Subgroups — C₆₇×#33141 #30712392

Order 17152.

810 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×#470×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2625	C₂	1	2	yes	maximal
4	C₆₇×#1971	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1969	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1943	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1941	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1935	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1933	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1892	C₂	1	2	yes	maximal
11	C₆₇×#1809	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
16	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
17	#470×C₆₇	—	2	4	no	—
18	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#103×C₆₇	C₂²	1	2	yes	—
22	C₂×#103×C₆₇	C₂²	1	2	yes	—
23	C₂×#103×C₆₇	C₂²	1	2	yes	—
24	#405×C₆₇	—	2	4	no	—
25	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
29	#400×C₆₇	—	2	2	no	—
30	#398×C₆₇	—	2	2	no	—
31	#397×C₆₇	—	2	2	no	—
32	#397×C₆₇	—	2	2	no	—
33	#391×C₆₇	C₂²	1	2	yes	—
34	#391×C₆₇	C₂²	1	2	yes	—
35	#389×C₆₇	C₂²	1	2	yes	—
36	#389×C₆₇	C₂²	1	2	yes	—
37	#380×C₆₇	C₂²	1	2	yes	—
38	#340×C₆₇	C₂²	1	2	yes	—
39	#340×C₆₇	C₂²	1	2	yes	—
40	#340×C₆₇	C₂²	1	2	yes	—
41	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
42	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
43	C₂×Q₁₆×C₆₇	—	2	2	no	—
44	C₂×Q₁₆×C₆₇	—	2	2	no	—
45	C₂×Q₁₆×C₆₇	—	2	2	no	—
46	C₂×M₁₆×C₆₇	—	2	2	no	—
47	C₂×M₁₆×C₆₇	—	2	2	no	—
48	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
49	C₂²×C₈×C₆₇	—	2	2	no	—
50	C₂²×C₈×C₆₇	—	2	2	no	—
51	C₂²×C₈×C₆₇	—	2	2	no	—
52	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
56	C₂²⋊Q₈×C₆₇	—	2	4	no	—
57	C₂²⋊Q₈×C₆₇	—	2	4	no	—
58	#117×C₆₇	—	2	2	no	—
59	#117×C₆₇	—	2	2	no	—
60	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
61	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
62	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
63	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
64	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
67	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
68	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
69	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
70	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
72	#107×C₆₇	—	2	2	no	—
73	#106×C₆₇	—	2	2	no	—
74	#106×C₆₇	—	2	2	no	—
75	#103×C₆₇	—	2	4	no	—
76	#103×C₆₇	—	2	4	no	—
77	#103×C₆₇	—	2	4	no	—
78	#97×C₆₇	—	2	2	no	—
79	#97×C₆₇	—	2	2	no	—
80	C₄×C₈×C₆₇	—	2	2	no	—
81	C₄×C₈×C₆₇	—	2	2	no	—
82	#95×C₆₇	—	2	2	no	—
83	#95×C₆₇	—	2	2	no	—
84	#95×C₆₇	—	2	2	no	—
85	#95×C₆₇	—	2	2	no	—
86	#95×C₆₇	—	2	2	no	—
87	#95×C₆₇	—	2	2	no	—
88	#95×C₆₇	—	2	2	no	—
