Subgroups — C₆₇×#21997 #30701248

Order 17152.

800 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2290	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1923	C₂	1	2	yes	maximal, complemented
3	C₆₇×#1916	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1869	C₂	1	2	yes	maximal
5	C₆₇×#1850	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1840	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1832	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1818	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1798	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₂²⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#336×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
22	#388×C₆₇	C₄	1	2	yes	—
23	#387×C₆₇	C₄	1	2	yes	—
24	#386×C₆₇	C₄	1	2	yes	—
25	#385×C₆₇	C₄	1	2	yes	—
26	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
27	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
28	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
29	#376×C₆₇	C₂²	1	1	yes	characteristic
30	#336×C₆₇	C₂²	1	1	yes	characteristic
31	#336×C₆₇	C₂²	1	1	yes	characteristic
32	#336×C₆₇	C₂²	1	1	yes	characteristic
33	#336×C₆₇	C₂²	1	1	yes	characteristic
34	#336×C₆₇	C₂²	1	2	yes	—
35	#336×C₆₇	C₂²	1	2	yes	—
36	#336×C₆₇	C₂²	1	2	yes	—
37	#336×C₆₇	C₂²	1	2	yes	—
38	#336×C₆₇	C₂²	1	2	yes	—
39	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
41	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
42	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₆₇	—	2	4	no	—
44	C₂²×C₈×C₆₇	—	4	2	no	—
45	C₂²×C₈×C₆₇	—	4	2	no	—
46	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
47	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
48	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
49	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
50	C₄×D₄×C₆₇	—	2	4	no	—
51	#117×C₆₇	—	4	2	no	—
52	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
54	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
55	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
58	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
59	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
60	C₂×C₄²×C₆₇	—	4	2	no	—
61	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
62	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
67	C₄⋊C₈×C₆₇	—	4	2	no	—
68	C₄⋊C₈×C₆₇	—	4	2	no	—
69	C₂²⋊C₈×C₆₇	—	2	4	no	—
70	C₂²⋊C₈×C₆₇	—	4	2	no	—
71	C₂²⋊C₈×C₆₇	—	4	2	no	—
72	C₄×C₈×C₆₇	—	4	2	no	—
73	#95×C₆₇	C₂×C₄	1	2	yes	—
74	#95×C₆₇	C₂×C₄	1	2	yes	—
75	#95×C₆₇	C₂×C₄	1	2	yes	—
76	#95×C₆₇	C₂×C₄	1	2	yes	—
77	#95×C₆₇	C₂×C₄	1	2	yes	—
78	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
79	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×D₄×C₆₇	—	2	2	no	—
81	C₂×D₄×C₆₇	—	2	4	no	—
82	C₂²×C₄×C₆₇	—	2	2	no	—
83	C₂²×C₄×C₆₇	—	2	1	no	—
84	C₂²×C₄×C₆₇	—	2	1	no	—
85	C₂²×C₄×C₆₇	—	2	4	no	—
86	C₂²×C₄×C₆₇	—	2	4	no	—
87	C₂²×C₄×C₆₇	—	2	2	no	—
88	C₂²×C₄×C₆₇	—	2	2	no	—
89	C₂²×C₄×C₆₇	—	2	1	no	—
90	C₂²×C₄×C₆₇	—	2	1	no	—
91	C₂²×C₄×C₆₇	—	4	2	no	—
92	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₆₇	#41	1	4	yes	—
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	C₂×C₈×C₆₇	—	2	2	no	—
109	C₂×C₈×C₆₇	—	2	2	no	—
110	C₂×C₈×C₆₇	—	2	2	no	—
111	C₂×C₈×C₆₇	—	2	2	no	—
112	C₂×C₈×C₆₇	—	2	2	no	—
113	C₂×C₈×C₆₇	—	2	2	no	—
114	C₂×C₈×C₆₇	—	4	8	no	—
115	C₄⋊C₄×C₆₇	—	2	2	no	—
116	C₄⋊C₄×C₆₇	—	2	4	no	—
