Subgroups — C₆₇×#25935 #30705186

Order 17152.

1106 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#557×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#486×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#460×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2548	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#2025	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×#1880	C₂	1	4	yes	maximal, complemented
9	C₄×#103×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
11	#557×C₆₇	—	2	4	no	—
12	#557×C₆₇	—	2	4	no	—
13	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
14	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#125×C₆₇	—	2	2	no	—
17	C₂×#125×C₆₇	C₂²	1	2	yes	—
18	C₂×#124×C₆₇	—	2	1	no	—
19	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
20	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
23	#486×C₆₇	—	2	4	no	—
24	#460×C₆₇	—	2	4	no	—
25	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#103×C₆₇	C₂²	1	2	yes	—
27	C₂×#102×C₆₇	C₂²	1	2	yes	—
28	C₂×#102×C₆₇	C₂²	1	4	yes	—
29	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
30	#394×C₆₇	—	2	2	no	—
31	#390×C₆₇	—	2	1	no	—
32	#390×C₆₇	C₂²	1	4	yes	—
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	1	no	—
38	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
39	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
40	C₂²×D₄×C₆₇	—	2	1	no	—
41	C₂²×D₄×C₆₇	—	2	1	no	—
42	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
43	C₂×SD₁₆×C₆₇	—	2	4	no	—
44	C₂×SD₁₆×C₆₇	—	2	8	no	—
45	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
46	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
47	C₄⋊Q₈×C₆₇	—	2	8	no	—
48	C₄⋊Q₈×C₆₇	—	2	2	no	—
49	C₄⋊Q₈×C₆₇	D₄	1	4	yes	—
50	#127×C₆₇	—	2	2	no	—
51	#127×C₆₇	—	2	2	no	—
52	#125×C₆₇	—	2	16	no	—
53	#125×C₆₇	—	2	4	no	—
54	#125×C₆₇	—	2	4	no	—
55	#124×C₆₇	—	4	4	no	—
56	C₄×Q₈×C₆₇	—	2	8	no	—
57	C₄×Q₈×C₆₇	—	2	4	no	—
58	C₄×Q₈×C₆₇	—	2	4	no	—
59	C₄×Q₈×C₆₇	—	2	2	no	—
60	C₄×Q₈×C₆₇	—	2	2	no	—
61	C₄×Q₈×C₆₇	—	2	2	no	—
62	C₄×Q₈×C₆₇	D₄	1	4	yes	—
63	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
64	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
65	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
66	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
67	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
68	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
70	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
71	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
73	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
74	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
75	C₂×C₄²×C₆₇	—	2	2	no	—
76	C₂×C₄²×C₆₇	—	2	1	no	—
77	C₂×C₄²×C₆₇	—	2	1	no	—
78	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
79	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
