Subgroups — C₆₇×#25930 #30705181

Order 17152.

1182 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#554×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#486×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#459×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2548	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2025	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1987	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1984	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1879	C₂	1	4	yes	maximal, complemented
9	C₄×#102×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₈×C₆₇	C₂²	1	2	yes	—
11	#554×C₆₇	—	2	4	no	—
12	#554×C₆₇	—	2	4	no	—
13	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#125×C₆₇	C₂²	1	2	yes	—
16	C₂×#124×C₆₇	—	2	1	no	—
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	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
23	#486×C₆₇	—	2	4	no	—
24	#459×C₆₇	—	2	4	no	—
25	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#102×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×D₄×C₆₇	—	2	1	no	—
40	C₂²×D₄×C₆₇	—	2	1	no	—
41	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
42	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
43	C₂³×C₄×C₆₇	—	2	1	no	—
44	C₂³×C₄×C₆₇	—	2	2	no	—
45	C₂×D₈×C₆₇	—	2	4	no	—
46	C₂×D₈×C₆₇	—	2	8	no	—
47	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
48	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
49	C₄⋊Q₈×C₆₇	—	2	2	no	—
50	C₄⋊Q₈×C₆₇	D₄	1	4	yes	—
51	#127×C₆₇	—	2	2	no	—
52	#127×C₆₇	—	2	2	no	—
53	#125×C₆₇	—	2	4	no	—
54	#125×C₆₇	—	2	4	no	—
55	#124×C₆₇	—	4	4	no	—
56	C₂²⋊Q₈×C₆₇	—	2	8	no	—
57	C₂²⋊Q₈×C₆₇	—	2	16	no	—
58	C₄×Q₈×C₆₇	—	2	2	no	—
59	C₄×Q₈×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	8	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	2	no	—
74	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
75	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
76	C₂×C₂²⋊C₄×C₆₇	—	2	4	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	#102×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	1	no	—
87	C₂⁴×C₆₇	—	2	2	no	—
88	C₂×Q₈×C₆₇	—	2	2	no	—
89	C₂×Q₈×C₆₇	—	2	4	no	—
90	C₂×Q₈×C₆₇	—	2	2	no	—
91	C₂×Q₈×C₆₇	—	2	2	no	—
92	C₂×Q₈×C₆₇	—	8	2	no	—
93	C₂×D₄×C₆₇	—	2	1	no	—
94	C₂×D₄×C₆₇	—	2	2	no	—
95	C₂×D₄×C₆₇	—	2	2	no	—
96	C₂×D₄×C₆₇	—	2	4	no	—
97	C₂×D₄×C₆₇	—	2	2	no	—
98	C₂×D₄×C₆₇	—	2	4	no	—
99	C₂×D₄×C₆₇	—	4	2	no	—
100	C₂×D₄×C₆₇	—	4	2	no	—
101	C₂×D₄×C₆₇	—	4	2	no	—
102	C₂×D₄×C₆₇	—	4	2	no	—
103	C₂×D₄×C₆₇	—	8	2	no	—
104	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
105	C₂×D₄×C₆₇	#41	1	4	yes	—
106	C₂²×C₄×C₆₇	—	2	1	no	—
107	C₂²×C₄×C₆₇	—	2	2	no	—
108	C₂²×C₄×C₆₇	—	2	8	no	—
109	C₂²×C₄×C₆₇	—	2	4	no	—
110	C₂²×C₄×C₆₇	—	2	2	no	—
111	C₂²×C₄×C₆₇	—	4	2	no	—
