Subgroups — C₆₇×#22235 #30701486

Order 17152.

686 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3135	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2960	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2931	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1995	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1963	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#1929	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1866	C₂	1	4	yes	maximal, complemented
8	C₆₇×#1849	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1746	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#131×C₆₇	C₂²	1	2	yes	—
11	#533×C₆₇	C₂²	1	1	yes	characteristic
12	#532×C₆₇	C₂²	1	1	yes	characteristic
13	#491×C₆₇	C₄	1	2	yes	complemented
14	#490×C₆₇	C₄	1	2	yes	complemented
15	#489×C₆₇	C₄	1	2	yes	complemented
16	#489×C₆₇	C₄	1	2	yes	complemented
17	#446×C₆₇	—	2	1	no	—
18	#446×C₆₇	—	2	1	no	—
19	C₈×Q₈×C₆₇	—	2	1	no	—
20	C₈×Q₈×C₆₇	—	2	1	no	—
21	#436×C₆₇	—	2	1	no	—
22	#436×C₆₇	—	2	1	no	—
23	C₈×D₄×C₆₇	—	2	1	no	—
24	C₈×D₄×C₆₇	—	2	1	no	—
25	C₂×#108×C₆₇	C₂²	1	4	yes	—
26	#424×C₆₇	C₂²	1	2	yes	—
27	#423×C₆₇	C₂²	1	1	yes	characteristic
28	#423×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×#104×C₆₇	C₂²	1	4	yes	—
30	#416×C₆₇	C₂²	1	4	yes	—
31	#408×C₆₇	C₂²	1	2	yes	—
32	#405×C₆₇	C₂²	1	2	yes	—
33	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
35	#376×C₆₇	C₂²	1	1	yes	characteristic
36	#339×C₆₇	C₂²	1	4	yes	—
37	#336×C₆₇	C₂²	1	2	yes	—
38	#336×C₆₇	C₂²	1	2	yes	—
39	C₂×#41×C₆₇	C₂³	1	2	yes	—
40	#131×C₆₇	—	4	4	no	—
41	C₂×M₁₆×C₆₇	—	2	2	no	—
42	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
43	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
44	C₂²×C₈×C₆₇	—	2	2	no	—
45	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
46	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
48	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
51	#127×C₆₇	C₂×C₄	1	1	yes	characteristic
52	#125×C₆₇	C₂×C₄	1	2	yes	—
53	#124×C₆₇	C₂×C₄	1	2	yes	—
54	C₂²⋊Q₈×C₆₇	—	2	1	no	—
55	C₂²⋊Q₈×C₆₇	—	2	1	no	—
56	#121×C₆₇	—	2	1	no	—
57	#121×C₆₇	—	2	1	no	—
58	C₄×Q₈×C₆₇	—	2	1	no	—
59	C₄×Q₈×C₆₇	—	2	1	no	—
60	C₄×D₄×C₆₇	—	2	1	no	—
61	C₄×D₄×C₆₇	—	2	1	no	—
62	#117×C₆₇	C₂³	1	2	yes	—
63	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
64	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
65	#108×C₆₇	—	4	4	no	—
66	C₄⋊C₈×C₆₇	—	2	2	no	—
67	C₄⋊C₈×C₆₇	—	2	2	no	—
68	C₄⋊C₈×C₆₇	—	2	1	no	—
69	C₄⋊C₈×C₆₇	—	2	1	no	—
70	C₄⋊C₈×C₆₇	—	2	1	no	—
71	C₄⋊C₈×C₆₇	—	2	1	no	—
72	#104×C₆₇	—	4	4	no	—
73	#103×C₆₇	C₂×C₄	1	4	yes	—
74	#103×C₆₇	C₂×C₄	1	4	yes	—
75	#102×C₆₇	C₂×C₄	1	4	yes	—
76	#102×C₆₇	C₂×C₄	1	4	yes	—
77	C₂²⋊C₈×C₆₇	—	2	2	no	—
78	C₂²⋊C₈×C₆₇	—	2	2	no	—
79	#97×C₆₇	—	2	2	no	—
80	#97×C₆₇	C₂×C₄	1	2	yes	—
81	#97×C₆₇	C₂×C₄	1	2	yes	—
82	C₄×C₈×C₆₇	—	2	2	no	—
83	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
84	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
85	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
86	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
87	#95×C₆₇	—	2	1	no	—
88	#95×C₆₇	—	2	1	no	—
89	#41×C₆₇	—	2	4	no	—
90	#41×C₆₇	—	2	4	no	—
91	C₂×Q₈×C₆₇	—	2	1	no	—
92	C₂×Q₈×C₆₇	—	2	1	no	—
93	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
94	C₂×D₄×C₆₇	—	2	2	no	—
95	C₂×D₄×C₆₇	—	2	1	no	—
96	C₂×D₄×C₆₇	—	2	1	no	—
