Subgroups — C₆₇×#19928 #30699179

Order 17152.

622 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3395	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3386	C₂	1	2	yes	maximal
3	D₄×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1713	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1694	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1670	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1669	C₂	1	1	yes	maximal, characteristic
8	C₆₇×#1649	C₂	1	2	yes	maximal
9	C₆₇×#1648	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1591	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₁₆×C₆₇	—	2	1	no	—
13	#557×C₆₇	C₂²	1	2	yes	—
14	D₄×Q₈×C₆₇	C₂²	1	2	yes	—
15	#501×C₆₇	—	2	2	no	—
16	#500×C₆₇	C₂²	1	2	yes	—
17	#500×C₆₇	C₂²	1	2	yes	—
18	#494×C₆₇	—	2	1	no	—
19	#494×C₆₇	C₂²	1	1	yes	characteristic
20	#494×C₆₇	C₂²	1	1	yes	characteristic
21	#478×C₆₇	—	2	2	no	—
22	#477×C₆₇	—	2	2	no	—
23	#475×C₆₇	C₂²	1	2	yes	—
24	#474×C₆₇	C₂²	1	2	yes	—
25	#467×C₆₇	—	2	1	no	—
26	#462×C₆₇	—	2	2	no	—
27	#462×C₆₇	C₂²	1	2	yes	—
28	#462×C₆₇	C₂²	1	2	yes	—
29	#451×C₆₇	—	2	2	no	—
30	#446×C₆₇	C₂²	1	2	yes	—
31	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
32	#440×C₆₇	—	2	2	no	—
33	C₄×Q₁₆×C₆₇	—	2	1	no	—
34	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
35	C₄×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
36	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
37	#335×C₆₇	C₂²	1	1	yes	characteristic
38	#334×C₆₇	C₂²	1	1	yes	characteristic
39	#332×C₆₇	C₂²	1	2	yes	—
40	#332×C₆₇	C₂²	1	2	yes	—
41	Q₈⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
42	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
43	D₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
44	C₈²×C₆₇	C₂²	1	1	yes	characteristic
45	C₂²×Q₈×C₆₇	—	2	2	no	—
46	C₂×Q₁₆×C₆₇	—	2	2	no	—
47	C₂×Q₁₆×C₆₇	—	2	2	no	—
48	C₂×Q₁₆×C₆₇	—	2	1	no	—
49	C₂×Q₁₆×C₆₇	—	2	1	no	—
50	C₂×Q₁₆×C₆₇	—	2	1	no	—
51	C₂×Q₁₆×C₆₇	—	2	1	no	—
52	C₂×Q₁₆×C₆₇	—	4	4	no	—
53	C₂×Q₁₆×C₆₇	D₄	1	1	yes	characteristic
54	C₂×SD₁₆×C₆₇	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₆₇	—	2	1	no	—
56	C₄⋊Q₈×C₆₇	—	2	2	no	—
57	C₄⋊Q₈×C₆₇	—	2	2	no	—
58	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
59	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
60	#125×C₆₇	—	2	2	no	—
61	#125×C₆₇	—	4	2	no	—
62	C₂²⋊Q₈×C₆₇	—	2	2	no	—
63	C₂²⋊Q₈×C₆₇	—	4	2	no	—
64	C₄×Q₈×C₆₇	—	2	2	no	—
65	C₄×Q₈×C₆₇	—	2	2	no	—
66	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
67	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
68	C₄×D₄×C₆₇	—	2	2	no	—
69	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
70	#107×C₆₇	—	2	2	no	—
71	#107×C₆₇	—	2	2	no	—
