Subgroups — C₆₇×#33062 #30712313

Order 17152.

764 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#488×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3080	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3040	C₂	1	1	yes	maximal, characteristic
4	C₄×#131×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1998	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1967	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1931	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1930	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1908	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1907	C₂	1	2	yes	maximal
11	C₆₇×#1837	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#137×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#131×C₆₇	C₂²	1	2	yes	—
15	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
17	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
18	#488×C₆₇	—	2	1	no	—
19	#488×C₆₇	—	2	1	no	—
20	#488×C₆₇	—	2	2	no	—
21	#479×C₆₇	—	2	2	no	—
22	#478×C₆₇	—	2	2	no	—
23	#464×C₆₇	—	2	2	no	—
24	#463×C₆₇	—	2	2	no	—
25	#428×C₆₇	C₂²	1	2	yes	—
26	#423×C₆₇	C₂²	1	1	yes	characteristic
27	#423×C₆₇	C₂²	1	1	yes	characteristic
28	#419×C₆₇	C₂²	1	1	yes	characteristic
29	#418×C₆₇	C₂²	1	1	yes	characteristic
30	#417×C₆₇	C₂²	1	2	yes	—
31	C₂×#103×C₆₇	C₂²	1	2	yes	—
32	C₂×#102×C₆₇	C₂²	1	2	yes	—
33	C₂×#101×C₆₇	C₂²	1	2	yes	—
34	C₂×#100×C₆₇	C₂²	1	2	yes	—
35	#405×C₆₇	—	2	2	no	—
36	#405×C₆₇	C₂²	1	2	yes	—
37	C₄×M₁₆×C₆₇	—	2	1	no	—
38	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₈×C₆₇	—	2	1	no	—
40	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
41	#343×C₆₇	C₂²	1	2	yes	—
42	#343×C₆₇	C₂²	1	2	yes	—
43	#341×C₆₇	C₂²	1	2	yes	—
44	#341×C₆₇	C₂²	1	2	yes	—
45	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
46	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
47	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
48	#137×C₆₇	—	2	4	no	—
49	AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
50	C₂×Q₁₆×C₆₇	—	2	1	no	—
51	C₂×SD₁₆×C₆₇	—	2	1	no	—
52	C₂×SD₁₆×C₆₇	—	2	1	no	—
53	C₂×D₈×C₆₇	—	2	1	no	—
54	#131×C₆₇	—	2	4	no	—
55	#131×C₆₇	—	2	4	no	—
56	C₂×M₁₆×C₆₇	—	2	2	no	—
57	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
58	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
59	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
60	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
61	C₂²×C₈×C₆₇	—	2	2	no	—
62	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
63	#125×C₆₇	—	2	2	no	—
64	#125×C₆₇	—	2	1	no	—
65	#125×C₆₇	—	2	1	no	—
66	#124×C₆₇	—	2	2	no	—
67	#124×C₆₇	—	2	1	no	—
68	#124×C₆₇	—	2	1	no	—
69	C₄×Q₈×C₆₇	D₄	1	2	yes	—
70	C₄×D₄×C₆₇	—	2	2	no	—
71	C₄×D₄×C₆₇	D₄	1	2	yes	—
72	#117×C₆₇	—	2	1	no	—
73	#117×C₆₇	C₂³	1	1	yes	characteristic
74	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
75	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