112	C₂²×C₄×C₆₇	—	4	2	no	—
113	C₂²×C₄×C₆₇	—	8	4	no	—
114	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
115	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
116	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
119	D₈×C₆₇	—	4	8	no	—
120	C₂×C₈×C₆₇	—	2	2	no	—
121	C₂×C₈×C₆₇	—	2	2	no	—
122	C₂×C₈×C₆₇	—	2	2	no	—
123	C₂×C₈×C₆₇	—	2	4	no	—
124	C₂×C₈×C₆₇	—	2	2	no	—
125	C₄⋊C₄×C₆₇	—	2	8	no	—
126	C₄⋊C₄×C₆₇	—	2	8	no	—
127	C₄⋊C₄×C₆₇	—	2	2	no	—
128	C₄⋊C₄×C₆₇	—	2	4	no	—
129	C₄⋊C₄×C₆₇	—	2	4	no	—
130	C₄⋊C₄×C₆₇	—	2	1	no	—
131	C₄⋊C₄×C₆₇	—	2	2	no	—
132	C₄⋊C₄×C₆₇	—	2	2	no	—
133	C₄⋊C₄×C₆₇	—	2	2	no	—
134	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
135	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
136	C₂²⋊C₄×C₆₇	—	2	8	no	—
137	C₂²⋊C₄×C₆₇	—	2	4	no	—
138	C₂²⋊C₄×C₆₇	—	4	8	no	—
139	C₂²⋊C₄×C₆₇	—	8	4	no	—
140	C₄²×C₆₇	—	2	2	no	—
141	C₄²×C₆₇	—	2	2	no	—
142	C₄²×C₆₇	—	4	2	no	—
143	C₄²×C₆₇	C₂×D₄	1	2	yes	—
144	C₄²×C₆₇	C₂×D₄	1	2	yes	—
145	C₂³×C₆₇	—	2	2	no	—
146	C₂³×C₆₇	—	2	4	no	—
147	C₂³×C₆₇	—	4	1	no	—
148	C₂³×C₆₇	—	4	1	no	—
149	C₂³×C₆₇	—	4	2	no	—
150	C₂³×C₆₇	—	4	2	no	—
151	C₂³×C₆₇	—	4	4	no	—
152	C₂³×C₆₇	—	4	2	no	—
153	C₂³×C₆₇	—	4	2	no	—
154	C₂³×C₆₇	—	4	4	no	—
155	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
156	Q₈×C₆₇	—	4	4	no	—
157	Q₈×C₆₇	—	8	4	no	—
158	D₄×C₆₇	—	2	2	no	—
159	D₄×C₆₇	—	2	4	no	—
160	D₄×C₆₇	—	4	2	no	—
161	D₄×C₆₇	—	4	2	no	—
162	D₄×C₆₇	—	4	4	no	—
163	D₄×C₆₇	—	8	4	no	—
164	D₄×C₆₇	—	8	4	no	—
165	D₄×C₆₇	#124	1	4	yes	complemented
166	C₂×C₄×C₆₇	—	2	2	no	—
167	C₂×C₄×C₆₇	—	2	2	no	—
168	C₂×C₄×C₆₇	—	2	2	no	—
169	C₂×C₄×C₆₇	—	2	2	no	—
170	C₂×C₄×C₆₇	—	2	4	no	—
171	C₂×C₄×C₆₇	—	2	2	no	—
172	C₂×C₄×C₆₇	—	2	2	no	—
173	C₂×C₄×C₆₇	—	2	4	no	—
174	C₂×C₄×C₆₇	—	4	1	no	—
175	C₂×C₄×C₆₇	—	4	1	no	—
176	C₂×C₄×C₆₇	—	4	4	no	—
177	C₂×C₄×C₆₇	—	4	8	no	—
178	C₂×C₄×C₆₇	—	4	2	no	—
179	C₂×C₄×C₆₇	—	4	2	no	—
180	C₂×C₄×C₆₇	—	4	2	no	—
181	C₂×C₄×C₆₇	—	8	8	no	—
182	C₂×C₄×C₆₇	#120	1	2	yes	—
183	C₂×C₄×C₆₇	#120	1	2	yes	—
184	C₂×C₄×C₆₇	#135	1	2	yes	—
185	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
186	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
187	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
188	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
189	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
190	C₈×C₆₇	—	4	2	no	—
191	C₈×C₆₇	—	4	2	no	—
192	C₂²×C₆₇	—	2	4	no	—
193	C₂²×C₆₇	—	4	4	no	—
194	C₂²×C₆₇	—	4	4	no	—
195	C₂²×C₆₇	—	4	2	no	—
196	C₂²×C₆₇	—	4	2	no	—
197	C₂²×C₆₇	—	8	4	no	—
198	C₂²×C₆₇	—	8	4	no	—
199	C₂²×C₆₇	—	8	2	no	—
200	C₂²×C₆₇	—	8	2	no	—
201	C₂²×C₆₇	—	8	2	no	—
202	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
203	C₂²×C₆₇	#548	1	2	yes	—
204	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
205	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
