Subgroups — #30864497

Order 17152.

822 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863525	C₂	1	2	yes	maximal
2	C₄×#14524042	C₂	1	1	yes	maximal, characteristic
3	#22561143	C₂	1	2	yes	maximal
4	#22560927	C₂	1	1	yes	maximal, characteristic
5	#22560924	C₂	1	2	yes	maximal
6	#22560360	C₂	1	1	yes	maximal, characteristic
7	#22560079	C₂	1	1	yes	maximal, characteristic
8	#22560078	C₂	1	1	yes	maximal, characteristic
9	#22560075	C₂	1	2	yes	maximal
10	C₄×C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#1939	C₂	1	1	yes	maximal, characteristic
12	C₂×#14524042	C₄	1	1	yes	characteristic
13	C₂×C₄×Dic₁₃₄	C₂²	1	2	yes	—
14	#16863525	—	2	8	no	—
15	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×Dic₆₇⋊C₈	C₂²	1	2	yes	—
17	C₂×Dic₆₇⋊C₈	C₂²	1	2	yes	—
18	C₂×Dic₆₇⋊C₈	C₂²	1	2	yes	—
19	C₂×#14523954	C₂²	1	2	yes	—
20	C₂×#14523953	C₂²	1	2	yes	—
21	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
26	#16863300	C₄	1	2	yes	—
27	#16863295	C₄	1	1	yes	characteristic
28	#16863293	C₄	1	2	yes	—
29	#16863292	C₄	1	1	yes	characteristic
30	#16863265	C₄	1	1	yes	characteristic
31	C₄×Dic₆₇⋊C₄	C₂²	1	2	yes	—
32	C₄×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
33	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
34	#16863151	C₂²	1	2	yes	—
35	#16863150	C₂²	1	2	yes	—
36	#16863150	C₂²	1	2	yes	—
37	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
39	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
40	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
41	#336×C₆₇	C₂²	1	2	yes	—
42	C₂²×Dic₁₃₄	C₂×C₄	1	2	yes	—
43	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
44	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
45	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
46	#14524042	—	2	4	no	—
47	C₄×Dic₁₃₄	—	2	4	no	—
48	C₄×Dic₁₃₄	—	2	4	no	—
49	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
52	C₂×Dic₆₇⋊C₄	C₂×C₄	1	2	yes	—
53	C₂×Dic₆₇⋊C₄	C₂×C₄	1	2	yes	—
54	C₂×Dic₆₇⋊C₄	C₂×C₄	1	2	yes	—
55	C₂×Dic₆₇⋊C₄	C₂×C₄	1	2	yes	—
56	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
61	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
62	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
63	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
64	Dic₆₇⋊C₈	—	2	4	no	—
65	Dic₆₇⋊C₈	—	2	4	no	—
66	Dic₆₇⋊C₈	—	2	4	no	—
67	#14523954	—	2	4	no	—
68	#14523953	—	2	4	no	—
69	C₄×C₆₇⋊C₈	Q₈	1	4	yes	complemented
70	C₈×Dic₆₇	—	2	2	no	—
71	C₈×Dic₆₇	—	2	2	no	—
72	C₈×Dic₆₇	D₄	1	4	yes	—
73	C₈×Dic₆₇	Q₈	1	4	yes	complemented
74	C₆₇⋊#95	C₂×C₄	1	2	yes	—
75	C₆₇⋊#95	C₂×C₄	1	2	yes	—
76	C₆₇⋊#95	C₂×C₄	1	2	yes	—
77	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
78	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
79	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
82	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
83	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