89	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
90	C₂×Q₈×C₆₇	—	2	2	no	—
91	C₂×Q₈×C₆₇	—	2	2	no	—
92	C₂×Q₈×C₆₇	—	2	2	no	—
93	C₂×Q₈×C₆₇	—	2	2	no	—
94	C₂×Q₈×C₆₇	—	2	2	no	—
95	C₂×Q₈×C₆₇	—	2	2	no	—
96	C₂²×C₄×C₆₇	—	2	1	no	—
97	C₂²×C₄×C₆₇	—	2	1	no	—
98	C₂²×C₄×C₆₇	—	2	1	no	—
99	C₂²×C₄×C₆₇	—	2	1	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	—	2	2	no	—
104	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
105	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
108	C₂²×C₄×C₆₇	#41	1	2	yes	—
109	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
110	Q₁₆×C₆₇	—	4	4	no	—
111	M₁₆×C₆₇	—	2	8	no	—
112	C₂×C₈×C₆₇	—	2	8	no	—
113	C₂×C₈×C₆₇	—	2	1	no	—
114	C₂×C₈×C₆₇	—	2	1	no	—
115	C₂×C₈×C₆₇	—	2	1	no	—
116	C₂×C₈×C₆₇	—	2	1	no	—
117	C₂×C₈×C₆₇	—	2	1	no	—
118	C₂×C₈×C₆₇	—	2	1	no	—
119	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
120	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
121	C₂×C₈×C₆₇	#41	1	2	yes	—
122	C₂×C₈×C₆₇	#41	1	2	yes	—
123	C₂×C₈×C₆₇	#41	1	2	yes	—
124	C₂×C₈×C₆₇	#41	1	2	yes	—
125	C₄⋊C₄×C₆₇	—	2	4	no	—
126	C₄⋊C₄×C₆₇	—	2	2	no	—
127	C₄⋊C₄×C₆₇	—	2	2	no	—
128	C₄⋊C₄×C₆₇	—	2	2	no	—
129	C₄⋊C₄×C₆₇	—	2	2	no	—
130	C₄⋊C₄×C₆₇	—	2	2	no	—
131	C₄⋊C₄×C₆₇	—	2	2	no	—
132	C₄⋊C₄×C₆₇	—	4	4	no	—
133	C₄⋊C₄×C₆₇	—	8	2	no	—
134	C₄⋊C₄×C₆₇	—	8	2	no	—
135	C₂²⋊C₄×C₆₇	—	2	4	no	—
136	C₂²⋊C₄×C₆₇	—	4	4	no	—
137	C₂²⋊C₄×C₆₇	—	8	2	no	—
138	C₄²×C₆₇	—	2	1	no	—
139	C₄²×C₆₇	—	2	1	no	—
140	C₄²×C₆₇	—	2	1	no	—
141	C₄²×C₆₇	—	2	1	no	—
142	C₂³×C₆₇	—	2	1	no	—
143	C₂³×C₆₇	—	2	1	no	—
144	C₂³×C₆₇	—	2	2	no	—
145	C₂³×C₆₇	—	2	2	no	—
146	C₂³×C₆₇	#121	1	2	yes	—
147	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
148	Q₈×C₆₇	—	4	2	no	—
149	Q₈×C₆₇	—	4	2	no	—
150	Q₈×C₆₇	—	4	2	no	—
151	Q₈×C₆₇	—	4	2	no	—
152	C₂×C₄×C₆₇	—	2	1	no	—
153	C₂×C₄×C₆₇	—	2	1	no	—
154	C₂×C₄×C₆₇	—	2	2	no	—
155	C₂×C₄×C₆₇	—	2	2	no	—
156	C₂×C₄×C₆₇	—	2	2	no	—
157	C₂×C₄×C₆₇	—	2	2	no	—
158	C₂×C₄×C₆₇	—	2	2	no	—
159	C₂×C₄×C₆₇	—	2	2	no	—
160	C₂×C₄×C₆₇	—	2	2	no	—
161	C₂×C₄×C₆₇	—	2	2	no	—
162	C₂×C₄×C₆₇	—	2	2	no	—
163	C₂×C₄×C₆₇	—	2	2	no	—
164	C₂×C₄×C₆₇	—	4	2	no	—
165	C₂×C₄×C₆₇	—	4	2	no	—
166	C₂×C₄×C₆₇	—	4	2	no	—
167	C₂×C₄×C₆₇	—	4	2	no	—
168	C₂×C₄×C₆₇	—	4	2	no	—
169	C₂×C₄×C₆₇	—	4	2	no	—
170	C₂×C₄×C₆₇	—	4	2	no	—
171	C₂×C₄×C₆₇	—	4	2	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	#121	1	2	yes	—
177	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
179	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
180	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
181	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
182	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
183	C₈×C₆₇	—	2	2	no	—
184	C₈×C₆₇	—	2	2	no	—
185	C₈×C₆₇	—	2	2	no	—
186	C₈×C₆₇	#123	1	4	yes	—
187	C₂²×C₆₇	—	2	2	no	—
