Subgroups — C₆₇×#33061 #30712312

Order 17152.

770 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#487×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3079	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#3039	C₂	1	1	yes	maximal, characteristic
4	C₄×#131×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1997	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1966	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1930	C₂	1	4	yes	maximal, complemented
8	C₆₇×#1908	C₂	1	4	yes	maximal
9	C₆₇×#1837	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#137×C₆₇	C₂²	1	2	yes	—
11	C₂×#131×C₆₇	C₂²	1	2	yes	—
12	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
15	#487×C₆₇	—	2	1	no	—
16	#487×C₆₇	—	2	1	no	—
17	#487×C₆₇	—	2	2	no	—
18	#477×C₆₇	—	2	1	no	—
19	#476×C₆₇	—	2	1	no	—
20	#475×C₆₇	—	2	1	no	—
21	#474×C₆₇	—	2	1	no	—
22	#462×C₆₇	—	2	1	no	—
23	#462×C₆₇	—	2	1	no	—
24	#461×C₆₇	—	2	1	no	—
25	#461×C₆₇	—	2	1	no	—
26	#428×C₆₇	C₂²	1	2	yes	—
27	#423×C₆₇	C₂²	1	1	yes	characteristic
28	#423×C₆₇	C₂²	1	1	yes	characteristic
29	#419×C₆₇	C₂²	1	2	yes	—
30	#417×C₆₇	C₂²	1	2	yes	—
31	C₂×#103×C₆₇	C₂²	1	4	yes	—
32	C₂×#101×C₆₇	C₂²	1	4	yes	—
33	#405×C₆₇	—	2	2	no	—
34	#405×C₆₇	C₂²	1	2	yes	—
35	C₄×M₁₆×C₆₇	—	2	1	no	—
36	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₆₇	—	2	1	no	—
38	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
39	#343×C₆₇	C₂²	1	4	yes	—
40	#341×C₆₇	C₂²	1	4	yes	—
41	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
42	C₂²×Q₈×C₆₇	—	2	1	no	—
43	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
44	#137×C₆₇	—	2	8	no	—
45	C₂×Q₁₆×C₆₇	—	2	2	no	—
46	C₂×SD₁₆×C₆₇	—	2	2	no	—
47	#131×C₆₇	—	2	4	no	—
48	#131×C₆₇	—	2	4	no	—
49	C₂×M₁₆×C₆₇	—	2	2	no	—
50	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
51	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
52	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
53	C₂²×C₈×C₆₇	—	2	2	no	—
54	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
55	C₄⋊Q₈×C₆₇	—	2	1	no	—
56	C₄⋊Q₈×C₆₇	—	2	1	no	—
57	C₄⋊Q₈×C₆₇	—	2	1	no	—
58	C₄⋊Q₈×C₆₇	—	2	1	no	—
59	C₄⋊Q₈×C₆₇	—	2	1	no	—
60	C₄⋊Q₈×C₆₇	—	2	1	no	—
61	C₄⋊Q₈×C₆₇	—	2	1	no	—
62	C₄⋊Q₈×C₆₇	—	2	1	no	—
63	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
64	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
65	C₄×D₄×C₆₇	—	2	1	no	—
66	C₄×D₄×C₆₇	—	2	1	no	—
67	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
68	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
69	#117×C₆₇	—	2	1	no	—
70	#117×C₆₇	C₂³	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
72	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
73	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
74	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
75	C₂×C₄²×C₆₇	—	2	1	no	—
76	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
77	#107×C₆₇	—	2	2	no	—
