Subgroups — C₆₇×#21042 #30700293

Order 17152.

822 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2911	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2905	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2873	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1922	C₂	1	8	yes	maximal, complemented
5	C₆₇×#1738	C₂	1	1	yes	maximal, characteristic
6	C₂×#339×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#339×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₆₇×#1728	C₂	1	1	yes	maximal, characteristic
9	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
12	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
16	#428×C₆₇	C₄	1	4	yes	complemented
17	#428×C₆₇	C₄	1	4	yes	complemented
18	#427×C₆₇	C₂²	1	4	yes	—
19	#424×C₆₇	—	2	2	no	—
20	#424×C₆₇	C₂²	1	4	yes	—
21	#423×C₆₇	—	2	2	no	—
22	#405×C₆₇	—	2	2	no	—
23	#405×C₆₇	C₂²	1	4	yes	—
24	C₄×M₁₆×C₆₇	—	2	2	no	—
25	#341×C₆₇	C₂²	1	4	yes	—
26	#339×C₆₇	C₂²	1	4	yes	—
27	#339×C₆₇	C₂²	1	4	yes	—
28	#336×C₆₇	C₂²	1	4	yes	—
29	C₂³×C₄×C₆₇	—	2	1	no	—
30	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
31	C₂×M₁₆×C₆₇	—	2	1	no	—
32	C₂×M₁₆×C₆₇	—	2	2	no	—
33	C₂×M₁₆×C₆₇	—	2	2	no	—
34	C₂×M₁₆×C₆₇	—	2	2	no	—
35	C₂×M₁₆×C₆₇	—	2	2	no	—
36	C₂×M₁₆×C₆₇	—	2	2	no	—
37	C₂×M₁₆×C₆₇	—	2	1	no	—
38	C₂×M₁₆×C₆₇	—	2	2	no	—
39	C₂×M₁₆×C₆₇	—	2	2	no	—
40	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
41	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
42	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
43	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
44	C₂×M₁₆×C₆₇	Q₈	1	4	yes	—
45	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
46	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
47	C₂²×C₈×C₆₇	—	2	1	no	—
48	C₂²×C₈×C₆₇	—	2	1	no	—
49	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
50	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
51	#117×C₆₇	—	2	1	no	—
52	#117×C₆₇	—	2	1	no	—
53	#117×C₆₇	—	2	1	no	—
54	#117×C₆₇	—	2	1	no	—
55	#117×C₆₇	C₂×C₄	1	2	yes	—
56	#117×C₆₇	C₂×C₄	1	2	yes	—
57	#117×C₆₇	C₂³	1	2	yes	—
58	#117×C₆₇	C₂³	1	2	yes	—
59	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
60	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
61	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
62	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
63	C₂×C₄²×C₆₇	—	2	1	no	—
64	C₂×C₄²×C₆₇	—	2	1	no	—
65	C₂×C₄²×C₆₇	—	2	2	no	—
66	C₂×C₄²×C₆₇	—	2	2	no	—
67	C₂×C₄²×C₆₇	—	2	1	no	—
68	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
69	#107×C₆₇	C₂×C₄	1	8	yes	complemented
70	#106×C₆₇	C₂×C₄	1	8	yes	complemented
71	C₄⋊C₈×C₆₇	—	2	8	no	—
72	#97×C₆₇	—	2	4	no	—
73	C₄×C₈×C₆₇	—	2	4	no	—
74	C₂⁴×C₆₇	#41	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	—	2	2	no	—
76	C₂²×C₄×C₆₇	—	2	2	no	—
77	C₂²×C₄×C₆₇	—	2	2	no	—
78	C₂²×C₄×C₆₇	—	4	4	no	—
79	C₂²×C₄×C₆₇	—	4	2	no	—
80	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
85	C₂²×C₄×C₆₇	#41	1	2	yes	—
86	C₂²×C₄×C₆₇	#41	1	2	yes	—
87	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	#41	1	2	yes	—
89	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
90	M₁₆×C₆₇	—	2	4	no	—
91	M₁₆×C₆₇	—	2	4	no	—
92	M₁₆×C₆₇	—	2	4	no	—
93	M₁₆×C₆₇	—	2	4	no	—
94	M₁₆×C₆₇	—	2	4	no	—
95	M₁₆×C₆₇	—	2	2	no	—
96	M₁₆×C₆₇	—	2	2	no	—
97	M₁₆×C₆₇	—	2	2	no	—
98	M₁₆×C₆₇	—	4	2	no	—
