Subgroups — C₆₇×#32799 #30712050

Order 17152.

814 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2863	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#2858	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2043	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2042	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2042	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2041	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#2016	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#2015	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2015	C₂	1	2	yes	maximal, complemented
10	C₆₇×#2014	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#1839	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
13	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#124×C₆₇	C₂²	1	2	yes	—
17	C₂×#124×C₆₇	C₂²	1	2	yes	—
18	C₂×#103×C₆₇	C₂²	1	2	yes	—
19	C₂×#103×C₆₇	C₂²	1	2	yes	—
20	C₂×#103×C₆₇	C₂²	1	2	yes	—
21	C₂×#102×C₆₇	C₂²	1	2	yes	—
22	C₂×#102×C₆₇	C₂²	1	2	yes	—
23	C₂×#102×C₆₇	C₂²	1	2	yes	—
24	#391×C₆₇	C₂²	1	1	yes	characteristic
25	#391×C₆₇	C₂²	1	1	yes	characteristic
26	#391×C₆₇	C₂²	1	1	yes	characteristic
27	#390×C₆₇	—	2	1	no	—
28	#390×C₆₇	—	2	1	no	—
29	#390×C₆₇	—	2	1	no	—
30	#390×C₆₇	—	2	1	no	—
31	#389×C₆₇	C₂²	1	1	yes	characteristic
32	#389×C₆₇	C₂²	1	1	yes	characteristic
33	#389×C₆₇	C₂²	1	1	yes	characteristic
34	#387×C₆₇	—	2	1	no	—
35	#387×C₆₇	—	2	1	no	—
36	#387×C₆₇	—	2	1	no	—
37	#383×C₆₇	—	2	1	no	—
38	C₄³×C₆₇	C₂²	1	1	yes	characteristic
39	#336×C₆₇	C₂²	1	1	yes	characteristic
40	#336×C₆₇	C₂²	1	2	yes	—
41	#336×C₆₇	C₂²	1	2	yes	—
42	#336×C₆₇	C₂²	1	1	yes	characteristic
43	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
44	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
45	C₂²×D₄×C₆₇	—	2	1	no	—
46	C₂²×D₄×C₆₇	—	2	1	no	—
47	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
48	C₂×Q₁₆×C₆₇	—	4	2	no	—
49	C₂×Q₁₆×C₆₇	—	4	2	no	—
50	C₂×Q₁₆×C₆₇	—	4	2	no	—
51	C₂×SD₁₆×C₆₇	—	4	2	no	—
52	C₂×SD₁₆×C₆₇	—	4	2	no	—
53	C₂×SD₁₆×C₆₇	—	4	2	no	—
54	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
55	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
56	C₄⋊Q₈×C₆₇	—	4	2	no	—
57	#127×C₆₇	—	4	2	no	—
58	#124×C₆₇	—	4	2	no	—
59	#124×C₆₇	—	4	2	no	—
60	#124×C₆₇	—	4	2	no	—
61	#124×C₆₇	—	4	2	no	—
62	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
63	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
64	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
65	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
66	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
67	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
70	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
