Subgroups — C₆₇×#25929 #30705180

Order 17152.

1262 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#487×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#461×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2568	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2024	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1988	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1984	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1877	C₂	1	4	yes	maximal
9	C₄×#102×C₆₇	C₂	1	1	yes	maximal, characteristic
10	C₂³×Q₈×C₆₇	—	2	1	no	—
11	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
12	D₄×Q₈×C₆₇	—	2	4	no	—
13	D₄×Q₈×C₆₇	—	2	4	no	—
14	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
18	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
19	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
22	#487×C₆₇	—	2	4	no	—
23	#461×C₆₇	—	2	4	no	—
24	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#103×C₆₇	C₂²	1	2	yes	—
26	C₂×#103×C₆₇	C₂²	1	4	yes	—
27	C₂×#102×C₆₇	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
29	#400×C₆₇	—	2	2	no	—
30	#391×C₆₇	—	2	1	no	—
31	#391×C₆₇	C₂²	1	4	yes	—
32	#382×C₆₇	—	2	1	no	—
33	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
34	#340×C₆₇	C₂²	1	4	yes	—
35	#340×C₆₇	C₂²	1	2	yes	—
36	#336×C₆₇	C₂²	1	4	yes	—
37	C₂²×Q₈×C₆₇	—	2	8	no	—
38	C₂²×Q₈×C₆₇	—	2	2	no	—
39	C₂²×Q₈×C₆₇	—	2	1	no	—
40	C₂²×Q₈×C₆₇	—	4	4	no	—
41	C₂²×Q₈×C₆₇	—	4	4	no	—
42	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
43	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
44	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₆₇	—	2	2	no	—
46	C₂³×C₄×C₆₇	—	2	1	no	—
47	C₂×SD₁₆×C₆₇	—	2	4	no	—
48	C₂×SD₁₆×C₆₇	—	2	8	no	—
49	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
50	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
51	C₄⋊Q₈×C₆₇	—	2	4	no	—
52	C₄⋊Q₈×C₆₇	—	2	4	no	—
53	C₄⋊Q₈×C₆₇	—	2	2	no	—
54	C₄⋊Q₈×C₆₇	—	2	2	no	—
55	C₄⋊Q₈×C₆₇	—	2	2	no	—
56	C₄⋊Q₈×C₆₇	D₄	1	4	yes	—
57	C₂²⋊Q₈×C₆₇	—	2	8	no	—
58	C₂²⋊Q₈×C₆₇	—	2	16	no	—
59	C₄×Q₈×C₆₇	—	2	2	no	—
60	C₄×Q₈×C₆₇	—	2	2	no	—
61	C₄×D₄×C₆₇	—	2	4	no	—
62	C₄×D₄×C₆₇	—	2	4	no	—
63	C₄×D₄×C₆₇	—	2	2	no	—
64	C₄×D₄×C₆₇	D₄	1	4	yes	—
65	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
66	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
67	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
68	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
69	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
71	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
72	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
73	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
74	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
75	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
76	C₂×C₄²×C₆₇	—	2	1	no	—
77	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
78	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
