇	—	2	2	no	—
13	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
14	#486×C₆₇	—	2	2	no	—
15	#486×C₆₇	—	2	2	no	—
16	#461×C₆₇	—	2	2	no	—
17	#461×C₆₇	—	2	2	no	—
18	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#103×C₆₇	—	2	1	no	—
21	C₂×#102×C₆₇	—	2	2	no	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	#392×C₆₇	—	2	2	no	—
24	#390×C₆₇	—	2	1	no	—
25	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
26	#340×C₆₇	—	2	2	no	—
27	#331×C₆₇	—	2	2	no	—
28	D₄⋊C₈×C₆₇	—	2	2	no	—
29	C₂²×Q₈×C₆₇	—	2	1	no	—
30	C₂²×D₄×C₆₇	—	2	4	no	—
31	C₂²×D₄×C₆₇	—	2	1	no	—
32	C₂²×D₄×C₆₇	—	2	1	no	—
33	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
34	C₂²×D₄×C₆₇	D₄	1	2	yes	—
35	C₂³×C₄×C₆₇	—	2	1	no	—
36	C₂×SD₁₆×C₆₇	—	2	4	no	—
37	C₂×SD₁₆×C₆₇	—	4	2	no	—
38	C₂×SD₁₆×C₆₇	—	4	2	no	—
39	C₂²×C₈×C₆₇	—	2	1	no	—
40	C₂²×C₈×C₆₇	—	2	1	no	—
41	C₂²×C₈×C₆₇	—	2	1	no	—
42	C₄⋊Q₈×C₆₇	—	2	1	no	—
43	C₄⋊Q₈×C₆₇	—	2	1	no	—
44	C₄⋊Q₈×C₆₇	—	2	1	no	—
45	C₄⋊Q₈×C₆₇	D₄	1	2	yes	complemented
46	#127×C₆₇	—	2	1	no	—
47	#127×C₆₇	—	2	1	no	—
48	#127×C₆₇	—	2	1	no	—
49	#127×C₆₇	—	2	1	no	—
50	#127×C₆₇	—	4	2	no	—
51	#127×C₆₇	—	4	2	no	—
52	#121×C₆₇	—	4	4	no	—
53	#121×C₆₇	—	4	4	no	—
54	C₄×D₄×C₆₇	—	2	1	no	—
55	C₄×D₄×C₆₇	—	2	1	no	—
56	C₄×D₄×C₆₇	—	2	1	no	—
57	C₄×D₄×C₆₇	—	2	1	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
59	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
60	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
62	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
63	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
64	C₂×C₄²×C₆₇	—	2	1	no	—
65	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
66	C₄⋊C₈×C₆₇	—	2	1	no	—
67	C₄⋊C₈×C₆₇	—	2	1	no	—
68	C₄⋊C₈×C₆₇	—	2	1	no	—
69	C₄⋊C₈×C₆₇	D₄	1	2	yes	complemented
70	#103×C₆₇	—	2	4	no	—
71	#102×C₆₇	—	2	8	no	—
72	C₄×C₈×C₆₇	—	2	1	no	—
73	C₄×C₈×C₆₇	D₄	1	2	yes	complemented
74	#95×C₆₇	—	2	1	no	—
75	C₂⁴×C₆₇	—	2	1	no	—
76	C₂⁴×C₆₇	—	2	2	no	—
77	C₂×Q₈×C₆₇	—	2	2	no	—
78	C₂×Q₈×C₆₇	—	4	1	no	—
79	C₂×Q₈×C₆₇	—	4	1	no	—
80	C₂×Q₈×C₆₇	—	4	1	no	—
81	C₂×Q₈×C₆₇	—	4	1	no	—
82	C₂×Q₈×C₆₇	—	4	1	no	—
83	C₂×D₄×C₆₇	—	2	1	no	—
84	C₂×D₄×C₆₇	—	2	2	no	—
85	C₂×D₄×C₆₇	—	2	1	no	—
86	C₂×D₄×C₆₇	—	2	2	no	—
87	C₂×D₄×C₆₇	—	2	1	no	—
88	C₂×D₄×C₆₇	—	2	1	no	—
89	C₂×D₄×C₆₇	—	2	1	no	—
90	C₂×D₄×C₆₇	—	4	4	no	—
91	C₂×D₄×C₆₇	—	4	8	no	—
92	C₂×D₄×C₆₇	—	4	8	no	—
93	C₂×D₄×C₆₇	—	4	1	no	—
