Subgroups — C₆₇×#32841 #30712092

Order 17152.

632 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2860	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2859	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2100	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2099	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2094	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#2093	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#2085	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2073	C₂	1	2	yes	maximal
9	C₆₇×#2059	C₂	1	2	yes	maximal, complemented
10	C₆₇×#2025	C₂	1	2	yes	maximal
11	C₆₇×#1839	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#126×C₆₇	—	2	1	no	—
13	C₂×#125×C₆₇	C₂²	1	2	yes	—
14	C₂×#107×C₆₇	C₂²	1	2	yes	—
15	C₂×#106×C₆₇	C₂²	1	2	yes	—
16	C₂×#102×C₆₇	C₂²	1	2	yes	—
17	C₂×#102×C₆₇	C₂²	1	2	yes	—
18	#390×C₆₇	C₂²	1	1	yes	characteristic
19	#390×C₆₇	C₂²	1	1	yes	characteristic
20	#390×C₆₇	C₂²	1	1	yes	characteristic
21	#389×C₆₇	—	2	1	no	—
22	#389×C₆₇	—	2	1	no	—
23	#389×C₆₇	—	2	1	no	—
24	#389×C₆₇	—	2	1	no	—
25	#389×C₆₇	C₂²	1	1	yes	characteristic
26	#389×C₆₇	C₂²	1	1	yes	characteristic
27	#389×C₆₇	C₂²	1	1	yes	characteristic
28	#388×C₆₇	—	2	1	no	—
29	#388×C₆₇	—	2	1	no	—
30	#388×C₆₇	—	2	1	no	—
31	#384×C₆₇	C₂²	1	1	yes	characteristic
32	#384×C₆₇	C₂²	1	1	yes	characteristic
33	#382×C₆₇	C₂²	1	2	yes	—
34	C₄³×C₆₇	C₂²	1	1	yes	characteristic
35	#340×C₆₇	C₂²	1	2	yes	—
36	#340×C₆₇	C₂²	1	2	yes	—
37	#340×C₆₇	C₂²	1	2	yes	—
38	#340×C₆₇	C₂²	1	2	yes	—
39	#336×C₆₇	C₂²	1	2	yes	—
40	#336×C₆₇	C₂²	1	2	yes	—
41	#336×C₆₇	C₂²	1	1	yes	characteristic
42	#336×C₆₇	C₂²	1	1	yes	characteristic
43	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
45	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
46	#126×C₆₇	—	4	4	no	—
47	#125×C₆₇	—	4	2	no	—
48	#125×C₆₇	—	4	2	no	—
49	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
50	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
51	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
52	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
53	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
54	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
55	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
56	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
57	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
58	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
59	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
60	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
64	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
65	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
66	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
67	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
68	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
69	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
70	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
