Subgroups — C₆₇×#33036 #30712287

Order 17152.

918 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×#474×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#463×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2640	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2275	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1987	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1983	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1928	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1925	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1879	C₂	1	2	yes	maximal
10	C₆₇×#1877	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1832	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
18	#474×C₆₇	—	2	4	no	—
19	#463×C₆₇	—	2	4	no	—
20	C₂×#107×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×#103×C₆₇	C₂²	1	2	yes	—
25	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#102×C₆₇	C₂²	1	2	yes	—
27	C₂×#102×C₆₇	C₂²	1	2	yes	—
28	#397×C₆₇	—	2	2	no	—
29	#393×C₆₇	—	2	1	no	—
30	#393×C₆₇	—	2	1	no	—
31	#391×C₆₇	C₂²	1	2	yes	—
32	#390×C₆₇	C₂²	1	2	yes	—
33	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
35	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
36	C₄×C₂²⋊C₄×C₆₇	—	2	2	no	—
37	#376×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×#95×C₆₇	—	2	1	no	—
39	#340×C₆₇	C₂²	1	2	yes	—
40	#340×C₆₇	C₂²	1	2	yes	—
41	#336×C₆₇	C₂²	1	2	yes	—
42	#336×C₆₇	C₂²	1	2	yes	—
43	C₂²×Q₈×C₆₇	—	2	1	no	—
44	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
45	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
46	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
47	C₂³×C₄×C₆₇	—	2	2	no	—
48	C₂³×C₄×C₆₇	—	2	1	no	—
49	C₂×SD₁₆×C₆₇	—	2	2	no	—
50	C₂×SD₁₆×C₆₇	—	2	4	no	—
51	C₂×D₈×C₆₇	—	2	2	no	—
52	C₂×D₈×C₆₇	—	2	4	no	—
53	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
56	C₄⋊Q₈×C₆₇	—	2	2	no	—
57	C₄⋊Q₈×C₆₇	—	2	2	no	—
58	#124×C₆₇	—	2	2	no	—
59	#124×C₆₇	—	2	2	no	—
60	C₄×D₄×C₆₇	—	2	4	no	—
61	C₄×D₄×C₆₇	—	2	4	no	—
62	C₄×D₄×C₆₇	—	2	2	no	—
63	C₄×D₄×C₆₇	D₄	1	4	yes	—
64	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
65	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
68	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
69	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
70	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
71	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
72	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
73	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
74	C₂×C₄²×C₆₇	—	2	2	no	—
75	C₂×C₄²×C₆₇	—	2	1	no	—
76	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
77	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
78	#107×C₆₇	—	2	4	no	—
79	C₄⋊C₈×C₆₇	—	2	2	no	—
80	C₄⋊C₈×C₆₇	—	2	2	no	—
81	#103×C₆₇	—	2	2	no	—
82	#103×C₆₇	—	2	4	no	—
83	#102×C₆₇	—	2	4	no	—
84	#102×C₆₇	—	2	2	no	—
85	#102×C₆₇	—	2	4	no	—
86	#95×C₆₇	—	2	2	no	—
87	#95×C₆₇	—	2	1	no	—
88	#95×C₆₇	—	2	2	no	—
89	#95×C₆₇	—	2	2	no	—