76	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
77	C₂×C₄²×C₆₇	—	2	1	no	—
78	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
79	#107×C₆₇	—	2	2	no	—
80	#106×C₆₇	—	2	2	no	—
81	C₄⋊C₈×C₆₇	—	2	2	no	—
82	C₄⋊C₈×C₆₇	—	2	2	no	—
83	#103×C₆₇	—	2	2	no	—
84	#103×C₆₇	—	2	2	no	—
85	#103×C₆₇	—	2	2	no	—
86	#103×C₆₇	—	2	2	no	—
87	#102×C₆₇	—	2	2	no	—
88	#102×C₆₇	—	2	2	no	—
89	#102×C₆₇	—	2	2	no	—
90	#102×C₆₇	—	2	2	no	—
91	#101×C₆₇	—	2	4	no	—
92	#100×C₆₇	—	2	4	no	—
93	#97×C₆₇	—	2	2	no	—
94	#97×C₆₇	—	2	1	no	—
95	C₄×C₈×C₆₇	—	2	2	no	—
96	C₄×C₈×C₆₇	—	2	1	no	—
97	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
98	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
99	C₄×C₈×C₆₇	D₄	1	2	yes	—
100	C₂⁴×C₆₇	—	2	1	no	—
101	#41×C₆₇	—	2	2	no	—
102	#41×C₆₇	#41	1	4	yes	—
103	C₂×Q₈×C₆₇	—	2	1	no	—
104	C₂×Q₈×C₆₇	—	2	1	no	—
105	C₂×Q₈×C₆₇	—	2	2	no	—
106	C₂×Q₈×C₆₇	—	2	2	no	—
107	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
108	C₂×D₄×C₆₇	—	2	1	no	—
109	C₂×D₄×C₆₇	—	2	1	no	—
110	C₂×D₄×C₆₇	—	2	1	no	—
111	C₂×D₄×C₆₇	—	2	2	no	—
112	C₂×D₄×C₆₇	—	2	2	no	—
113	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₆₇	—	2	1	no	—
115	C₂²×C₄×C₆₇	—	2	2	no	—
116	C₂²×C₄×C₆₇	—	2	1	no	—
117	C₂²×C₄×C₆₇	—	2	2	no	—
118	C₂²×C₄×C₆₇	#41	1	2	yes	—
119	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
120	Q₁₆×C₆₇	—	4	2	no	—
121	SD₁₆×C₆₇	—	4	2	no	—
122	SD₁₆×C₆₇	—	4	2	no	—
123	D₈×C₆₇	—	4	2	no	—
124	M₁₆×C₆₇	—	2	8	no	—
125	M₁₆×C₆₇	—	2	2	no	—
126	M₁₆×C₆₇	—	2	2	no	—
127	M₁₆×C₆₇	—	2	2	no	—
128	M₁₆×C₆₇	—	2	2	no	—
129	M₁₆×C₆₇	—	2	2	no	—
130	M₁₆×C₆₇	#41	1	4	yes	—
131	C₂×C₈×C₆₇	—	2	8	no	—
132	C₂×C₈×C₆₇	—	2	2	no	—
133	C₂×C₈×C₆₇	—	2	2	no	—
134	C₂×C₈×C₆₇	—	2	2	no	—
135	C₂×C₈×C₆₇	—	2	1	no	—
136	C₂×C₈×C₆₇	—	2	2	no	—
137	C₂×C₈×C₆₇	—	2	1	no	—
138	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
139	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
140	C₄⋊C₄×C₆₇	—	2	2	no	—
141	C₄⋊C₄×C₆₇	—	2	2	no	—
142	C₄⋊C₄×C₆₇	—	2	2	no	—
143	C₄⋊C₄×C₆₇	—	4	2	no	—
144	C₄⋊C₄×C₆₇	—	4	2	no	—
145	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
146	C₂²⋊C₄×C₆₇	—	2	1	no	—
147	C₂²⋊C₄×C₆₇	—	2	2	no	—
148	C₂²⋊C₄×C₆₇	—	4	2	no	—
149	C₂²⋊C₄×C₆₇	—	4	2	no	—
150	C₄²×C₆₇	—	2	2	no	—
151	C₄²×C₆₇	—	2	1	no	—
152	C₄²×C₆₇	C₂×D₄	1	2	yes	—
153	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
154	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
155	C₂³×C₆₇	—	2	1	no	—
156	C₂³×C₆₇	—	4	1	no	—
157	C₂³×C₆₇	—	4	2	no	—
158	C₂³×C₆₇	—	8	2	no	—
159	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
160	Q₈×C₆₇	—	2	1	no	—
161	Q₈×C₆₇	—	4	2	no	—
162	Q₈×C₆₇	—	4	1	no	—
163	Q₈×C₆₇	—	4	1	no	—
164	Q₈×C₆₇	#124	1	2	yes	—
165	D₄×C₆₇	—	2	4	no	—
166	D₄×C₆₇	—	2	1	no	—
167	D₄×C₆₇	—	4	2	no	—
168	D₄×C₆₇	—	4	1	no	—
169	D₄×C₆₇	—	4	1	no	—
170	D₄×C₆₇	#124	1	2	yes	—
171	C₂×C₄×C₆₇	—	2	1	no	—
172	C₂×C₄×C₆₇	—	2	4	no	—
173	C₂×C₄×C₆₇	—	2	2	no	—