71	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
72	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
73	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
74	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
76	#107×C₆₇	—	4	2	no	—
77	#106×C₆₇	—	4	2	no	—
78	#102×C₆₇	—	4	2	no	—
79	#102×C₆₇	—	4	2	no	—
80	#95×C₆₇	—	2	2	no	—
81	#95×C₆₇	—	2	2	no	—
82	#95×C₆₇	—	2	1	no	—
83	#95×C₆₇	—	2	1	no	—
84	#95×C₆₇	—	2	1	no	—
85	C₂⁴×C₆₇	—	2	1	no	—
86	C₂×D₄×C₆₇	—	2	2	no	—
87	C₂×D₄×C₆₇	—	2	2	no	—
88	C₂×D₄×C₆₇	—	2	1	no	—
89	C₂×D₄×C₆₇	—	2	1	no	—
90	C₂²×C₄×C₆₇	—	2	1	no	—
91	C₂²×C₄×C₆₇	—	2	1	no	—
92	C₂²×C₄×C₆₇	—	2	1	no	—
93	C₂²×C₄×C₆₇	—	2	2	no	—
94	C₂²×C₄×C₆₇	—	2	2	no	—
95	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
102	C₂×C₈×C₆₇	—	2	2	no	—
103	C₂×C₈×C₆₇	—	2	2	no	—
104	C₂×C₈×C₆₇	—	2	2	no	—
105	C₂×C₈×C₆₇	—	2	2	no	—
106	C₂×C₈×C₆₇	—	2	2	no	—
107	C₂×C₈×C₆₇	—	2	2	no	—
108	C₄⋊C₄×C₆₇	—	2	2	no	—
109	C₄⋊C₄×C₆₇	—	2	2	no	—
110	C₄⋊C₄×C₆₇	—	2	2	no	—
111	C₄⋊C₄×C₆₇	—	2	2	no	—
112	C₄⋊C₄×C₆₇	—	2	2	no	—
113	C₄⋊C₄×C₆₇	—	2	2	no	—
114	C₄⋊C₄×C₆₇	—	2	1	no	—
115	C₄⋊C₄×C₆₇	—	2	1	no	—
116	C₄⋊C₄×C₆₇	—	4	2	no	—
117	C₄⋊C₄×C₆₇	—	4	2	no	—
118	C₄⋊C₄×C₆₇	—	4	4	no	—
119	C₄⋊C₄×C₆₇	—	4	4	no	—
120	C₄⋊C₄×C₆₇	—	4	2	no	—
121	C₄⋊C₄×C₆₇	—	4	2	no	—
122	C₄⋊C₄×C₆₇	—	4	2	no	—
123	C₄⋊C₄×C₆₇	—	4	2	no	—
124	C₄⋊C₄×C₆₇	—	4	2	no	—
125	C₂²⋊C₄×C₆₇	—	4	2	no	—
126	C₂²⋊C₄×C₆₇	—	4	2	no	—
127	C₂²⋊C₄×C₆₇	—	4	2	no	—
128	C₂²⋊C₄×C₆₇	—	4	2	no	—
129	C₂²⋊C₄×C₆₇	—	4	2	no	—
130	C₂²⋊C₄×C₆₇	—	4	2	no	—
131	C₄²×C₆₇	—	2	2	no	—
132	C₄²×C₆₇	—	2	2	no	—
133	C₄²×C₆₇	—	2	2	no	—
134	C₄²×C₆₇	—	2	2	no	—
135	C₄²×C₆₇	—	2	1	no	—
136	C₄²×C₆₇	—	2	1	no	—
137	C₄²×C₆₇	—	4	1	no	—
138	C₄²×C₆₇	—	4	1	no	—
139	C₂³×C₆₇	—	4	1	no	—
140	C₂³×C₆₇	—	4	1	no	—
141	C₂³×C₆₇	—	4	1	no	—
142	C₂³×C₆₇	—	4	1	no	—
143	C₂³×C₆₇	—	4	1	no	—
144	C₂³×C₆₇	—	4	1	no	—
145	C₂³×C₆₇	—	4	1	no	—
146	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
147	D₄×C₆₇	—	4	2	no	—
148	D₄×C₆₇	—	4	2	no	—
149	C₂×C₄×C₆₇	—	2	1	no	—
150	C₂×C₄×C₆₇	—	2	1	no	—
151	C₂×C₄×C₆₇	—	2	1	no	—
152	C₂×C₄×C₆₇	—	2	1	no	—
153	C₂×C₄×C₆₇	—	2	1	no	—
154	C₂×C₄×C₆₇	—	2	1	no	—
155	C₂×C₄×C₆₇	—	2	1	no	—
156	C₂×C₄×C₆₇	—	2	1	no	—
157	C₂×C₄×C₆₇	—	2	1	no	—
158	C₂×C₄×C₆₇	—	2	1	no	—
159	C₂×C₄×C₆₇	—	2	1	no	—
160	C₂×C₄×C₆₇	—	2	1	no	—
161	C₂×C₄×C₆₇	—	2	1	no	—
162	C₂×C₄×C₆₇	—	2	1	no	—
163	C₂×C₄×C₆₇	—	2	1	no	—
164	C₂×C₄×C₆₇	—	2	1	no	—
165	C₂×C₄×C₆₇	—	4	2	no	—
166	C₂×C₄×C₆₇	—	4	2	no	—
167	C₂×C₄×C₆₇	—	4	1	no	—
168	C₂×C₄×C₆₇	—	4	1	no	—
169	C₂×C₄×C₆₇	—	4	1	no	—
170	C₂×C₄×C₆₇	—	4	1	no	—
171	C₂×C₄×C₆₇	—	4	2	no	—
172	C₂×C₄×C₆₇	—	4	1	no	—
173	C₂×C₄×C₆₇	—	4	1	no	—
174	C₂×C₄×C₆₇	—	4	1	no	—
175	C₂×C₄×C₆₇	—	4	2	no	—
176	C₂×C₄×C₆₇	—	4	1	no	—
177	C₂×C₄×C₆₇	—	4	1	no	—
178	C₂×C₄×C₆₇	—	4	2	no	—
179	C₂×C₄×C₆₇	—	4	2	no	—
180	C₂×C₄×C₆₇	#135	1	2	yes	—
181	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
182	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