206	C₂²×C₆₇	#576	1	2	yes	—
207	C₄×C₆₇	—	2	2	no	—
208	C₄×C₆₇	—	4	2	no	—
209	C₄×C₆₇	—	4	2	no	—
210	C₄×C₆₇	—	4	2	no	—
211	C₄×C₆₇	—	4	4	no	—
212	C₄×C₆₇	—	8	2	no	—
213	C₄×C₆₇	#449	1	4	yes	—
214	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
215	C₄×C₆₇	#536	1	1	yes	characteristic
216	C₄×C₆₇	#548	1	2	yes	—
217	#25930	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
218	C₂×C₆₇	—	4	4	no	—
219	C₂×C₆₇	—	16	2	no	—
220	C₂×C₆₇	#2436	1	1	yes	characteristic
221	C₂×C₆₇	C₂×#449	1	1	yes	characteristic
222	C₂×C₆₇	#3008	1	1	yes	characteristic
223	C₂×C₆₇	#3361	1	2	yes	—
224	C₂×C₆₇	#3366	1	2	yes	—
225	C₂×#554	C₂×C₆₇	1	1	yes	characteristic, complemented
226	C₂×#486	C₂×C₆₇	1	1	yes	characteristic, complemented
227	C₂×#459	C₂×C₆₇	1	1	yes	characteristic
228	#2548	C₂×C₆₇	1	1	yes	characteristic, complemented
229	#2025	C₂×C₆₇	1	4	yes	complemented
230	#1987	C₂×C₆₇	1	1	yes	characteristic, complemented
231	#1984	C₂×C₆₇	1	1	yes	characteristic, complemented
232	#1879	C₂×C₆₇	1	4	yes	complemented
233	C₄×#102	C₂×C₆₇	1	1	yes	characteristic, complemented
234	C₆₇	#25930	1	1	yes	characteristic, direct product factor, Sylow
235	C₂²×D₈	C₂²×C₆₇	1	2	yes	—
236	#554	—	2	4	no	—
237	#554	—	2	4	no	—
238	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
239	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
240	C₂×#125	C₂²×C₆₇	1	2	yes	—
241	C₂×#124	—	2	1	no	—
242	C₂×C₂²⋊Q₈	—	2	1	no	—
243	C₂×C₂²⋊Q₈	—	2	2	no	—
244	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
245	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
246	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
247	C₂²×C₄⋊C₄	—	2	1	no	—
248	#486	—	2	4	no	—
249	#459	—	2	4	no	—
250	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
251	C₂×#102	C₂²×C₆₇	1	4	yes	—
252	C₂×#102	C₂²×C₆₇	1	2	yes	—
253	C₂×#102	C₂²×C₆₇	1	2	yes	—
254	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
255	#394	—	2	2	no	—
256	#390	—	2	1	no	—
257	#390	C₂²×C₆₇	1	4	yes	—
258	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
259	#340	C₂²×C₆₇	1	4	yes	—
260	#340	C₂²×C₆₇	1	2	yes	—
261	#336	C₂²×C₆₇	1	4	yes	—
262	C₂²×Q₈	—	2	1	no	—
263	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
264	C₂²×D₄	—	2	1	no	—
265	C₂²×D₄	—	2	1	no	—
266	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
267	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
268	C₂³×C₄	—	2	1	no	—
269	C₂³×C₄	—	2	2	no	—
270	C₂×D₈	—	2	4	no	—
271	C₂×D₈	—	2	8	no	—
272	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
273	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
274	C₄⋊Q₈	—	2	2	no	—
275	C₄⋊Q₈	D₄×C₆₇	1	4	yes	—
276	#127	—	2	2	no	—
277	#127	—	2	2	no	—
278	#125	—	2	4	no	—
279	#125	—	2	4	no	—
280	#124	—	4	4	no	—
281	C₂²⋊Q₈	—	2	8	no	—
282	C₂²⋊Q₈	—	2	16	no	—
283	C₄×Q₈	—	2	2	no	—