174	C₂×C₄×C₆₇	—	4	2	no	—
175	C₂×C₄×C₆₇	—	4	4	no	—
176	C₂×C₄×C₆₇	—	4	1	no	—
177	C₂×C₄×C₆₇	—	4	2	no	—
178	C₂×C₄×C₆₇	—	4	2	no	—
179	C₂×C₄×C₆₇	—	8	2	no	—
180	C₂×C₄×C₆₇	#121	1	2	yes	—
181	C₂×C₄×C₆₇	#121	1	2	yes	—
182	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
183	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
184	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
185	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
186	C₈×C₆₇	—	2	2	no	—
187	C₈×C₆₇	—	2	4	no	—
188	C₈×C₆₇	—	2	2	no	—
189	C₈×C₆₇	—	4	1	no	—
190	C₈×C₆₇	—	4	1	no	—
191	C₈×C₆₇	—	4	2	no	—
192	C₂²×C₆₇	—	2	2	no	—
193	C₂²×C₆₇	—	2	2	no	—
194	C₂²×C₆₇	—	4	2	no	—
195	C₂²×C₆₇	—	8	1	no	—
196	C₂²×C₆₇	—	8	2	no	—
197	C₂²×C₆₇	—	16	2	no	—
198	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
199	C₂²×C₆₇	#529	1	2	yes	—
200	C₄×C₆₇	—	2	2	no	—
201	C₄×C₆₇	—	2	2	no	—
202	C₄×C₆₇	—	2	2	no	—
203	C₄×C₆₇	—	4	2	no	—
204	C₄×C₆₇	—	8	1	no	—
205	C₄×C₆₇	—	8	2	no	—
206	C₄×C₆₇	#525	1	2	yes	—
207	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
208	C₄×C₆₇	#532	1	1	yes	characteristic
209	#33062	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
210	C₂×C₆₇	—	2	2	no	—
211	C₂×C₆₇	—	4	2	no	—
212	C₂×C₆₇	—	16	1	no	—
213	C₂×C₆₇	#2850	1	1	yes	characteristic
214	C₂×C₆₇	#3067	1	2	yes	—
215	C₂×#488	C₂×C₆₇	1	1	yes	characteristic, complemented
216	#3080	C₂×C₆₇	1	1	yes	characteristic, complemented
217	#3040	C₂×C₆₇	1	1	yes	characteristic
218	C₄×#131	C₂×C₆₇	1	1	yes	characteristic, complemented
219	#1998	C₂×C₆₇	1	1	yes	characteristic, complemented
220	#1967	C₂×C₆₇	1	1	yes	characteristic, complemented
221	#1931	C₂×C₆₇	1	2	yes	complemented
222	#1930	C₂×C₆₇	1	2	yes	complemented
223	#1908	C₂×C₆₇	1	2	yes	complemented
224	#1907	C₂×C₆₇	1	2	yes	—
225	#1837	C₂×C₆₇	1	1	yes	characteristic, complemented
226	C₆₇	#33062	1	1	yes	characteristic, direct product factor, Sylow
227	C₂×#137	C₂²×C₆₇	1	1	yes	characteristic
228	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	1	yes	characteristic
229	C₂×#131	C₂²×C₆₇	1	2	yes	—
230	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
231	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
232	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
233	#488	—	2	1	no	—
234	#488	—	2	1	no	—
235	#488	—	2	2	no	—
236	#479	—	2	2	no	—
237	#478	—	2	2	no	—
238	#464	—	2	2	no	—
239	#463	—	2	2	no	—
240	#428	C₂²×C₆₇	1	2	yes	—
241	#423	C₂²×C₆₇	1	1	yes	characteristic
242	#423	C₂²×C₆₇	1	1	yes	characteristic
243	#419	C₂²×C₆₇	1	1	yes	characteristic
244	#418	C₂²×C₆₇	1	1	yes	characteristic
245	#417	C₂²×C₆₇	1	2	yes	—
246	C₂×#103	C₂²×C₆₇	1	2	yes	—
247	C₂×#102	C₂²×C₆₇	1	2	yes	—
248	C₂×#101	C₂²×C₆₇	1	2	yes	—
249	C₂×#100	C₂²×C₆₇	1	2	yes	—
250	#405	—	2	2	no	—
251	#405	C₂²×C₆₇	1	2	yes	—
252	C₄×M₁₆	—	2	1	no	—
253	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
254	C₂×C₄×C₈	—	2	1	no	—