79	C₄⋊C₈×C₆₇	—	2	2	no	—
80	#103×C₆₇	—	2	4	no	—
81	#103×C₆₇	—	2	8	no	—
82	#102×C₆₇	—	2	4	no	—
83	C₄×C₈×C₆₇	—	2	2	no	—
84	#95×C₆₇	—	2	1	no	—
85	#95×C₆₇	—	2	4	no	—
86	#95×C₆₇	—	2	4	no	—
87	C₂⁴×C₆₇	—	2	1	no	—
88	C₂×Q₈×C₆₇	—	2	8	no	—
89	C₂×Q₈×C₆₇	—	2	2	no	—
90	C₂×Q₈×C₆₇	—	2	4	no	—
91	C₂×Q₈×C₆₇	—	2	2	no	—
92	C₂×Q₈×C₆₇	—	2	2	no	—
93	C₂×Q₈×C₆₇	—	2	2	no	—
94	C₂×Q₈×C₆₇	—	2	4	no	—
95	C₂×Q₈×C₆₇	—	2	4	no	—
96	C₂×Q₈×C₆₇	—	4	4	no	—
97	C₂×Q₈×C₆₇	—	4	32	no	—
98	C₂×Q₈×C₆₇	—	4	4	no	—
99	C₂×Q₈×C₆₇	—	4	2	no	—
100	C₂×Q₈×C₆₇	—	4	2	no	—
101	C₂×Q₈×C₆₇	SD₁₆	1	4	yes	complemented
102	C₂×D₄×C₆₇	—	2	1	no	—
103	C₂×D₄×C₆₇	—	2	2	no	—
104	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
105	C₂×D₄×C₆₇	#41	1	4	yes	—
106	C₂²×C₄×C₆₇	—	2	8	no	—
107	C₂²×C₄×C₆₇	—	2	2	no	—
108	C₂²×C₄×C₆₇	—	2	1	no	—
109	C₂²×C₄×C₆₇	—	2	4	no	—
110	C₂²×C₄×C₆₇	—	2	2	no	—
111	C₂²×C₄×C₆₇	—	2	2	no	—
112	C₂²×C₄×C₆₇	—	4	4	no	—
113	C₂²×C₄×C₆₇	—	4	4	no	—
114	C₂²×C₄×C₆₇	—	4	2	no	—
115	C₂²×C₄×C₆₇	—	4	2	no	—
116	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
117	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
119	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
121	SD₁₆×C₆₇	—	4	8	no	—
122	C₂×C₈×C₆₇	—	2	2	no	—
123	C₂×C₈×C₆₇	—	2	2	no	—
124	C₂×C₈×C₆₇	—	2	2	no	—
125	C₂×C₈×C₆₇	—	2	4	no	—
126	C₂×C₈×C₆₇	—	2	2	no	—
127	C₄⋊C₄×C₆₇	—	2	8	no	—
128	C₄⋊C₄×C₆₇	—	2	4	no	—
129	C₄⋊C₄×C₆₇	—	2	4	no	—
130	C₄⋊C₄×C₆₇	—	2	2	no	—
131	C₄⋊C₄×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	—	4	2	no	—
136	C₄⋊C₄×C₆₇	—	4	2	no	—
137	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
138	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
139	C₂²⋊C₄×C₆₇	—	2	8	no	—
140	C₂²⋊C₄×C₆₇	—	2	4	no	—
141	C₄²×C₆₇	—	2	2	no	—
142	C₄²×C₆₇	—	2	2	no	—
143	C₄²×C₆₇	—	4	2	no	—
144	C₄²×C₆₇	C₂×D₄	1	2	yes	—
145	C₄²×C₆₇	C₂×D₄	1	2	yes	—
146	C₂³×C₆₇	—	2	2	no	—
147	C₂³×C₆₇	—	2	4	no	—
148	C₂³×C₆₇	—	4	1	no	—
149	C₂³×C₆₇	—	4	1	no	—
150	C₂³×C₆₇	—	4	2	no	—
151	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
152	Q₈×C₆₇	—	2	8	no	—
153	Q₈×C₆₇	—	4	16	no	—
154	Q₈×C₆₇	—	4	8	no	—
155	Q₈×C₆₇	—	4	2	no	—
156	Q₈×C₆₇	—	4	2	no	—
157	Q₈×C₆₇	—	4	4	no	—
158	D₄×C₆₇	—	2	2	no	—
159	D₄×C₆₇	—	2	4	no	—
160	D₄×C₆₇	#124	1	4	yes	—
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	—	2	2	no	—
165	C₂×C₄×C₆₇	—	2	4	no	—
166	C₂×C₄×C₆₇	—	2	2	no	—
167	C₂×C₄×C₆₇	—	2	4	no	—
168	C₂×C₄×C₆₇	—	4	16	no	—
169	C₂×C₄×C₆₇	—	4	2	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	8	no	—
174	C₂×C₄×C₆₇	—	4	2	no	—
175	C₂×C₄×C₆₇	—	4	2	no	—
176	C₂×C₄×C₆₇	—	4	4	no	—
177	C₂×C₄×C₆₇	—	4	2	no	—
178	C₂×C₄×C₆₇	—	4	2	no	—
179	C₂×C₄×C₆₇	#120	1	2	yes	—
180	C₂×C₄×C₆₇	#120	1	2	yes	—
181	C₂×C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
182	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
183	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
184	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