117	C₄⋊C₄×C₆₇	—	4	2	no	—
118	C₂²⋊C₄×C₆₇	—	2	2	no	—
119	C₂²⋊C₄×C₆₇	—	2	4	no	—
120	C₂²⋊C₄×C₆₇	—	2	2	no	—
121	C₂²⋊C₄×C₆₇	—	2	4	no	—
122	C₂²⋊C₄×C₆₇	—	4	2	no	—
123	C₄²×C₆₇	—	2	2	no	—
124	C₄²×C₆₇	—	2	1	no	—
125	C₄²×C₆₇	—	2	2	no	—
126	C₄²×C₆₇	—	2	2	no	—
127	C₄²×C₆₇	—	2	2	no	—
128	C₄²×C₆₇	—	2	1	no	—
129	C₄²×C₆₇	—	4	2	no	—
130	C₂³×C₆₇	—	2	1	no	—
131	C₂³×C₆₇	—	2	1	no	—
132	C₂³×C₆₇	—	2	2	no	—
133	C₂³×C₆₇	—	2	1	no	—
134	C₂³×C₆₇	—	2	1	no	—
135	C₂³×C₆₇	—	2	1	no	—
136	C₂³×C₆₇	—	2	2	no	—
137	C₂³×C₆₇	—	2	2	no	—
138	C₂³×C₆₇	#117	1	2	yes	—
139	C₂³×C₆₇	#131	1	4	yes	—
140	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
141	D₄×C₆₇	—	4	4	no	—
142	C₂×C₄×C₆₇	—	2	2	no	—
143	C₂×C₄×C₆₇	—	2	1	no	—
144	C₂×C₄×C₆₇	—	2	1	no	—
145	C₂×C₄×C₆₇	—	2	2	no	—
146	C₂×C₄×C₆₇	—	2	4	no	—
147	C₂×C₄×C₆₇	—	2	4	no	—
148	C₂×C₄×C₆₇	—	2	1	no	—
149	C₂×C₄×C₆₇	—	2	1	no	—
150	C₂×C₄×C₆₇	—	2	1	no	—
151	C₂×C₄×C₆₇	—	2	2	no	—
152	C₂×C₄×C₆₇	—	2	2	no	—
153	C₂×C₄×C₆₇	—	2	1	no	—
154	C₂×C₄×C₆₇	—	2	2	no	—
155	C₂×C₄×C₆₇	—	2	1	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	8	no	—
165	C₂×C₄×C₆₇	—	4	2	no	—
166	C₂×C₄×C₆₇	—	4	2	no	—
167	C₂×C₄×C₆₇	#117	1	2	yes	—
168	C₂×C₄×C₆₇	#131	1	4	yes	—
169	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
170	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
171	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
172	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
173	C₈×C₆₇	—	4	2	no	—
174	C₈×C₆₇	—	4	2	no	—
175	C₈×C₆₇	—	4	2	no	—
176	C₈×C₆₇	—	4	2	no	—
177	C₂²×C₆₇	—	2	2	no	—
178	C₂²×C₆₇	—	2	2	no	—
179	C₂²×C₆₇	—	2	4	no	—
180	C₂²×C₆₇	—	2	4	no	—
181	C₂²×C₆₇	—	4	1	no	—
182	C₂²×C₆₇	—	4	1	no	—
183	C₂²×C₆₇	—	4	1	no	—
184	C₂²×C₆₇	—	4	2	no	—
185	C₂²×C₆₇	—	4	2	no	—
186	C₂²×C₆₇	#433	1	4	yes	—
187	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
188	C₂²×C₆₇	#518	1	1	yes	characteristic
189	C₂²×C₆₇	#520	1	1	yes	characteristic
190	C₂²×C₆₇	C₂×#131	1	2	yes	—
191	C₂²×C₆₇	C₂×#131	1	2	yes	—
192	C₄×C₆₇	—	2	2	no	—
193	C₄×C₆₇	—	2	4	no	—
194	C₄×C₆₇	—	4	1	no	—
195	C₄×C₆₇	—	4	1	no	—
196	C₄×C₆₇	—	4	1	no	—
197	C₄×C₆₇	—	4	2	no	—
198	C₄×C₆₇	—	4	2	no	—
199	C₄×C₆₇	—	4	2	no	—
200	C₄×C₆₇	#538	1	2	yes	—
201	C₄×C₆₇	#539	1	2	yes	—
202	#21997	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
203	C₂×C₆₇	—	2	4	no	—
204	C₂×C₆₇	—	8	1	no	—
205	C₂×C₆₇	#2318	1	1	yes	characteristic
206	C₂×C₆₇	C₂×#433	1	1	yes	characteristic
207	C₂×C₆₇	#2922	1	1	yes	characteristic
208	C₂×C₆₇	#2977	1	2	yes	—
209	C₂×C₆₇	#2994	1	2	yes	—
210	#2290	C₂×C₆₇	1	1	yes	characteristic
211	#1923	C₂×C₆₇	1	2	yes	complemented
212	#1916	C₂×C₆₇	1	2	yes	complemented
213	#1869	C₂×C₆₇	1	2	yes	—
214	#1850	C₂×C₆₇	1	2	yes	complemented
215	#1840	C₂×C₆₇	1	1	yes	characteristic, complemented
216	#1832	C₂×C₆₇	1	1	yes	characteristic, complemented
217	#1818	C₂×C₆₇	1	1	yes	characteristic, complemented
218	#1798	C₂×C₆₇	1	1	yes	characteristic, complemented
219	C₄×C₂²⋊C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
220	C₂×#336	C₂×C₆₇	1	1	yes	characteristic, complemented
221	C₆₇	#21997	1	1	yes	characteristic, direct product factor, Sylow
222	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
223	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
224	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
225	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
226	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