90	#95×C₆₇	D₄	1	1	yes	characteristic
91	#95×C₆₇	D₄	1	1	yes	characteristic
92	#95×C₆₇	D₄	1	2	yes	—
93	C₂⁴×C₆₇	—	2	1	no	—
94	C₂⁴×C₆₇	—	2	1	no	—
95	C₂×Q₈×C₆₇	—	2	1	no	—
96	C₂×Q₈×C₆₇	—	2	2	no	—
97	C₂×Q₈×C₆₇	—	2	1	no	—
98	C₂×Q₈×C₆₇	—	2	2	no	—
99	C₂×Q₈×C₆₇	—	4	2	no	—
100	C₂×Q₈×C₆₇	—	8	2	no	—
101	C₂×D₄×C₆₇	—	2	1	no	—
102	C₂×D₄×C₆₇	—	2	1	no	—
103	C₂×D₄×C₆₇	—	2	2	no	—
104	C₂×D₄×C₆₇	—	2	2	no	—
105	C₂×D₄×C₆₇	—	2	1	no	—
106	C₂×D₄×C₆₇	—	2	2	no	—
107	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
108	C₂×D₄×C₆₇	#41	1	4	yes	—
109	C₂²×C₄×C₆₇	—	2	1	no	—
110	C₂²×C₄×C₆₇	—	2	2	no	—
111	C₂²×C₄×C₆₇	—	2	8	no	—
112	C₂²×C₄×C₆₇	—	2	4	no	—
113	C₂²×C₄×C₆₇	—	2	2	no	—
114	C₂²×C₄×C₆₇	—	2	1	no	—
115	C₂²×C₄×C₆₇	—	8	2	no	—
116	C₂²×C₄×C₆₇	—	8	2	no	—
117	C₂²×C₄×C₆₇	—	8	2	no	—
118	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
119	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
120	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
123	SD₁₆×C₆₇	—	4	4	no	—
124	D₈×C₆₇	—	4	4	no	—
125	C₂×C₈×C₆₇	—	2	2	no	—
126	C₂×C₈×C₆₇	—	2	1	no	—
127	C₂×C₈×C₆₇	—	2	1	no	—
128	C₂×C₈×C₆₇	—	2	2	no	—
129	C₂×C₈×C₆₇	—	2	2	no	—
130	C₂×C₈×C₆₇	—	2	2	no	—
131	C₂×C₈×C₆₇	—	2	2	no	—
132	C₄⋊C₄×C₆₇	—	2	1	no	—
133	C₄⋊C₄×C₆₇	—	2	2	no	—
134	C₄⋊C₄×C₆₇	—	2	2	no	—
135	C₄⋊C₄×C₆₇	—	2	2	no	—
136	C₄⋊C₄×C₆₇	—	4	2	no	—
137	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
138	C₄⋊C₄×C₆₇	#41	1	4	yes	—
139	C₂²⋊C₄×C₆₇	—	2	8	no	—
140	C₂²⋊C₄×C₆₇	—	2	4	no	—
141	C₂²⋊C₄×C₆₇	—	4	2	no	—
142	C₂²⋊C₄×C₆₇	—	4	2	no	—
143	C₂²⋊C₄×C₆₇	—	8	2	no	—
144	C₂²⋊C₄×C₆₇	—	8	2	no	—
145	C₄²×C₆₇	—	2	2	no	—
146	C₄²×C₆₇	—	2	2	no	—
147	C₄²×C₆₇	—	8	1	no	—
148	C₄²×C₆₇	—	8	2	no	—
149	C₄²×C₆₇	C₂×D₄	1	2	yes	—
150	C₄²×C₆₇	#41	1	2	yes	—
151	C₂³×C₆₇	—	2	2	no	—
152	C₂³×C₆₇	—	2	4	no	—
153	C₂³×C₆₇	—	4	2	no	—
154	C₂³×C₆₇	—	4	1	no	—
155	C₂³×C₆₇	—	4	1	no	—
156	C₂³×C₆₇	—	4	1	no	—
157	C₂³×C₆₇	—	4	2	no	—
158	C₂³×C₆₇	—	4	2	no	—
159	C₂³×C₆₇	—	4	1	no	—
160	C₂³×C₆₇	—	4	1	no	—
161	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
162	Q₈×C₆₇	—	4	1	no	—
163	Q₈×C₆₇	—	4	2	no	—
164	Q₈×C₆₇	—	4	1	no	—
165	Q₈×C₆₇	—	8	4	no	—
166	D₄×C₆₇	—	2	4	no	—
167	D₄×C₆₇	—	2	2	no	—
168	D₄×C₆₇	—	4	1	no	—
169	D₄×C₆₇	—	4	2	no	—
170	D₄×C₆₇	—	4	1	no	—
171	D₄×C₆₇	#124	1	4	yes	—
172	C₂×C₄×C₆₇	—	2	2	no	—
173	C₂×C₄×C₆₇	—	2	1	no	—
174	C₂×C₄×C₆₇	—	2	1	no	—
175	C₂×C₄×C₆₇	—	2	2	no	—
176	C₂×C₄×C₆₇	—	2	2	no	—
177	C₂×C₄×C₆₇	—	2	2	no	—
178	C₂×C₄×C₆₇	—	2	4	no	—
179	C₂×C₄×C₆₇	—	2	2	no	—
180	C₂×C₄×C₆₇	—	2	2	no	—
181	C₂×C₄×C₆₇	—	2	2	no	—
182	C₂×C₄×C₆₇	—	2	2	no	—
183	C₂×C₄×C₆₇	—	4	2	no	—
184	C₂×C₄×C₆₇	—	4	2	no	—
185	C₂×C₄×C₆₇	—	4	2	no	—
186	C₂×C₄×C₆₇	—	4	1	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	—	8	4	no	—
192	C₂×C₄×C₆₇	—	8	8	no	—
193	C₂×C₄×C₆₇	#121	1	2	yes	—
194	C₂×C₄×C₆₇	#121	1	2	yes	—
195	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
196	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
197	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
198	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