78	#106×C₆₇	—	2	2	no	—
79	C₄⋊C₈×C₆₇	—	2	1	no	—
80	C₄⋊C₈×C₆₇	—	2	1	no	—
81	C₄⋊C₈×C₆₇	—	2	1	no	—
82	C₄⋊C₈×C₆₇	—	2	1	no	—
83	#103×C₆₇	—	2	2	no	—
84	#103×C₆₇	—	2	4	no	—
85	#103×C₆₇	—	2	4	no	—
86	#103×C₆₇	—	2	2	no	—
87	#103×C₆₇	—	2	2	no	—
88	#102×C₆₇	—	2	2	no	—
89	#101×C₆₇	—	2	8	no	—
90	#97×C₆₇	—	2	2	no	—
91	#97×C₆₇	—	2	1	no	—
92	C₄×C₈×C₆₇	—	2	2	no	—
93	C₄×C₈×C₆₇	—	2	1	no	—
94	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
95	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
96	C₄×C₈×C₆₇	D₄	1	2	yes	—
97	#41×C₆₇	—	2	2	no	—
98	#41×C₆₇	#41	1	4	yes	—
99	C₂×Q₈×C₆₇	—	2	2	no	—
100	C₂×Q₈×C₆₇	—	2	2	no	—
101	C₂×Q₈×C₆₇	—	2	4	no	—
102	C₂×Q₈×C₆₇	—	2	4	no	—
103	C₂×Q₈×C₆₇	—	4	1	no	—
104	C₂×Q₈×C₆₇	—	4	1	no	—
105	C₂×Q₈×C₆₇	—	8	2	no	—
106	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
107	C₂×D₄×C₆₇	—	2	1	no	—
108	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₆₇	—	2	1	no	—
110	C₂²×C₄×C₆₇	—	2	1	no	—
111	C₂²×C₄×C₆₇	—	2	1	no	—
112	C₂²×C₄×C₆₇	—	2	2	no	—
113	C₂²×C₄×C₆₇	—	2	2	no	—
114	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
115	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
116	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
117	Q₁₆×C₆₇	—	4	4	no	—
118	SD₁₆×C₆₇	—	4	4	no	—
119	M₁₆×C₆₇	—	2	8	no	—
120	M₁₆×C₆₇	—	2	2	no	—
121	M₁₆×C₆₇	—	2	4	no	—
122	M₁₆×C₆₇	—	2	4	no	—
123	M₁₆×C₆₇	#41	1	4	yes	—
124	C₂×C₈×C₆₇	—	2	8	no	—
125	C₂×C₈×C₆₇	—	2	2	no	—
126	C₂×C₈×C₆₇	—	2	2	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	C₂×D₄	1	2	yes	—
131	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
132	C₄⋊C₄×C₆₇	—	2	1	no	—
133	C₄⋊C₄×C₆₇	—	2	1	no	—
134	C₄⋊C₄×C₆₇	—	2	2	no	—
135	C₄⋊C₄×C₆₇	—	2	2	no	—
136	C₄⋊C₄×C₆₇	—	4	1	no	—
137	C₄⋊C₄×C₆₇	—	4	1	no	—
138	C₄⋊C₄×C₆₇	—	4	1	no	—
139	C₄⋊C₄×C₆₇	—	4	1	no	—
140	C₄⋊C₄×C₆₇	—	4	1	no	—
141	C₄⋊C₄×C₆₇	—	4	1	no	—
142	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
143	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
144	C₂²⋊C₄×C₆₇	—	2	1	no	—
145	C₂²⋊C₄×C₆₇	—	2	1	no	—
146	C₂²⋊C₄×C₆₇	—	2	1	no	—
147	C₄²×C₆₇	—	2	1	no	—
148	C₄²×C₆₇	—	2	1	no	—
149	C₄²×C₆₇	—	2	1	no	—
150	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
151	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
152	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
153	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
154	C₂³×C₆₇	—	2	1	no	—
155	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
156	Q₈×C₆₇	—	2	1	no	—
157	Q₈×C₆₇	—	4	4	no	—
158	Q₈×C₆₇	—	4	2	no	—
159	Q₈×C₆₇	—	4	2	no	—
160	Q₈×C₆₇	—	8	2	no	—
161	Q₈×C₆₇	—	8	2	no	—
162	Q₈×C₆₇	#124	1	2	yes	—
163	D₄×C₆₇	—	2	4	no	—
164	D₄×C₆₇	—	2	1	no	—
165	D₄×C₆₇	#124	1	2	yes	—
166	C₂×C₄×C₆₇	—	2	1	no	—
167	C₂×C₄×C₆₇	—	2	4	no	—
168	C₂×C₄×C₆₇	—	2	2	no	—
169	C₂×C₄×C₆₇	—	2	2	no	—
170	C₂×C₄×C₆₇	—	2	1	no	—
171	C₂×C₄×C₆₇	—	2	1	no	—
172	C₂×C₄×C₆₇	—	4	2	no	—