71	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
72	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
73	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
74	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
75	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
76	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
77	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
78	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
80	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
81	#103×C₆₇	—	4	2	no	—
82	#103×C₆₇	—	4	2	no	—
83	#103×C₆₇	—	4	2	no	—
84	#102×C₆₇	—	4	2	no	—
85	#102×C₆₇	—	4	2	no	—
86	#102×C₆₇	—	4	2	no	—
87	#95×C₆₇	—	2	1	no	—
88	#95×C₆₇	—	2	1	no	—
89	#95×C₆₇	—	2	1	no	—
90	#95×C₆₇	—	2	1	no	—
91	#95×C₆₇	—	2	1	no	—
92	#95×C₆₇	—	2	1	no	—
93	#95×C₆₇	—	2	1	no	—
94	#95×C₆₇	—	2	1	no	—
95	#95×C₆₇	—	2	1	no	—
96	C₂⁴×C₆₇	—	2	2	no	—
97	C₂×Q₈×C₆₇	—	2	2	no	—
98	C₂×Q₈×C₆₇	—	2	2	no	—
99	C₂×Q₈×C₆₇	—	2	2	no	—
100	C₂×Q₈×C₆₇	—	2	2	no	—
101	C₂×Q₈×C₆₇	—	2	2	no	—
102	C₂×Q₈×C₆₇	—	2	2	no	—
103	C₂×Q₈×C₆₇	—	2	1	no	—
104	C₂×Q₈×C₆₇	—	2	2	no	—
105	C₂×Q₈×C₆₇	—	2	2	no	—
106	C₂×Q₈×C₆₇	—	2	1	no	—
107	C₂×D₄×C₆₇	—	2	2	no	—
108	C₂×D₄×C₆₇	—	2	2	no	—
109	C₂×D₄×C₆₇	—	2	2	no	—
110	C₂×D₄×C₆₇	—	2	2	no	—
111	C₂×D₄×C₆₇	—	2	2	no	—
112	C₂×D₄×C₆₇	—	2	2	no	—
113	C₂×D₄×C₆₇	—	4	2	no	—
114	C₂×D₄×C₆₇	—	4	2	no	—
115	C₂×D₄×C₆₇	—	4	2	no	—
116	C₂×D₄×C₆₇	—	4	2	no	—
117	C₂×D₄×C₆₇	—	4	2	no	—
118	C₂×D₄×C₆₇	—	4	2	no	—
119	C₂²×C₄×C₆₇	—	2	2	no	—
120	C₂²×C₄×C₆₇	—	2	1	no	—
121	C₂²×C₄×C₆₇	—	2	1	no	—
122	C₂²×C₄×C₆₇	—	2	1	no	—
123	C₂²×C₄×C₆₇	—	2	1	no	—
124	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
131	Q₁₆×C₆₇	—	4	8	no	—
132	SD₁₆×C₆₇	—	4	8	no	—
133	C₂×C₈×C₆₇	—	2	2	no	—
134	C₂×C₈×C₆₇	—	2	2	no	—
135	C₂×C₈×C₆₇	—	2	2	no	—
136	C₂×C₈×C₆₇	—	2	2	no	—
137	C₂×C₈×C₆₇	—	2	2	no	—
138	C₂×C₈×C₆₇	—	2	2	no	—
139	C₄⋊C₄×C₆₇	—	2	1	no	—
140	C₄⋊C₄×C₆₇	—	2	2	no	—
141	C₄⋊C₄×C₆₇	—	2	2	no	—
142	C₄⋊C₄×C₆₇	—	2	1	no	—
143	C₄⋊C₄×C₆₇	—	4	2	no	—
144	C₄⋊C₄×C₆₇	—	4	2	no	—
145	C₄⋊C₄×C₆₇	—	4	2	no	—
146	C₄⋊C₄×C₆₇	—	4	2	no	—
147	C₄⋊C₄×C₆₇	—	4	2	no	—
148	C₂²⋊C₄×C₆₇	—	4	4	no	—
149	C₂²⋊C₄×C₆₇	—	4	4	no	—
150	C₂²⋊C₄×C₆₇	—	4	4	no	—
151	C₂²⋊C₄×C₆₇	—	4	4	no	—
152	C₄²×C₆₇	—	2	2	no	—
153	C₄²×C₆₇	—	2	2	no	—
154	C₄²×C₆₇	—	2	2	no	—
155	C₄²×C₆₇	—	2	1	no	—
156	C₄²×C₆₇	—	2	1	no	—
157	C₄²×C₆₇	—	2	2	no	—
158	C₄²×C₆₇	—	4	1	no	—
159	C₄²×C₆₇	—	4	1	no	—
160	C₂³×C₆₇	—	4	2	no	—
161	C₂³×C₆₇	—	4	2	no	—
162	C₂³×C₆₇	—	4	2	no	—
163	C₂³×C₆₇	—	4	2	no	—
164	C₂³×C₆₇	—	4	2	no	—
165	C₂³×C₆₇	—	4	2	no	—
166	C₂³×C₆₇	—	4	2	no	—
167	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
168	Q₈×C₆₇	—	4	2	no	—
169	Q₈×C₆₇	—	4	2	no	—
170	Q₈×C₆₇	—	4	2	no	—
171	Q₈×C₆₇	—	4	2	no	—
172	Q₈×C₆₇	—	4	2	no	—