94	C₂×D₄×C₆₇	—	4	1	no	—
95	C₂×D₄×C₆₇	—	4	1	no	—
96	C₂×D₄×C₆₇	—	4	1	no	—
97	C₂×D₄×C₆₇	—	4	1	no	—
98	C₂×D₄×C₆₇	—	4	2	no	—
99	C₂×D₄×C₆₇	—	4	1	no	—
100	C₂×D₄×C₆₇	—	4	2	no	—
101	C₂×D₄×C₆₇	—	4	1	no	—
102	C₂×D₄×C₆₇	—	4	1	no	—
103	C₂×D₄×C₆₇	—	8	4	no	—
104	C₂×D₄×C₆₇	D₈	1	2	yes	—
105	C₂²×C₄×C₆₇	—	2	2	no	—
106	C₂²×C₄×C₆₇	—	2	1	no	—
107	C₂²×C₄×C₆₇	—	4	1	no	—
108	C₂²×C₄×C₆₇	—	4	1	no	—
109	C₂²×C₄×C₆₇	—	4	1	no	—
110	C₂²×C₄×C₆₇	—	4	2	no	—
111	C₂²×C₄×C₆₇	—	4	2	no	—
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	SD₁₆×C₆₇	—	4	8	no	—
116	C₂×C₈×C₆₇	—	2	2	no	—
117	C₂×C₈×C₆₇	—	2	2	no	—
118	C₂×C₈×C₆₇	—	4	2	no	—
119	C₂×C₈×C₆₇	—	4	1	no	—
120	C₂×C₈×C₆₇	—	4	1	no	—
121	C₂×C₈×C₆₇	—	4	1	no	—
122	C₂×C₈×C₆₇	—	4	2	no	—
123	C₄⋊C₄×C₆₇	—	2	2	no	—
124	C₄⋊C₄×C₆₇	—	2	2	no	—
125	C₄⋊C₄×C₆₇	—	2	1	no	—
126	C₄⋊C₄×C₆₇	—	2	2	no	—
127	C₄⋊C₄×C₆₇	—	2	1	no	—
128	C₄⋊C₄×C₆₇	—	4	1	no	—
129	C₄⋊C₄×C₆₇	—	4	1	no	—
130	C₄⋊C₄×C₆₇	SD₁₆	1	4	yes	complemented
131	C₂²⋊C₄×C₆₇	—	4	8	no	—
132	C₂²⋊C₄×C₆₇	—	4	1	no	—
133	C₂²⋊C₄×C₆₇	—	4	1	no	—
134	C₄²×C₆₇	—	4	1	no	—
135	C₄²×C₆₇	—	4	1	no	—
136	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
137	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
138	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
139	C₄²×C₆₇	#41	1	1	yes	characteristic
140	C₂³×C₆₇	—	2	2	no	—
141	C₂³×C₆₇	—	4	1	no	—
142	C₂³×C₆₇	—	4	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	4	no	—
149	C₂³×C₆₇	—	4	4	no	—
150	C₂³×C₆₇	#120	1	1	yes	characteristic
151	Q₈×C₆₇	—	4	2	no	—
152	Q₈×C₆₇	—	4	2	no	—
153	Q₈×C₆₇	—	8	1	no	—
154	Q₈×C₆₇	—	8	1	no	—
155	D₄×C₆₇	—	2	2	no	—
156	D₄×C₆₇	—	2	2	no	—
157	D₄×C₆₇	—	4	4	no	—
158	D₄×C₆₇	—	4	1	no	—
159	D₄×C₆₇	—	4	1	no	—
160	D₄×C₆₇	—	8	2	no	—
161	D₄×C₆₇	—	8	16	no	—
162	D₄×C₆₇	—	8	1	no	—
163	D₄×C₆₇	—	8	1	no	—
164	D₄×C₆₇	—	8	2	no	—
165	D₄×C₆₇	—	8	1	no	—
166	D₄×C₆₇	—	8	1	no	—
167	D₄×C₆₇	—	8	1	no	—
168	D₄×C₆₇	—	8	1	no	—
169	C₂×C₄×C₆₇	—	2	2	no	—
170	C₂×C₄×C₆₇	—	2	4	no	—
171	C₂×C₄×C₆₇	—	4	2	no	—
172	C₂×C₄×C₆₇	—	4	2	no	—
173	C₂×C₄×C₆₇	—	4	1	no	—
174	C₂×C₄×C₆₇	—	4	2	no	—
175	C₂×C₄×C₆₇	—	4	2	no	—
176	C₂×C₄×C₆₇	—	8	1	no	—
177	C₂×C₄×C₆₇	—	8	1	no	—
178	C₂×C₄×C₆₇	—	8	2	no	—
179	C₂×C₄×C₆₇	—	8	1	no	—
180	C₂×C₄×C₆₇	—	8	1	no	—
181	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
182	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
183	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