227	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
228	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
229	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
230	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
231	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
232	#388	C₄×C₆₇	1	2	yes	—
233	#387	C₄×C₆₇	1	2	yes	—
234	#386	C₄×C₆₇	1	2	yes	—
235	#385	C₄×C₆₇	1	2	yes	—
236	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
237	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
238	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
239	#376	C₂²×C₆₇	1	1	yes	characteristic
240	#336	C₂²×C₆₇	1	1	yes	characteristic
241	#336	C₂²×C₆₇	1	1	yes	characteristic
242	#336	C₂²×C₆₇	1	1	yes	characteristic
243	#336	C₂²×C₆₇	1	1	yes	characteristic
244	#336	C₂²×C₆₇	1	2	yes	—
245	#336	C₂²×C₆₇	1	2	yes	—
246	#336	C₂²×C₆₇	1	2	yes	—
247	#336	C₂²×C₆₇	1	2	yes	—
248	#336	C₂²×C₆₇	1	2	yes	—
249	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
250	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
251	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
252	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
253	C₂²×C₈	—	2	4	no	—
254	C₂²×C₈	—	4	2	no	—
255	C₂²×C₈	—	4	2	no	—
256	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
257	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
258	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
259	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
260	C₄×D₄	—	2	4	no	—
261	#117	—	4	2	no	—
262	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
263	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
264	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
265	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
266	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
267	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
268	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
269	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
270	C₂×C₄²	—	4	2	no	—
271	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
272	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
273	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
274	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
275	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
276	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
277	C₄⋊C₈	—	4	2	no	—
278	C₄⋊C₈	—	4	2	no	—
279	C₂²⋊C₈	—	2	4	no	—
280	C₂²⋊C₈	—	4	2	no	—
281	C₂²⋊C₈	—	4	2	no	—
282	C₄×C₈	—	4	2	no	—
283	#95	C₂×C₄×C₆₇	1	2	yes	—
284	#95	C₂×C₄×C₆₇	1	2	yes	—
285	#95	C₂×C₄×C₆₇	1	2	yes	—
286	#95	C₂×C₄×C₆₇	1	2	yes	—
287	#95	C₂×C₄×C₆₇	1	2	yes	—
288	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
289	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
290	C₂×D₄	—	2	2	no	—
291	C₂×D₄	—	2	4	no	—
292	C₂²×C₄	—	2	2	no	—
293	C₂²×C₄	—	2	1	no	—
294	C₂²×C₄	—	2	1	no	—
295	C₂²×C₄	—	2	4	no	—
296	C₂²×C₄	—	2	4	no	—
297	C₂²×C₄	—	2	2	no	—
298	C₂²×C₄	—	2	1	no	—
299	C₂²×C₄	—	2	2	no	—
300	C₂²×C₄	—	2	1	no	—
301	C₂²×C₄	—	4	2	no	—
302	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
303	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
304	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
305	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
306	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