188	C₂²×C₆₇	—	2	4	no	—
189	C₂²×C₆₇	—	4	1	no	—
190	C₂²×C₆₇	—	4	1	no	—
191	C₂²×C₆₇	—	4	2	no	—
192	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
193	C₂²×C₆₇	#532	1	1	yes	characteristic
194	C₂²×C₆₇	#532	1	1	yes	characteristic
195	C₂²×C₆₇	#577	1	2	yes	—
196	C₂²×C₆₇	#578	1	2	yes	—
197	C₄×C₆₇	—	2	2	no	—
198	C₄×C₆₇	—	4	2	no	—
199	C₄×C₆₇	—	4	2	no	—
200	C₄×C₆₇	—	8	2	no	—
201	C₄×C₆₇	—	8	2	no	—
202	C₄×C₆₇	—	8	2	no	—
203	C₄×C₆₇	—	8	2	no	—
204	C₄×C₆₇	C₂×#123	1	1	yes	characteristic
205	C₄×C₆₇	#525	1	1	yes	characteristic
206	C₄×C₆₇	#529	1	1	yes	characteristic
207	C₄×C₆₇	#529	1	1	yes	characteristic
208	#33141	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
209	C₂×C₆₇	—	4	2	no	—
210	C₂×C₆₇	#2854	1	1	yes	characteristic
211	C₂×C₆₇	#3063	1	2	yes	—
212	C₂×C₆₇	#3173	1	1	yes	characteristic
213	C₂×C₆₇	#3177	1	1	yes	characteristic
214	C₂×C₆₇	#3178	1	2	yes	—
215	C₂×#470	C₂×C₆₇	1	1	yes	characteristic
216	C₂×#405	C₂×C₆₇	1	1	yes	characteristic
217	#2625	C₂×C₆₇	1	2	yes	—
218	#1971	C₂×C₆₇	1	1	yes	characteristic, complemented
219	#1969	C₂×C₆₇	1	1	yes	characteristic, complemented
220	#1943	C₂×C₆₇	1	1	yes	characteristic, complemented
221	#1941	C₂×C₆₇	1	1	yes	characteristic, complemented
222	#1935	C₂×C₆₇	1	2	yes	complemented
223	#1933	C₂×C₆₇	1	2	yes	complemented
224	#1892	C₂×C₆₇	1	2	yes	—
225	#1809	C₂×C₆₇	1	1	yes	characteristic
226	C₆₇	#33141	1	1	yes	characteristic, direct product factor, Sylow
227	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
228	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
229	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
230	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
231	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
232	#470	—	2	4	no	—
233	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
234	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
235	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
236	C₂×#103	C₂²×C₆₇	1	2	yes	—
237	C₂×#103	C₂²×C₆₇	1	2	yes	—
238	C₂×#103	C₂²×C₆₇	1	2	yes	—
239	#405	—	2	4	no	—
240	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
241	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
242	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
243	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
244	#400	—	2	2	no	—
245	#398	—	2	2	no	—
246	#397	—	2	2	no	—
247	#397	—	2	2	no	—
248	#391	C₂²×C₆₇	1	2	yes	—
249	#391	C₂²×C₆₇	1	2	yes	—
250	#389	C₂²×C₆₇	1	2	yes	—
251	#389	C₂²×C₆₇	1	2	yes	—
252	#380	C₂²×C₆₇	1	2	yes	—
253	#340	C₂²×C₆₇	1	2	yes	—
254	#340	C₂²×C₆₇	1	2	yes	—
255	#340	C₂²×C₆₇	1	2	yes	—
256	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
257	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
258	C₂×Q₁₆	—	2	2	no	—