183	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
184	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
185	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
186	C₈×C₆₇	—	4	2	no	—
187	C₈×C₆₇	—	4	2	no	—
188	C₂²×C₆₇	—	8	1	no	—
189	C₂²×C₆₇	—	8	1	no	—
190	C₂²×C₆₇	—	8	1	no	—
191	C₂²×C₆₇	—	8	1	no	—
192	C₂²×C₆₇	—	8	1	no	—
193	C₂²×C₆₇	—	8	1	no	—
194	C₂²×C₆₇	—	8	1	no	—
195	C₂²×C₆₇	#535	1	1	yes	characteristic
196	C₂²×C₆₇	#536	1	1	yes	characteristic
197	C₂²×C₆₇	#536	1	1	yes	characteristic
198	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
199	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
200	C₂²×C₆₇	#575	1	1	yes	characteristic
201	C₂²×C₆₇	#575	1	1	yes	characteristic
202	C₄×C₆₇	—	2	2	no	—
203	C₄×C₆₇	—	2	2	no	—
204	C₄×C₆₇	—	4	1	no	—
205	C₄×C₆₇	—	4	1	no	—
206	C₄×C₆₇	—	4	1	no	—
207	C₄×C₆₇	—	4	1	no	—
208	C₄×C₆₇	—	8	2	no	—
209	C₄×C₆₇	—	8	1	no	—
210	C₄×C₆₇	—	8	2	no	—
211	C₄×C₆₇	—	8	1	no	—
212	C₄×C₆₇	—	8	1	no	—
213	C₄×C₆₇	#563	1	2	yes	—
214	C₄×C₆₇	#563	1	2	yes	—
215	#32841	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
216	C₂×C₆₇	—	16	1	no	—
217	C₂×C₆₇	#2821	1	1	yes	characteristic
218	C₂×C₆₇	#3207	1	1	yes	characteristic
219	C₂×C₆₇	#3219	1	1	yes	characteristic
220	C₂×C₆₇	#3232	1	1	yes	characteristic
221	C₂×C₆₇	#3233	1	1	yes	characteristic
222	C₂×C₆₇	#3234	1	1	yes	characteristic
223	C₂×C₆₇	#3235	1	1	yes	characteristic
224	#2860	C₂×C₆₇	1	1	yes	characteristic
225	#2859	C₂×C₆₇	1	1	yes	characteristic, complemented
226	#2100	C₂×C₆₇	1	1	yes	characteristic, complemented
227	#2099	C₂×C₆₇	1	1	yes	characteristic, complemented
228	#2094	C₂×C₆₇	1	1	yes	characteristic
229	#2093	C₂×C₆₇	1	1	yes	characteristic
230	#2085	C₂×C₆₇	1	2	yes	complemented
231	#2073	C₂×C₆₇	1	2	yes	—
232	#2059	C₂×C₆₇	1	2	yes	complemented
233	#2025	C₂×C₆₇	1	2	yes	—
234	#1839	C₂×C₆₇	1	1	yes	characteristic, complemented
235	C₆₇	#32841	1	1	yes	characteristic, direct product factor, Sylow
236	C₂×#126	—	2	1	no	—
237	C₂×#125	C₂²×C₆₇	1	2	yes	—
238	C₂×#107	C₂²×C₆₇	1	2	yes	—
239	C₂×#106	C₂²×C₆₇	1	2	yes	—
240	C₂×#102	C₂²×C₆₇	1	2	yes	—
241	C₂×#102	C₂²×C₆₇	1	2	yes	—
242	#390	C₂²×C₆₇	1	1	yes	characteristic
243	#390	C₂²×C₆₇	1	1	yes	characteristic
244	#390	C₂²×C₆₇	1	1	yes	characteristic
245	#389	—	2	1	no	—
246	#389	—	2	1	no	—
247	#389	—	2	1	no	—
248	#389	—	2	1	no	—
249	#389	C₂²×C₆₇	1	1	yes	characteristic
250	#389	C₂²×C₆₇	1	1	yes	characteristic
251	#389	C₂²×C₆₇	1	1	yes	characteristic
252	#388	—	2	1	no	—
253	#388	—	2	1	no	—
254	#388	—	2	1	no	—
255	#384	C₂²×C₆₇	1	1	yes	characteristic
256	#384	C₂²×C₆₇	1	1	yes	characteristic
257	#382	C₂²×C₆₇	1	2	yes	—
258	C₄³	C₂²×C₆₇	1	1	yes	characteristic
259	#340	C₂²×C₆₇	1	2	yes	—
260	#340	C₂²×C₆₇	1	2	yes	—
261	#340	C₂²×C₆₇	1	2	yes	—
262	#340	C₂²×C₆₇	1	2	yes	—
263	#336	C₂²×C₆₇	1	2	yes	—
264	#336	C₂²×C₆₇	1	2	yes	—
265	#336	C₂²×C₆₇	1	1	yes	characteristic
266	#336	C₂²×C₆₇	1	1	yes	characteristic
267	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