84	C₄⋊C₈×C₆₇	—	2	4	no	—
85	C₄⋊C₈×C₆₇	—	2	2	no	—
86	C₄×C₈×C₆₇	—	2	2	no	—
87	#95×C₆₇	C₂×C₄	1	2	yes	—
88	C₂×Dic₁₃₄	—	2	4	no	—
89	C₂×Dic₁₃₄	—	2	4	no	—
90	C₂×Dic₁₃₄	—	2	4	no	—
91	C₂²×Dic₆₇	C₂²×C₄	1	2	yes	—
92	C₂²×Dic₆₇	C₂²×C₄	1	2	yes	—
93	C₂²×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
94	C₂²×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
95	C₂²×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
96	C₂²×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₆₇⋊C₈	—	2	1	no	—
98	C₂×C₆₇⋊C₈	—	2	1	no	—
99	C₂×C₆₇⋊C₈	—	2	2	no	—
100	C₂×C₆₇⋊C₈	—	2	2	no	—
101	C₂×C₆₇⋊C₈	—	2	2	no	—
102	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
103	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
104	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
105	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
106	C₄⋊Dic₆₇	—	2	2	no	—
107	C₄⋊Dic₆₇	—	2	2	no	—
108	C₄⋊Dic₆₇	—	2	4	no	—
109	Dic₆₇⋊C₄	—	2	4	no	—
110	Dic₆₇⋊C₄	—	2	4	no	—
111	Dic₆₇⋊C₄	—	2	4	no	—
112	Dic₆₇⋊C₄	—	2	4	no	—
113	C₄×Dic₆₇	—	2	2	no	—
114	C₄×Dic₆₇	—	2	2	no	—
115	C₄×Dic₆₇	—	2	2	no	—
116	C₄×Dic₆₇	—	2	2	no	—
117	C₄×Dic₆₇	—	2	2	no	—
118	C₄×Dic₆₇	M₁₆	1	4	yes	—
119	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
120	C₄×Dic₆₇	C₂×Q₈	1	2	yes	—
121	C₄×Dic₆₇	#41	1	2	yes	—
122	C₄×Dic₆₇	#41	1	2	yes	—
123	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
124	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
125	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
126	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
127	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
128	C₂×C₈×C₆₇	—	2	1	no	—
129	C₂×C₈×C₆₇	—	2	1	no	—
130	C₂×C₈×C₆₇	—	2	1	no	—
131	C₂×C₈×C₆₇	—	2	1	no	—
132	C₂×C₈×C₆₇	—	2	2	no	—
133	C₂×C₈×C₆₇	—	2	2	no	—
134	C₂×C₈×C₆₇	—	2	2	no	—
135	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
136	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
137	C₄⋊C₄×C₆₇	—	2	2	no	—
138	C₄⋊C₄×C₆₇	—	2	2	no	—
139	C₄²×C₆₇	—	2	2	no	—
140	C₄²×C₆₇	—	2	2	no	—
141	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
142	C₄²×C₆₇	#41	1	2	yes	—
143	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₆₇	—	2	1	no	—
145	C₂×C₄×C₆₇	—	2	1	no	—
146	C₂×C₄×C₆₇	—	2	2	no	—
147	C₂×C₄×C₆₇	—	2	2	no	—
148	C₂×C₄×C₆₇	—	2	2	no	—
149	C₂×C₄×C₆₇	—	2	2	no	—
150	C₂×C₄×C₆₇	—	2	2	no	—
151	C₂×C₄×C₆₇	—	2	2	no	—
152	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
153	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
154	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
155	C₂×C₄×C₆₇	#131	1	2	yes	—
156	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
158	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
159	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
160	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
161	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
162	C₂×Dic₆₇	—	2	1	no	—