185	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
186	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
187	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
188	C₈×C₆₇	—	4	2	no	—
189	C₈×C₆₇	—	4	2	no	—
190	C₂²×C₆₇	—	2	4	no	—
191	C₂²×C₆₇	—	4	4	no	—
192	C₂²×C₆₇	—	4	4	no	—
193	C₂²×C₆₇	—	4	2	no	—
194	C₂²×C₆₇	—	4	2	no	—
195	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
196	C₂²×C₆₇	#548	1	2	yes	—
197	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
198	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
199	C₂²×C₆₇	#578	1	2	yes	—
200	C₄×C₆₇	—	2	4	no	—
201	C₄×C₆₇	—	2	2	no	—
202	C₄×C₆₇	—	4	4	no	—
203	C₄×C₆₇	—	4	2	no	—
204	C₄×C₆₇	—	4	2	no	—
205	C₄×C₆₇	—	8	2	no	—
206	C₄×C₆₇	—	8	2	no	—
207	C₄×C₆₇	#450	1	4	yes	—
208	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
209	C₄×C₆₇	#536	1	1	yes	characteristic
210	C₄×C₆₇	#548	1	2	yes	—
211	#25929	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
212	C₂×C₆₇	—	4	4	no	—
213	C₂×C₆₇	#2436	1	1	yes	characteristic
214	C₂×C₆₇	C₂×#450	1	1	yes	characteristic
215	C₂×C₆₇	#3011	1	1	yes	characteristic
216	C₂×C₆₇	#3359	1	2	yes	—
217	C₂×C₆₇	#3367	1	2	yes	—
218	C₂×D₄×Q₈	C₂×C₆₇	1	1	yes	characteristic
219	C₂×#487	C₂×C₆₇	1	1	yes	characteristic, complemented
220	C₂×#461	C₂×C₆₇	1	1	yes	characteristic
221	#2568	C₂×C₆₇	1	1	yes	characteristic, complemented
222	#2024	C₂×C₆₇	1	4	yes	complemented
223	#1988	C₂×C₆₇	1	1	yes	characteristic, complemented
224	#1984	C₂×C₆₇	1	1	yes	characteristic, complemented
225	#1877	C₂×C₆₇	1	4	yes	—
226	C₄×#102	C₂×C₆₇	1	1	yes	characteristic
227	C₆₇	#25929	1	1	yes	characteristic, direct product factor, Sylow
228	C₂³×Q₈	—	2	1	no	—
229	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
230	D₄×Q₈	—	2	4	no	—
231	D₄×Q₈	—	2	4	no	—
232	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
233	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
234	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
235	C₂×C₂²⋊Q₈	—	2	1	no	—
236	C₂×C₂²⋊Q₈	—	2	2	no	—
237	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
238	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
239	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
240	#487	—	2	4	no	—
241	#461	—	2	4	no	—
242	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
243	C₂×#103	C₂²×C₆₇	1	4	yes	—
244	C₂×#103	C₂²×C₆₇	1	2	yes	—
245	C₂×#102	C₂²×C₆₇	1	2	yes	—
246	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
247	#400	—	2	2	no	—
248	#391	—	2	1	no	—
249	#391	C₂²×C₆₇	1	4	yes	—
250	#382	—	2	1	no	—
251	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
252	#340	C₂²×C₆₇	1	4	yes	—
253	#340	C₂²×C₆₇	1	2	yes	—
254	#336	C₂²×C₆₇	1	4	yes	—
255	C₂²×Q₈	—	2	8	no	—
256	C₂²×Q₈	—	2	2	no	—
257	C₂²×Q₈	—	2	1	no	—
258	C₂²×Q₈	—	4	4	no	—
259	C₂²×Q₈	—	4	4	no	—
260	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
261	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