80	C₄⋊C₈×C₆₇	—	2	2	no	—
81	#103×C₆₇	—	2	4	no	—
82	#102×C₆₇	—	2	4	no	—
83	#102×C₆₇	—	2	8	no	—
84	C₄×C₈×C₆₇	—	2	2	no	—
85	#95×C₆₇	—	2	1	no	—
86	C₂⁴×C₆₇	—	2	2	no	—
87	C₂×Q₈×C₆₇	—	2	2	no	—
88	C₂×Q₈×C₆₇	—	2	4	no	—
89	C₂×Q₈×C₆₇	—	2	2	no	—
90	C₂×Q₈×C₆₇	—	2	2	no	—
91	C₂×Q₈×C₆₇	—	2	1	no	—
92	C₂×Q₈×C₆₇	—	2	2	no	—
93	C₂×Q₈×C₆₇	—	4	2	no	—
94	C₂×Q₈×C₆₇	—	4	2	no	—
95	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
96	C₂×Q₈×C₆₇	#41	1	4	yes	—
97	C₂×D₄×C₆₇	—	2	2	no	—
98	C₂×D₄×C₆₇	—	2	2	no	—
99	C₂×D₄×C₆₇	—	2	4	no	—
100	C₂×D₄×C₆₇	—	2	4	no	—
101	C₂×D₄×C₆₇	—	4	2	no	—
102	C₂×D₄×C₆₇	—	4	2	no	—
103	C₂×D₄×C₆₇	—	4	2	no	—
104	C₂×D₄×C₆₇	—	4	2	no	—
105	C₂×D₄×C₆₇	—	8	2	no	—
106	C₂²×C₄×C₆₇	—	2	1	no	—
107	C₂²×C₄×C₆₇	—	2	1	no	—
108	C₂²×C₄×C₆₇	—	2	2	no	—
109	C₂²×C₄×C₆₇	—	2	2	no	—
110	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
112	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
115	SD₁₆×C₆₇	—	4	8	no	—
116	C₂×C₈×C₆₇	—	2	2	no	—
117	C₂×C₈×C₆₇	—	2	2	no	—
118	C₂×C₈×C₆₇	—	2	2	no	—
119	C₂×C₈×C₆₇	—	2	4	no	—
120	C₂×C₈×C₆₇	—	2	2	no	—
121	C₄⋊C₄×C₆₇	—	2	8	no	—
122	C₄⋊C₄×C₆₇	—	2	8	no	—
123	C₄⋊C₄×C₆₇	—	2	8	no	—
124	C₄⋊C₄×C₆₇	—	2	4	no	—
125	C₄⋊C₄×C₆₇	—	2	4	no	—
126	C₄⋊C₄×C₆₇	—	2	2	no	—
127	C₄⋊C₄×C₆₇	—	2	4	no	—
128	C₄⋊C₄×C₆₇	—	2	1	no	—
129	C₄⋊C₄×C₆₇	—	2	2	no	—
130	C₄⋊C₄×C₆₇	—	2	2	no	—
131	C₄⋊C₄×C₆₇	—	2	2	no	—
132	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
133	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
134	C₂²⋊C₄×C₆₇	—	4	8	no	—
135	C₂²⋊C₄×C₆₇	—	8	4	no	—
136	C₄²×C₆₇	—	2	8	no	—
137	C₄²×C₆₇	—	2	4	no	—
138	C₄²×C₆₇	—	2	2	no	—
139	C₄²×C₆₇	—	2	2	no	—
140	C₄²×C₆₇	—	4	2	no	—
141	C₄²×C₆₇	C₂×D₄	1	2	yes	—
142	C₄²×C₆₇	C₂×D₄	1	2	yes	—
143	C₂³×C₆₇	—	4	2	no	—
144	C₂³×C₆₇	—	4	4	no	—
145	C₂³×C₆₇	—	4	2	no	—
146	C₂³×C₆₇	—	4	2	no	—
147	C₂³×C₆₇	—	4	4	no	—
148	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
149	Q₈×C₆₇	—	2	2	no	—
150	Q₈×C₆₇	—	2	4	no	—
151	Q₈×C₆₇	—	4	4	no	—
152	Q₈×C₆₇	—	4	8	no	—
153	Q₈×C₆₇	#124	1	4	yes	complemented
154	D₄×C₆₇	—	4	2	no	—
155	D₄×C₆₇	—	4	2	no	—
156	D₄×C₆₇	—	4	4	no	—
157	D₄×C₆₇	—	8	4	no	—
158	D₄×C₆₇	—	8	4	no	—
159	C₂×C₄×C₆₇	—	2	2	no	—
160	C₂×C₄×C₆₇	—	2	4	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	—	2	2	no	—
165	C₂×C₄×C₆₇	—	2	4	no	—
166	C₂×C₄×C₆₇	—	2	2	no	—
167	C₂×C₄×C₆₇	—	2	2	no	—
168	C₂×C₄×C₆₇	—	2	4	no	—
169	C₂×C₄×C₆₇	—	4	2	no	—
170	C₂×C₄×C₆₇	—	4	2	no	—
171	C₂×C₄×C₆₇	—	4	2	no	—
172	C₂×C₄×C₆₇	—	4	2	no	—
173	C₂×C₄×C₆₇	—	4	2	no	—
174	C₂×C₄×C₆₇	—	4	2	no	—
175	C₂×C₄×C₆₇	#120	1	2	yes	—
176	C₂×C₄×C₆₇	#120	1	2	yes	—
177	C₂×C₄×C₆₇	#135	1	2	yes	—
178	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
179	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
180	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
181	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