163	C₂×Dic₆₇	—	2	1	no	—
164	C₂×Dic₆₇	—	2	1	no	—
165	C₂×Dic₆₇	—	2	1	no	—
166	C₂×Dic₆₇	—	2	2	no	—
167	C₂×Dic₆₇	—	2	2	no	—
168	C₂×Dic₆₇	—	2	2	no	—
169	C₂×Dic₆₇	—	2	2	no	—
170	C₂×Dic₆₇	—	2	2	no	—
171	C₂×Dic₆₇	—	2	2	no	—
172	C₂×Dic₆₇	C₄×D₄	1	2	yes	—
173	C₂×Dic₆₇	C₄×D₄	1	2	yes	—
174	C₂×Dic₆₇	C₄×Q₈	1	2	yes	—
175	C₂×Dic₆₇	C₄×Q₈	1	2	yes	—
176	C₂×Dic₆₇	C₂×M₁₆	1	2	yes	—
177	C₂×Dic₆₇	C₂×M₁₆	1	2	yes	—
178	C₂×Dic₆₇	#131	1	2	yes	—
179	C₂×Dic₆₇	#131	1	2	yes	—
180	Dic₁₃₄	—	4	8	no	—
181	C₈×C₆₇	—	2	2	no	—
182	C₈×C₆₇	—	2	2	no	—
183	C₈×C₆₇	—	4	2	no	—
184	C₆₇⋊C₈	—	2	2	no	—
185	C₆₇⋊C₈	—	4	2	no	—
186	C₆₇⋊C₈	C₄⋊Q₈	1	4	yes	complemented
187	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
188	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
189	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
190	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
191	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
192	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
193	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
194	C₄×C₆₇	—	2	2	no	—
195	C₄×C₆₇	—	4	2	no	—
196	C₄×C₆₇	—	4	2	no	—
197	C₄×C₆₇	#446	1	4	yes	complemented
198	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
199	C₄×C₆₇	#532	1	1	yes	characteristic
200	C₄×C₆₇	#533	1	1	yes	characteristic
201	C₄×C₆₇	#533	1	1	yes	characteristic
202	Dic₆₇	—	2	2	no	—
203	Dic₆₇	—	2	2	no	—
204	Dic₆₇	—	4	2	no	—
205	Dic₆₇	—	4	2	no	—
206	Dic₆₇	#436	1	4	yes	—
207	Dic₆₇	#446	1	4	yes	complemented
208	#21712	—	67	1	no	maximal, Sylow
209	C₂×C₆₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
210	C₂×C₆₇	C₂×#436	1	1	yes	characteristic
211	C₂×C₆₇	#2931	1	1	yes	characteristic
212	C₂×C₆₇	C₂×#446	1	1	yes	characteristic
213	C₂×C₆₇	C₂×#446	1	1	yes	characteristic
214	C₂×C₆₇	#2960	1	1	yes	characteristic
215	C₂×C₆₇	#2960	1	1	yes	characteristic
216	C₂×#446	—	67	2	no	—
217	C₄×C₄⋊Q₈	—	67	1	no	—
218	#1984	—	67	2	no	—
219	#1939	—	67	2	no	—
220	#1939	—	67	1	no	—
221	#1939	D₆₇	1	1	yes	characteristic
222	#1842	—	67	1	no	—
223	#1770	—	67	1	no	—
224	#1770	—	67	1	no	—
225	#1770	—	67	2	no	—
226	C₄²×C₈	—	67	1	no	—
227	C₆₇	#21712	1	1	yes	characteristic, complemented, Sylow
228	C₂×C₄⋊Q₈	—	67	1	no	—
229	C₂×C₄×Q₈	—	67	2	no	—
230	#446	—	134	8	no	—
231	C₂×C₄⋊C₈	—	67	2	no	—
232	C₂×C₄⋊C₈	—	67	2	no	—
233	C₂×C₄⋊C₈	—	67	2	no	—
234	C₂×C₄⋊C₈	—	67	1	no	—
235	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
236	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
237	C₂×#97	—	67	2	no	—
238	C₂×#97	—	67	2	no	—
239	C₂×C₄×C₈	—	67	1	no	—
240	C₂×C₄×C₈	—	67	1	no	—
241	C₂×C₄×C₈	—	67	1	no	—
242	C₂×C₄×C₈	—	67	1	no	—
243	C₂×C₄×C₈	—	67	1	no	—
244	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
245	#391	—	67	2	no	—
246	#389	—	67	1	no	—
247	#389	—	67	1	no	—
248	#389	—	67	2	no	—
249	#384	—	67	1	no	—
250	C₄×C₄⋊C₄	—	67	1	no	—
251	C₄×C₄⋊C₄	—	67	2	no	—
