Subgroups — C₆₇×#33019 #30712270

Order 17152.

1216 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#519×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#485×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#451×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2625	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1878	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1878	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1865	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1863	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1825	C₂	1	1	yes	maximal, characteristic
10	C₆₇×#1818	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#1787	C₂	1	1	yes	maximal, characteristic
12	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
14	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	#519×C₆₇	—	2	4	no	—
16	#519×C₆₇	—	2	4	no	—
17	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
18	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
19	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
21	C₂×#117×C₆₇	—	2	2	no	—
22	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
24	#485×C₆₇	—	2	4	no	—
25	#451×C₆₇	—	2	4	no	—
26	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#103×C₆₇	C₂²	1	2	yes	—
29	C₂×#103×C₆₇	C₂²	1	2	yes	—
30	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×#103×C₆₇	C₂²	1	2	yes	—
33	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
35	#400×C₆₇	—	2	1	no	—
36	#398×C₆₇	—	2	1	no	—
37	#397×C₆₇	—	2	2	no	—
38	#391×C₆₇	C₂²	1	2	yes	—
39	#389×C₆₇	C₂²	1	2	yes	—
40	#380×C₆₇	C₂²	1	1	yes	characteristic
41	#340×C₆₇	C₂²	1	2	yes	—
42	#340×C₆₇	C₂²	1	2	yes	—
43	#336×C₆₇	C₂²	1	2	yes	—
44	#336×C₆₇	C₂²	1	2	yes	—
45	C₂²×Q₈×C₆₇	—	2	4	no	—
46	C₂²×Q₈×C₆₇	—	2	4	no	—
47	C₂²×Q₈×C₆₇	—	2	1	no	—
48	C₂²×Q₈×C₆₇	—	2	2	no	—
49	C₂²×Q₈×C₆₇	D₄	1	4	yes	—
50	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
51	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
52	C₂³×C₄×C₆₇	—	2	1	no	—
53	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
54	C₂×Q₁₆×C₆₇	—	2	4	no	—
55	C₂×Q₁₆×C₆₇	—	2	4	no	—
56	C₂×Q₁₆×C₆₇	—	2	4	no	—
57	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
58	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
60	C₂²⋊Q₈×C₆₇	—	2	2	no	—
61	C₂²⋊Q₈×C₆₇	—	2	2	no	—
62	C₄×Q₈×C₆₇	—	2	8	no	—
63	C₄×Q₈×C₆₇	—	2	8	no	—
64	C₄×Q₈×C₆₇	—	2	4	no	—
65	C₄×Q₈×C₆₇	—	2	4	no	—
66	C₄×Q₈×C₆₇	—	2	4	no	—
67	C₄×Q₈×C₆₇	—	2	4	no	—
68	#117×C₆₇	—	2	16	no	—
69	#117×C₆₇	—	2	2	no	—
70	#117×C₆₇	—	2	2	no	—
71	#117×C₆₇	—	2	2	no	—
72	#117×C₆₇	D₄	1	4	yes	—
73	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
74	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
75	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
76	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
77	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
81	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
83	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
84	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
85	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
86	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