99	M₁₆×C₆₇	C₂×Q₈	1	4	yes	—
100	C₂×C₈×C₆₇	—	2	4	no	—
101	C₂×C₈×C₆₇	—	2	2	no	—
102	C₂×C₈×C₆₇	—	2	2	no	—
103	C₂×C₈×C₆₇	—	2	2	no	—
104	C₂×C₈×C₆₇	—	2	2	no	—
105	C₂×C₈×C₆₇	—	2	2	no	—
106	C₂×C₈×C₆₇	—	2	2	no	—
107	C₂×C₈×C₆₇	—	4	2	no	—
108	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
109	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
110	C₄⋊C₄×C₆₇	—	2	2	no	—
111	C₄⋊C₄×C₆₇	—	2	2	no	—
112	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
113	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
114	C₂²⋊C₄×C₆₇	—	2	2	no	—
115	C₂²⋊C₄×C₆₇	—	2	2	no	—
116	C₂²⋊C₄×C₆₇	—	2	2	no	—
117	C₂²⋊C₄×C₆₇	—	2	2	no	—
118	C₄²×C₆₇	—	2	2	no	—
119	C₄²×C₆₇	—	2	2	no	—
120	C₄²×C₆₇	—	2	2	no	—
121	C₄²×C₆₇	—	2	2	no	—
122	C₄²×C₆₇	—	2	2	no	—
123	C₄²×C₆₇	—	2	2	no	—
124	C₄²×C₆₇	—	4	2	no	—
125	C₂³×C₆₇	—	4	2	no	—
126	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
127	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
128	C₂³×C₆₇	#125	1	2	yes	—
129	C₂³×C₆₇	#126	1	2	yes	—
130	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
131	C₂×C₄×C₆₇	—	2	4	no	—
132	C₂×C₄×C₆₇	—	2	4	no	—
133	C₂×C₄×C₆₇	—	2	4	no	—
134	C₂×C₄×C₆₇	—	4	2	no	—
135	C₂×C₄×C₆₇	—	4	8	no	—
136	C₂×C₄×C₆₇	—	4	2	no	—
137	C₂×C₄×C₆₇	—	8	2	no	—
138	C₂×C₄×C₆₇	—	8	2	no	—
139	C₂×C₄×C₆₇	#117	1	2	yes	—
140	C₂×C₄×C₆₇	#117	1	2	yes	—
141	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
144	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
145	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
146	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
147	C₂×C₄×C₆₇	#123	1	2	yes	—
148	C₂×C₄×C₆₇	#123	1	2	yes	—
149	C₂×C₄×C₆₇	#125	1	2	yes	—
150	C₂×C₄×C₆₇	#126	1	2	yes	—
151	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
152	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
154	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
155	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
156	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
157	C₈×C₆₇	—	2	4	no	—
158	C₈×C₆₇	—	2	4	no	—
159	C₈×C₆₇	—	2	4	no	—
160	C₈×C₆₇	—	4	2	no	—
161	C₂²×C₆₇	—	2	2	no	—
162	C₂²×C₆₇	—	2	2	no	—
163	C₂²×C₆₇	—	2	2	no	—
164	C₂²×C₆₇	—	4	2	no	—
165	C₂²×C₆₇	—	4	2	no	—
166	C₂²×C₆₇	#387	1	2	yes	—
167	C₂²×C₆₇	#387	1	2	yes	—
168	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
169	C₂²×C₆₇	C₂×#125	1	1	yes	characteristic
170	C₂²×C₆₇	C₂×#126	1	1	yes	characteristic
171	C₄×C₆₇	—	4	4	no	—
172	C₄×C₆₇	—	4	4	no	—
173	C₄×C₆₇	—	4	4	no	—
174	C₄×C₆₇	#387	1	2	yes	—
175	C₄×C₆₇	#387	1	2	yes	—
176	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
177	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
178	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
179	C₄×C₆₇	C₂×#123	1	1	yes	characteristic
180	#21042	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
181	C₂×C₆₇	—	2	1	no	—
182	C₂×C₆₇	—	2	1	no	—
183	C₂×C₆₇	—	2	2	no	—
184	C₂×C₆₇	—	2	2	no	—
185	C₂×C₆₇	C₂×#387	1	1	yes	characteristic
186	C₂×C₆₇	#2956	1	2	yes	—
187	#2911	C₂×C₆₇	1	1	yes	characteristic
188	#2905	C₂×C₆₇	1	1	yes	characteristic
189	#2873	C₂×C₆₇	1	1	yes	characteristic
190	#1922	C₂×C₆₇	1	8	yes	complemented
191	#1738	C₂×C₆₇	1	1	yes	characteristic
192	C₂×#339	C₂×C₆₇	1	1	yes	characteristic
193	C₂×#339	C₂×C₆₇	1	1	yes	characteristic
194	#1728	C₂×C₆₇	1	1	yes	characteristic