184	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
185	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
186	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
187	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
188	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
189	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
190	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
191	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
192	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
193	C₈×C₆₇	—	4	2	no	—
194	C₈×C₆₇	—	4	2	no	—
195	C₈×C₆₇	—	8	1	no	—
196	C₂²×C₆₇	—	4	4	no	—
197	C₂²×C₆₇	—	4	2	no	—
198	C₂²×C₆₇	—	8	2	no	—
199	C₂²×C₆₇	—	8	1	no	—
200	C₂²×C₆₇	—	8	1	no	—
201	C₂²×C₆₇	—	8	4	no	—
202	C₂²×C₆₇	—	8	4	no	—
203	C₂²×C₆₇	—	8	2	no	—
204	C₂²×C₆₇	—	8	2	no	—
205	C₂²×C₆₇	—	8	2	no	—
206	C₂²×C₆₇	#392	1	1	yes	characteristic
207	C₂²×C₆₇	#447	1	1	yes	characteristic
208	C₂²×C₆₇	#449	1	1	yes	characteristic
209	C₂²×C₆₇	#450	1	2	yes	—
210	C₂²×C₆₇	#455	1	2	yes	—
211	C₄×C₆₇	—	2	1	no	—
212	C₄×C₆₇	—	2	1	no	—
213	C₄×C₆₇	—	4	4	no	—
214	C₄×C₆₇	—	8	2	no	—
215	C₄×C₆₇	—	8	1	no	—
216	C₄×C₆₇	#447	1	1	yes	characteristic
217	C₄×C₆₇	#449	1	1	yes	characteristic
218	C₄×C₆₇	#460	1	2	yes	—
219	C₄×C₆₇	#466	1	1	yes	characteristic
220	C₄×C₆₇	#469	1	1	yes	characteristic
221	C₄×C₆₇	#492	1	1	yes	characteristic
222	C₄×C₆₇	#496	1	1	yes	characteristic
223	#10062	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
224	C₂×C₆₇	—	8	2	no	—
225	C₂×C₆₇	—	16	2	no	—
226	C₂×C₆₇	#1639	1	2	yes	—
227	C₂×C₆₇	#1642	1	2	yes	—
228	C₂×C₆₇	#1999	1	1	yes	characteristic
229	C₂×C₆₇	#2004	1	1	yes	characteristic
230	C₂×C₆₇	#2014	1	1	yes	characteristic
231	C₂×#486	C₂×C₆₇	1	1	yes	characteristic, complemented
232	C₂×#461	C₂×C₆₇	1	1	yes	characteristic, complemented
233	#2541	C₂×C₆₇	1	1	yes	characteristic
234	#1988	C₂×C₆₇	1	1	yes	characteristic, complemented
235	C₂×#331	C₂×C₆₇	1	1	yes	characteristic, complemented
236	C₂×D₄⋊C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
237	#1279	C₂×C₆₇	1	1	yes	characteristic, complemented
238	C₆₇	#10062	1	1	yes	characteristic, direct product factor, Sylow
239	C₂²×SD₁₆	—	2	1	no	—
240	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
241	C₂×#127	—	2	1	no	—
242	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