72	#107×C₆₇	—	2	2	no	—
73	#107×C₆₇	—	4	2	no	—
74	#107×C₆₇	D₄	1	1	yes	characteristic
75	#106×C₆₇	—	2	2	no	—
76	#106×C₆₇	D₄	1	1	yes	characteristic
77	C₄⋊C₈×C₆₇	—	2	1	no	—
78	C₄⋊C₈×C₆₇	—	2	2	no	—
79	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
80	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
81	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
82	#103×C₆₇	—	2	2	no	—
83	#103×C₆₇	—	2	2	no	—
84	#103×C₆₇	—	2	2	no	—
85	#103×C₆₇	D₄	1	2	yes	—
86	#103×C₆₇	D₄	1	1	yes	characteristic
87	#102×C₆₇	—	2	2	no	—
88	#102×C₆₇	D₄	1	1	yes	characteristic
89	C₂²⋊C₈×C₆₇	—	2	1	no	—
90	#97×C₆₇	—	2	2	no	—
91	C₄×C₈×C₆₇	—	2	1	no	—
92	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₆₇	—	2	2	no	—
96	C₂×Q₈×C₆₇	—	2	2	no	—
97	C₂×Q₈×C₆₇	—	4	2	no	—
98	C₂×Q₈×C₆₇	—	4	4	no	—
99	C₂×Q₈×C₆₇	—	8	2	no	—
100	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
101	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₆₇	—	2	1	no	—
104	C₂²×C₄×C₆₇	—	4	2	no	—
105	Q₁₆×C₆₇	—	2	2	no	—
106	Q₁₆×C₆₇	—	4	4	no	—
107	Q₁₆×C₆₇	—	4	2	no	—
108	Q₁₆×C₆₇	—	4	2	no	—
109	Q₁₆×C₆₇	—	8	1	no	—
110	Q₁₆×C₆₇	—	8	1	no	—
111	Q₁₆×C₆₇	—	8	2	no	—
112	SD₁₆×C₆₇	Q₁₆	1	4	yes	complemented
113	C₂×C₈×C₆₇	—	2	1	no	—
114	C₂×C₈×C₆₇	—	2	2	no	—
115	C₂×C₈×C₆₇	—	2	1	no	—
116	C₂×C₈×C₆₇	—	2	1	no	—
117	C₂×C₈×C₆₇	—	4	2	no	—
118	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
122	C₄⋊C₄×C₆₇	—	2	2	no	—
123	C₄⋊C₄×C₆₇	—	2	1	no	—
124	C₄⋊C₄×C₆₇	—	2	2	no	—
125	C₄⋊C₄×C₆₇	—	2	2	no	—
126	C₄⋊C₄×C₆₇	—	2	2	no	—
127	C₄⋊C₄×C₆₇	—	2	2	no	—
128	C₄⋊C₄×C₆₇	—	4	2	no	—
129	C₄⋊C₄×C₆₇	—	4	2	no	—
130	C₄⋊C₄×C₆₇	—	4	2	no	—
131	C₄⋊C₄×C₆₇	—	4	2	no	—
132	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
133	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
134	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
135	C₂²⋊C₄×C₆₇	—	2	1	no	—
136	C₂²⋊C₄×C₆₇	—	4	2	no	—
137	C₄²×C₆₇	—	2	2	no	—
138	C₄²×C₆₇	—	2	1	no	—
139	C₄²×C₆₇	—	4	2	no	—
140	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
141	C₂³×C₆₇	—	2	1	no	—
142	Q₈×C₆₇	—	2	1	no	—
143	Q₈×C₆₇	—	2	2	no	—
144	Q₈×C₆₇	—	2	2	no	—
145	Q₈×C₆₇	—	4	2	no	—
146	Q₈×C₆₇	—	4	2	no	—
147	Q₈×C₆₇	—	4	2	no	—
148	Q₈×C₆₇	—	4	2	no	—
149	Q₈×C₆₇	—	8	4	no	—
150	Q₈×C₆₇	C₂×Q₁₆	1	2	yes	complemented
151	D₄×C₆₇	—	2	1	no	—
152	D₄×C₆₇	C₂×Q₁₆	1	2	yes	—
153	C₂×C₄×C₆₇	—	2	1	no	—
154	C₂×C₄×C₆₇	—	2	2	no	—
155	C₂×C₄×C₆₇	—	2	1	no	—
156	C₂×C₄×C₆₇	—	2	2	no	—
157	C₂×C₄×C₆₇	—	2	1	no	—
158	C₂×C₄×C₆₇	—	4	1	no	—
159	C₂×C₄×C₆₇	—	4	1	no	—
160	C₂×C₄×C₆₇	—	4	2	no	—
161	C₂×C₄×C₆₇	—	4	2	no	—
162	C₂×C₄×C₆₇	—	8	4	no	—
163	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
165	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