262	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
263	C₂³×C₄	—	2	2	no	—
264	C₂³×C₄	—	2	1	no	—
265	C₂×SD₁₆	—	2	4	no	—
266	C₂×SD₁₆	—	2	8	no	—
267	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
268	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
269	C₄⋊Q₈	—	2	4	no	—
270	C₄⋊Q₈	—	2	4	no	—
271	C₄⋊Q₈	—	2	2	no	—
272	C₄⋊Q₈	—	2	2	no	—
273	C₄⋊Q₈	—	2	2	no	—
274	C₄⋊Q₈	D₄×C₆₇	1	4	yes	—
275	C₂²⋊Q₈	—	2	8	no	—
276	C₂²⋊Q₈	—	2	16	no	—
277	C₄×Q₈	—	2	2	no	—
278	C₄×Q₈	—	2	2	no	—
279	C₄×D₄	—	2	4	no	—
280	C₄×D₄	—	2	4	no	—
281	C₄×D₄	—	2	2	no	—
282	C₄×D₄	D₄×C₆₇	1	4	yes	—
283	C₂×C₄⋊C₄	—	2	2	no	—
284	C₂×C₄⋊C₄	—	2	1	no	—
285	C₂×C₄⋊C₄	—	2	1	no	—
286	C₂×C₄⋊C₄	—	2	1	no	—
287	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
288	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
289	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
290	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
291	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
292	C₂×C₂²⋊C₄	—	2	2	no	—
293	C₂×C₂²⋊C₄	—	2	1	no	—
294	C₂×C₄²	—	2	1	no	—
295	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
296	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
297	C₄⋊C₈	—	2	2	no	—
298	#103	—	2	4	no	—
299	#103	—	2	8	no	—
300	#102	—	2	4	no	—
301	C₄×C₈	—	2	2	no	—
302	#95	—	2	1	no	—
303	#95	—	2	4	no	—
304	#95	—	2	4	no	—
305	C₂⁴	—	2	1	no	—
306	C₂×Q₈	—	2	8	no	—
307	C₂×Q₈	—	2	2	no	—
308	C₂×Q₈	—	2	4	no	—
309	C₂×Q₈	—	2	2	no	—
310	C₂×Q₈	—	2	2	no	—
311	C₂×Q₈	—	2	2	no	—
312	C₂×Q₈	—	2	4	no	—
313	C₂×Q₈	—	2	4	no	—
314	C₂×Q₈	—	4	4	no	—
315	C₂×Q₈	—	4	32	no	—
316	C₂×Q₈	—	4	4	no	—
317	C₂×Q₈	—	4	2	no	—
318	C₂×Q₈	—	4	2	no	—
319	C₂×Q₈	SD₁₆×C₆₇	1	4	yes	complemented
320	C₂×D₄	—	2	1	no	—
321	C₂×D₄	—	2	2	no	—
322	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
323	C₂×D₄	#41×C₆₇	1	4	yes	—
324	C₂²×C₄	—	2	8	no	—
325	C₂²×C₄	—	2	2	no	—
326	C₂²×C₄	—	2	1	no	—
327	C₂²×C₄	—	2	4	no	—
328	C₂²×C₄	—	2	2	no	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	—	4	4	no	—
331	C₂²×C₄	—	4	4	no	—
332	C₂²×C₄	—	4	2	no	—
333	C₂²×C₄	—	4	2	no	—
334	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
335	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
336	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
337	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
338	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
339	SD₁₆	—	4	8	no	—
340	C₂×C₈	—	2	2	no	—
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	—	2	2	no	—
343	C₂×C₈	—	2	4	no	—
344	C₂×C₈	—	2	2	no	—
345	C₄⋊C₄	—	2	8	no	—
346	C₄⋊C₄	—	2	4	no	—
347	C₄⋊C₄	—	2	4	no	—
348	C₄⋊C₄	—	2	2	no	—
349	C₄⋊C₄	—	2	2	no	—
350	C₄⋊C₄	—	2	1	no	—
351	C₄⋊C₄	—	2	2	no	—
352	C₄⋊C₄	—	2	2	no	—
353	C₄⋊C₄	—	4	2	no	—
354	C₄⋊C₄	—	4	2	no	—
355	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
356	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
357	C₂²⋊C₄	—	2	8	no	—
358	C₂²⋊C₄	—	2	4	no	—
359	C₄²	—	2	2	no	—
360	C₄²	—	2	2	no	—