182	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
183	C₈×C₆₇	—	4	2	no	—
184	C₈×C₆₇	—	4	2	no	—
185	C₂²×C₆₇	—	8	4	no	—
186	C₂²×C₆₇	—	8	4	no	—
187	C₂²×C₆₇	—	8	2	no	—
188	C₂²×C₆₇	—	8	2	no	—
189	C₂²×C₆₇	—	8	2	no	—
190	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
191	C₂²×C₆₇	#548	1	2	yes	—
192	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
193	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
194	C₂²×C₆₇	#577	1	2	yes	—
195	C₄×C₆₇	—	2	4	no	—
196	C₄×C₆₇	—	2	2	no	—
197	C₄×C₆₇	—	4	2	no	—
198	C₄×C₆₇	—	4	4	no	—
199	C₄×C₆₇	—	4	2	no	—
200	C₄×C₆₇	—	4	2	no	—
201	C₄×C₆₇	—	8	2	no	—
202	C₄×C₆₇	#449	1	4	yes	—
203	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
204	C₄×C₆₇	#536	1	1	yes	characteristic
205	C₄×C₆₇	#548	1	2	yes	—
206	#25935	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
207	C₂×C₆₇	—	16	2	no	—
208	C₂×C₆₇	#2436	1	1	yes	characteristic
209	C₂×C₆₇	C₂×#449	1	1	yes	characteristic
210	C₂×C₆₇	#3010	1	1	yes	characteristic
211	C₂×C₆₇	#3365	1	2	yes	—
212	C₂×C₆₇	#3368	1	2	yes	—
213	C₂×#557	C₂×C₆₇	1	1	yes	characteristic, complemented
214	C₂×#486	C₂×C₆₇	1	1	yes	characteristic
215	C₂×#460	C₂×C₆₇	1	1	yes	characteristic
216	#2548	C₂×C₆₇	1	1	yes	characteristic
217	#2025	C₂×C₆₇	1	4	yes	complemented
218	#1988	C₂×C₆₇	1	1	yes	characteristic, complemented
219	#1984	C₂×C₆₇	1	1	yes	characteristic, complemented
220	#1880	C₂×C₆₇	1	4	yes	complemented
221	C₄×#103	C₂×C₆₇	1	1	yes	characteristic, complemented
222	C₆₇	#25935	1	1	yes	characteristic, direct product factor, Sylow
223	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
224	#557	—	2	4	no	—
225	#557	—	2	4	no	—
226	C₂×C₄⋊Q₈	—	2	1	no	—
227	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
228	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
229	C₂×#125	—	2	2	no	—
230	C₂×#125	C₂²×C₆₇	1	2	yes	—
231	C₂×#124	—	2	1	no	—
232	C₂×C₄×Q₈	—	2	1	no	—
233	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
234	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
235	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
236	#486	—	2	4	no	—
237	#460	—	2	4	no	—
238	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
239	C₂×#103	C₂²×C₆₇	1	2	yes	—
240	C₂×#102	C₂²×C₆₇	1	4	yes	—
241	C₂×#102	C₂²×C₆₇	1	2	yes	—
242	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
243	#394	—	2	2	no	—
244	#390	—	2	1	no	—
245	#390	C₂²×C₆₇	1	4	yes	—
246	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
247	#340	C₂²×C₆₇	1	4	yes	—
248	#340	C₂²×C₆₇	1	2	yes	—
249	#336	C₂²×C₆₇	1	4	yes	—
250	C₂²×Q₈	—	2	1	no	—
251	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
252	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
253	C₂²×D₄	—	2	1	no	—
254	C₂²×D₄	—	2	1	no	—
255	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
256	C₂×SD₁₆	—	2	4	no	—
