Subgroups — C₆₇×#33181 #30712432

Order 17152.

1110 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#495×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#470×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#469×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#465×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2620	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1972	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1971	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1968	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1959	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1844	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
14	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#124×C₆₇	C₂²	1	2	yes	—
18	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
19	C₂×#121×C₆₇	C₂²	1	2	yes	—
20	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
21	#495×C₆₇	—	2	4	no	—
22	#470×C₆₇	—	2	4	no	—
23	#469×C₆₇	—	2	4	no	—
24	#465×C₆₇	—	2	8	no	—
25	C₂×#106×C₆₇	C₂²	1	2	yes	—
26	C₂×#106×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×#102×C₆₇	C₂²	1	2	yes	—
31	C₂×#102×C₆₇	C₂²	1	2	yes	—
32	#405×C₆₇	—	2	4	no	—
33	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
37	#396×C₆₇	—	2	4	no	—
38	#394×C₆₇	—	2	4	no	—
39	#391×C₆₇	C₂²	1	2	yes	—
40	#390×C₆₇	C₂²	1	2	yes	—
41	#384×C₆₇	C₂²	1	2	yes	—
42	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
43	C₂²×D₄×C₆₇	—	2	2	no	—
44	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
45	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
46	C₂×Q₁₆×C₆₇	—	2	2	no	—
47	C₂×Q₁₆×C₆₇	—	2	2	no	—
48	C₂×Q₁₆×C₆₇	—	2	2	no	—
49	C₂×SD₁₆×C₆₇	—	2	4	no	—
50	C₂×SD₁₆×C₆₇	—	2	4	no	—
51	C₂×SD₁₆×C₆₇	—	2	4	no	—
52	C₂×D₈×C₆₇	—	2	2	no	—
53	C₂×D₈×C₆₇	—	2	2	no	—
54	C₂×D₈×C₆₇	—	2	2	no	—
55	C₂×M₁₆×C₆₇	—	2	2	no	—
56	C₂×M₁₆×C₆₇	—	2	2	no	—
57	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
58	C₂²×C₈×C₆₇	—	2	2	no	—
59	C₂²×C₈×C₆₇	—	2	2	no	—
60	C₂²×C₈×C₆₇	D₄	1	4	yes	—
61	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
65	#124×C₆₇	—	2	4	no	—
66	#124×C₆₇	—	2	4	no	—
67	C₂²⋊Q₈×C₆₇	—	2	4	no	—
68	C₂²⋊Q₈×C₆₇	—	2	4	no	—
69	#121×C₆₇	—	2	4	no	—
70	#121×C₆₇	—	2	4	no	—
71	#117×C₆₇	—	2	2	no	—
72	#117×C₆₇	—	2	2	no	—
73	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
74	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
75	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
76	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
77	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
78	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
79	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
80	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
81	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
82	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
83	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
84	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
85	#106×C₆₇	—	2	4	no	—
86	#106×C₆₇	—	2	2	no	—
87	#106×C₆₇	—	2	2	no	—
88	#103×C₆₇	—	2	4	no	—
89	#103×C₆₇	—	2	4	no	—
90	#102×C₆₇	—	2	4	no	—
91	#102×C₆₇	—	2	4	no	—
92	#97×C₆₇	—	2	2	no	—
93	#97×C₆₇	—	2	2	no	—
94	C₄×C₈×C₆₇	—	2	2	no	—