252	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
253	C₄³	—	67	1	no	—
254	#336	—	67	2	no	—
255	#336	—	67	2	no	—
256	#336	—	67	2	no	—
257	#336	C₂×D₆₇	1	2	yes	—
258	C₂²×Q₈	—	67	2	no	—
259	C₂²×C₈	—	67	1	no	—
260	C₂²×C₈	—	67	1	no	—
261	C₂²×C₈	—	67	2	no	—
262	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
263	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
264	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
265	C₄⋊Q₈	—	134	4	no	—
266	C₄×Q₈	—	134	4	no	—
267	C₄×Q₈	—	134	4	no	—
268	C₂×C₄⋊C₄	—	67	2	no	—
269	C₂×C₄⋊C₄	—	67	2	no	—
270	C₂×C₄⋊C₄	—	67	2	no	—
271	C₂×C₄⋊C₄	—	67	2	no	—
272	C₂×C₄⋊C₄	—	67	1	no	—
273	C₂×C₄⋊C₄	—	67	2	no	—
274	C₂×C₄⋊C₄	—	67	1	no	—
275	C₂×C₄⋊C₄	C₄×D₆₇	1	1	yes	characteristic
276	C₂×C₄⋊C₄	C₄×D₆₇	1	1	yes	characteristic
277	C₂×C₄²	—	67	1	no	—
278	C₂×C₄²	—	67	1	no	—
279	C₂×C₄²	—	67	1	no	—
280	C₂×C₄²	—	67	1	no	—
281	C₂×C₄²	—	67	1	no	—
282	C₂×C₄²	—	67	1	no	—
283	C₂×C₄²	—	67	2	no	—
284	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
285	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
286	C₄⋊C₈	—	2	4	no	—
287	C₄⋊C₈	—	2	2	no	—
288	C₄⋊C₈	—	134	4	no	—
289	C₄⋊C₈	—	134	4	no	—
290	C₄⋊C₈	—	134	4	no	—
291	C₄⋊C₈	—	134	2	no	—
292	#97	—	134	4	no	—
293	#97	—	134	4	no	—
294	C₄×C₈	—	2	2	no	—
295	C₄×C₈	—	67	4	no	—
296	C₄×C₈	—	67	4	no	—
297	C₄×C₈	—	67	4	no	—
298	C₄×C₈	—	134	2	no	—
299	C₄×C₈	—	134	2	no	—
300	#95	—	67	2	no	—
301	#95	—	67	2	no	—
302	#95	—	67	2	no	—
303	#95	C₄×D₆₇	1	2	yes	—
304	C₂×Q₈	—	134	4	no	—
305	C₂×Q₈	—	134	4	no	—
306	C₂×Q₈	—	134	4	no	—
307	C₂²×C₄	—	67	1	no	—
308	C₂²×C₄	—	67	1	no	—
309	C₂²×C₄	—	67	1	no	—
310	C₂²×C₄	—	67	1	no	—
311	C₂²×C₄	—	67	2	no	—
312	C₂²×C₄	—	67	2	no	—
313	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
314	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
315	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
316	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
317	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
318	C₂×C₈	—	2	1	no	—
319	C₂×C₈	—	2	1	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	2	1	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	2	2	no	—
325	C₂×C₈	—	67	2	no	—
326	C₂×C₈	—	67	2	no	—
327	C₂×C₈	—	67	2	no	—
328	C₂×C₈	—	67	2	no	—
329	C₂×C₈	—	134	2	no	—
330	C₂×C₈	—	134	2	no	—
331	C₂×C₈	—	134	2	no	—
332	C₂×C₈	—	134	1	no	—
333	C₂×C₈	—	134	1	no	—
334	C₂×C₈	D₄×D₆₇	1	2	yes	—
335	C₂×C₈	Q₈×D₆₇	1	2	yes	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	2	2	no	—
338	C₄⋊C₄	—	134	4	no	—
339	C₄⋊C₄	—	134	4	no	—
340	C₄⋊C₄	—	134	4	no	—
341	C₄⋊C₄	—	134	4	no	—
342	C₄⋊C₄	—	134	4	no	—
343	C₄⋊C₄	—	134	2	no	—
344	C₄⋊C₄	—	134	2	no	—
345	C₄²	—	2	2	no	—
346	C₄²	—	2	2	no	—
347	C₄²	—	67	4	no	—
348	C₄²	—	67	2	no	—
349	C₄²	—	67	2	no	—
350	C₄²	—	67	2	no	—
351	C₄²	—	67	2	no	—
352	C₄²	—	134	2	no	—
353	C₄²	—	134	2	no	—
354	C₄²	—	134	2	no	—
355	C₄²	—	134	2	no	—
356	C₄²	—	134	2	no	—