173	C₂×C₄×C₆₇	—	4	2	no	—
174	C₂×C₄×C₆₇	—	4	2	no	—
175	C₂×C₄×C₆₇	—	4	2	no	—
176	C₂×C₄×C₆₇	—	4	4	no	—
177	C₂×C₄×C₆₇	—	8	2	no	—
178	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
179	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
180	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
181	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
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	2	no	—
190	C₈×C₆₇	—	4	2	no	—
191	C₂²×C₆₇	—	2	1	no	—
192	C₂²×C₆₇	—	2	1	no	—
193	C₂²×C₆₇	—	2	2	no	—
194	C₂²×C₆₇	—	4	1	no	—
195	C₂²×C₆₇	—	4	1	no	—
196	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
197	C₂²×C₆₇	#529	1	2	yes	—
198	C₄×C₆₇	—	2	2	no	—
199	C₄×C₆₇	—	2	1	no	—
200	C₄×C₆₇	—	2	1	no	—
201	C₄×C₆₇	—	4	1	no	—
202	C₄×C₆₇	—	4	1	no	—
203	C₄×C₆₇	—	8	2	no	—
204	C₄×C₆₇	—	8	2	no	—
205	C₄×C₆₇	#471	1	2	yes	—
206	C₄×C₆₇	#473	1	2	yes	—
207	C₄×C₆₇	#525	1	2	yes	—
208	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
209	C₄×C₆₇	#532	1	1	yes	characteristic
210	#33061	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
211	C₂×C₆₇	—	2	2	no	—
212	C₂×C₆₇	—	4	2	no	—
213	C₂×C₆₇	#2850	1	1	yes	characteristic
214	C₂×C₆₇	C₂×#471	1	1	yes	characteristic
215	C₂×C₆₇	C₂×#473	1	1	yes	characteristic
216	C₂×#487	C₂×C₆₇	1	1	yes	characteristic, complemented
217	#3079	C₂×C₆₇	1	1	yes	characteristic
218	#3039	C₂×C₆₇	1	1	yes	characteristic
219	C₄×#131	C₂×C₆₇	1	1	yes	characteristic
220	#1997	C₂×C₆₇	1	1	yes	characteristic, complemented
221	#1966	C₂×C₆₇	1	1	yes	characteristic, complemented
222	#1930	C₂×C₆₇	1	4	yes	complemented
223	#1908	C₂×C₆₇	1	4	yes	—
224	#1837	C₂×C₆₇	1	1	yes	characteristic, complemented
225	C₆₇	#33061	1	1	yes	characteristic, direct product factor, Sylow
226	C₂×#137	C₂²×C₆₇	1	2	yes	—
227	C₂×#131	C₂²×C₆₇	1	2	yes	—
228	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
229	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
230	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
231	#487	—	2	1	no	—
232	#487	—	2	1	no	—
233	#487	—	2	2	no	—
234	#477	—	2	1	no	—
235	#476	—	2	1	no	—
236	#475	—	2	1	no	—
237	#474	—	2	1	no	—
238	#462	—	2	1	no	—
239	#462	—	2	1	no	—
240	#461	—	2	1	no	—
241	#461	—	2	1	no	—
242	#428	C₂²×C₆₇	1	2	yes	—
243	#423	C₂²×C₆₇	1	1	yes	characteristic
244	#423	C₂²×C₆₇	1	1	yes	characteristic
245	#419	C₂²×C₆₇	1	2	yes	—
246	#417	C₂²×C₆₇	1	2	yes	—
247	C₂×#103	C₂²×C₆₇	1	4	yes	—
248	C₂×#101	C₂²×C₆₇	1	4	yes	—
249	#405	—	2	2	no	—
250	#405	C₂²×C₆₇	1	2	yes	—
251	C₄×M₁₆	—	2	1	no	—
252	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
253	C₂×C₄×C₈	—	2	1	no	—
254	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
255	#343	C₂²×C₆₇	1	4	yes	—
256	#341	C₂²×C₆₇	1	4	yes	—
257	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
258	C₂²×Q₈	—	2	1	no	—
259	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
260	#137	—	2	8	no	—