95	C₄×C₈×C₆₇	D₄	1	4	yes	complemented
96	#95×C₆₇	—	2	4	no	—
97	C₂⁴×C₆₇	—	2	2	no	—
98	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
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	2	no	—
104	C₂×Q₈×C₆₇	—	2	2	no	—
105	C₂×D₄×C₆₇	—	2	2	no	—
106	C₂×D₄×C₆₇	—	2	2	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	—	4	4	no	—
112	C₂×D₄×C₆₇	—	8	4	no	—
113	C₂²×C₄×C₆₇	—	2	1	no	—
114	C₂²×C₄×C₆₇	—	2	1	no	—
115	C₂²×C₄×C₆₇	—	2	1	no	—
116	C₂²×C₄×C₆₇	—	2	1	no	—
117	C₂²×C₄×C₆₇	—	2	2	no	—
118	C₂²×C₄×C₆₇	—	2	2	no	—
119	C₂²×C₄×C₆₇	—	2	2	no	—
120	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
121	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
122	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
124	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
125	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
126	Q₁₆×C₆₇	—	4	4	no	—
127	SD₁₆×C₆₇	—	4	8	no	—
128	D₈×C₆₇	—	4	4	no	—
129	M₁₆×C₆₇	—	2	8	no	—
130	C₂×C₈×C₆₇	—	2	1	no	—
131	C₂×C₈×C₆₇	—	2	1	no	—
132	C₂×C₈×C₆₇	—	2	1	no	—
133	C₂×C₈×C₆₇	—	2	1	no	—
134	C₂×C₈×C₆₇	—	2	8	no	—
135	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
136	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
137	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
138	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
139	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
140	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
141	C₂×C₈×C₆₇	#41	1	2	yes	—
142	C₂×C₈×C₆₇	#41	1	2	yes	—
143	C₄⋊C₄×C₆₇	—	2	4	no	—
144	C₄⋊C₄×C₆₇	—	2	2	no	—
145	C₄⋊C₄×C₆₇	—	2	2	no	—
146	C₄⋊C₄×C₆₇	—	2	2	no	—
147	C₄⋊C₄×C₆₇	—	2	2	no	—
148	C₄⋊C₄×C₆₇	—	4	4	no	—
149	C₄⋊C₄×C₆₇	—	8	4	no	—
150	C₂²⋊C₄×C₆₇	—	2	4	no	—
151	C₂²⋊C₄×C₆₇	—	4	4	no	—
152	C₂²⋊C₄×C₆₇	—	4	8	no	—
153	C₂²⋊C₄×C₆₇	—	4	4	no	—
154	C₂²⋊C₄×C₆₇	—	8	4	no	—
155	C₄²×C₆₇	—	2	1	no	—
156	C₄²×C₆₇	—	2	1	no	—
157	C₄²×C₆₇	C₂×D₄	1	2	yes	—
158	C₄²×C₆₇	#41	1	2	yes	—
159	C₂³×C₆₇	—	2	1	no	—
160	C₂³×C₆₇	—	2	1	no	—
161	C₂³×C₆₇	—	2	4	no	—
162	C₂³×C₆₇	—	4	2	no	—
163	C₂³×C₆₇	—	4	2	no	—
164	C₂³×C₆₇	—	4	2	no	—
165	C₂³×C₆₇	—	4	4	no	—
166	C₂³×C₆₇	—	4	4	no	—
167	C₂³×C₆₇	#127	1	2	yes	—
168	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
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	D₄×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	—	8	8	no	—
178	C₂×C₄×C₆₇	—	2	1	no	—
179	C₂×C₄×C₆₇	—	2	1	no	—
180	C₂×C₄×C₆₇	—	2	4	no	—
181	C₂×C₄×C₆₇	—	2	2	no	—
182	C₂×C₄×C₆₇	—	2	2	no	—
183	C₂×C₄×C₆₇	—	2	2	no	—
184	C₂×C₄×C₆₇	—	2	2	no	—
185	C₂×C₄×C₆₇	—	2	2	no	—
186	C₂×C₄×C₆₇	—	2	4	no	—
187	C₂×C₄×C₆₇	—	4	2	no	—
188	C₂×C₄×C₆₇	—	4	2	no	—
189	C₂×C₄×C₆₇	—	4	2	no	—
190	C₂×C₄×C₆₇	—	4	2	no	—
191	C₂×C₄×C₆₇	—	4	2	no	—
192	C₂×C₄×C₆₇	—	4	4	no	—
193	C₂×C₄×C₆₇	—	4	4	no	—
194	C₂×C₄×C₆₇	#121	1	4	yes	—
195	C₂×C₄×C₆₇	#127	1	2	yes	—
196	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
197	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
198	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
199	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
200	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
201	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
202	C₈×C₆₇	—	2	2	no	—
203	C₈×C₆₇	—	2	2	no	—