307	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
308	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
309	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
310	C₂²×C₄	#41×C₆₇	1	4	yes	—
311	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	2	2	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	4	8	no	—
325	C₄⋊C₄	—	2	2	no	—
326	C₄⋊C₄	—	2	4	no	—
327	C₄⋊C₄	—	4	2	no	—
328	C₂²⋊C₄	—	2	2	no	—
329	C₂²⋊C₄	—	2	4	no	—
330	C₂²⋊C₄	—	2	2	no	—
331	C₂²⋊C₄	—	2	4	no	—
332	C₂²⋊C₄	—	4	2	no	—
333	C₄²	—	2	2	no	—
334	C₄²	—	2	2	no	—
335	C₄²	—	2	1	no	—
336	C₄²	—	2	2	no	—
337	C₄²	—	2	2	no	—
338	C₄²	—	2	1	no	—
339	C₄²	—	4	2	no	—
340	C₂³	—	2	1	no	—
341	C₂³	—	2	1	no	—
342	C₂³	—	2	2	no	—
343	C₂³	—	2	1	no	—
344	C₂³	—	2	1	no	—
345	C₂³	—	2	1	no	—
346	C₂³	—	2	2	no	—
347	C₂³	—	2	2	no	—
348	C₂³	#117×C₆₇	1	2	yes	—
349	C₂³	#131×C₆₇	1	4	yes	—
350	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
351	D₄	—	4	4	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	1	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	4	no	—
357	C₂×C₄	—	2	4	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	2	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	—	4	8	no	—
375	C₂×C₄	—	4	2	no	—
376	C₂×C₄	—	4	2	no	—
377	C₂×C₄	#117×C₆₇	1	2	yes	—
378	C₂×C₄	#131×C₆₇	1	4	yes	—
379	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
380	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
381	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
382	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
383	C₈	—	4	2	no	—
384	C₈	—	4	2	no	—
385	C₈	—	4	2	no	—
386	C₈	—	4	2	no	—
387	C₂²	—	2	2	no	—
388	C₂²	—	2	2	no	—
389	C₂²	—	2	4	no	—
390	C₂²	—	2	4	no	—
391	C₂²	—	4	1	no	—
392	C₂²	—	4	1	no	—
393	C₂²	—	4	1	no	—
394	C₂²	—	4	2	no	—
395	C₂²	—	4	2	no	—
396	C₂²	#433×C₆₇	1	4	yes	—
397	C₂²	C₂×#117×C₆₇	1	1	yes	characteristic
398	C₂²	#518×C₆₇	1	1	yes	characteristic
399	C₂²	#520×C₆₇	1	1	yes	characteristic
400	C₂²	C₂×#131×C₆₇	1	2	yes	—
401	C₂²	C₂×#131×C₆₇	1	2	yes	—
402	C₄	—	2	2	no	—
403	C₄	—	2	4	no	—
404	C₄	—	4	1	no	—
405	C₄	—	4	1	no	—
406	C₄	—	4	1	no	—
407	C₄	—	4	2	no	—
408	C₄	—	4	2	no	—
409	C₄	—	4	2	no	—
410	C₄	#538×C₆₇	1	2	yes	—
411	C₄	#539×C₆₇	1	2	yes	—
412	C₂	—	2	4	no	—
413	C₂	—	8	1	no	—
414	C₂	C₆₇×#2318	1	1	yes	characteristic
415	C₂	C₂×#433×C₆₇	1	1	yes	characteristic
416	C₂	C₆₇×#2922	1	1	yes	characteristic
417	C₂	C₆₇×#2977	1	2	yes	—
418	C₂	C₆₇×#2994	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₆₇×#1923	C₂	semidirect
3. C₆₇×#1916	C₂	semidirect
5. C₆₇×#1850	C₂	semidirect
6. C₆₇×#1840	C₂	semidirect
7. C₆₇×#1832	C₂	semidirect
8. C₆₇×#1818	C₂	semidirect
9. C₆₇×#1798	C₂	semidirect
10. C₄×C₂²⋊C₈×C₆₇	C₂	semidirect
11. C₂×#336×C₆₇	C₂	semidirect
202. #21997	C₆₇	direct (N × H)
211. #1923	C₂×C₆₇	semidirect
212. #1916	C₂×C₆₇	semidirect
214. #1850	C₂×C₆₇	semidirect
215. #1840	C₂×C₆₇	semidirect
216. #1832	C₂×C₆₇	semidirect
217. #1818	C₂×C₆₇	semidirect
218. #1798	C₂×C₆₇	semidirect
219. C₄×C₂²⋊C₈	C₂×C₆₇	semidirect
220. C₂×#336	C₂×C₆₇	semidirect
221. C₆₇	#21997	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×#2290	2
C₆₇×#1923	2
C₆₇×#1916	2
C₆₇×#1869	2
C₆₇×#1850	2
C₆₇×#1840	2
C₆₇×#1832	2
C₆₇×#1818	2
C₆₇×#1798	2
C₄×C₂²⋊C₈×C₆₇	2
C₂×#336×C₆₇	2
#21997	67

Subgroups — C67×#21997 #30701248

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21997 #30701248