199	C₈×C₆₇	—	4	1	no	—
200	C₈×C₆₇	—	4	1	no	—
201	C₈×C₆₇	—	4	2	no	—
202	C₂²×C₆₇	—	2	4	no	—
203	C₂²×C₆₇	—	4	4	no	—
204	C₂²×C₆₇	—	4	4	no	—
205	C₂²×C₆₇	—	4	2	no	—
206	C₂²×C₆₇	—	4	2	no	—
207	C₂²×C₆₇	—	8	1	no	—
208	C₂²×C₆₇	—	8	2	no	—
209	C₂²×C₆₇	—	8	2	no	—
210	C₂²×C₆₇	—	8	1	no	—
211	C₂²×C₆₇	—	8	1	no	—
212	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
213	C₂²×C₆₇	#525	1	2	yes	—
214	C₂²×C₆₇	#576	1	2	yes	—
215	C₂²×C₆₇	#577	1	1	yes	characteristic
216	C₂²×C₆₇	#578	1	1	yes	characteristic
217	C₄×C₆₇	—	2	2	no	—
218	C₄×C₆₇	—	2	2	no	—
219	C₄×C₆₇	—	4	2	no	—
220	C₄×C₆₇	—	4	4	no	—
221	C₄×C₆₇	—	4	2	no	—
222	C₄×C₆₇	—	4	2	no	—
223	C₄×C₆₇	—	8	1	no	—
224	C₄×C₆₇	—	8	2	no	—
225	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
226	C₄×C₆₇	#529	1	2	yes	—
227	C₄×C₆₇	#532	1	1	yes	characteristic
228	#33036	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
229	C₂×C₆₇	—	4	4	no	—
230	C₂×C₆₇	—	16	1	no	—
231	C₂×C₆₇	#2850	1	1	yes	characteristic
232	C₂×C₆₇	#3041	1	1	yes	characteristic
233	C₂×C₆₇	#3046	1	1	yes	characteristic
234	C₂×C₆₇	#3118	1	2	yes	—
235	C₂×C₆₇	#3124	1	2	yes	—
236	C₂×#474	C₂×C₆₇	1	1	yes	characteristic, complemented
237	C₂×#463	C₂×C₆₇	1	1	yes	characteristic
238	#2640	C₂×C₆₇	1	1	yes	characteristic, complemented
239	#2275	C₂×C₆₇	1	1	yes	characteristic, complemented
240	#1987	C₂×C₆₇	1	1	yes	characteristic, complemented
241	#1983	C₂×C₆₇	1	1	yes	characteristic, complemented
242	#1928	C₂×C₆₇	1	2	yes	complemented
243	#1925	C₂×C₆₇	1	2	yes	complemented
244	#1879	C₂×C₆₇	1	2	yes	—
245	#1877	C₂×C₆₇	1	2	yes	complemented
246	#1832	C₂×C₆₇	1	1	yes	characteristic, complemented
247	C₆₇	#33036	1	1	yes	characteristic, direct product factor, Sylow
248	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
249	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
250	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
251	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
252	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
253	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
254	#474	—	2	4	no	—
255	#463	—	2	4	no	—
256	C₂×#107	C₂²×C₆₇	1	2	yes	—
257	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
258	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
259	C₂×#103	C₂²×C₆₇	1	2	yes	—
260	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
261	C₂×#102	C₂²×C₆₇	1	2	yes	—
262	C₂×#102	C₂²×C₆₇	1	2	yes	—
263	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
264	#397	—	2	2	no	—
265	#393	—	2	1	no	—
266	#393	—	2	1	no	—
267	#391	C₂²×C₆₇	1	2	yes	—
268	#390	C₂²×C₆₇	1	2	yes	—
269	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
270	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
271	C₄×C₂²⋊C₄	—	2	1	no	—
272	C₄×C₂²⋊C₄	—	2	2	no	—
273	#376	C₂²×C₆₇	1	1	yes	characteristic
274	C₂×#95	—	2	1	no	—
275	#340	C₂²×C₆₇	1	2	yes	—
276	#340	C₂²×C₆₇	1	2	yes	—
277	#336	C₂²×C₆₇	1	2	yes	—
278	#336	C₂²×C₆₇	1	2	yes	—
279	C₂²×Q₈	—	2	1	no	—
280	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