173	Q₈×C₆₇	—	4	2	no	—
174	D₄×C₆₇	—	4	2	no	—
175	D₄×C₆₇	—	4	2	no	—
176	D₄×C₆₇	—	4	2	no	—
177	D₄×C₆₇	—	4	2	no	—
178	D₄×C₆₇	—	8	4	no	—
179	D₄×C₆₇	—	8	4	no	—
180	C₂×C₄×C₆₇	—	2	1	no	—
181	C₂×C₄×C₆₇	—	2	1	no	—
182	C₂×C₄×C₆₇	—	2	1	no	—
183	C₂×C₄×C₆₇	—	2	1	no	—
184	C₂×C₄×C₆₇	—	2	1	no	—
185	C₂×C₄×C₆₇	—	2	1	no	—
186	C₂×C₄×C₆₇	—	2	1	no	—
187	C₂×C₄×C₆₇	—	2	1	no	—
188	C₂×C₄×C₆₇	—	2	1	no	—
189	C₂×C₄×C₆₇	—	2	1	no	—
190	C₂×C₄×C₆₇	—	2	1	no	—
191	C₂×C₄×C₆₇	—	2	1	no	—
192	C₂×C₄×C₆₇	—	2	1	no	—
193	C₂×C₄×C₆₇	—	2	1	no	—
194	C₂×C₄×C₆₇	—	2	1	no	—
195	C₂×C₄×C₆₇	—	2	1	no	—
196	C₂×C₄×C₆₇	—	4	2	no	—
197	C₂×C₄×C₆₇	—	4	2	no	—
198	C₂×C₄×C₆₇	—	4	1	no	—
199	C₂×C₄×C₆₇	—	4	1	no	—
200	C₂×C₄×C₆₇	—	4	1	no	—
201	C₂×C₄×C₆₇	—	4	2	no	—
202	C₂×C₄×C₆₇	—	4	1	no	—
203	C₂×C₄×C₆₇	—	4	1	no	—
204	C₂×C₄×C₆₇	—	4	1	no	—
205	C₂×C₄×C₆₇	—	4	1	no	—
206	C₂×C₄×C₆₇	—	4	1	no	—
207	C₂×C₄×C₆₇	—	4	1	no	—
208	C₂×C₄×C₆₇	—	4	1	no	—
209	C₂×C₄×C₆₇	—	4	1	no	—
210	C₂×C₄×C₆₇	—	4	1	no	—
211	C₂×C₄×C₆₇	#135	1	2	yes	—
212	C₂×C₄×C₆₇	#137	1	2	yes	—
213	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
214	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
215	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
216	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
217	C₈×C₆₇	—	4	2	no	—
218	C₈×C₆₇	—	4	2	no	—
219	C₂²×C₆₇	—	8	2	no	—
220	C₂²×C₆₇	—	8	2	no	—
221	C₂²×C₆₇	—	8	2	no	—
222	C₂²×C₆₇	—	8	2	no	—
223	C₂²×C₆₇	—	8	2	no	—
224	C₂²×C₆₇	—	8	2	no	—
225	C₂²×C₆₇	—	8	2	no	—
226	C₂²×C₆₇	#535	1	1	yes	characteristic
227	C₂²×C₆₇	#535	1	1	yes	characteristic
228	C₂²×C₆₇	#535	1	1	yes	characteristic
229	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
230	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
231	C₂²×C₆₇	#575	1	1	yes	characteristic
232	C₂²×C₆₇	#575	1	1	yes	characteristic
233	C₄×C₆₇	—	2	2	no	—
234	C₄×C₆₇	—	2	2	no	—
235	C₄×C₆₇	—	4	1	no	—
236	C₄×C₆₇	—	4	1	no	—
237	C₄×C₆₇	—	4	1	no	—
238	C₄×C₆₇	—	4	1	no	—
239	C₄×C₆₇	—	8	2	no	—
240	C₄×C₆₇	—	8	1	no	—
241	C₄×C₆₇	—	8	1	no	—
242	C₄×C₆₇	—	8	1	no	—
243	C₄×C₆₇	—	8	1	no	—
244	C₄×C₆₇	#560	1	2	yes	—
245	C₄×C₆₇	#560	1	2	yes	—
246	#32799	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
247	C₂×C₆₇	—	16	2	no	—
248	C₂×C₆₇	#2818	1	1	yes	characteristic
249	C₂×C₆₇	#3210	1	1	yes	characteristic
250	C₂×C₆₇	#3211	1	1	yes	characteristic
251	C₂×C₆₇	#3212	1	1	yes	characteristic
252	C₂×C₆₇	#3215	1	1	yes	characteristic
253	C₂×C₆₇	#3216	1	1	yes	characteristic
254	C₂×C₆₇	#3219	1	1	yes	characteristic
255	#2863	C₂×C₆₇	1	1	yes	characteristic, complemented
256	#2858	C₂×C₆₇	1	1	yes	characteristic
257	#2043	C₂×C₆₇	1	1	yes	characteristic, complemented
258	#2042	C₂×C₆₇	1	2	yes	complemented
259	#2042	C₂×C₆₇	1	2	yes	complemented