261	C₂×Q₁₆	—	2	2	no	—
262	C₂×SD₁₆	—	2	2	no	—
263	#131	—	2	4	no	—
264	#131	—	2	4	no	—
265	C₂×M₁₆	—	2	2	no	—
266	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
267	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
268	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
269	C₂²×C₈	—	2	2	no	—
270	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
271	C₄⋊Q₈	—	2	1	no	—
272	C₄⋊Q₈	—	2	1	no	—
273	C₄⋊Q₈	—	2	1	no	—
274	C₄⋊Q₈	—	2	1	no	—
275	C₄⋊Q₈	—	2	1	no	—
276	C₄⋊Q₈	—	2	1	no	—
277	C₄⋊Q₈	—	2	1	no	—
278	C₄⋊Q₈	—	2	1	no	—
279	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
280	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
281	C₄×D₄	—	2	1	no	—
282	C₄×D₄	—	2	1	no	—
283	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
284	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
285	#117	—	2	1	no	—
286	#117	C₂³×C₆₇	1	1	yes	characteristic
287	C₂×C₄⋊C₄	—	2	1	no	—
288	C₂×C₄⋊C₄	—	2	1	no	—
289	C₂×C₄⋊C₄	—	2	1	no	—
290	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
291	C₂×C₄²	—	2	1	no	—
292	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
293	#107	—	2	2	no	—
294	#106	—	2	2	no	—
295	C₄⋊C₈	—	2	1	no	—
296	C₄⋊C₈	—	2	1	no	—
297	C₄⋊C₈	—	2	1	no	—
298	C₄⋊C₈	—	2	1	no	—
299	#103	—	2	2	no	—
300	#103	—	2	4	no	—
301	#103	—	2	4	no	—
302	#103	—	2	2	no	—
303	#103	—	2	2	no	—
304	#102	—	2	2	no	—
305	#101	—	2	8	no	—
306	#97	—	2	2	no	—
307	#97	—	2	1	no	—
308	C₄×C₈	—	2	2	no	—
309	C₄×C₈	—	2	1	no	—
310	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
311	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
312	C₄×C₈	D₄×C₆₇	1	2	yes	—
313	#41	—	2	2	no	—
314	#41	#41×C₆₇	1	4	yes	—
315	C₂×Q₈	—	2	2	no	—
316	C₂×Q₈	—	2	2	no	—
317	C₂×Q₈	—	2	4	no	—
318	C₂×Q₈	—	2	4	no	—
319	C₂×Q₈	—	4	1	no	—
320	C₂×Q₈	—	4	1	no	—
321	C₂×Q₈	—	8	2	no	—
322	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
323	C₂×D₄	—	2	1	no	—
324	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
325	C₂²×C₄	—	2	1	no	—
326	C₂²×C₄	—	2	1	no	—
327	C₂²×C₄	—	2	1	no	—
328	C₂²×C₄	—	2	2	no	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
331	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
332	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
333	Q₁₆	—	4	4	no	—
334	SD₁₆	—	4	4	no	—
335	M₁₆	—	2	8	no	—
336	M₁₆	—	2	2	no	—
337	M₁₆	—	2	4	no	—
338	M₁₆	—	2	4	no	—
339	M₁₆	#41×C₆₇	1	4	yes	—
340	C₂×C₈	—	2	8	no	—
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	—	2	2	no	—
343	C₂×C₈	—	2	2	no	—
344	C₂×C₈	—	2	2	no	—
345	C₂×C₈	—	2	2	no	—
346	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
347	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
348	C₄⋊C₄	—	2	1	no	—
349	C₄⋊C₄	—	2	1	no	—
350	C₄⋊C₄	—	2	2	no	—
351	C₄⋊C₄	—	2	2	no	—
352	C₄⋊C₄	—	4	1	no	—
353	C₄⋊C₄	—	4	1	no	—
354	C₄⋊C₄	—	4	1	no	—
355	C₄⋊C₄	—	4	1	no	—
356	C₄⋊C₄	—	4	1	no	—
357	C₄⋊C₄	—	4	1	no	—