268	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
269	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
270	#126	—	4	4	no	—
271	#125	—	4	2	no	—
272	#125	—	4	2	no	—
273	C₂×C₄⋊C₄	—	2	2	no	—
274	C₂×C₄⋊C₄	—	2	2	no	—
275	C₂×C₄⋊C₄	—	2	1	no	—
276	C₂×C₄⋊C₄	—	2	1	no	—
277	C₂×C₄⋊C₄	—	2	1	no	—
278	C₂×C₄⋊C₄	—	2	1	no	—
279	C₂×C₄⋊C₄	—	2	1	no	—
280	C₂×C₄⋊C₄	—	2	1	no	—
281	C₂×C₄⋊C₄	—	2	1	no	—
282	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
283	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
284	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
285	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
286	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
287	C₂×C₂²⋊C₄	—	2	1	no	—
288	C₂×C₂²⋊C₄	—	2	1	no	—
289	C₂×C₂²⋊C₄	—	2	1	no	—
290	C₂×C₂²⋊C₄	—	2	1	no	—
291	C₂×C₂²⋊C₄	—	2	1	no	—
292	C₂×C₂²⋊C₄	—	2	1	no	—
293	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
294	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
295	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
296	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
297	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
298	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
299	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
300	#107	—	4	2	no	—
301	#106	—	4	2	no	—
302	#102	—	4	2	no	—
303	#102	—	4	2	no	—
304	#95	—	2	2	no	—
305	#95	—	2	2	no	—
306	#95	—	2	1	no	—
307	#95	—	2	1	no	—
308	#95	—	2	1	no	—
309	C₂⁴	—	2	1	no	—
310	C₂×D₄	—	2	2	no	—
311	C₂×D₄	—	2	2	no	—
312	C₂×D₄	—	2	1	no	—
313	C₂×D₄	—	2	1	no	—
314	C₂²×C₄	—	2	1	no	—
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	—	2	1	no	—
317	C₂²×C₄	—	2	2	no	—
318	C₂²×C₄	—	2	2	no	—
319	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
320	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
321	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
322	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
323	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
324	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
325	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	—	2	2	no	—
331	C₂×C₈	—	2	2	no	—
332	C₄⋊C₄	—	2	2	no	—
333	C₄⋊C₄	—	2	2	no	—
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	2	2	no	—
338	C₄⋊C₄	—	2	1	no	—
339	C₄⋊C₄	—	2	1	no	—
340	C₄⋊C₄	—	4	4	no	—
341	C₄⋊C₄	—	4	4	no	—
342	C₄⋊C₄	—	4	2	no	—
343	C₄⋊C₄	—	4	2	no	—
344	C₄⋊C₄	—	4	2	no	—
345	C₄⋊C₄	—	4	2	no	—
346	C₄⋊C₄	—	4	2	no	—
347	C₄⋊C₄	—	4	2	no	—
348	C₄⋊C₄	—	4	2	no	—
349	C₂²⋊C₄	—	4	2	no	—
350	C₂²⋊C₄	—	4	2	no	—
351	C₂²⋊C₄	—	4	2	no	—
352	C₂²⋊C₄	—	4	2	no	—
353	C₂²⋊C₄	—	4	2	no	—
354	C₂²⋊C₄	—	4	2	no	—
355	C₄²	—	2	2	no	—
356	C₄²	—	2	2	no	—
357	C₄²	—	2	2	no	—
358	C₄²	—	2	2	no	—
359	C₄²	—	2	1	no	—
360	C₄²	—	2	1	no	—
361	C₄²	—	4	1	no	—
362	C₄²	—	4	1	no	—
363	C₂³	—	4	1	no	—
364	C₂³	—	4	1	no	—
365	C₂³	—	4	1	no	—
366	C₂³	—	4	1	no	—
367	C₂³	—	4	1	no	—
368	C₂³	—	4	1	no	—
369	C₂³	—	4	1	no	—
370	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