243	C₂×#121	—	2	2	no	—
244	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
245	#486	—	2	2	no	—
246	#486	—	2	2	no	—
247	#461	—	2	2	no	—
248	#461	—	2	2	no	—
249	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
250	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
251	C₂×#103	—	2	1	no	—
252	C₂×#102	—	2	2	no	—
253	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
254	#392	—	2	2	no	—
255	#390	—	2	1	no	—
256	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
257	#340	—	2	2	no	—
258	#331	—	2	2	no	—
259	D₄⋊C₈	—	2	2	no	—
260	C₂²×Q₈	—	2	1	no	—
261	C₂²×D₄	—	2	4	no	—
262	C₂²×D₄	—	2	1	no	—
263	C₂²×D₄	—	2	1	no	—
264	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
265	C₂²×D₄	D₄×C₆₇	1	2	yes	—
266	C₂³×C₄	—	2	1	no	—
267	C₂×SD₁₆	—	2	4	no	—
268	C₂×SD₁₆	—	4	2	no	—
269	C₂×SD₁₆	—	4	2	no	—
270	C₂²×C₈	—	2	1	no	—
271	C₂²×C₈	—	2	1	no	—
272	C₂²×C₈	—	2	1	no	—
273	C₄⋊Q₈	—	2	1	no	—
274	C₄⋊Q₈	—	2	1	no	—
275	C₄⋊Q₈	—	2	1	no	—
276	C₄⋊Q₈	D₄×C₆₇	1	2	yes	complemented
277	#127	—	2	1	no	—
278	#127	—	2	1	no	—
279	#127	—	2	1	no	—
280	#127	—	2	1	no	—
281	#127	—	4	2	no	—
282	#127	—	4	2	no	—
283	#121	—	4	4	no	—
284	#121	—	4	4	no	—
285	C₄×D₄	—	2	1	no	—
286	C₄×D₄	—	2	1	no	—
287	C₄×D₄	—	2	1	no	—
288	C₄×D₄	—	2	1	no	—
289	C₂×C₄⋊C₄	—	2	1	no	—
290	C₂×C₄⋊C₄	—	2	1	no	—
291	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
292	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
293	C₂×C₂²⋊C₄	—	2	4	no	—
294	C₂×C₂²⋊C₄	—	2	1	no	—
295	C₂×C₄²	—	2	1	no	—
296	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
297	C₄⋊C₈	—	2	1	no	—
298	C₄⋊C₈	—	2	1	no	—
299	C₄⋊C₈	—	2	1	no	—
300	C₄⋊C₈	D₄×C₆₇	1	2	yes	complemented
301	#103	—	2	4	no	—
302	#102	—	2	8	no	—
303	C₄×C₈	—	2	1	no	—
304	C₄×C₈	D₄×C₆₇	1	2	yes	complemented
305	#95	—	2	1	no	—
306	C₂⁴	—	2	1	no	—
307	C₂⁴	—	2	2	no	—
308	C₂×Q₈	—	2	2	no	—
309	C₂×Q₈	—	4	1	no	—
310	C₂×Q₈	—	4	1	no	—
311	C₂×Q₈	—	4	1	no	—
312	C₂×Q₈	—	4	1	no	—
313	C₂×Q₈	—	4	1	no	—
314	C₂×D₄	—	2	1	no	—
315	C₂×D₄	—	2	2	no	—
316	C₂×D₄	—	2	1	no	—
317	C₂×D₄	—	2	2	no	—
318	C₂×D₄	—	2	1	no	—
319	C₂×D₄	—	2	1	no	—
320	C₂×D₄	—	2	1	no	—
321	C₂×D₄	—	4	4	no	—
322	C₂×D₄	—	4	8	no	—
323	C₂×D₄	—	4	8	no	—
324	C₂×D₄	—	4	1	no	—
325	C₂×D₄	—	4	1	no	—
326	C₂×D₄	—	4	1	no	—
327	C₂×D₄	—	4	1	no	—
328	C₂×D₄	—	4	1	no	—
329	C₂×D₄	—	4	2	no	—
330	C₂×D₄	—	4	1	no	—
331	C₂×D₄	—	4	2	no	—
332	C₂×D₄	—	4	1	no	—