255	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
256	#343	C₂²×C₆₇	1	2	yes	—
257	#343	C₂²×C₆₇	1	2	yes	—
258	#341	C₂²×C₆₇	1	2	yes	—
259	#341	C₂²×C₆₇	1	2	yes	—
260	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
261	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
262	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
263	#137	—	2	4	no	—
264	AGL(1,ℤ/8)	—	2	4	no	—
265	C₂×Q₁₆	—	2	1	no	—
266	C₂×SD₁₆	—	2	1	no	—
267	C₂×SD₁₆	—	2	1	no	—
268	C₂×D₈	—	2	1	no	—
269	#131	—	2	4	no	—
270	#131	—	2	4	no	—
271	C₂×M₁₆	—	2	2	no	—
272	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
273	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
274	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
275	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
276	C₂²×C₈	—	2	2	no	—
277	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
278	#125	—	2	2	no	—
279	#125	—	2	1	no	—
280	#125	—	2	1	no	—
281	#124	—	2	2	no	—
282	#124	—	2	1	no	—
283	#124	—	2	1	no	—
284	C₄×Q₈	D₄×C₆₇	1	2	yes	—
285	C₄×D₄	—	2	2	no	—
286	C₄×D₄	D₄×C₆₇	1	2	yes	—
287	#117	—	2	1	no	—
288	#117	C₂³×C₆₇	1	1	yes	characteristic
289	C₂×C₄⋊C₄	—	2	2	no	—
290	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
291	C₂×C₂²⋊C₄	—	2	2	no	—
292	C₂×C₄²	—	2	1	no	—
293	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
294	#107	—	2	2	no	—
295	#106	—	2	2	no	—
296	C₄⋊C₈	—	2	2	no	—
297	C₄⋊C₈	—	2	2	no	—
298	#103	—	2	2	no	—
299	#103	—	2	2	no	—
300	#103	—	2	2	no	—
301	#103	—	2	2	no	—
302	#102	—	2	2	no	—
303	#102	—	2	2	no	—
304	#102	—	2	2	no	—
305	#102	—	2	2	no	—
306	#101	—	2	4	no	—
307	#100	—	2	4	no	—
308	#97	—	2	2	no	—
309	#97	—	2	1	no	—
310	C₄×C₈	—	2	2	no	—
311	C₄×C₈	—	2	1	no	—
312	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
313	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
314	C₄×C₈	D₄×C₆₇	1	2	yes	—
315	C₂⁴	—	2	1	no	—
316	#41	—	2	2	no	—
317	#41	#41×C₆₇	1	4	yes	—
318	C₂×Q₈	—	2	1	no	—
319	C₂×Q₈	—	2	1	no	—
320	C₂×Q₈	—	2	2	no	—
321	C₂×Q₈	—	2	2	no	—
322	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
323	C₂×D₄	—	2	1	no	—
324	C₂×D₄	—	2	1	no	—
325	C₂×D₄	—	2	1	no	—
326	C₂×D₄	—	2	2	no	—
327	C₂×D₄	—	2	2	no	—
328	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
329	C₂²×C₄	—	2	1	no	—
330	C₂²×C₄	—	2	2	no	—
331	C₂²×C₄	—	2	1	no	—
332	C₂²×C₄	—	2	2	no	—
333	C₂²×C₄	#41×C₆₇	1	2	yes	—
334	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
335	Q₁₆	—	4	2	no	—
336	SD₁₆	—	4	2	no	—
337	SD₁₆	—	4	2	no	—
338	D₈	—	4	2	no	—
339	M₁₆	—	2	8	no	—
340	M₁₆	—	2	2	no	—
341	M₁₆	—	2	2	no	—
342	M₁₆	—	2	2	no	—
343	M₁₆	—	2	2	no	—
344	M₁₆	—	2	2	no	—
345	M₁₆	#41×C₆₇	1	4	yes	—
346	C₂×C₈	—	2	8	no	—
347	C₂×C₈	—	2	2	no	—
348	C₂×C₈	—	2	2	no	—
349	C₂×C₈	—	2	2	no	—
350	C₂×C₈	—	2	1	no	—
351	C₂×C₈	—	2	2	no	—
352	C₂×C₈	—	2	1	no	—
353	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
354	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