361	C₄²	—	4	2	no	—
362	C₄²	C₂×D₄×C₆₇	1	2	yes	—
363	C₄²	C₂×D₄×C₆₇	1	2	yes	—
364	C₂³	—	2	2	no	—
365	C₂³	—	2	4	no	—
366	C₂³	—	4	1	no	—
367	C₂³	—	4	1	no	—
368	C₂³	—	4	2	no	—
369	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
370	Q₈	—	2	8	no	—
371	Q₈	—	4	16	no	—
372	Q₈	—	4	8	no	—
373	Q₈	—	4	2	no	—
374	Q₈	—	4	2	no	—
375	Q₈	—	4	4	no	—
376	D₄	—	2	2	no	—
377	D₄	—	2	4	no	—
378	D₄	#124×C₆₇	1	4	yes	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	2	4	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	2	4	no	—
386	C₂×C₄	—	4	16	no	—
387	C₂×C₄	—	4	2	no	—
388	C₂×C₄	—	4	1	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	—	4	1	no	—
391	C₂×C₄	—	4	8	no	—
392	C₂×C₄	—	4	2	no	—
393	C₂×C₄	—	4	2	no	—
394	C₂×C₄	—	4	4	no	—
395	C₂×C₄	—	4	2	no	—
396	C₂×C₄	—	4	2	no	—
397	C₂×C₄	#120×C₆₇	1	2	yes	—
398	C₂×C₄	#120×C₆₇	1	2	yes	—
399	C₂×C₄	C₂×SD₁₆×C₆₇	1	4	yes	—
400	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
401	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
402	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
403	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
404	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
405	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
406	C₈	—	4	2	no	—
407	C₈	—	4	2	no	—
408	C₂²	—	2	4	no	—
409	C₂²	—	4	2	no	—
410	C₂²	—	4	4	no	—
411	C₂²	—	4	4	no	—
412	C₂²	—	4	2	no	—
413	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
414	C₂²	#548×C₆₇	1	2	yes	—
415	C₂²	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
416	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
417	C₂²	#578×C₆₇	1	2	yes	—
418	C₄	—	2	2	no	—
419	C₄	—	2	4	no	—
420	C₄	—	4	2	no	—
421	C₄	—	4	4	no	—
422	C₄	—	4	2	no	—
423	C₄	—	8	2	no	—
424	C₄	—	8	2	no	—
425	C₄	#450×C₆₇	1	4	yes	—
426	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
427	C₄	#536×C₆₇	1	1	yes	characteristic
428	C₄	#548×C₆₇	1	2	yes	—
429	C₂	—	4	4	no	—
430	C₂	C₆₇×#2436	1	1	yes	characteristic
431	C₂	C₂×#450×C₆₇	1	1	yes	characteristic
432	C₂	C₆₇×#3011	1	1	yes	characteristic
433	C₂	C₆₇×#3359	1	2	yes	—
434	C₂	C₆₇×#3367	1	2	yes	—

	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 ◁ C₂×C₄×Q₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#487×C₆₇	C₂	semidirect
4. C₆₇×#2568	C₂	semidirect
5. C₆₇×#2024	C₂	semidirect
6. C₆₇×#1988	C₂	semidirect
7. C₆₇×#1984	C₂	semidirect
101. C₂×Q₈×C₆₇	SD₁₆	semidirect
211. #25929	C₆₇	direct (N × H)
219. C₂×#487	C₂×C₆₇	semidirect
221. #2568	C₂×C₆₇	semidirect
222. #2024	C₂×C₆₇	semidirect
223. #1988	C₂×C₆₇	semidirect
224. #1984	C₂×C₆₇	semidirect
227. C₆₇	#25929	direct (N × H)
319. C₂×Q₈	SD₁₆×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×Q₈×C₆₇	2
C₂×#487×C₆₇	2
C₂×#461×C₆₇	2
C₆₇×#2568	2
C₆₇×#2024	2
C₆₇×#1988	2
C₆₇×#1984	2
C₆₇×#1877	2
C₄×#102×C₆₇	2
#25929	67

Subgroups — C67×#25929 #30705180

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#25929 #30705180