204	C₈×C₆₇	#121	1	4	yes	—
205	C₈×C₆₇	#121	1	4	yes	—
206	C₂²×C₆₇	—	2	2	no	—
207	C₂²×C₆₇	—	2	4	no	—
208	C₂²×C₆₇	—	4	2	no	—
209	C₂²×C₆₇	—	4	2	no	—
210	C₂²×C₆₇	—	8	2	no	—
211	C₂²×C₆₇	—	8	2	no	—
212	C₂²×C₆₇	—	8	2	no	—
213	C₂²×C₆₇	—	8	4	no	—
214	C₂²×C₆₇	—	8	4	no	—
215	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
216	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
217	C₂²×C₆₇	#532	1	1	yes	characteristic
218	C₂²×C₆₇	#577	1	4	yes	—
219	C₄×C₆₇	—	2	2	no	—
220	C₄×C₆₇	—	2	4	no	—
221	C₄×C₆₇	—	4	2	no	—
222	C₄×C₆₇	—	8	2	no	—
223	C₄×C₆₇	—	8	2	no	—
224	C₄×C₆₇	—	8	2	no	—
225	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
226	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
227	C₄×C₆₇	#532	1	1	yes	characteristic
228	C₄×C₆₇	#532	1	1	yes	characteristic
229	#33181	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
230	C₂×C₆₇	—	4	2	no	—
231	C₂×C₆₇	—	16	2	no	—
232	C₂×C₆₇	#2856	1	1	yes	characteristic
233	C₂×C₆₇	#3060	1	2	yes	—
234	C₂×C₆₇	#3154	1	2	yes	—
235	C₂×C₆₇	#3173	1	2	yes	—
236	C₂×#495	C₂×C₆₇	1	1	yes	characteristic, complemented
237	C₂×#470	C₂×C₆₇	1	1	yes	characteristic, complemented
238	C₂×#469	C₂×C₆₇	1	1	yes	characteristic
239	C₂×#465	C₂×C₆₇	1	2	yes	complemented
240	C₂×#405	C₂×C₆₇	1	1	yes	characteristic, complemented
241	#2620	C₂×C₆₇	1	2	yes	complemented
242	#1972	C₂×C₆₇	1	1	yes	characteristic, complemented
243	#1971	C₂×C₆₇	1	1	yes	characteristic, complemented
244	#1968	C₂×C₆₇	1	2	yes	complemented
245	#1959	C₂×C₆₇	1	2	yes	complemented
246	#1844	C₂×C₆₇	1	1	yes	characteristic, complemented
247	C₆₇	#33181	1	1	yes	characteristic, direct product factor, Sylow
248	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
249	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
250	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
251	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
252	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
253	C₂×#124	C₂²×C₆₇	1	2	yes	—
254	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
255	C₂×#121	C₂²×C₆₇	1	2	yes	—
256	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
257	#495	—	2	4	no	—
258	#470	—	2	4	no	—
259	#469	—	2	4	no	—
260	#465	—	2	8	no	—
261	C₂×#106	C₂²×C₆₇	1	2	yes	—
262	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
263	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
264	C₂×#103	C₂²×C₆₇	1	2	yes	—
265	C₂×#103	C₂²×C₆₇	1	2	yes	—
266	C₂×#102	C₂²×C₆₇	1	2	yes	—
267	C₂×#102	C₂²×C₆₇	1	2	yes	—
268	#405	—	2	4	no	—
269	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
270	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
271	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
272	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
273	#396	—	2	4	no	—
274	#394	—	2	4	no	—
275	#391	C₂²×C₆₇	1	2	yes	—
276	#390	C₂²×C₆₇	1	2	yes	—
277	#384	C₂²×C₆₇	1	2	yes	—
278	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
279	C₂²×D₄	—	2	2	no	—
280	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
281	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
282	C₂×Q₁₆	—	2	2	no	—
283	C₂×Q₁₆	—	2	2	no	—
284	C₂×Q₁₆	—	2	2	no	—
285	C₂×SD₁₆	—	2	4	no	—
286	C₂×SD₁₆	—	2	4	no	—
287	C₂×SD₁₆	—	2	4	no	—
288	C₂×D₈	—	2	2	no	—