260	#2041	C₂×C₆₇	1	1	yes	characteristic
261	#2016	C₂×C₆₇	1	1	yes	characteristic, complemented
262	#2015	C₂×C₆₇	1	2	yes	complemented
263	#2015	C₂×C₆₇	1	2	yes	complemented
264	#2014	C₂×C₆₇	1	1	yes	characteristic
265	#1839	C₂×C₆₇	1	1	yes	characteristic, complemented
266	C₆₇	#32799	1	1	yes	characteristic, direct product factor, Sylow
267	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	—
268	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
269	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
270	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
271	C₂×#124	C₂²×C₆₇	1	2	yes	—
272	C₂×#124	C₂²×C₆₇	1	2	yes	—
273	C₂×#103	C₂²×C₆₇	1	2	yes	—
274	C₂×#103	C₂²×C₆₇	1	2	yes	—
275	C₂×#103	C₂²×C₆₇	1	2	yes	—
276	C₂×#102	C₂²×C₆₇	1	2	yes	—
277	C₂×#102	C₂²×C₆₇	1	2	yes	—
278	C₂×#102	C₂²×C₆₇	1	2	yes	—
279	#391	C₂²×C₆₇	1	1	yes	characteristic
280	#391	C₂²×C₆₇	1	1	yes	characteristic
281	#391	C₂²×C₆₇	1	1	yes	characteristic
282	#390	—	2	1	no	—
283	#390	—	2	1	no	—
284	#390	—	2	1	no	—
285	#390	—	2	1	no	—
286	#389	C₂²×C₆₇	1	1	yes	characteristic
287	#389	C₂²×C₆₇	1	1	yes	characteristic
288	#389	C₂²×C₆₇	1	1	yes	characteristic
289	#387	—	2	1	no	—
290	#387	—	2	1	no	—
291	#387	—	2	1	no	—
292	#383	—	2	1	no	—
293	C₄³	C₂²×C₆₇	1	1	yes	characteristic
294	#336	C₂²×C₆₇	1	1	yes	characteristic
295	#336	C₂²×C₆₇	1	2	yes	—
296	#336	C₂²×C₆₇	1	2	yes	—
297	#336	C₂²×C₆₇	1	1	yes	characteristic
298	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
299	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
300	C₂²×D₄	—	2	1	no	—
301	C₂²×D₄	—	2	1	no	—
302	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
303	C₂×Q₁₆	—	4	2	no	—
304	C₂×Q₁₆	—	4	2	no	—
305	C₂×Q₁₆	—	4	2	no	—
306	C₂×SD₁₆	—	4	2	no	—
307	C₂×SD₁₆	—	4	2	no	—
308	C₂×SD₁₆	—	4	2	no	—
309	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
310	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
311	C₄⋊Q₈	—	4	2	no	—
312	#127	—	4	2	no	—
313	#124	—	4	2	no	—
314	#124	—	4	2	no	—
315	#124	—	4	2	no	—
316	#124	—	4	2	no	—
317	C₂×C₄⋊C₄	—	2	1	no	—
318	C₂×C₄⋊C₄	—	2	1	no	—
319	C₂×C₄⋊C₄	—	2	1	no	—
320	C₂×C₄⋊C₄	—	2	1	no	—
321	C₂×C₄⋊C₄	—	2	1	no	—
322	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
323	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
324	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
325	C₂×C₂²⋊C₄	—	2	2	no	—
326	C₂×C₂²⋊C₄	—	2	2	no	—
327	C₂×C₂²⋊C₄	—	2	2	no	—
328	C₂×C₂²⋊C₄	—	2	2	no	—
329	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
330	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
331	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
332	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
333	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