166	C₈×C₆₇	—	2	2	no	—
167	C₈×C₆₇	—	2	1	no	—
168	C₈×C₆₇	—	2	1	no	—
169	C₈×C₆₇	—	4	1	no	—
170	C₈×C₆₇	—	4	1	no	—
171	C₈×C₆₇	—	4	2	no	—
172	C₈×C₆₇	C₂×Q₁₆	1	2	yes	complemented
173	C₈×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
174	C₂²×C₆₇	—	2	2	no	—
175	C₂²×C₆₇	—	4	1	no	—
176	C₂²×C₆₇	—	4	1	no	—
177	C₂²×C₆₇	D₄²	1	1	yes	characteristic
178	C₄×C₆₇	—	2	2	no	—
179	C₄×C₆₇	—	2	1	no	—
180	C₄×C₆₇	—	4	1	no	—
181	C₄×C₆₇	—	4	2	no	—
182	C₄×C₆₇	—	4	2	no	—
183	C₄×C₆₇	—	4	1	no	—
184	C₄×C₆₇	—	8	2	no	—
185	C₄×C₆₇	—	8	2	no	—
186	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
187	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
188	C₄×C₆₇	#576	1	1	yes	characteristic
189	C₄×C₆₇	#578	1	1	yes	characteristic
190	#19928	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
191	C₂×C₆₇	—	4	2	no	—
192	C₂×C₆₇	#3274	1	1	yes	characteristic
193	C₂×C₆₇	D₄×Q₁₆	1	1	yes	characteristic
194	C₂×C₆₇	#3293	1	1	yes	characteristic
195	#3395	C₂×C₆₇	1	1	yes	characteristic, complemented
196	#3386	C₂×C₆₇	1	2	yes	—
197	D₄×Q₁₆	C₂×C₆₇	1	1	yes	characteristic
198	#1713	C₂×C₆₇	1	1	yes	characteristic, complemented
199	#1694	C₂×C₆₇	1	2	yes	complemented
200	#1670	C₂×C₆₇	1	1	yes	characteristic, complemented
201	#1669	C₂×C₆₇	1	1	yes	characteristic
202	#1649	C₂×C₆₇	1	2	yes	—
203	#1648	C₂×C₆₇	1	2	yes	complemented
204	#1591	C₂×C₆₇	1	1	yes	characteristic, complemented
205	C₈×SD₁₆	C₂×C₆₇	1	1	yes	characteristic
206	C₆₇	#19928	1	1	yes	characteristic, direct product factor, Sylow
207	C₂²×Q₁₆	—	2	1	no	—
208	#557	C₂²×C₆₇	1	2	yes	—
209	D₄×Q₈	C₂²×C₆₇	1	2	yes	—
210	#501	—	2	2	no	—
211	#500	C₂²×C₆₇	1	2	yes	—
212	#500	C₂²×C₆₇	1	2	yes	—
213	#494	—	2	1	no	—
214	#494	C₂²×C₆₇	1	1	yes	characteristic
215	#494	C₂²×C₆₇	1	1	yes	characteristic
216	#478	—	2	2	no	—
217	#477	—	2	2	no	—
218	#475	C₂²×C₆₇	1	2	yes	—
219	#474	C₂²×C₆₇	1	2	yes	—
220	#467	—	2	1	no	—
221	#462	—	2	2	no	—
222	#462	C₂²×C₆₇	1	2	yes	—
223	#462	C₂²×C₆₇	1	2	yes	—
224	#451	—	2	2	no	—
225	#446	C₂²×C₆₇	1	2	yes	—
226	C₈×Q₈	C₂²×C₆₇	1	1	yes	characteristic
227	#440	—	2	2	no	—
228	C₄×Q₁₆	—	2	1	no	—
229	C₄×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
230	C₄×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
231	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic
232	#335	C₂²×C₆₇	1	1	yes	characteristic
233	#334	C₂²×C₆₇	1	1	yes	characteristic
234	#332	C₂²×C₆₇	1	2	yes	—
235	#332	C₂²×C₆₇	1	2	yes	—
236	Q₈⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
237	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
238	D₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
239	C₈²	C₂²×C₆₇	1	1	yes	characteristic
240	C₂²×Q₈	—	2	2	no	—
241	C₂×Q₁₆	—	2	2	no	—