357	C₄²	Q₈×D₆₇	1	2	yes	—
358	C₄²	#11761627	1	2	yes	—
359	C₂³	C₂²×C₄×D₆₇	1	1	yes	characteristic
360	Q₈	—	268	8	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	67	2	no	—
370	C₂×C₄	—	67	2	no	—
371	C₂×C₄	—	67	2	no	—
372	C₂×C₄	—	67	2	no	—
373	C₂×C₄	—	67	2	no	—
374	C₂×C₄	—	67	2	no	—
375	C₂×C₄	—	67	2	no	—
376	C₂×C₄	—	67	2	no	—
377	C₂×C₄	—	134	2	no	—
378	C₂×C₄	—	134	2	no	—
379	C₂×C₄	—	134	2	no	—
380	C₂×C₄	—	134	2	no	—
381	C₂×C₄	—	134	2	no	—
382	C₂×C₄	—	134	2	no	—
383	C₂×C₄	—	134	1	no	—
384	C₂×C₄	—	134	1	no	—
385	C₂×C₄	—	134	1	no	—
386	C₂×C₄	—	134	1	no	—
387	C₂×C₄	#14524008	1	2	yes	—
388	C₂×C₄	#14524008	1	2	yes	—
389	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
390	C₂×C₄	#14524049	1	2	yes	—
391	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
392	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
393	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
394	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
395	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
396	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
397	C₈	—	2	2	no	—
398	C₈	—	2	2	no	—
399	C₈	—	4	2	no	—
400	C₈	—	268	2	no	—
401	C₈	—	67	4	no	—
402	C₈	—	134	2	no	—
403	C₂²	C₄×D₄×D₆₇	1	1	yes	characteristic
404	C₂²	C₄×Q₈×D₆₇	1	1	yes	characteristic
405	C₂²	C₂×#14524008	1	1	yes	characteristic
406	C₂²	C₂×M₁₆×D₆₇	1	1	yes	characteristic
407	C₂²	#131×D₆₇	1	1	yes	characteristic
408	C₂²	#131×D₆₇	1	1	yes	characteristic
409	C₂²	C₂×#14524049	1	1	yes	characteristic
410	C₄	—	2	2	no	—
411	C₄	—	4	2	no	—
412	C₄	—	4	2	no	—
413	C₄	—	268	2	no	—
414	C₄	—	268	2	no	—
415	C₄	—	67	4	no	—
416	C₄	—	67	4	no	—
417	C₄	—	134	2	no	—
418	C₄	—	134	2	no	—
419	C₄	#16863525	1	4	yes	complemented
420	C₄	C₄⋊Q₈×D₆₇	1	1	yes	characteristic
421	C₄	#16863941	1	1	yes	characteristic
422	C₄	#16863948	1	1	yes	characteristic
423	C₄	#16863952	1	1	yes	characteristic
424	C₂	#22562780	1	1	yes	characteristic
425	C₂	#436×D₆₇	1	1	yes	characteristic
426	C₂	#22568916	1	1	yes	characteristic
427	C₂	#446×D₆₇	1	1	yes	characteristic
428	C₂	C₂×#16863525	1	1	yes	characteristic
429	C₂	#22569138	1	1	yes	characteristic
430	C₂	#22569142	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator, socle, prime radical	C₂³×C₆₇	C₂³×C₄
Fitting	C₆₇×#1939	C₂
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1939	D₆₇
hypocenter	C₆₇	#21712
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ #1939

Complements

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

N	H ≅ G/N	type
69. C₄×C₆₇⋊C₈	Q₈	semidirect
73. C₈×Dic₆₇	Q₈	semidirect
186. C₆₇⋊C₈	C₄⋊Q₈	semidirect
197. C₄×C₆₇	#446	semidirect
207. Dic₆₇	#446	semidirect
227. C₆₇	#21712	semidirect
419. C₄	#16863525	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863525	2
C₄×#14524042	2
#22561143	2
#22560927	2
#22560924	2
#22560360	2
#22560079	2
#22560078	2
#22560075	2
C₄×C₈×Dic₆₇	2
C₆₇×#1939	2
#21712	67