97	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
98	C₂²×C₄×C₆₇	—	2	2	no	—
99	C₂²×C₄×C₆₇	—	2	2	no	—
100	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
103	M₁₆×C₆₇	—	2	4	no	—
104	M₁₆×C₆₇	—	2	4	no	—
105	M₁₆×C₆₇	—	4	4	no	—
106	C₂×C₈×C₆₇	—	2	2	no	—
107	C₂×C₈×C₆₇	—	2	2	no	—
108	C₂×C₈×C₆₇	—	2	2	no	—
109	C₂×C₈×C₆₇	—	2	2	no	—
110	C₂×C₈×C₆₇	—	2	2	no	—
111	C₂×C₈×C₆₇	—	2	2	no	—
112	C₂×C₈×C₆₇	—	4	4	no	—
113	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
114	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
118	C₄⋊C₄×C₆₇	—	2	1	no	—
119	C₄⋊C₄×C₆₇	—	2	1	no	—
120	C₄⋊C₄×C₆₇	—	2	2	no	—
121	C₄⋊C₄×C₆₇	—	2	2	no	—
122	C₄⋊C₄×C₆₇	—	2	1	no	—
123	C₄⋊C₄×C₆₇	—	2	1	no	—
124	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
125	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
126	C₂²⋊C₄×C₆₇	—	2	2	no	—
127	C₂²⋊C₄×C₆₇	—	2	2	no	—
128	C₂²⋊C₄×C₆₇	—	2	2	no	—
129	C₄²×C₆₇	—	2	2	no	—
130	C₄²×C₆₇	—	2	2	no	—
131	C₄²×C₆₇	—	2	2	no	—
132	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
133	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
134	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
135	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
136	C₂³×C₆₇	—	2	2	no	—
137	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
138	Q₈×C₆₇	—	2	4	no	—
139	Q₈×C₆₇	—	4	2	no	—
140	Q₈×C₆₇	—	4	2	no	—
141	D₄×C₆₇	—	2	4	no	—
142	D₄×C₆₇	—	4	2	no	—
143	D₄×C₆₇	—	4	2	no	—
144	D₄×C₆₇	—	4	4	no	—
145	C₂×C₄×C₆₇	—	2	2	no	—
146	C₂×C₄×C₆₇	—	2	2	no	—
147	C₂×C₄×C₆₇	—	2	2	no	—
148	C₂×C₄×C₆₇	—	4	1	no	—
149	C₂×C₄×C₆₇	—	4	1	no	—
150	C₂×C₄×C₆₇	—	4	2	no	—
151	C₂×C₄×C₆₇	—	4	2	no	—
152	C₂×C₄×C₆₇	—	4	2	no	—
153	C₂×C₄×C₆₇	—	4	2	no	—
154	C₂×C₄×C₆₇	—	4	2	no	—
155	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
156	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
157	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
158	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
159	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
160	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
162	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
163	C₈×C₆₇	—	2	4	no	—
164	C₈×C₆₇	—	2	1	no	—
165	C₈×C₆₇	—	2	1	no	—
166	C₈×C₆₇	—	2	1	no	—
167	C₈×C₆₇	—	2	1	no	—
168	C₈×C₆₇	—	4	2	no	—
169	C₈×C₆₇	—	4	2	no	—
170	C₂²×C₆₇	—	2	1	no	—
171	C₂²×C₆₇	—	2	1	no	—
172	C₂²×C₆₇	—	4	2	no	—
173	C₂²×C₆₇	—	4	2	no	—
174	C₂²×C₆₇	—	4	2	no	—
175	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
176	C₂²×C₆₇	#542	1	2	yes	—
177	C₄×C₆₇	—	2	1	no	—
178	C₄×C₆₇	—	2	1	no	—
179	C₄×C₆₇	—	2	1	no	—
180	C₄×C₆₇	—	2	1	no	—
181	C₄×C₆₇	—	4	2	no	—
182	C₄×C₆₇	—	4	2	no	—
183	C₄×C₆₇	—	4	2	no	—
184	C₄×C₆₇	—	8	2	no	—
185	C₄×C₆₇	#520	1	2	yes	—
186	C₄×C₆₇	#535	1	1	yes	characteristic
187	C₄×C₆₇	#536	1	1	yes	characteristic
188	#22235	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
189	C₂×C₆₇	—	2	2	no	—
190	C₂×C₆₇	—	8	2	no	—
191	C₂×C₆₇	#2326	1	1	yes	characteristic
192	C₂×C₆₇	#2955	1	1	yes	characteristic
193	C₂×C₆₇	#2955	1	1	yes	characteristic
194	#3135	C₂×C₆₇	1	1	yes	characteristic
195	#2960	C₂×C₆₇	1	1	yes	characteristic, complemented
196	#2931	C₂×C₆₇	1	1	yes	characteristic
197	#1995	C₂×C₆₇	1	1	yes	characteristic, complemented
198	#1963	C₂×C₆₇	1	1	yes	characteristic
199	#1929	C₂×C₆₇	1	4	yes	complemented
200	#1866	C₂×C₆₇	1	4	yes	complemented
201	#1849	C₂×C₆₇	1	1	yes	characteristic, complemented