257	C₂×SD₁₆	—	2	8	no	—
258	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
259	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
260	C₄⋊Q₈	—	2	8	no	—
261	C₄⋊Q₈	—	2	2	no	—
262	C₄⋊Q₈	D₄×C₆₇	1	4	yes	—
263	#127	—	2	2	no	—
264	#127	—	2	2	no	—
265	#125	—	2	16	no	—
266	#125	—	2	4	no	—
267	#125	—	2	4	no	—
268	#124	—	4	4	no	—
269	C₄×Q₈	—	2	8	no	—
270	C₄×Q₈	—	2	4	no	—
271	C₄×Q₈	—	2	4	no	—
272	C₄×Q₈	—	2	2	no	—
273	C₄×Q₈	—	2	2	no	—
274	C₄×Q₈	—	2	2	no	—
275	C₄×Q₈	D₄×C₆₇	1	4	yes	—
276	C₂×C₄⋊C₄	—	2	2	no	—
277	C₂×C₄⋊C₄	—	2	1	no	—
278	C₂×C₄⋊C₄	—	2	2	no	—
279	C₂×C₄⋊C₄	—	2	1	no	—
280	C₂×C₄⋊C₄	—	2	2	no	—
281	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
282	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
283	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
284	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
285	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
286	C₂×C₂²⋊C₄	—	2	4	no	—
287	C₂×C₂²⋊C₄	—	2	4	no	—
288	C₂×C₄²	—	2	2	no	—
289	C₂×C₄²	—	2	1	no	—
290	C₂×C₄²	—	2	1	no	—
291	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
292	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
293	C₄⋊C₈	—	2	2	no	—
294	#103	—	2	4	no	—
295	#102	—	2	4	no	—
296	#102	—	2	8	no	—
297	C₄×C₈	—	2	2	no	—
298	#95	—	2	1	no	—
299	C₂⁴	—	2	2	no	—
300	C₂×Q₈	—	2	2	no	—
301	C₂×Q₈	—	2	4	no	—
302	C₂×Q₈	—	2	2	no	—
303	C₂×Q₈	—	2	2	no	—
304	C₂×Q₈	—	2	1	no	—
305	C₂×Q₈	—	2	2	no	—
306	C₂×Q₈	—	4	2	no	—
307	C₂×Q₈	—	4	2	no	—
308	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
309	C₂×Q₈	#41×C₆₇	1	4	yes	—
310	C₂×D₄	—	2	2	no	—
311	C₂×D₄	—	2	2	no	—
312	C₂×D₄	—	2	4	no	—
313	C₂×D₄	—	2	4	no	—
314	C₂×D₄	—	4	2	no	—
315	C₂×D₄	—	4	2	no	—
316	C₂×D₄	—	4	2	no	—
317	C₂×D₄	—	4	2	no	—
318	C₂×D₄	—	8	2	no	—
319	C₂²×C₄	—	2	1	no	—
320	C₂²×C₄	—	2	1	no	—
321	C₂²×C₄	—	2	2	no	—
322	C₂²×C₄	—	2	2	no	—
323	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
324	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
325	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
326	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
327	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
328	SD₁₆	—	4	8	no	—
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	—	2	2	no	—
331	C₂×C₈	—	2	2	no	—
332	C₂×C₈	—	2	4	no	—
333	C₂×C₈	—	2	2	no	—
334	C₄⋊C₄	—	2	8	no	—
335	C₄⋊C₄	—	2	8	no	—
336	C₄⋊C₄	—	2	8	no	—
337	C₄⋊C₄	—	2	4	no	—
338	C₄⋊C₄	—	2	4	no	—
339	C₄⋊C₄	—	2	2	no	—
340	C₄⋊C₄	—	2	4	no	—
341	C₄⋊C₄	—	2	1	no	—
342	C₄⋊C₄	—	2	2	no	—
343	C₄⋊C₄	—	2	2	no	—
344	C₄⋊C₄	—	2	2	no	—
345	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
346	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
347	C₂²⋊C₄	—	4	8	no	—
348	C₂²⋊C₄	—	8	4	no	—
349	C₄²	—	2	8	no	—
350	C₄²	—	2	4	no	—
351	C₄²	—	2	2	no	—
352	C₄²	—	2	2	no	—
353	C₄²	—	4	2	no	—
354	C₄²	C₂×D₄×C₆₇	1	2	yes	—