333	C₂×D₄	—	4	1	no	—
334	C₂×D₄	—	8	4	no	—
335	C₂×D₄	D₈×C₆₇	1	2	yes	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	2	1	no	—
338	C₂²×C₄	—	4	1	no	—
339	C₂²×C₄	—	4	1	no	—
340	C₂²×C₄	—	4	1	no	—
341	C₂²×C₄	—	4	2	no	—
342	C₂²×C₄	—	4	2	no	—
343	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
344	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
345	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
346	SD₁₆	—	4	8	no	—
347	C₂×C₈	—	2	2	no	—
348	C₂×C₈	—	2	2	no	—
349	C₂×C₈	—	4	2	no	—
350	C₂×C₈	—	4	1	no	—
351	C₂×C₈	—	4	1	no	—
352	C₂×C₈	—	4	1	no	—
353	C₂×C₈	—	4	2	no	—
354	C₄⋊C₄	—	2	2	no	—
355	C₄⋊C₄	—	2	2	no	—
356	C₄⋊C₄	—	2	1	no	—
357	C₄⋊C₄	—	2	2	no	—
358	C₄⋊C₄	—	2	1	no	—
359	C₄⋊C₄	—	4	1	no	—
360	C₄⋊C₄	—	4	1	no	—
361	C₄⋊C₄	SD₁₆×C₆₇	1	4	yes	complemented
362	C₂²⋊C₄	—	4	8	no	—
363	C₂²⋊C₄	—	4	1	no	—
364	C₂²⋊C₄	—	4	1	no	—
365	C₄²	—	4	1	no	—
366	C₄²	—	4	1	no	—
367	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
368	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
369	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
370	C₄²	#41×C₆₇	1	1	yes	characteristic
371	C₂³	—	2	2	no	—
372	C₂³	—	4	1	no	—
373	C₂³	—	4	2	no	—
374	C₂³	—	4	2	no	—
375	C₂³	—	4	1	no	—
376	C₂³	—	4	2	no	—
377	C₂³	—	4	4	no	—
378	C₂³	—	4	4	no	—
379	C₂³	—	4	2	no	—
380	C₂³	—	4	2	no	—
381	C₂³	#120×C₆₇	1	1	yes	characteristic
382	Q₈	—	4	2	no	—
383	Q₈	—	4	2	no	—
384	Q₈	—	8	1	no	—
385	Q₈	—	8	1	no	—
386	D₄	—	2	2	no	—
387	D₄	—	2	2	no	—
388	D₄	—	4	4	no	—
389	D₄	—	4	1	no	—
390	D₄	—	4	1	no	—
391	D₄	—	8	2	no	—
392	D₄	—	8	16	no	—
393	D₄	—	8	1	no	—
394	D₄	—	8	2	no	—
395	D₄	—	8	1	no	—
396	D₄	—	8	1	no	—
397	D₄	—	8	1	no	—
398	D₄	—	8	1	no	—
399	D₄	—	8	1	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	4	no	—
402	C₂×C₄	—	4	2	no	—
403	C₂×C₄	—	4	2	no	—
404	C₂×C₄	—	4	2	no	—
405	C₂×C₄	—	4	1	no	—
406	C₂×C₄	—	4	2	no	—
407	C₂×C₄	—	8	1	no	—
408	C₂×C₄	—	8	1	no	—
409	C₂×C₄	—	8	2	no	—
410	C₂×C₄	—	8	1	no	—
411	C₂×C₄	—	8	1	no	—
412	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
413	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
414	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
415	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
416	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
417	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