202	#1746	C₂×C₆₇	1	1	yes	characteristic, complemented
203	C₆₇	#22235	1	1	yes	characteristic, direct product factor, Sylow
204	C₂×#131	C₂²×C₆₇	1	2	yes	—
205	#533	C₂²×C₆₇	1	1	yes	characteristic
206	#532	C₂²×C₆₇	1	1	yes	characteristic
207	#491	C₄×C₆₇	1	2	yes	complemented
208	#490	C₄×C₆₇	1	2	yes	complemented
209	#489	C₄×C₆₇	1	2	yes	complemented
210	#489	C₄×C₆₇	1	2	yes	complemented
211	#446	—	2	1	no	—
212	#446	—	2	1	no	—
213	C₈×Q₈	—	2	1	no	—
214	C₈×Q₈	—	2	1	no	—
215	#436	—	2	1	no	—
216	#436	—	2	1	no	—
217	C₈×D₄	—	2	1	no	—
218	C₈×D₄	—	2	1	no	—
219	C₂×#108	C₂²×C₆₇	1	4	yes	—
220	#424	C₂²×C₆₇	1	2	yes	—
221	#423	C₂²×C₆₇	1	1	yes	characteristic
222	#423	C₂²×C₆₇	1	1	yes	characteristic
223	C₂×#104	C₂²×C₆₇	1	4	yes	—
224	#416	C₂²×C₆₇	1	4	yes	—
225	#408	C₂²×C₆₇	1	2	yes	—
226	#405	C₂²×C₆₇	1	2	yes	—
227	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
228	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
229	#376	C₂²×C₆₇	1	1	yes	characteristic
230	#339	C₂²×C₆₇	1	4	yes	—
231	#336	C₂²×C₆₇	1	2	yes	—
232	#336	C₂²×C₆₇	1	2	yes	—
233	C₂×#41	C₂³×C₆₇	1	2	yes	—
234	#131	—	4	4	no	—
235	C₂×M₁₆	—	2	2	no	—
236	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
237	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
238	C₂²×C₈	—	2	2	no	—
239	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
240	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
241	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
242	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
243	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
244	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
245	#127	C₂×C₄×C₆₇	1	1	yes	characteristic
246	#125	C₂×C₄×C₆₇	1	2	yes	—
247	#124	C₂×C₄×C₆₇	1	2	yes	—
248	C₂²⋊Q₈	—	2	1	no	—
249	C₂²⋊Q₈	—	2	1	no	—
250	#121	—	2	1	no	—
251	#121	—	2	1	no	—
252	C₄×Q₈	—	2	1	no	—
253	C₄×Q₈	—	2	1	no	—
254	C₄×D₄	—	2	1	no	—
255	C₄×D₄	—	2	1	no	—
256	#117	C₂³×C₆₇	1	2	yes	—
257	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
258	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
259	#108	—	4	4	no	—
260	C₄⋊C₈	—	2	2	no	—
261	C₄⋊C₈	—	2	2	no	—
262	C₄⋊C₈	—	2	1	no	—
263	C₄⋊C₈	—	2	1	no	—
264	C₄⋊C₈	—	2	1	no	—
265	C₄⋊C₈	—	2	1	no	—
266	#104	—	4	4	no	—
267	#103	C₂×C₄×C₆₇	1	4	yes	—
268	#103	C₂×C₄×C₆₇	1	4	yes	—
269	#102	C₂×C₄×C₆₇	1	4	yes	—
270	#102	C₂×C₄×C₆₇	1	4	yes	—
271	C₂²⋊C₈	—	2	2	no	—
272	C₂²⋊C₈	—	2	2	no	—
273	#97	—	2	2	no	—
274	#97	C₂×C₄×C₆₇	1	2	yes	—
275	#97	C₂×C₄×C₆₇	1	2	yes	—
276	C₄×C₈	—	2	2	no	—
277	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
278	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
279	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
280	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
281	#95	—	2	1	no	—
282	#95	—	2	1	no	—
283	#41	—	2	4	no	—
284	#41	—	2	4	no	—
285	C₂×Q₈	—	2	1	no	—
286	C₂×Q₈	—	2	1	no	—
287	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
288	C₂×D₄	—	2	2	no	—
289	C₂×D₄	—	2	1	no	—
290	C₂×D₄	—	2	1	no	—
291	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
292	C₂²×C₄	—	2	2	no	—
293	C₂²×C₄	—	2	2	no	—
294	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
295	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
296	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
297	M₁₆	—	2	4	no	—
298	M₁₆	—	2	4	no	—
299	M₁₆	—	4	4	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	4	4	no	—