358	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
359	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
360	C₂²⋊C₄	—	2	1	no	—
361	C₂²⋊C₄	—	2	1	no	—
362	C₂²⋊C₄	—	2	1	no	—
363	C₄²	—	2	1	no	—
364	C₄²	—	2	1	no	—
365	C₄²	—	2	1	no	—
366	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
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	C₂³	—	2	1	no	—
371	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
372	Q₈	—	2	1	no	—
373	Q₈	—	4	4	no	—
374	Q₈	—	4	2	no	—
375	Q₈	—	4	2	no	—
376	Q₈	—	8	2	no	—
377	Q₈	—	8	2	no	—
378	Q₈	#124×C₆₇	1	2	yes	—
379	D₄	—	2	4	no	—
380	D₄	—	2	1	no	—
381	D₄	#124×C₆₇	1	2	yes	—
382	C₂×C₄	—	2	1	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	2	4	no	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	2	1	no	—
388	C₂×C₄	—	4	2	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	—	4	2	no	—
391	C₂×C₄	—	4	2	no	—
392	C₂×C₄	—	4	4	no	—
393	C₂×C₄	—	8	2	no	—
394	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
395	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
396	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
397	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
398	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
399	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
400	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
401	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
402	C₈	—	2	2	no	—
403	C₈	—	2	4	no	—
404	C₈	—	2	2	no	—
405	C₈	—	4	2	no	—
406	C₈	—	4	2	no	—
407	C₂²	—	2	1	no	—
408	C₂²	—	2	1	no	—
409	C₂²	—	2	2	no	—
410	C₂²	—	4	1	no	—
411	C₂²	—	4	1	no	—
412	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
413	C₂²	#529×C₆₇	1	2	yes	—
414	C₄	—	2	1	no	—
415	C₄	—	2	1	no	—
416	C₄	—	2	2	no	—
417	C₄	—	4	1	no	—
418	C₄	—	4	1	no	—
419	C₄	—	8	2	no	—
420	C₄	—	8	2	no	—
421	C₄	#471×C₆₇	1	2	yes	—
422	C₄	#473×C₆₇	1	2	yes	—
423	C₄	#525×C₆₇	1	2	yes	—
424	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
425	C₄	#532×C₆₇	1	1	yes	characteristic
426	C₂	—	2	2	no	—
427	C₂	—	4	2	no	—
428	C₂	C₆₇×#2850	1	1	yes	characteristic
429	C₂	C₂×#471×C₆₇	1	1	yes	characteristic
430	C₂	C₂×#473×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#41×C₆₇	C₂³
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×#487×C₆₇	C₂	semidirect
5. C₆₇×#1997	C₂	semidirect
6. C₆₇×#1966	C₂	semidirect
7. C₆₇×#1930	C₂	semidirect
9. C₆₇×#1837	C₂	semidirect
210. #33061	C₆₇	direct (N × H)
216. C₂×#487	C₂×C₆₇	semidirect
220. #1997	C₂×C₆₇	semidirect
221. #1966	C₂×C₆₇	semidirect
222. #1930	C₂×C₆₇	semidirect
224. #1837	C₂×C₆₇	semidirect
225. C₆₇	#33061	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×#487×C₆₇	2
C₆₇×#3079	2
C₆₇×#3039	2
C₄×#131×C₆₇	2
C₆₇×#1997	2
C₆₇×#1966	2
C₆₇×#1930	2
C₆₇×#1908	2
C₆₇×#1837	2
#33061	67

Subgroups — C67×#33061 #30712312

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33061 #30712312