242	C₂×Q₁₆	—	2	2	no	—
243	C₂×Q₁₆	—	2	1	no	—
244	C₂×Q₁₆	—	2	1	no	—
245	C₂×Q₁₆	—	2	1	no	—
246	C₂×Q₁₆	—	2	1	no	—
247	C₂×Q₁₆	—	4	4	no	—
248	C₂×Q₁₆	D₄×C₆₇	1	1	yes	characteristic
249	C₂×SD₁₆	D₄×C₆₇	1	1	yes	characteristic
250	C₂²×C₈	—	2	1	no	—
251	C₄⋊Q₈	—	2	2	no	—
252	C₄⋊Q₈	—	2	2	no	—
253	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
254	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
255	#125	—	2	2	no	—
256	#125	—	4	2	no	—
257	C₂²⋊Q₈	—	2	2	no	—
258	C₂²⋊Q₈	—	4	2	no	—
259	C₄×Q₈	—	2	2	no	—
260	C₄×Q₈	—	2	2	no	—
261	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
262	C₄×Q₈	C₂³×C₆₇	1	1	yes	characteristic
263	C₄×D₄	—	2	2	no	—
264	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
265	#107	—	2	2	no	—
266	#107	—	2	2	no	—
267	#107	—	2	2	no	—
268	#107	—	4	2	no	—
269	#107	D₄×C₆₇	1	1	yes	characteristic
270	#106	—	2	2	no	—
271	#106	D₄×C₆₇	1	1	yes	characteristic
272	C₄⋊C₈	—	2	1	no	—
273	C₄⋊C₈	—	2	2	no	—
274	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
275	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
276	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
277	#103	—	2	2	no	—
278	#103	—	2	2	no	—
279	#103	—	2	2	no	—
280	#103	D₄×C₆₇	1	1	yes	characteristic
281	#103	D₄×C₆₇	1	2	yes	—
282	#102	—	2	2	no	—
283	#102	D₄×C₆₇	1	1	yes	characteristic
284	C₂²⋊C₈	—	2	1	no	—
285	#97	—	2	2	no	—
286	C₄×C₈	—	2	1	no	—
287	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
288	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
289	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
290	C₂×Q₈	—	2	2	no	—
291	C₂×Q₈	—	2	2	no	—
292	C₂×Q₈	—	4	2	no	—
293	C₂×Q₈	—	4	4	no	—
294	C₂×Q₈	—	8	2	no	—
295	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
296	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
297	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
298	C₂²×C₄	—	2	1	no	—
299	C₂²×C₄	—	4	2	no	—
300	Q₁₆	—	2	2	no	—
301	Q₁₆	—	4	4	no	—
302	Q₁₆	—	4	2	no	—
303	Q₁₆	—	4	2	no	—
304	Q₁₆	—	8	1	no	—
305	Q₁₆	—	8	1	no	—
306	Q₁₆	—	8	2	no	—
307	SD₁₆	Q₁₆×C₆₇	1	4	yes	complemented
308	C₂×C₈	—	2	1	no	—
309	C₂×C₈	—	2	1	no	—
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	—	2	1	no	—
312	C₂×C₈	—	4	2	no	—
313	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
314	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
315	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
316	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	—	2	1	no	—
319	C₄⋊C₄	—	2	2	no	—
320	C₄⋊C₄	—	2	2	no	—
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	2	2	no	—
323	C₄⋊C₄	—	4	2	no	—
324	C₄⋊C₄	—	4	2	no	—
325	C₄⋊C₄	—	4	2	no	—
326	C₄⋊C₄	—	4	2	no	—
327	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