334	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
335	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
336	#103	—	4	2	no	—
337	#103	—	4	2	no	—
338	#103	—	4	2	no	—
339	#102	—	4	2	no	—
340	#102	—	4	2	no	—
341	#102	—	4	2	no	—
342	#95	—	2	1	no	—
343	#95	—	2	1	no	—
344	#95	—	2	1	no	—
345	#95	—	2	1	no	—
346	#95	—	2	1	no	—
347	#95	—	2	1	no	—
348	#95	—	2	1	no	—
349	#95	—	2	1	no	—
350	#95	—	2	1	no	—
351	C₂⁴	—	2	2	no	—
352	C₂×Q₈	—	2	2	no	—
353	C₂×Q₈	—	2	2	no	—
354	C₂×Q₈	—	2	2	no	—
355	C₂×Q₈	—	2	2	no	—
356	C₂×Q₈	—	2	2	no	—
357	C₂×Q₈	—	2	2	no	—
358	C₂×Q₈	—	2	1	no	—
359	C₂×Q₈	—	2	2	no	—
360	C₂×Q₈	—	2	2	no	—
361	C₂×Q₈	—	2	1	no	—
362	C₂×D₄	—	2	2	no	—
363	C₂×D₄	—	2	2	no	—
364	C₂×D₄	—	2	2	no	—
365	C₂×D₄	—	2	2	no	—
366	C₂×D₄	—	2	2	no	—
367	C₂×D₄	—	2	2	no	—
368	C₂×D₄	—	4	2	no	—
369	C₂×D₄	—	4	2	no	—
370	C₂×D₄	—	4	2	no	—
371	C₂×D₄	—	4	2	no	—
372	C₂×D₄	—	4	2	no	—
373	C₂×D₄	—	4	2	no	—
374	C₂²×C₄	—	2	1	no	—
375	C₂²×C₄	—	2	1	no	—
376	C₂²×C₄	—	2	1	no	—
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	—	2	2	no	—
379	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
380	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
381	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
382	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
383	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
384	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
385	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
386	Q₁₆	—	4	8	no	—
387	SD₁₆	—	4	8	no	—
388	C₂×C₈	—	2	2	no	—
389	C₂×C₈	—	2	2	no	—
390	C₂×C₈	—	2	2	no	—
391	C₂×C₈	—	2	2	no	—
392	C₂×C₈	—	2	2	no	—
393	C₂×C₈	—	2	2	no	—
394	C₄⋊C₄	—	2	1	no	—
395	C₄⋊C₄	—	2	2	no	—
396	C₄⋊C₄	—	2	2	no	—
397	C₄⋊C₄	—	2	1	no	—
398	C₄⋊C₄	—	4	2	no	—
399	C₄⋊C₄	—	4	2	no	—
400	C₄⋊C₄	—	4	2	no	—
401	C₄⋊C₄	—	4	2	no	—
402	C₄⋊C₄	—	4	2	no	—
403	C₂²⋊C₄	—	4	4	no	—
404	C₂²⋊C₄	—	4	4	no	—
405	C₂²⋊C₄	—	4	4	no	—
406	C₂²⋊C₄	—	4	4	no	—
407	C₄²	—	2	2	no	—
408	C₄²	—	2	2	no	—
409	C₄²	—	2	2	no	—
410	C₄²	—	2	1	no	—
411	C₄²	—	2	1	no	—
412	C₄²	—	2	2	no	—
413	C₄²	—	4	1	no	—
414	C₄²	—	4	1	no	—
415	C₂³	—	4	2	no	—
416	C₂³	—	4	2	no	—
417	C₂³	—	4	2	no	—
418	C₂³	—	4	2	no	—
419	C₂³	—	4	2	no	—
420	C₂³	—	4	2	no	—
421	C₂³	—	4	2	no	—
422	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
423	Q₈	—	4	2	no	—
424	Q₈	—	4	2	no	—
425	Q₈	—	4	2	no	—
426	Q₈	—	4	2	no	—
427	Q₈	—	4	2	no	—
428	Q₈	—	4	2	no	—
429	D₄	—	4	2	no	—
430	D₄	—	4	2	no	—
431	D₄	—	4	2	no	—
432	D₄	—	4	2	no	—
433	D₄	—	8	4	no	—
434	D₄	—	8	4	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	2	1	no	—
437	C₂×C₄	—	2	1	no	—
438	C₂×C₄	—	2	1	no	—
439	C₂×C₄	—	2	1	no	—
440	C₂×C₄	—	2	1	no	—
441	C₂×C₄	—	2	1	no	—
442	C₂×C₄	—	2	1	no	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	2	1	no	—