259	C₂×Q₁₆	—	2	2	no	—
260	C₂×Q₁₆	—	2	2	no	—
261	C₂×M₁₆	—	2	2	no	—
262	C₂×M₁₆	—	2	2	no	—
263	C₂×M₁₆	D₄×C₆₇	1	4	yes	—
264	C₂²×C₈	—	2	2	no	—
265	C₂²×C₈	—	2	2	no	—
266	C₂²×C₈	—	2	2	no	—
267	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
268	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
269	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
270	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
271	C₂²⋊Q₈	—	2	4	no	—
272	C₂²⋊Q₈	—	2	4	no	—
273	#117	—	2	2	no	—
274	#117	—	2	2	no	—
275	C₂×C₄⋊C₄	—	2	2	no	—
276	C₂×C₄⋊C₄	—	2	2	no	—
277	C₂×C₄⋊C₄	—	2	2	no	—
278	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
279	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
280	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
281	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
282	C₂×C₂²⋊C₄	—	2	2	no	—
283	C₂×C₂²⋊C₄	—	2	2	no	—
284	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
285	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
286	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
287	#107	—	2	2	no	—
288	#106	—	2	2	no	—
289	#106	—	2	2	no	—
290	#103	—	2	4	no	—
291	#103	—	2	4	no	—
292	#103	—	2	4	no	—
293	#97	—	2	2	no	—
294	#97	—	2	2	no	—
295	C₄×C₈	—	2	2	no	—
296	C₄×C₈	—	2	2	no	—
297	#95	—	2	2	no	—
298	#95	—	2	2	no	—
299	#95	—	2	2	no	—
300	#95	—	2	2	no	—
301	#95	—	2	2	no	—
302	#95	—	2	2	no	—
303	#95	—	2	2	no	—
304	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
305	C₂×Q₈	—	2	2	no	—
306	C₂×Q₈	—	2	2	no	—
307	C₂×Q₈	—	2	2	no	—
308	C₂×Q₈	—	2	2	no	—
309	C₂×Q₈	—	2	2	no	—
310	C₂×Q₈	—	2	2	no	—
311	C₂²×C₄	—	2	1	no	—
312	C₂²×C₄	—	2	1	no	—
313	C₂²×C₄	—	2	1	no	—
314	C₂²×C₄	—	2	1	no	—
315	C₂²×C₄	—	2	2	no	—
316	C₂²×C₄	—	2	2	no	—
317	C₂²×C₄	—	2	2	no	—
318	C₂²×C₄	—	2	2	no	—
319	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
320	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
321	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
322	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
323	C₂²×C₄	#41×C₆₇	1	2	yes	—
324	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
325	Q₁₆	—	4	4	no	—
326	M₁₆	—	2	8	no	—
327	C₂×C₈	—	2	8	no	—
328	C₂×C₈	—	2	1	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	2	1	no	—
331	C₂×C₈	—	2	1	no	—
332	C₂×C₈	—	2	1	no	—
333	C₂×C₈	—	2	1	no	—
334	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
335	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
336	C₂×C₈	#41×C₆₇	1	2	yes	—
337	C₂×C₈	#41×C₆₇	1	2	yes	—
338	C₂×C₈	#41×C₆₇	1	2	yes	—
339	C₂×C₈	#41×C₆₇	1	2	yes	—
340	C₄⋊C₄	—	2	4	no	—
341	C₄⋊C₄	—	2	2	no	—
342	C₄⋊C₄	—	2	2	no	—
343	C₄⋊C₄	—	2	2	no	—
344	C₄⋊C₄	—	2	2	no	—
345	C₄⋊C₄	—	2	2	no	—
346	C₄⋊C₄	—	2	2	no	—
347	C₄⋊C₄	—	4	4	no	—
348	C₄⋊C₄	—	8	2	no	—
349	C₄⋊C₄	—	8	2	no	—