87	C₂×C₄²×C₆₇	—	2	2	no	—
88	C₂×C₄²×C₆₇	—	2	2	no	—
89	C₂×C₄²×C₆₇	—	2	1	no	—
90	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
91	#107×C₆₇	—	2	2	no	—
92	#106×C₆₇	—	2	2	no	—
93	#103×C₆₇	—	2	4	no	—
94	#103×C₆₇	—	2	4	no	—
95	#103×C₆₇	—	2	4	no	—
96	#103×C₆₇	—	2	2	no	—
97	#103×C₆₇	—	2	2	no	—
98	C₂²⋊C₈×C₆₇	—	2	2	no	—
99	C₂²⋊C₈×C₆₇	—	2	2	no	—
100	#95×C₆₇	—	2	1	no	—
101	#95×C₆₇	—	2	2	no	—
102	#95×C₆₇	—	2	2	no	—
103	#95×C₆₇	—	2	2	no	—
104	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
105	C₂×Q₈×C₆₇	—	2	8	no	—
106	C₂×Q₈×C₆₇	—	2	8	no	—
107	C₂×Q₈×C₆₇	—	2	8	no	—
108	C₂×Q₈×C₆₇	—	2	16	no	—
109	C₂×Q₈×C₆₇	—	2	4	no	—
110	C₂×Q₈×C₆₇	—	2	2	no	—
111	C₂×Q₈×C₆₇	—	2	2	no	—
112	C₂×Q₈×C₆₇	—	2	2	no	—
113	C₂×Q₈×C₆₇	—	2	1	no	—
114	C₂×Q₈×C₆₇	—	2	1	no	—
115	C₂×Q₈×C₆₇	—	2	2	no	—
116	C₂×Q₈×C₆₇	—	2	2	no	—
117	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
118	C₂×Q₈×C₆₇	#41	1	4	yes	—
119	C₂×Q₈×C₆₇	#41	1	4	yes	—
120	C₂²×C₄×C₆₇	—	2	8	no	—
121	C₂²×C₄×C₆₇	—	2	4	no	—
122	C₂²×C₄×C₆₇	—	2	2	no	—
123	C₂²×C₄×C₆₇	—	2	2	no	—
124	C₂²×C₄×C₆₇	—	2	2	no	—
125	C₂²×C₄×C₆₇	—	2	1	no	—
126	C₂²×C₄×C₆₇	—	2	1	no	—
127	C₂²×C₄×C₆₇	—	2	1	no	—
128	C₂²×C₄×C₆₇	—	2	2	no	—
129	C₂²×C₄×C₆₇	—	2	1	no	—
130	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
131	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
132	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
134	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
135	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
136	Q₁₆×C₆₇	—	4	8	no	—
137	C₂×C₈×C₆₇	—	2	1	no	—
138	C₂×C₈×C₆₇	—	2	1	no	—
139	C₂×C₈×C₆₇	—	2	2	no	—
140	C₂×C₈×C₆₇	—	2	2	no	—
141	C₂×C₈×C₆₇	—	2	2	no	—
142	C₂×C₈×C₆₇	—	2	2	no	—
143	C₂×C₈×C₆₇	—	2	2	no	—
144	C₄⋊C₄×C₆₇	—	2	8	no	—
145	C₄⋊C₄×C₆₇	—	2	8	no	—
146	C₄⋊C₄×C₆₇	—	2	2	no	—
147	C₄⋊C₄×C₆₇	—	2	2	no	—
148	C₄⋊C₄×C₆₇	—	2	1	no	—
149	C₄⋊C₄×C₆₇	—	2	1	no	—
150	C₄⋊C₄×C₆₇	—	2	1	no	—
151	C₄⋊C₄×C₆₇	—	2	1	no	—
152	C₄⋊C₄×C₆₇	—	2	2	no	—
153	C₄⋊C₄×C₆₇	—	2	2	no	—
154	C₄⋊C₄×C₆₇	—	4	2	no	—
155	C₄⋊C₄×C₆₇	—	8	2	no	—
156	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
157	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
158	C₂²⋊C₄×C₆₇	—	2	4	no	—
159	C₂²⋊C₄×C₆₇	—	2	4	no	—
160	C₂²⋊C₄×C₆₇	—	4	2	no	—
161	C₂²⋊C₄×C₆₇	—	8	1	no	—
162	C₄²×C₆₇	—	2	8	no	—
163	C₄²×C₆₇	—	2	8	no	—
164	C₄²×C₆₇	—	2	4	no	—
165	C₄²×C₆₇	—	2	2	no	—
166	C₄²×C₆₇	—	2	2	no	—
167	C₂³×C₆₇	—	2	2	no	—
168	C₂³×C₆₇	—	2	1	no	—
169	C₂³×C₆₇	—	2	2	no	—
170	C₂³×C₆₇	—	2	1	no	—
171	C₂³×C₆₇	#120	1	2	yes	—
172	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
173	Q₈×C₆₇	—	2	16	no	—
174	Q₈×C₆₇	—	2	4	no	—
175	Q₈×C₆₇	—	2	4	no	—
176	Q₈×C₆₇	—	2	4	no	—
177	Q₈×C₆₇	—	4	2	no	—
178	Q₈×C₆₇	—	4	1	no	—
179	Q₈×C₆₇	—	4	1	no	—
180	Q₈×C₆₇	#123	1	8	yes	—
181	C₂×C₄×C₆₇	—	2	16	no	—
182	C₂×C₄×C₆₇	—	2	4	no	—
183	C₂×C₄×C₆₇	—	2	4	no	—
184	C₂×C₄×C₆₇	—	2	4	no	—
185	C₂×C₄×C₆₇	—	2	2	no	—
186	C₂×C₄×C₆₇	—	2	4	no	—
187	C₂×C₄×C₆₇	—	2	2	no	—
188	C₂×C₄×C₆₇	—	2	2	no	—
189	C₂×C₄×C₆₇	—	2	2	no	—
190	C₂×C₄×C₆₇	—	2	2	no	—
191	C₂×C₄×C₆₇	—	2	2	no	—
192	C₂×C₄×C₆₇	—	2	2	no	—
193	C₂×C₄×C₆₇	—	2	2	no	—
194	C₂×C₄×C₆₇	—	2	4	no	—
195	C₂×C₄×C₆₇	—	2	2	no	—