355	C₄⋊C₄	—	2	2	no	—
356	C₄⋊C₄	—	2	2	no	—
357	C₄⋊C₄	—	2	2	no	—
358	C₄⋊C₄	—	4	2	no	—
359	C₄⋊C₄	—	4	2	no	—
360	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
361	C₂²⋊C₄	—	2	1	no	—
362	C₂²⋊C₄	—	2	2	no	—
363	C₂²⋊C₄	—	4	2	no	—
364	C₂²⋊C₄	—	4	2	no	—
365	C₄²	—	2	2	no	—
366	C₄²	—	2	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	2	yes	—
370	C₂³	—	2	1	no	—
371	C₂³	—	4	1	no	—
372	C₂³	—	4	2	no	—
373	C₂³	—	8	2	no	—
374	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
375	Q₈	—	2	1	no	—
376	Q₈	—	4	2	no	—
377	Q₈	—	4	1	no	—
378	Q₈	—	4	1	no	—
379	Q₈	#124×C₆₇	1	2	yes	—
380	D₄	—	2	4	no	—
381	D₄	—	2	1	no	—
382	D₄	—	4	2	no	—
383	D₄	—	4	1	no	—
384	D₄	—	4	1	no	—
385	D₄	#124×C₆₇	1	2	yes	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	2	4	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	—	4	4	no	—
391	C₂×C₄	—	4	1	no	—
392	C₂×C₄	—	4	2	no	—
393	C₂×C₄	—	4	2	no	—
394	C₂×C₄	—	8	2	no	—
395	C₂×C₄	#121×C₆₇	1	2	yes	—
396	C₂×C₄	#121×C₆₇	1	2	yes	—
397	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
398	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
399	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
400	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
401	C₈	—	2	2	no	—
402	C₈	—	2	4	no	—
403	C₈	—	2	2	no	—
404	C₈	—	4	1	no	—
405	C₈	—	4	1	no	—
406	C₈	—	4	2	no	—
407	C₂²	—	2	2	no	—
408	C₂²	—	2	2	no	—
409	C₂²	—	4	2	no	—
410	C₂²	—	8	2	no	—
411	C₂²	—	8	1	no	—
412	C₂²	—	16	2	no	—
413	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
414	C₂²	#529×C₆₇	1	2	yes	—
415	C₄	—	2	2	no	—
416	C₄	—	2	2	no	—
417	C₄	—	2	2	no	—
418	C₄	—	4	2	no	—
419	C₄	—	8	2	no	—
420	C₄	—	8	1	no	—
421	C₄	#525×C₆₇	1	2	yes	—
422	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
423	C₄	#532×C₆₇	1	1	yes	characteristic
424	C₂	—	2	2	no	—
425	C₂	—	4	2	no	—
426	C₂	—	16	1	no	—
427	C₂	C₆₇×#2850	1	1	yes	characteristic
428	C₂	C₆₇×#3067	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×AGL(1,ℤ/8)×C₆₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#488×C₆₇	C₂	semidirect
2. C₆₇×#3080	C₂	semidirect
4. C₄×#131×C₆₇	C₂	semidirect
5. C₆₇×#1998	C₂	semidirect
6. C₆₇×#1967	C₂	semidirect
7. C₆₇×#1931	C₂	semidirect
8. C₆₇×#1930	C₂	semidirect
9. C₆₇×#1908	C₂	semidirect
11. C₆₇×#1837	C₂	semidirect
209. #33062	C₆₇	direct (N × H)
215. C₂×#488	C₂×C₆₇	semidirect
216. #3080	C₂×C₆₇	semidirect
218. C₄×#131	C₂×C₆₇	semidirect
219. #1998	C₂×C₆₇	semidirect
220. #1967	C₂×C₆₇	semidirect
221. #1931	C₂×C₆₇	semidirect
222. #1930	C₂×C₆₇	semidirect
223. #1908	C₂×C₆₇	semidirect
225. #1837	C₂×C₆₇	semidirect
226. C₆₇	#33062	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#488×C₆₇	2
C₆₇×#3080	2
C₆₇×#3040	2
C₄×#131×C₆₇	2
C₆₇×#1998	2
C₆₇×#1967	2
C₆₇×#1931	2
C₆₇×#1930	2
C₆₇×#1908	2
C₆₇×#1907	2
C₆₇×#1837	2
#33062	67

Subgroups — C67×#33062 #30712313

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33062 #30712313