445	C₂×C₄	—	2	1	no	—
446	C₂×C₄	—	2	1	no	—
447	C₂×C₄	—	2	1	no	—
448	C₂×C₄	—	2	1	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	4	2	no	—
452	C₂×C₄	—	4	2	no	—
453	C₂×C₄	—	4	1	no	—
454	C₂×C₄	—	4	1	no	—
455	C₂×C₄	—	4	1	no	—
456	C₂×C₄	—	4	2	no	—
457	C₂×C₄	—	4	1	no	—
458	C₂×C₄	—	4	1	no	—
459	C₂×C₄	—	4	1	no	—
460	C₂×C₄	—	4	1	no	—
461	C₂×C₄	—	4	1	no	—
462	C₂×C₄	—	4	1	no	—
463	C₂×C₄	—	4	1	no	—
464	C₂×C₄	—	4	1	no	—
465	C₂×C₄	—	4	1	no	—
466	C₂×C₄	#135×C₆₇	1	2	yes	—
467	C₂×C₄	#137×C₆₇	1	2	yes	—
468	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
469	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
470	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
471	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
472	C₈	—	4	2	no	—
473	C₈	—	4	2	no	—
474	C₂²	—	8	2	no	—
475	C₂²	—	8	2	no	—
476	C₂²	—	8	2	no	—
477	C₂²	—	8	2	no	—
478	C₂²	—	8	2	no	—
479	C₂²	—	8	2	no	—
480	C₂²	—	8	2	no	—
481	C₂²	#535×C₆₇	1	1	yes	characteristic
482	C₂²	#535×C₆₇	1	1	yes	characteristic
483	C₂²	#535×C₆₇	1	1	yes	characteristic
484	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
485	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
486	C₂²	#575×C₆₇	1	1	yes	characteristic
487	C₂²	#575×C₆₇	1	1	yes	characteristic
488	C₄	—	2	2	no	—
489	C₄	—	2	2	no	—
490	C₄	—	4	1	no	—
491	C₄	—	4	1	no	—
492	C₄	—	4	1	no	—
493	C₄	—	4	1	no	—
494	C₄	—	8	2	no	—
495	C₄	—	8	1	no	—
496	C₄	—	8	1	no	—
497	C₄	—	8	1	no	—
498	C₄	—	8	1	no	—
499	C₄	#560×C₆₇	1	2	yes	—
500	C₄	#560×C₆₇	1	2	yes	—
501	C₂	—	16	2	no	—
502	C₂	C₆₇×#2818	1	1	yes	characteristic
503	C₂	C₆₇×#3210	1	1	yes	characteristic
504	C₂	C₆₇×#3211	1	1	yes	characteristic
505	C₂	C₆₇×#3212	1	1	yes	characteristic
506	C₂	C₆₇×#3215	1	1	yes	characteristic
507	C₂	C₆₇×#3216	1	1	yes	characteristic
508	C₂	C₆₇×#3219	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#127×C₆₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#2863	C₂	semidirect
3. C₆₇×#2043	C₂	semidirect
4. C₆₇×#2042	C₂	semidirect
5. C₆₇×#2042	C₂	semidirect
7. C₆₇×#2016	C₂	semidirect
8. C₆₇×#2015	C₂	semidirect
9. C₆₇×#2015	C₂	semidirect
11. C₆₇×#1839	C₂	semidirect
246. #32799	C₆₇	direct (N × H)
255. #2863	C₂×C₆₇	semidirect
257. #2043	C₂×C₆₇	semidirect
258. #2042	C₂×C₆₇	semidirect
259. #2042	C₂×C₆₇	semidirect
261. #2016	C₂×C₆₇	semidirect
262. #2015	C₂×C₆₇	semidirect
263. #2015	C₂×C₆₇	semidirect
265. #1839	C₂×C₆₇	semidirect
266. C₆₇	#32799	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#2863	2
C₆₇×#2858	2
C₆₇×#2043	2
C₆₇×#2042	2
C₆₇×#2041	2
C₆₇×#2016	2
C₆₇×#2015	2
C₆₇×#2014	2
C₆₇×#1839	2
#32799	67

Subgroups — C67×#32799 #30712050

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32799 #30712050