281	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
282	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
283	C₂³×C₄	—	2	2	no	—
284	C₂³×C₄	—	2	1	no	—
285	C₂×SD₁₆	—	2	2	no	—
286	C₂×SD₁₆	—	2	4	no	—
287	C₂×D₈	—	2	2	no	—
288	C₂×D₈	—	2	4	no	—
289	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
290	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
291	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
292	C₄⋊Q₈	—	2	2	no	—
293	C₄⋊Q₈	—	2	2	no	—
294	#124	—	2	2	no	—
295	#124	—	2	2	no	—
296	C₄×D₄	—	2	4	no	—
297	C₄×D₄	—	2	4	no	—
298	C₄×D₄	—	2	2	no	—
299	C₄×D₄	D₄×C₆₇	1	4	yes	—
300	C₂×C₄⋊C₄	—	2	1	no	—
301	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
302	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
303	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
304	C₂×C₂²⋊C₄	—	2	2	no	—
305	C₂×C₂²⋊C₄	—	2	1	no	—
306	C₂×C₂²⋊C₄	—	2	1	no	—
307	C₂×C₂²⋊C₄	—	2	1	no	—
308	C₂×C₂²⋊C₄	—	2	2	no	—
309	C₂×C₂²⋊C₄	—	2	2	no	—
310	C₂×C₄²	—	2	2	no	—
311	C₂×C₄²	—	2	1	no	—
312	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
313	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
314	#107	—	2	4	no	—
315	C₄⋊C₈	—	2	2	no	—
316	C₄⋊C₈	—	2	2	no	—
317	#103	—	2	2	no	—
318	#103	—	2	4	no	—
319	#102	—	2	4	no	—
320	#102	—	2	2	no	—
321	#102	—	2	4	no	—
322	#95	—	2	2	no	—
323	#95	—	2	1	no	—
324	#95	—	2	2	no	—
325	#95	—	2	2	no	—
326	#95	D₄×C₆₇	1	1	yes	characteristic
327	#95	D₄×C₆₇	1	1	yes	characteristic
328	#95	D₄×C₆₇	1	2	yes	—
329	C₂⁴	—	2	1	no	—
330	C₂⁴	—	2	1	no	—
331	C₂×Q₈	—	2	1	no	—
332	C₂×Q₈	—	2	2	no	—
333	C₂×Q₈	—	2	1	no	—
334	C₂×Q₈	—	2	2	no	—
335	C₂×Q₈	—	4	2	no	—
336	C₂×Q₈	—	8	2	no	—
337	C₂×D₄	—	2	1	no	—
338	C₂×D₄	—	2	1	no	—
339	C₂×D₄	—	2	2	no	—
340	C₂×D₄	—	2	2	no	—
341	C₂×D₄	—	2	1	no	—
342	C₂×D₄	—	2	2	no	—
343	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
344	C₂×D₄	#41×C₆₇	1	4	yes	—
345	C₂²×C₄	—	2	1	no	—
346	C₂²×C₄	—	2	2	no	—
347	C₂²×C₄	—	2	8	no	—
348	C₂²×C₄	—	2	4	no	—
349	C₂²×C₄	—	2	1	no	—
350	C₂²×C₄	—	2	2	no	—
351	C₂²×C₄	—	8	2	no	—
352	C₂²×C₄	—	8	2	no	—
353	C₂²×C₄	—	8	2	no	—
354	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
355	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
356	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
357	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
358	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
359	SD₁₆	—	4	4	no	—
360	D₈	—	4	4	no	—
361	C₂×C₈	—	2	2	no	—
362	C₂×C₈	—	2	1	no	—
363	C₂×C₈	—	2	1	no	—
364	C₂×C₈	—	2	2	no	—
365	C₂×C₈	—	2	2	no	—
366	C₂×C₈	—	2	2	no	—
367	C₂×C₈	—	2	2	no	—
368	C₄⋊C₄	—	2	1	no	—
369	C₄⋊C₄	—	2	2	no	—
370	C₄⋊C₄	—	2	2	no	—
371	C₄⋊C₄	—	2	2	no	—
372	C₄⋊C₄	—	4	2	no	—
373	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
374	C₄⋊C₄	#41×C₆₇	1	4	yes	—
375	C₂²⋊C₄	—	2	8	no	—
376	C₂²⋊C₄	—	2	4	no	—
377	C₂²⋊C₄	—	4	2	no	—
378	C₂²⋊C₄	—	4	2	no	—
379	C₂²⋊C₄	—	8	2	no	—
380	C₂²⋊C₄	—	8	2	no	—
381	C₄²	—	2	2	no	—
382	C₄²	—	2	2	no	—
383	C₄²	—	8	1	no	—
384	C₄²	—	8	2	no	—
385	C₄²	C₂×D₄×C₆₇	1	2	yes	—