284	C₄×Q₈	—	2	2	no	—
285	C₄×D₄	—	2	4	no	—
286	C₄×D₄	—	2	4	no	—
287	C₄×D₄	—	2	2	no	—
288	C₄×D₄	D₄×C₆₇	1	4	yes	—
289	C₂×C₄⋊C₄	—	2	8	no	—
290	C₂×C₄⋊C₄	—	2	2	no	—
291	C₂×C₄⋊C₄	—	2	1	no	—
292	C₂×C₄⋊C₄	—	2	2	no	—
293	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
294	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
295	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
296	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
297	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
298	C₂×C₂²⋊C₄	—	2	2	no	—
299	C₂×C₂²⋊C₄	—	2	1	no	—
300	C₂×C₂²⋊C₄	—	2	4	no	—
301	C₂×C₂²⋊C₄	—	2	4	no	—
302	C₂×C₄²	—	2	1	no	—
303	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
304	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
305	C₄⋊C₈	—	2	2	no	—
306	#102	—	2	4	no	—
307	#102	—	2	4	no	—
308	#102	—	2	8	no	—
309	C₄×C₈	—	2	2	no	—
310	#95	—	2	1	no	—
311	C₂⁴	—	2	1	no	—
312	C₂⁴	—	2	2	no	—
313	C₂×Q₈	—	2	2	no	—
314	C₂×Q₈	—	2	4	no	—
315	C₂×Q₈	—	2	2	no	—
316	C₂×Q₈	—	2	2	no	—
317	C₂×Q₈	—	8	2	no	—
318	C₂×D₄	—	2	1	no	—
319	C₂×D₄	—	2	2	no	—
320	C₂×D₄	—	2	2	no	—
321	C₂×D₄	—	2	4	no	—
322	C₂×D₄	—	2	2	no	—
323	C₂×D₄	—	2	4	no	—
324	C₂×D₄	—	4	2	no	—
325	C₂×D₄	—	4	2	no	—
326	C₂×D₄	—	4	2	no	—
327	C₂×D₄	—	4	2	no	—
328	C₂×D₄	—	8	2	no	—
329	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
330	C₂×D₄	#41×C₆₇	1	4	yes	—
331	C₂²×C₄	—	2	1	no	—
332	C₂²×C₄	—	2	2	no	—
333	C₂²×C₄	—	2	8	no	—
334	C₂²×C₄	—	2	4	no	—
335	C₂²×C₄	—	2	2	no	—
336	C₂²×C₄	—	4	2	no	—
337	C₂²×C₄	—	4	2	no	—
338	C₂²×C₄	—	8	4	no	—
339	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
340	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
341	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
342	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
343	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
344	D₈	—	4	8	no	—
345	C₂×C₈	—	2	2	no	—
346	C₂×C₈	—	2	2	no	—
347	C₂×C₈	—	2	2	no	—
348	C₂×C₈	—	2	4	no	—
349	C₂×C₈	—	2	2	no	—
350	C₄⋊C₄	—	2	8	no	—
351	C₄⋊C₄	—	2	8	no	—
352	C₄⋊C₄	—	2	2	no	—
353	C₄⋊C₄	—	2	4	no	—
354	C₄⋊C₄	—	2	4	no	—
355	C₄⋊C₄	—	2	1	no	—
356	C₄⋊C₄	—	2	2	no	—
357	C₄⋊C₄	—	2	2	no	—
358	C₄⋊C₄	—	2	2	no	—
359	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
360	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
361	C₂²⋊C₄	—	2	8	no	—
362	C₂²⋊C₄	—	2	4	no	—
363	C₂²⋊C₄	—	4	8	no	—
364	C₂²⋊C₄	—	8	4	no	—
365	C₄²	—	2	2	no	—
366	C₄²	—	2	2	no	—
367	C₄²	—	4	2	no	—
368	C₄²	C₂×D₄×C₆₇	1	2	yes	—
369	C₄²	C₂×D₄×C₆₇	1	2	yes	—
370	C₂³	—	2	2	no	—
371	C₂³	—	2	4	no	—
372	C₂³	—	4	1	no	—
373	C₂³	—	4	1	no	—
374	C₂³	—	4	2	no	—
375	C₂³	—	4	2	no	—
376	C₂³	—	4	4	no	—
377	C₂³	—	4	2	no	—
378	C₂³	—	4	2	no	—
379	C₂³	—	4	4	no	—
380	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
381	Q₈	—	4	4	no	—
382	Q₈	—	8	4	no	—
383	D₄	—	2	2	no	—