371	D₄	—	4	2	no	—
372	D₄	—	4	2	no	—
373	C₂×C₄	—	2	1	no	—
374	C₂×C₄	—	2	1	no	—
375	C₂×C₄	—	2	1	no	—
376	C₂×C₄	—	2	1	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	2	1	no	—
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	2	1	no	—
383	C₂×C₄	—	2	1	no	—
384	C₂×C₄	—	2	1	no	—
385	C₂×C₄	—	2	1	no	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	2	1	no	—
388	C₂×C₄	—	2	1	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	—	4	2	no	—
391	C₂×C₄	—	4	1	no	—
392	C₂×C₄	—	4	1	no	—
393	C₂×C₄	—	4	1	no	—
394	C₂×C₄	—	4	1	no	—
395	C₂×C₄	—	4	2	no	—
396	C₂×C₄	—	4	1	no	—
397	C₂×C₄	—	4	1	no	—
398	C₂×C₄	—	4	1	no	—
399	C₂×C₄	—	4	2	no	—
400	C₂×C₄	—	4	1	no	—
401	C₂×C₄	—	4	1	no	—
402	C₂×C₄	—	4	2	no	—
403	C₂×C₄	—	4	2	no	—
404	C₂×C₄	#135×C₆₇	1	2	yes	—
405	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
406	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
407	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
408	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
409	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
410	C₈	—	4	2	no	—
411	C₈	—	4	2	no	—
412	C₂²	—	8	1	no	—
413	C₂²	—	8	1	no	—
414	C₂²	—	8	1	no	—
415	C₂²	—	8	1	no	—
416	C₂²	—	8	1	no	—
417	C₂²	—	8	1	no	—
418	C₂²	—	8	1	no	—
419	C₂²	#535×C₆₇	1	1	yes	characteristic
420	C₂²	#536×C₆₇	1	1	yes	characteristic
421	C₂²	#536×C₆₇	1	1	yes	characteristic
422	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
423	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
424	C₂²	#575×C₆₇	1	1	yes	characteristic
425	C₂²	#575×C₆₇	1	1	yes	characteristic
426	C₄	—	2	2	no	—
427	C₄	—	2	2	no	—
428	C₄	—	4	1	no	—
429	C₄	—	4	1	no	—
430	C₄	—	4	1	no	—
431	C₄	—	4	1	no	—
432	C₄	—	8	2	no	—
433	C₄	—	8	1	no	—
434	C₄	—	8	2	no	—
435	C₄	—	8	1	no	—
436	C₄	—	8	1	no	—
437	C₄	#563×C₆₇	1	2	yes	—
438	C₄	#563×C₆₇	1	2	yes	—
439	C₂	—	16	1	no	—
440	C₂	C₆₇×#2821	1	1	yes	characteristic
441	C₂	C₆₇×#3207	1	1	yes	characteristic
442	C₂	C₆₇×#3219	1	1	yes	characteristic
443	C₂	C₆₇×#3232	1	1	yes	characteristic
444	C₂	C₆₇×#3233	1	1	yes	characteristic
445	C₂	C₆₇×#3234	1	1	yes	characteristic
446	C₂	C₆₇×#3235	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×D₄×C₆₇	C₂³
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
2. C₆₇×#2859	C₂	semidirect
3. C₆₇×#2100	C₂	semidirect
4. C₆₇×#2099	C₂	semidirect
7. C₆₇×#2085	C₂	semidirect
9. C₆₇×#2059	C₂	semidirect
11. C₆₇×#1839	C₂	semidirect
215. #32841	C₆₇	direct (N × H)
225. #2859	C₂×C₆₇	semidirect
226. #2100	C₂×C₆₇	semidirect
227. #2099	C₂×C₆₇	semidirect
230. #2085	C₂×C₆₇	semidirect
232. #2059	C₂×C₆₇	semidirect
234. #1839	C₂×C₆₇	semidirect
235. C₆₇	#32841	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×#2860	2
C₆₇×#2859	2
C₆₇×#2100	2
C₆₇×#2099	2
C₆₇×#2094	2
C₆₇×#2093	2
C₆₇×#2085	2
C₆₇×#2073	2
C₆₇×#2059	2
C₆₇×#2025	2
C₆₇×#1839	2
#32841	67

Subgroups — C67×#32841 #30712092

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32841 #30712092