328	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
329	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
330	C₂²⋊C₄	—	2	1	no	—
331	C₂²⋊C₄	—	4	2	no	—
332	C₄²	—	2	1	no	—
333	C₄²	—	2	2	no	—
334	C₄²	—	4	2	no	—
335	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
336	C₂³	—	2	1	no	—
337	Q₈	—	2	1	no	—
338	Q₈	—	2	2	no	—
339	Q₈	—	2	2	no	—
340	Q₈	—	4	2	no	—
341	Q₈	—	4	2	no	—
342	Q₈	—	4	2	no	—
343	Q₈	—	4	2	no	—
344	Q₈	—	8	4	no	—
345	Q₈	C₂×Q₁₆×C₆₇	1	2	yes	complemented
346	D₄	—	2	1	no	—
347	D₄	C₂×Q₁₆×C₆₇	1	2	yes	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	4	1	no	—
354	C₂×C₄	—	4	1	no	—
355	C₂×C₄	—	4	2	no	—
356	C₂×C₄	—	4	2	no	—
357	C₂×C₄	—	8	4	no	—
358	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
359	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
360	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
361	C₈	—	2	2	no	—
362	C₈	—	2	1	no	—
363	C₈	—	2	1	no	—
364	C₈	—	4	1	no	—
365	C₈	—	4	1	no	—
366	C₈	—	4	2	no	—
367	C₈	C₂×Q₁₆×C₆₇	1	2	yes	complemented
368	C₈	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
369	C₂²	—	2	2	no	—
370	C₂²	—	4	1	no	—
371	C₂²	—	4	1	no	—
372	C₂²	D₄²×C₆₇	1	1	yes	characteristic
373	C₄	—	2	2	no	—
374	C₄	—	2	1	no	—
375	C₄	—	4	1	no	—
376	C₄	—	4	2	no	—
377	C₄	—	4	2	no	—
378	C₄	—	4	1	no	—
379	C₄	—	8	2	no	—
380	C₄	—	8	2	no	—
381	C₄	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
382	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
383	C₄	#576×C₆₇	1	1	yes	characteristic
384	C₄	#578×C₆₇	1	1	yes	characteristic
385	C₂	—	4	2	no	—
386	C₂	C₆₇×#3274	1	1	yes	characteristic
387	C₂	D₄×Q₁₆×C₆₇	1	1	yes	characteristic
388	C₂	C₆₇×#3293	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	C₂×D₄×C₆₇	C₂×D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3395	C₂	semidirect
4. C₆₇×#1713	C₂	semidirect
5. C₆₇×#1694	C₂	semidirect
6. C₆₇×#1670	C₂	semidirect
9. C₆₇×#1648	C₂	semidirect
10. C₆₇×#1591	C₂	semidirect
112. SD₁₆×C₆₇	Q₁₆	semidirect
150. Q₈×C₆₇	C₂×Q₁₆	semidirect
172. C₈×C₆₇	C₂×Q₁₆	semidirect
190. #19928	C₆₇	direct (N × H)
195. #3395	C₂×C₆₇	semidirect
198. #1713	C₂×C₆₇	semidirect
199. #1694	C₂×C₆₇	semidirect
200. #1670	C₂×C₆₇	semidirect
203. #1648	C₂×C₆₇	semidirect
204. #1591	C₂×C₆₇	semidirect
206. C₆₇	#19928	direct (N × H)
307. SD₁₆	Q₁₆×C₆₇	semidirect
345. Q₈	C₂×Q₁₆×C₆₇	semidirect
367. C₈	C₂×Q₁₆×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3395	2
C₆₇×#3386	2
D₄×Q₁₆×C₆₇	2
C₆₇×#1713	2
C₆₇×#1694	2
C₆₇×#1670	2
C₆₇×#1669	2
C₆₇×#1649	2
C₆₇×#1648	2
C₆₇×#1591	2
C₈×SD₁₆×C₆₇	2
#19928	67

Subgroups — C67×#19928 #30699179

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#19928 #30699179