289	C₂×D₈	—	2	2	no	—
290	C₂×D₈	—	2	2	no	—
291	C₂×M₁₆	—	2	2	no	—
292	C₂×M₁₆	—	2	2	no	—
293	C₂×M₁₆	D₄×C₆₇	1	4	yes	—
294	C₂²×C₈	—	2	2	no	—
295	C₂²×C₈	—	2	2	no	—
296	C₂²×C₈	D₄×C₆₇	1	4	yes	—
297	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
298	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
299	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
300	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
301	#124	—	2	4	no	—
302	#124	—	2	4	no	—
303	C₂²⋊Q₈	—	2	4	no	—
304	C₂²⋊Q₈	—	2	4	no	—
305	#121	—	2	4	no	—
306	#121	—	2	4	no	—
307	#117	—	2	2	no	—
308	#117	—	2	2	no	—
309	C₂×C₄⋊C₄	—	2	2	no	—
310	C₂×C₄⋊C₄	—	2	4	no	—
311	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
312	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
313	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
314	C₂×C₂²⋊C₄	—	2	2	no	—
315	C₂×C₂²⋊C₄	—	2	2	no	—
316	C₂×C₂²⋊C₄	—	2	4	no	—
317	C₂×C₂²⋊C₄	—	2	4	no	—
318	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
319	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
320	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
321	#106	—	2	4	no	—
322	#106	—	2	2	no	—
323	#106	—	2	2	no	—
324	#103	—	2	4	no	—
325	#103	—	2	4	no	—
326	#102	—	2	4	no	—
327	#102	—	2	4	no	—
328	#97	—	2	2	no	—
329	#97	—	2	2	no	—
330	C₄×C₈	—	2	2	no	—
331	C₄×C₈	D₄×C₆₇	1	4	yes	complemented
332	#95	—	2	4	no	—
333	C₂⁴	—	2	2	no	—
334	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
335	C₂×Q₈	—	2	2	no	—
336	C₂×Q₈	—	2	2	no	—
337	C₂×Q₈	—	2	2	no	—
338	C₂×Q₈	—	2	2	no	—
339	C₂×Q₈	—	2	2	no	—
340	C₂×Q₈	—	2	2	no	—
341	C₂×D₄	—	2	2	no	—
342	C₂×D₄	—	2	2	no	—
343	C₂×D₄	—	2	2	no	—
344	C₂×D₄	—	2	2	no	—
345	C₂×D₄	—	2	2	no	—
346	C₂×D₄	—	2	2	no	—
347	C₂×D₄	—	4	4	no	—
348	C₂×D₄	—	8	4	no	—
349	C₂²×C₄	—	2	1	no	—
350	C₂²×C₄	—	2	1	no	—
351	C₂²×C₄	—	2	1	no	—
352	C₂²×C₄	—	2	1	no	—
353	C₂²×C₄	—	2	2	no	—
354	C₂²×C₄	—	2	2	no	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
357	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
358	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
359	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
360	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
361	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
362	Q₁₆	—	4	4	no	—
363	SD₁₆	—	4	8	no	—
364	D₈	—	4	4	no	—
365	M₁₆	—	2	8	no	—
366	C₂×C₈	—	2	1	no	—
367	C₂×C₈	—	2	1	no	—
368	C₂×C₈	—	2	1	no	—
369	C₂×C₈	—	2	1	no	—
370	C₂×C₈	—	2	8	no	—
371	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
372	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
373	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
374	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
375	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
376	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
377	C₂×C₈	#41×C₆₇	1	2	yes	—
378	C₂×C₈	#41×C₆₇	1	2	yes	—
379	C₄⋊C₄	—	2	4	no	—
380	C₄⋊C₄	—	2	2	no	—
381	C₄⋊C₄	—	2	2	no	—
382	C₄⋊C₄	—	2	2	no	—
383	C₄⋊C₄	—	2	2	no	—
384	C₄⋊C₄	—	4	4	no	—
385	C₄⋊C₄	—	8	4	no	—
386	C₂²⋊C₄	—	2	4	no	—
387	C₂²⋊C₄	—	4	4	no	—
388	C₂²⋊C₄	—	4	8	no	—
389	C₂²⋊C₄	—	4	4	no	—
390	C₂²⋊C₄	—	8	4	no	—
391	C₄²	—	2	1	no	—
392	C₄²	—	2	1	no	—
393	C₄²	C₂×D₄×C₆₇	1	2	yes	—
394	C₄²	#41×C₆₇	1	2	yes	—