196	C₂×C₄×C₆₇	—	4	1	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	2	no	—
203	C₂×C₄×C₆₇	—	4	2	no	—
204	C₂×C₄×C₆₇	—	4	1	no	—
205	C₂×C₄×C₆₇	—	4	2	no	—
206	C₂×C₄×C₆₇	—	4	1	no	—
207	C₂×C₄×C₆₇	#120	1	2	yes	—
208	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
209	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
210	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
211	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
212	C₈×C₆₇	—	4	2	no	—
213	C₈×C₆₇	—	4	1	no	—
214	C₈×C₆₇	—	4	1	no	—
215	C₂²×C₆₇	—	2	2	no	—
216	C₂²×C₆₇	—	2	2	no	—
217	C₂²×C₆₇	—	2	2	no	—
218	C₂²×C₆₇	—	4	2	no	—
219	C₂²×C₆₇	—	4	2	no	—
220	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
221	C₂²×C₆₇	#525	1	2	yes	—
222	C₂²×C₆₇	#577	1	1	yes	characteristic
223	C₂²×C₆₇	#578	1	2	yes	—
224	C₂²×C₆₇	#578	1	1	yes	characteristic
225	C₄×C₆₇	—	2	8	no	—
226	C₄×C₆₇	—	2	2	no	—
227	C₄×C₆₇	—	4	2	no	—
228	C₄×C₆₇	—	4	2	no	—
229	C₄×C₆₇	—	4	2	no	—
230	C₄×C₆₇	—	4	2	no	—
231	C₄×C₆₇	—	8	1	no	—
232	C₄×C₆₇	—	8	2	no	—
233	C₄×C₆₇	—	8	1	no	—
234	C₄×C₆₇	C₂×#123	1	2	yes	—
235	C₄×C₆₇	#525	1	1	yes	characteristic
236	C₄×C₆₇	#525	1	1	yes	characteristic
237	#33019	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
238	C₂×C₆₇	—	4	2	no	—
239	C₂×C₆₇	#2849	1	1	yes	characteristic
240	C₂×C₆₇	#3011	1	1	yes	characteristic
241	C₂×C₆₇	#3013	1	1	yes	characteristic
242	C₂×C₆₇	#3121	1	2	yes	—
243	C₂×C₆₇	#3127	1	2	yes	—
244	C₂×#519	C₂×C₆₇	1	1	yes	characteristic
245	C₂×#485	C₂×C₆₇	1	1	yes	characteristic
246	C₂×#451	C₂×C₆₇	1	1	yes	characteristic
247	#2625	C₂×C₆₇	1	1	yes	characteristic
248	#1878	C₂×C₆₇	1	2	yes	complemented
249	#1878	C₂×C₆₇	1	2	yes	complemented
250	#1865	C₂×C₆₇	1	2	yes	complemented
251	#1863	C₂×C₆₇	1	2	yes	complemented
252	#1825	C₂×C₆₇	1	1	yes	characteristic
253	#1818	C₂×C₆₇	1	1	yes	characteristic
254	#1787	C₂×C₆₇	1	1	yes	characteristic
255	C₆₇	#33019	1	1	yes	characteristic, direct product factor, Sylow
256	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
257	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	—
258	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
259	#519	—	2	4	no	—
260	#519	—	2	4	no	—
261	C₂×C₄×Q₈	—	2	1	no	—
262	C₂×C₄×Q₈	—	2	1	no	—
263	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
264	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
265	C₂×#117	—	2	2	no	—
266	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
267	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
268	#485	—	2	4	no	—
269	#451	—	2	4	no	—
270	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
271	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
272	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
273	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
274	C₂×#103	C₂²×C₆₇	1	2	yes	—
275	C₂×#103	C₂²×C₆₇	1	2	yes	—
276	C₂×#103	C₂²×C₆₇	1	2	yes	—
277	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
278	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
279	#400	—	2	1	no	—
280	#398	—	2	1	no	—
281	#397	—	2	2	no	—
282	#391	C₂²×C₆₇	1	2	yes	—
283	#389	C₂²×C₆₇	1	2	yes	—
284	#380	C₂²×C₆₇	1	1	yes	characteristic