355	C₄²	C₂×D₄×C₆₇	1	2	yes	—
356	C₂³	—	4	2	no	—
357	C₂³	—	4	4	no	—
358	C₂³	—	4	2	no	—
359	C₂³	—	4	2	no	—
360	C₂³	—	4	4	no	—
361	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
362	Q₈	—	2	2	no	—
363	Q₈	—	2	4	no	—
364	Q₈	—	4	4	no	—
365	Q₈	—	4	8	no	—
366	Q₈	#124×C₆₇	1	4	yes	complemented
367	D₄	—	4	2	no	—
368	D₄	—	4	2	no	—
369	D₄	—	4	4	no	—
370	D₄	—	8	4	no	—
371	D₄	—	8	4	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	4	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	4	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	2	4	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	4	2	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	4	2	no	—
387	C₂×C₄	—	4	2	no	—
388	C₂×C₄	#120×C₆₇	1	2	yes	—
389	C₂×C₄	#120×C₆₇	1	2	yes	—
390	C₂×C₄	#135×C₆₇	1	2	yes	—
391	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
392	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
393	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
394	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
395	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
396	C₈	—	4	2	no	—
397	C₈	—	4	2	no	—
398	C₂²	—	8	2	no	—
399	C₂²	—	8	4	no	—
400	C₂²	—	8	4	no	—
401	C₂²	—	8	2	no	—
402	C₂²	—	8	2	no	—
403	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
404	C₂²	#548×C₆₇	1	2	yes	—
405	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
406	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
407	C₂²	#577×C₆₇	1	2	yes	—
408	C₄	—	2	2	no	—
409	C₄	—	2	4	no	—
410	C₄	—	4	2	no	—
411	C₄	—	4	2	no	—
412	C₄	—	4	2	no	—
413	C₄	—	4	4	no	—
414	C₄	—	8	2	no	—
415	C₄	#449×C₆₇	1	4	yes	—
416	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
417	C₄	#536×C₆₇	1	1	yes	characteristic
418	C₄	#548×C₆₇	1	2	yes	—
419	C₂	—	16	2	no	—
420	C₂	C₆₇×#2436	1	1	yes	characteristic
421	C₂	C₂×#449×C₆₇	1	1	yes	characteristic
422	C₂	C₆₇×#3010	1	1	yes	characteristic
423	C₂	C₆₇×#3365	1	2	yes	—
424	C₂	C₆₇×#3368	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#127×C₆₇	C₂²
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
1. C₂×#557×C₆₇	C₂	semidirect
5. C₆₇×#2025	C₂	semidirect
6. C₆₇×#1988	C₂	semidirect
7. C₆₇×#1984	C₂	semidirect
8. C₆₇×#1880	C₂	semidirect
9. C₄×#103×C₆₇	C₂	semidirect
153. Q₈×C₆₇	#124	semidirect
206. #25935	C₆₇	direct (N × H)
213. C₂×#557	C₂×C₆₇	semidirect
217. #2025	C₂×C₆₇	semidirect
218. #1988	C₂×C₆₇	semidirect
219. #1984	C₂×C₆₇	semidirect
220. #1880	C₂×C₆₇	semidirect
221. C₄×#103	C₂×C₆₇	semidirect
222. C₆₇	#25935	direct (N × H)
366. Q₈	#124×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#557×C₆₇	2
C₂×#486×C₆₇	2
C₂×#460×C₆₇	2
C₆₇×#2548	2
C₆₇×#2025	2
C₆₇×#1988	2
C₆₇×#1984	2
C₆₇×#1880	2
C₄×#103×C₆₇	2
#25935	67

Subgroups — C67×#25935 #30705186

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#25935 #30705186