395	C₂³	—	2	1	no	—
396	C₂³	—	2	1	no	—
397	C₂³	—	2	4	no	—
398	C₂³	—	4	2	no	—
399	C₂³	—	4	2	no	—
400	C₂³	—	4	2	no	—
401	C₂³	—	4	4	no	—
402	C₂³	—	4	4	no	—
403	C₂³	#127×C₆₇	1	2	yes	—
404	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
405	Q₈	—	4	2	no	—
406	Q₈	—	4	2	no	—
407	Q₈	—	4	2	no	—
408	Q₈	—	4	2	no	—
409	D₄	—	4	2	no	—
410	D₄	—	4	2	no	—
411	D₄	—	4	2	no	—
412	D₄	—	4	2	no	—
413	D₄	—	8	8	no	—
414	C₂×C₄	—	2	1	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	2	4	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	4	no	—
423	C₂×C₄	—	4	2	no	—
424	C₂×C₄	—	4	2	no	—
425	C₂×C₄	—	4	2	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	2	no	—
428	C₂×C₄	—	4	4	no	—
429	C₂×C₄	—	4	4	no	—
430	C₂×C₄	#121×C₆₇	1	4	yes	—
431	C₂×C₄	#127×C₆₇	1	2	yes	—
432	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
433	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
434	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
435	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
436	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
437	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
438	C₈	—	2	2	no	—
439	C₈	—	2	2	no	—
440	C₈	#121×C₆₇	1	4	yes	—
441	C₈	#121×C₆₇	1	4	yes	—
442	C₂²	—	2	2	no	—
443	C₂²	—	2	4	no	—
444	C₂²	—	4	2	no	—
445	C₂²	—	4	2	no	—
446	C₂²	—	8	4	no	—
447	C₂²	—	8	2	no	—
448	C₂²	—	8	2	no	—
449	C₂²	—	8	2	no	—
450	C₂²	—	8	4	no	—
451	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
452	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
453	C₂²	#532×C₆₇	1	1	yes	characteristic
454	C₂²	#577×C₆₇	1	4	yes	—
455	C₄	—	2	4	no	—
456	C₄	—	2	2	no	—
457	C₄	—	4	2	no	—
458	C₄	—	8	2	no	—
459	C₄	—	8	2	no	—
460	C₄	—	8	2	no	—
461	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
462	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
463	C₄	#532×C₆₇	1	1	yes	characteristic
464	C₄	#532×C₆₇	1	1	yes	characteristic
465	C₂	—	4	2	no	—
466	C₂	—	16	2	no	—
467	C₂	C₆₇×#2856	1	1	yes	characteristic
468	C₂	C₆₇×#3060	1	2	yes	—
469	C₂	C₆₇×#3154	1	2	yes	—
470	C₂	C₆₇×#3173	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#469×C₆₇	C₂
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
1. C₂×#495×C₆₇	C₂	semidirect
2. C₂×#470×C₆₇	C₂	semidirect
4. C₂×#465×C₆₇	C₂	semidirect
5. C₂×#405×C₆₇	C₂	semidirect
6. C₆₇×#2620	C₂	semidirect
7. C₆₇×#1972	C₂	semidirect
8. C₆₇×#1971	C₂	semidirect
9. C₆₇×#1968	C₂	semidirect
10. C₆₇×#1959	C₂	semidirect
11. C₆₇×#1844	C₂	semidirect
95. C₄×C₈×C₆₇	D₄	semidirect
229. #33181	C₆₇	direct (N × H)
236. C₂×#495	C₂×C₆₇	semidirect
237. C₂×#470	C₂×C₆₇	semidirect
239. C₂×#465	C₂×C₆₇	semidirect
240. C₂×#405	C₂×C₆₇	semidirect
241. #2620	C₂×C₆₇	semidirect
242. #1972	C₂×C₆₇	semidirect
243. #1971	C₂×C₆₇	semidirect
244. #1968	C₂×C₆₇	semidirect
245. #1959	C₂×C₆₇	semidirect
246. #1844	C₂×C₆₇	semidirect
247. C₆₇	#33181	direct (N × H)
331. C₄×C₈	D₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#495×C₆₇	2
C₂×#470×C₆₇	2
C₂×#469×C₆₇	2
C₂×#465×C₆₇	2
C₂×#405×C₆₇	2
C₆₇×#2620	2
C₆₇×#1972	2
C₆₇×#1971	2
C₆₇×#1968	2
C₆₇×#1959	2
C₆₇×#1844	2
#33181	67

Subgroups — C67×#33181 #30712432

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33181 #30712432