285	#340	C₂²×C₆₇	1	2	yes	—
286	#340	C₂²×C₆₇	1	2	yes	—
287	#336	C₂²×C₆₇	1	2	yes	—
288	#336	C₂²×C₆₇	1	2	yes	—
289	C₂²×Q₈	—	2	4	no	—
290	C₂²×Q₈	—	2	4	no	—
291	C₂²×Q₈	—	2	1	no	—
292	C₂²×Q₈	—	2	2	no	—
293	C₂²×Q₈	D₄×C₆₇	1	4	yes	—
294	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
295	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
296	C₂³×C₄	—	2	1	no	—
297	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
298	C₂×Q₁₆	—	2	4	no	—
299	C₂×Q₁₆	—	2	4	no	—
300	C₂×Q₁₆	—	2	4	no	—
301	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
302	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
303	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
304	C₂²⋊Q₈	—	2	2	no	—
305	C₂²⋊Q₈	—	2	2	no	—
306	C₄×Q₈	—	2	8	no	—
307	C₄×Q₈	—	2	8	no	—
308	C₄×Q₈	—	2	4	no	—
309	C₄×Q₈	—	2	4	no	—
310	C₄×Q₈	—	2	4	no	—
311	C₄×Q₈	—	2	4	no	—
312	#117	—	2	16	no	—
313	#117	—	2	2	no	—
314	#117	—	2	2	no	—
315	#117	—	2	2	no	—
316	#117	D₄×C₆₇	1	4	yes	—
317	C₂×C₄⋊C₄	—	2	2	no	—
318	C₂×C₄⋊C₄	—	2	2	no	—
319	C₂×C₄⋊C₄	—	2	1	no	—
320	C₂×C₄⋊C₄	—	2	2	no	—
321	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
322	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
323	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
324	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
325	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
326	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
327	C₂×C₂²⋊C₄	—	2	1	no	—
328	C₂×C₂²⋊C₄	—	2	1	no	—
329	C₂×C₂²⋊C₄	—	2	1	no	—
330	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
331	C₂×C₄²	—	2	2	no	—
332	C₂×C₄²	—	2	2	no	—
333	C₂×C₄²	—	2	1	no	—
334	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
335	#107	—	2	2	no	—
336	#106	—	2	2	no	—
337	#103	—	2	4	no	—
338	#103	—	2	4	no	—
339	#103	—	2	4	no	—
340	#103	—	2	2	no	—
341	#103	—	2	2	no	—
342	C₂²⋊C₈	—	2	2	no	—
343	C₂²⋊C₈	—	2	2	no	—
344	#95	—	2	1	no	—
345	#95	—	2	2	no	—
346	#95	—	2	2	no	—
347	#95	—	2	2	no	—
348	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
349	C₂×Q₈	—	2	8	no	—
350	C₂×Q₈	—	2	8	no	—
351	C₂×Q₈	—	2	8	no	—
352	C₂×Q₈	—	2	16	no	—
353	C₂×Q₈	—	2	4	no	—
354	C₂×Q₈	—	2	2	no	—
355	C₂×Q₈	—	2	2	no	—
356	C₂×Q₈	—	2	2	no	—
357	C₂×Q₈	—	2	1	no	—
358	C₂×Q₈	—	2	1	no	—
359	C₂×Q₈	—	2	2	no	—
360	C₂×Q₈	—	2	2	no	—
361	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
362	C₂×Q₈	#41×C₆₇	1	4	yes	—
363	C₂×Q₈	#41×C₆₇	1	4	yes	—
364	C₂²×C₄	—	2	8	no	—
365	C₂²×C₄	—	2	4	no	—
366	C₂²×C₄	—	2	2	no	—
367	C₂²×C₄	—	2	2	no	—
368	C₂²×C₄	—	2	2	no	—
369	C₂²×C₄	—	2	1	no	—
370	C₂²×C₄	—	2	1	no	—
371	C₂²×C₄	—	2	2	no	—
372	C₂²×C₄	—	2	1	no	—
373	C₂²×C₄	—	2	1	no	—
374	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
375	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
376	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
377	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
378	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
379	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
380	Q₁₆	—	4	8	no	—
381	C₂×C₈	—	2	1	no	—
382	C₂×C₈	—	2	1	no	—
383	C₂×C₈	—	2	2	no	—
384	C₂×C₈	—	2	2	no	—
385	C₂×C₈	—	2	2	no	—
386	C₂×C₈	—	2	2	no	—
387	C₂×C₈	—	2	2	no	—