307	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
308	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
309	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
310	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
311	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
312	C₄⋊C₄	—	2	1	no	—
313	C₄⋊C₄	—	2	1	no	—
314	C₄⋊C₄	—	2	2	no	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	2	1	no	—
317	C₄⋊C₄	—	2	1	no	—
318	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
319	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
320	C₂²⋊C₄	—	2	2	no	—
321	C₂²⋊C₄	—	2	2	no	—
322	C₂²⋊C₄	—	2	2	no	—
323	C₄²	—	2	2	no	—
324	C₄²	—	2	2	no	—
325	C₄²	—	2	2	no	—
326	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
327	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
328	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
329	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
330	C₂³	—	2	2	no	—
331	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
332	Q₈	—	2	4	no	—
333	Q₈	—	4	2	no	—
334	Q₈	—	4	2	no	—
335	D₄	—	2	4	no	—
336	D₄	—	4	2	no	—
337	D₄	—	4	2	no	—
338	D₄	—	4	4	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
350	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
351	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
352	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
353	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
354	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
355	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
356	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
357	C₈	—	2	4	no	—
358	C₈	—	2	1	no	—
359	C₈	—	2	1	no	—
360	C₈	—	2	1	no	—
361	C₈	—	2	1	no	—
362	C₈	—	4	2	no	—
363	C₈	—	4	2	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	4	2	no	—
367	C₂²	—	4	2	no	—
368	C₂²	—	4	2	no	—
369	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
370	C₂²	#542×C₆₇	1	2	yes	—
371	C₄	—	2	1	no	—
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	4	2	no	—
376	C₄	—	4	2	no	—
377	C₄	—	4	2	no	—
378	C₄	—	8	2	no	—
379	C₄	#520×C₆₇	1	2	yes	—
380	C₄	#535×C₆₇	1	1	yes	characteristic
381	C₄	#536×C₆₇	1	1	yes	characteristic
382	C₂	—	2	2	no	—
383	C₂	—	8	2	no	—
384	C₂	C₆₇×#2326	1	1	yes	characteristic
385	C₂	C₆₇×#2955	1	1	yes	characteristic
386	C₂	C₆₇×#2955	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#532×C₆₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2960	C₂	semidirect
4. C₆₇×#1995	C₂	semidirect
6. C₆₇×#1929	C₂	semidirect
7. C₆₇×#1866	C₂	semidirect
8. C₆₇×#1849	C₂	semidirect
9. C₆₇×#1746	C₂	semidirect
13. #491×C₆₇	C₄	semidirect
14. #490×C₆₇	C₄	semidirect
15. #489×C₆₇	C₄	semidirect
16. #489×C₆₇	C₄	semidirect
188. #22235	C₆₇	direct (N × H)
195. #2960	C₂×C₆₇	semidirect
197. #1995	C₂×C₆₇	semidirect
199. #1929	C₂×C₆₇	semidirect
200. #1866	C₂×C₆₇	semidirect
201. #1849	C₂×C₆₇	semidirect
202. #1746	C₂×C₆₇	semidirect
203. C₆₇	#22235	direct (N × H)
207. #491	C₄×C₆₇	semidirect
208. #490	C₄×C₆₇	semidirect
209. #489	C₄×C₆₇	semidirect
210. #489	C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3135	2
C₆₇×#2960	2
C₆₇×#2931	2
C₆₇×#1995	2
C₆₇×#1963	2
C₆₇×#1929	2
C₆₇×#1866	2
C₆₇×#1849	2
C₆₇×#1746	2
#22235	67

Subgroups — C67×#22235 #30701486

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22235 #30701486