418	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
419	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
420	C₂×C₄	#135×C₆₇	1	1	yes	characteristic
421	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
422	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
423	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
424	C₈	—	4	2	no	—
425	C₈	—	4	2	no	—
426	C₈	—	8	1	no	—
427	C₂²	—	4	2	no	—
428	C₂²	—	4	4	no	—
429	C₂²	—	8	1	no	—
430	C₂²	—	8	1	no	—
431	C₂²	—	8	2	no	—
432	C₂²	—	8	2	no	—
433	C₂²	—	8	2	no	—
434	C₂²	—	8	2	no	—
435	C₂²	—	8	4	no	—
436	C₂²	—	8	4	no	—
437	C₂²	#392×C₆₇	1	1	yes	characteristic
438	C₂²	#447×C₆₇	1	1	yes	characteristic
439	C₂²	#449×C₆₇	1	1	yes	characteristic
440	C₂²	#450×C₆₇	1	2	yes	—
441	C₂²	#455×C₆₇	1	2	yes	—
442	C₄	—	2	1	no	—
443	C₄	—	2	1	no	—
444	C₄	—	4	4	no	—
445	C₄	—	8	2	no	—
446	C₄	—	8	1	no	—
447	C₄	#447×C₆₇	1	1	yes	characteristic
448	C₄	#449×C₆₇	1	1	yes	characteristic
449	C₄	#460×C₆₇	1	2	yes	—
450	C₄	#466×C₆₇	1	1	yes	characteristic
451	C₄	#469×C₆₇	1	1	yes	characteristic
452	C₄	#492×C₆₇	1	1	yes	characteristic
453	C₄	#496×C₆₇	1	1	yes	characteristic
454	C₂	—	8	2	no	—
455	C₂	—	16	2	no	—
456	C₂	C₆₇×#1639	1	2	yes	—
457	C₂	C₆₇×#1642	1	2	yes	—
458	C₂	C₆₇×#1999	1	1	yes	characteristic
459	C₂	C₆₇×#2004	1	1	yes	characteristic
460	C₂	C₆₇×#2014	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₆₇
prime radical	C₆₇×#2541	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#486×C₆₇	C₂	semidirect
2. C₂×#461×C₆₇	C₂	semidirect
4. C₆₇×#1988	C₂	semidirect
5. C₂×#331×C₆₇	C₂	semidirect
6. C₂×D₄⋊C₈×C₆₇	C₂	semidirect
7. C₆₇×#1279	C₂	semidirect
45. C₄⋊Q₈×C₆₇	D₄	semidirect
69. C₄⋊C₈×C₆₇	D₄	semidirect
73. C₄×C₈×C₆₇	D₄	semidirect
130. C₄⋊C₄×C₆₇	SD₁₆	semidirect
223. #10062	C₆₇	direct (N × H)
231. C₂×#486	C₂×C₆₇	semidirect
232. C₂×#461	C₂×C₆₇	semidirect
234. #1988	C₂×C₆₇	semidirect
235. C₂×#331	C₂×C₆₇	semidirect
236. C₂×D₄⋊C₈	C₂×C₆₇	semidirect
237. #1279	C₂×C₆₇	semidirect
238. C₆₇	#10062	direct (N × H)
276. C₄⋊Q₈	D₄×C₆₇	semidirect
300. C₄⋊C₈	D₄×C₆₇	semidirect
304. C₄×C₈	D₄×C₆₇	semidirect
361. C₄⋊C₄	SD₁₆×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#486×C₆₇	2
C₂×#461×C₆₇	2
C₆₇×#2541	2
C₆₇×#1988	2
C₂×#331×C₆₇	2
C₂×D₄⋊C₈×C₆₇	2
C₆₇×#1279	2
#10062	67

Subgroups — C67×#10062 #30689313

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#10062 #30689313