195	C₆₇	#21042	1	1	yes	characteristic, direct product factor, Sylow
196	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
197	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
198	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
199	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
200	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
201	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
202	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
203	#428	C₄×C₆₇	1	4	yes	complemented
204	#428	C₄×C₆₇	1	4	yes	complemented
205	#427	C₂²×C₆₇	1	4	yes	—
206	#424	—	2	2	no	—
207	#424	C₂²×C₆₇	1	4	yes	—
208	#423	—	2	2	no	—
209	#405	—	2	2	no	—
210	#405	C₂²×C₆₇	1	4	yes	—
211	C₄×M₁₆	—	2	2	no	—
212	#341	C₂²×C₆₇	1	4	yes	—
213	#339	C₂²×C₆₇	1	4	yes	—
214	#339	C₂²×C₆₇	1	4	yes	—
215	#336	C₂²×C₆₇	1	4	yes	—
216	C₂³×C₄	—	2	1	no	—
217	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
218	C₂×M₁₆	—	2	1	no	—
219	C₂×M₁₆	—	2	2	no	—
220	C₂×M₁₆	—	2	2	no	—
221	C₂×M₁₆	—	2	2	no	—
222	C₂×M₁₆	—	2	2	no	—
223	C₂×M₁₆	—	2	2	no	—
224	C₂×M₁₆	—	2	1	no	—
225	C₂×M₁₆	—	2	2	no	—
226	C₂×M₁₆	—	2	2	no	—
227	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
228	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
229	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
230	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
231	C₂×M₁₆	Q₈×C₆₇	1	4	yes	—
232	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
233	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
234	C₂²×C₈	—	2	1	no	—
235	C₂²×C₈	—	2	1	no	—
236	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
237	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
238	#117	—	2	1	no	—
239	#117	—	2	1	no	—
240	#117	—	2	1	no	—
241	#117	—	2	1	no	—
242	#117	C₂×C₄×C₆₇	1	2	yes	—
243	#117	C₂×C₄×C₆₇	1	2	yes	—
244	#117	C₂³×C₆₇	1	2	yes	—
245	#117	C₂³×C₆₇	1	2	yes	—
246	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
247	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
248	C₂×C₂²⋊C₄	—	2	1	no	—
249	C₂×C₂²⋊C₄	—	2	1	no	—
250	C₂×C₄²	—	2	1	no	—
251	C₂×C₄²	—	2	1	no	—
252	C₂×C₄²	—	2	2	no	—
253	C₂×C₄²	—	2	2	no	—
254	C₂×C₄²	—	2	1	no	—
255	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
256	#107	C₂×C₄×C₆₇	1	8	yes	complemented
257	#106	C₂×C₄×C₆₇	1	8	yes	complemented
258	C₄⋊C₈	—	2	8	no	—
259	#97	—	2	4	no	—
260	C₄×C₈	—	2	4	no	—
261	C₂⁴	#41×C₆₇	1	1	yes	characteristic
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	4	4	no	—
266	C₂²×C₄	—	4	2	no	—
267	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
268	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
269	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
270	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
271	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
272	C₂²×C₄	#41×C₆₇	1	2	yes	—
273	C₂²×C₄	#41×C₆₇	1	2	yes	—
274	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
275	C₂²×C₄	#41×C₆₇	1	2	yes	—
276	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
277	M₁₆	—	2	4	no	—
278	M₁₆	—	2	4	no	—
279	M₁₆	—	2	4	no	—
280	M₁₆	—	2	4	no	—
281	M₁₆	—	2	4	no	—
282	M₁₆	—	2	2	no	—
283	M₁₆	—	2	2	no	—
284	M₁₆	—	2	2	no	—
285	M₁₆	—	4	2	no	—
286	M₁₆	C₂×Q₈×C₆₇	1	4	yes	—
287	C₂×C₈	—	2	4	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	4	2	no	—