388	C₄⋊C₄	—	2	8	no	—
389	C₄⋊C₄	—	2	8	no	—
390	C₄⋊C₄	—	2	2	no	—
391	C₄⋊C₄	—	2	2	no	—
392	C₄⋊C₄	—	2	1	no	—
393	C₄⋊C₄	—	2	1	no	—
394	C₄⋊C₄	—	2	1	no	—
395	C₄⋊C₄	—	2	1	no	—
396	C₄⋊C₄	—	2	2	no	—
397	C₄⋊C₄	—	2	2	no	—
398	C₄⋊C₄	—	4	2	no	—
399	C₄⋊C₄	—	8	2	no	—
400	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
401	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
402	C₂²⋊C₄	—	2	4	no	—
403	C₂²⋊C₄	—	2	4	no	—
404	C₂²⋊C₄	—	4	2	no	—
405	C₂²⋊C₄	—	8	1	no	—
406	C₄²	—	2	8	no	—
407	C₄²	—	2	8	no	—
408	C₄²	—	2	4	no	—
409	C₄²	—	2	2	no	—
410	C₄²	—	2	2	no	—
411	C₂³	—	2	2	no	—
412	C₂³	—	2	1	no	—
413	C₂³	—	2	2	no	—
414	C₂³	—	2	1	no	—
415	C₂³	#120×C₆₇	1	2	yes	—
416	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
417	Q₈	—	2	16	no	—
418	Q₈	—	2	4	no	—
419	Q₈	—	2	4	no	—
420	Q₈	—	2	4	no	—
421	Q₈	—	4	2	no	—
422	Q₈	—	4	1	no	—
423	Q₈	—	4	1	no	—
424	Q₈	#123×C₆₇	1	8	yes	—
425	C₂×C₄	—	2	16	no	—
426	C₂×C₄	—	2	4	no	—
427	C₂×C₄	—	2	4	no	—
428	C₂×C₄	—	2	4	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	4	no	—
431	C₂×C₄	—	2	2	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	4	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	4	1	no	—
441	C₂×C₄	—	4	2	no	—
442	C₂×C₄	—	4	1	no	—
443	C₂×C₄	—	4	1	no	—
444	C₂×C₄	—	4	1	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	4	2	no	—
447	C₂×C₄	—	4	2	no	—
448	C₂×C₄	—	4	1	no	—
449	C₂×C₄	—	4	1	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	#120×C₆₇	1	2	yes	—
452	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
453	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
454	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
455	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
456	C₈	—	4	2	no	—
457	C₈	—	4	1	no	—
458	C₈	—	4	1	no	—
459	C₂²	—	2	2	no	—
460	C₂²	—	2	2	no	—
461	C₂²	—	2	2	no	—
462	C₂²	—	4	2	no	—
463	C₂²	—	4	2	no	—
464	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
465	C₂²	#525×C₆₇	1	2	yes	—
466	C₂²	#577×C₆₇	1	1	yes	characteristic
467	C₂²	#578×C₆₇	1	2	yes	—
468	C₂²	#578×C₆₇	1	1	yes	characteristic
469	C₄	—	2	8	no	—
470	C₄	—	2	2	no	—
471	C₄	—	4	2	no	—
472	C₄	—	4	2	no	—
473	C₄	—	4	2	no	—
474	C₄	—	4	2	no	—
475	C₄	—	8	1	no	—
476	C₄	—	8	1	no	—
477	C₄	—	8	2	no	—
478	C₄	C₂×#123×C₆₇	1	2	yes	—
479	C₄	#525×C₆₇	1	1	yes	characteristic
480	C₄	#525×C₆₇	1	1	yes	characteristic
481	C₂	—	4	2	no	—
482	C₂	C₆₇×#2849	1	1	yes	characteristic
483	C₂	C₆₇×#3011	1	1	yes	characteristic
484	C₂	C₆₇×#3013	1	1	yes	characteristic
485	C₂	C₆₇×#3121	1	2	yes	—
486	C₂	C₆₇×#3127	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
5. C₆₇×#1878	C₂	semidirect
6. C₆₇×#1878	C₂	semidirect
7. C₆₇×#1865	C₂	semidirect
8. C₆₇×#1863	C₂	semidirect
237. #33019	C₆₇	direct (N × H)
248. #1878	C₂×C₆₇	semidirect
249. #1878	C₂×C₆₇	semidirect
250. #1865	C₂×C₆₇	semidirect
251. #1863	C₂×C₆₇	semidirect
255. C₆₇	#33019	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×#519×C₆₇	2
C₂×#485×C₆₇	2
C₂×#451×C₆₇	2
C₆₇×#2625	2
C₆₇×#1878	2
C₆₇×#1865	2
C₆₇×#1863	2
C₆₇×#1825	2
C₆₇×#1818	2
C₆₇×#1787	2
#33019	67

Subgroups — C67×#33019 #30712270

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33019 #30712270