350	C₂²⋊C₄	—	2	4	no	—
351	C₂²⋊C₄	—	4	4	no	—
352	C₂²⋊C₄	—	8	2	no	—
353	C₄²	—	2	1	no	—
354	C₄²	—	2	1	no	—
355	C₄²	—	2	1	no	—
356	C₄²	—	2	1	no	—
357	C₂³	—	2	1	no	—
358	C₂³	—	2	1	no	—
359	C₂³	—	2	2	no	—
360	C₂³	—	2	2	no	—
361	C₂³	#121×C₆₇	1	2	yes	—
362	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
363	Q₈	—	4	2	no	—
364	Q₈	—	4	2	no	—
365	Q₈	—	4	2	no	—
366	Q₈	—	4	2	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	4	2	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	4	2	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	4	2	no	—
387	C₂×C₄	—	4	2	no	—
388	C₂×C₄	—	4	2	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	—	4	2	no	—
391	C₂×C₄	#121×C₆₇	1	2	yes	—
392	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
393	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
394	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
395	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
396	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
397	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
398	C₈	—	2	2	no	—
399	C₈	—	2	2	no	—
400	C₈	—	2	2	no	—
401	C₈	#123×C₆₇	1	4	yes	—
402	C₂²	—	2	4	no	—
403	C₂²	—	2	2	no	—
404	C₂²	—	4	1	no	—
405	C₂²	—	4	1	no	—
406	C₂²	—	4	2	no	—
407	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
408	C₂²	#532×C₆₇	1	1	yes	characteristic
409	C₂²	#532×C₆₇	1	1	yes	characteristic
410	C₂²	#577×C₆₇	1	2	yes	—
411	C₂²	#578×C₆₇	1	2	yes	—
412	C₄	—	2	2	no	—
413	C₄	—	4	2	no	—
414	C₄	—	4	2	no	—
415	C₄	—	8	2	no	—
416	C₄	—	8	2	no	—
417	C₄	—	8	2	no	—
418	C₄	—	8	2	no	—
419	C₄	C₂×#123×C₆₇	1	1	yes	characteristic
420	C₄	#525×C₆₇	1	1	yes	characteristic
421	C₄	#529×C₆₇	1	1	yes	characteristic
422	C₄	#529×C₆₇	1	1	yes	characteristic
423	C₂	—	4	2	no	—
424	C₂	C₆₇×#2854	1	1	yes	characteristic
425	C₂	C₆₇×#3063	1	2	yes	—
426	C₂	C₆₇×#3173	1	1	yes	characteristic
427	C₂	C₆₇×#3177	1	1	yes	characteristic
428	C₂	C₆₇×#3178	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂⁴×C₆₇	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₆₇×#1971	C₂	semidirect
5. C₆₇×#1969	C₂	semidirect
6. C₆₇×#1943	C₂	semidirect
7. C₆₇×#1941	C₂	semidirect
8. C₆₇×#1935	C₂	semidirect
9. C₆₇×#1933	C₂	semidirect
208. #33141	C₆₇	direct (N × H)
218. #1971	C₂×C₆₇	semidirect
219. #1969	C₂×C₆₇	semidirect
220. #1943	C₂×C₆₇	semidirect
221. #1941	C₂×C₆₇	semidirect
222. #1935	C₂×C₆₇	semidirect
223. #1933	C₂×C₆₇	semidirect
226. C₆₇	#33141	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×#470×C₆₇	2
C₂×#405×C₆₇	2
C₆₇×#2625	2
C₆₇×#1971	2
C₆₇×#1969	2
C₆₇×#1943	2
C₆₇×#1941	2
C₆₇×#1935	2
C₆₇×#1933	2
C₆₇×#1892	2
C₆₇×#1809	2
#33141	67

Subgroups — C67×#33141 #30712392

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33141 #30712392