295	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
296	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
300	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
301	C₂²⋊C₄	—	2	2	no	—
302	C₂²⋊C₄	—	2	2	no	—
303	C₂²⋊C₄	—	2	2	no	—
304	C₂²⋊C₄	—	2	2	no	—
305	C₄²	—	2	2	no	—
306	C₄²	—	2	2	no	—
307	C₄²	—	2	2	no	—
308	C₄²	—	2	2	no	—
309	C₄²	—	2	2	no	—
310	C₄²	—	2	2	no	—
311	C₄²	—	4	2	no	—
312	C₂³	—	4	2	no	—
313	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
314	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
315	C₂³	#125×C₆₇	1	2	yes	—
316	C₂³	#126×C₆₇	1	2	yes	—
317	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
318	C₂×C₄	—	2	4	no	—
319	C₂×C₄	—	2	4	no	—
320	C₂×C₄	—	2	4	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	8	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	8	2	no	—
325	C₂×C₄	—	8	2	no	—
326	C₂×C₄	#117×C₆₇	1	2	yes	—
327	C₂×C₄	#117×C₆₇	1	2	yes	—
328	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
329	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
330	C₂×C₄	C₄×Q₈×C₆₇	1	2	yes	—
331	C₂×C₄	C₄×Q₈×C₆₇	1	2	yes	—
332	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
333	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
334	C₂×C₄	#123×C₆₇	1	2	yes	—
335	C₂×C₄	#123×C₆₇	1	2	yes	—
336	C₂×C₄	#125×C₆₇	1	2	yes	—
337	C₂×C₄	#126×C₆₇	1	2	yes	—
338	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
339	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
340	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
341	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
342	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
343	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
344	C₈	—	2	4	no	—
345	C₈	—	2	4	no	—
346	C₈	—	2	4	no	—
347	C₈	—	4	2	no	—
348	C₂²	—	2	2	no	—
349	C₂²	—	2	2	no	—
350	C₂²	—	2	2	no	—
351	C₂²	—	4	2	no	—
352	C₂²	—	4	2	no	—
353	C₂²	#387×C₆₇	1	2	yes	—
354	C₂²	#387×C₆₇	1	2	yes	—
355	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
356	C₂²	C₂×#125×C₆₇	1	1	yes	characteristic
357	C₂²	C₂×#126×C₆₇	1	1	yes	characteristic
358	C₄	—	4	4	no	—
359	C₄	—	4	4	no	—
360	C₄	—	4	4	no	—
361	C₄	#387×C₆₇	1	2	yes	—
362	C₄	#387×C₆₇	1	2	yes	—
363	C₄	C₂×#117×C₆₇	1	1	yes	characteristic
364	C₄	C₂×C₄×Q₈×C₆₇	1	1	yes	characteristic
365	C₄	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
366	C₄	C₂×#123×C₆₇	1	1	yes	characteristic
367	C₂	—	2	1	no	—
368	C₂	—	2	1	no	—
369	C₂	—	2	2	no	—
370	C₂	—	2	2	no	—
371	C₂	C₂×#387×C₆₇	1	1	yes	characteristic
372	C₂	C₆₇×#2956	1	2	yes	—

	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	C₂⁴×C₆₇	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂²×M₁₆×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₆₇×#1922	C₂	semidirect
16. #428×C₆₇	C₄	semidirect
17. #428×C₆₇	C₄	semidirect
69. #107×C₆₇	C₂×C₄	semidirect
70. #106×C₆₇	C₂×C₄	semidirect
180. #21042	C₆₇	direct (N × H)
190. #1922	C₂×C₆₇	semidirect
195. C₆₇	#21042	direct (N × H)
203. #428	C₄×C₆₇	semidirect
204. #428	C₄×C₆₇	semidirect
256. #107	C₂×C₄×C₆₇	semidirect
257. #106	C₂×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×#2911	2
C₆₇×#2905	2
C₆₇×#2873	2
C₆₇×#1922	2
C₆₇×#1738	2
C₂×#339×C₆₇	2
C₆₇×#1728	2
#21042	67

Subgroups — C67×#21042 #30700293

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21042 #30700293