384	D₄	—	2	4	no	—
385	D₄	—	4	2	no	—
386	D₄	—	4	2	no	—
387	D₄	—	4	4	no	—
388	D₄	—	8	4	no	—
389	D₄	—	8	4	no	—
390	D₄	#124×C₆₇	1	4	yes	complemented
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	2	no	—
395	C₂×C₄	—	2	4	no	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	4	no	—
399	C₂×C₄	—	4	1	no	—
400	C₂×C₄	—	4	1	no	—
401	C₂×C₄	—	4	4	no	—
402	C₂×C₄	—	4	8	no	—
403	C₂×C₄	—	4	2	no	—
404	C₂×C₄	—	4	2	no	—
405	C₂×C₄	—	4	2	no	—
406	C₂×C₄	—	8	8	no	—
407	C₂×C₄	#120×C₆₇	1	2	yes	—
408	C₂×C₄	#120×C₆₇	1	2	yes	—
409	C₂×C₄	#135×C₆₇	1	2	yes	—
410	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
411	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
412	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
413	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
414	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
415	C₈	—	4	2	no	—
416	C₈	—	4	2	no	—
417	C₂²	—	2	4	no	—
418	C₂²	—	4	4	no	—
419	C₂²	—	4	4	no	—
420	C₂²	—	4	2	no	—
421	C₂²	—	4	2	no	—
422	C₂²	—	8	4	no	—
423	C₂²	—	8	4	no	—
424	C₂²	—	8	2	no	—
425	C₂²	—	8	2	no	—
426	C₂²	—	8	2	no	—
427	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
428	C₂²	#548×C₆₇	1	2	yes	—
429	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
430	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
431	C₂²	#576×C₆₇	1	2	yes	—
432	C₄	—	2	2	no	—
433	C₄	—	4	2	no	—
434	C₄	—	4	2	no	—
435	C₄	—	4	2	no	—
436	C₄	—	4	4	no	—
437	C₄	—	8	2	no	—
438	C₄	#449×C₆₇	1	4	yes	—
439	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
440	C₄	#536×C₆₇	1	1	yes	characteristic
441	C₄	#548×C₆₇	1	2	yes	—
442	C₂	—	4	4	no	—
443	C₂	—	16	2	no	—
444	C₂	C₆₇×#2436	1	1	yes	characteristic
445	C₂	C₂×#449×C₆₇	1	1	yes	characteristic
446	C₂	C₆₇×#3008	1	1	yes	characteristic
447	C₂	C₆₇×#3361	1	2	yes	—
448	C₂	C₆₇×#3366	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#459×C₆₇	C₂
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×#554×C₆₇	C₂	semidirect
2. C₂×#486×C₆₇	C₂	semidirect
4. C₆₇×#2548	C₂	semidirect
5. C₆₇×#2025	C₂	semidirect
6. C₆₇×#1987	C₂	semidirect
7. C₆₇×#1984	C₂	semidirect
8. C₆₇×#1879	C₂	semidirect
9. C₄×#102×C₆₇	C₂	semidirect
165. D₄×C₆₇	#124	semidirect
217. #25930	C₆₇	direct (N × H)
225. C₂×#554	C₂×C₆₇	semidirect
226. C₂×#486	C₂×C₆₇	semidirect
228. #2548	C₂×C₆₇	semidirect
229. #2025	C₂×C₆₇	semidirect
230. #1987	C₂×C₆₇	semidirect
231. #1984	C₂×C₆₇	semidirect
232. #1879	C₂×C₆₇	semidirect
233. C₄×#102	C₂×C₆₇	semidirect
234. C₆₇	#25930	direct (N × H)
390. D₄	#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×#554×C₆₇	2
C₂×#486×C₆₇	2
C₂×#459×C₆₇	2
C₆₇×#2548	2
C₆₇×#2025	2
C₆₇×#1987	2
C₆₇×#1984	2
C₆₇×#1879	2
C₄×#102×C₆₇	2
#25930	67

Subgroups — C67×#25930 #30705181

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#25930 #30705181