386	C₄²	#41×C₆₇	1	2	yes	—
387	C₂³	—	2	2	no	—
388	C₂³	—	2	4	no	—
389	C₂³	—	4	2	no	—
390	C₂³	—	4	1	no	—
391	C₂³	—	4	1	no	—
392	C₂³	—	4	1	no	—
393	C₂³	—	4	2	no	—
394	C₂³	—	4	2	no	—
395	C₂³	—	4	1	no	—
396	C₂³	—	4	1	no	—
397	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
398	Q₈	—	4	1	no	—
399	Q₈	—	4	2	no	—
400	Q₈	—	4	1	no	—
401	Q₈	—	8	4	no	—
402	D₄	—	2	4	no	—
403	D₄	—	2	2	no	—
404	D₄	—	4	1	no	—
405	D₄	—	4	2	no	—
406	D₄	—	4	1	no	—
407	D₄	#124×C₆₇	1	4	yes	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	4	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	4	2	no	—
421	C₂×C₄	—	4	2	no	—
422	C₂×C₄	—	4	1	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	—	8	4	no	—
428	C₂×C₄	—	8	8	no	—
429	C₂×C₄	#121×C₆₇	1	2	yes	—
430	C₂×C₄	#121×C₆₇	1	2	yes	—
431	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
432	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
433	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
434	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
435	C₈	—	4	1	no	—
436	C₈	—	4	1	no	—
437	C₈	—	4	2	no	—
438	C₂²	—	2	4	no	—
439	C₂²	—	4	2	no	—
440	C₂²	—	4	2	no	—
441	C₂²	—	4	4	no	—
442	C₂²	—	4	4	no	—
443	C₂²	—	8	1	no	—
444	C₂²	—	8	1	no	—
445	C₂²	—	8	1	no	—
446	C₂²	—	8	2	no	—
447	C₂²	—	8	2	no	—
448	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
449	C₂²	#525×C₆₇	1	2	yes	—
450	C₂²	#576×C₆₇	1	2	yes	—
451	C₂²	#577×C₆₇	1	1	yes	characteristic
452	C₂²	#578×C₆₇	1	1	yes	characteristic
453	C₄	—	2	2	no	—
454	C₄	—	2	2	no	—
455	C₄	—	4	2	no	—
456	C₄	—	4	2	no	—
457	C₄	—	4	4	no	—
458	C₄	—	4	2	no	—
459	C₄	—	8	2	no	—
460	C₄	—	8	1	no	—
461	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
462	C₄	#529×C₆₇	1	2	yes	—
463	C₄	#532×C₆₇	1	1	yes	characteristic
464	C₂	—	4	4	no	—
465	C₂	—	16	1	no	—
466	C₂	C₆₇×#2850	1	1	yes	characteristic
467	C₂	C₆₇×#3041	1	1	yes	characteristic
468	C₂	C₆₇×#3046	1	1	yes	characteristic
469	C₂	C₆₇×#3118	1	2	yes	—
470	C₂	C₆₇×#3124	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×D₈×C₆₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ #376×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#474×C₆₇	C₂	semidirect
3. C₆₇×#2640	C₂	semidirect
4. C₆₇×#2275	C₂	semidirect
5. C₆₇×#1987	C₂	semidirect
6. C₆₇×#1983	C₂	semidirect
7. C₆₇×#1928	C₂	semidirect
8. C₆₇×#1925	C₂	semidirect
10. C₆₇×#1877	C₂	semidirect
11. C₆₇×#1832	C₂	semidirect
228. #33036	C₆₇	direct (N × H)
236. C₂×#474	C₂×C₆₇	semidirect
238. #2640	C₂×C₆₇	semidirect
239. #2275	C₂×C₆₇	semidirect
240. #1987	C₂×C₆₇	semidirect
241. #1983	C₂×C₆₇	semidirect
242. #1928	C₂×C₆₇	semidirect
243. #1925	C₂×C₆₇	semidirect
245. #1877	C₂×C₆₇	semidirect
246. #1832	C₂×C₆₇	semidirect
247. C₆₇	#33036	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×#474×C₆₇	2
C₂×#463×C₆₇	2
C₆₇×#2640	2
C₆₇×#2275	2
C₆₇×#1987	2
C₆₇×#1983	2
C₆₇×#1928	2
C₆₇×#1925	2
C₆₇×#1879	2
C₆₇×#1877	2
C₆₇×#1832	2
#33036	67

Subgroups — C67×#33036 #30712287

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33036 #30712287