Subgroups — C₆₇×#19391 #30698642

Order 17152.

696 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3227	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#476×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3036	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2972	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1677	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1671	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1570	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1548	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1540	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×AGL(1,ℤ/8)×C₆₇	—	2	1	no	—
11	#532×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
14	#498×C₆₇	C₂²	1	2	yes	complemented
15	#486×C₆₇	C₂²	1	2	yes	complemented
16	#481×C₆₇	—	2	2	no	—
17	#476×C₆₇	—	2	2	no	—
18	#476×C₆₇	—	2	2	no	—
19	#476×C₆₇	C₂²	1	4	yes	—
20	#463×C₆₇	—	2	2	no	—
21	#461×C₆₇	C₂²	1	2	yes	—
22	#459×C₆₇	C₂²	1	2	yes	—
23	C₈×D₄×C₆₇	—	2	2	no	—
24	C₈×D₄×C₆₇	C₂²	1	4	yes	—
25	#431×C₆₇	C₂²	1	1	yes	characteristic
26	#431×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#106×C₆₇	—	2	1	no	—
28	#423×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
30	#417×C₆₇	—	2	1	no	—
31	C₂×#102×C₆₇	—	2	1	no	—
32	C₂×C₂²⋊C₈×C₆₇	—	2	2	no	—
33	#334×C₆₇	C₂²	1	4	yes	complemented
34	#333×C₆₇	C₂²	1	4	yes	complemented
35	#332×C₆₇	C₂²	1	2	yes	complemented
36	#331×C₆₇	C₂²	1	2	yes	complemented
37	C₂×#41×C₆₇	—	2	1	no	—
38	C₂²×Q₈×C₆₇	—	2	1	no	—
39	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₆₇	—	2	1	no	—
41	AGL(1,ℤ/8)×C₆₇	—	4	4	no	—
42	C₂×SD₁₆×C₆₇	—	2	2	no	—
43	C₂×D₈×C₆₇	—	2	2	no	—
44	C₂×M₁₆×C₆₇	—	2	1	no	—
45	C₂²×C₈×C₆₇	—	2	1	no	—
46	C₂²×C₈×C₆₇	—	2	4	no	—
47	C₄⋊Q₈×C₆₇	—	2	1	no	—
48	C₄⋊Q₈×C₆₇	—	2	1	no	—
49	C₄⋊Q₈×C₆₇	—	2	1	no	—
50	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
51	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
52	#127×C₆₇	C₂³	1	1	yes	characteristic
53	#124×C₆₇	—	2	1	no	—
54	#121×C₆₇	—	2	1	no	—
55	#121×C₆₇	—	2	1	no	—
56	C₄×D₄×C₆₇	—	2	1	no	—
57	C₄×D₄×C₆₇	—	2	1	no	—
58	C₄×D₄×C₆₇	—	2	1	no	—
59	C₄×D₄×C₆₇	—	2	1	no	—
60	C₄×D₄×C₆₇	D₄	1	2	yes	—
61	C₄×D₄×C₆₇	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
63	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
64	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
65	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
66	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
67	#106×C₆₇	—	2	4	no	—
68	#106×C₆₇	—	2	4	no	—
69	C₄⋊C₈×C₆₇	—	2	1	no	—
70	C₄⋊C₈×C₆₇	—	2	1	no	—
71	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
72	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
73	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
74	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
75	#103×C₆₇	—	2	2	no	—
76	#102×C₆₇	—	2	4	no	—
77	#102×C₆₇	—	2	4	no	—
78	#102×C₆₇	—	2	2	no	—
79	C₂²⋊C₈×C₆₇	—	2	1	no	—
80	C₂²⋊C₈×C₆₇	—	2	4	no	—
81	C₂²⋊C₈×C₆₇	D₄	1	2	yes	—
82	C₄×C₈×C₆₇	—	2	1	no	—
83	C₄×C₈×C₆₇	C₂³	1	2	yes	—
84	C₂⁴×C₆₇	—	2	1	no	—
85	#41×C₆₇	—	4	2	no	—
86	#41×C₆₇	—	4	2	no	—
87	C₂×Q₈×C₆₇	—	2	1	no	—
88	C₂×Q₈×C₆₇	—	2	2	no	—
89	C₂×Q₈×C₆₇	—	4	1	no	—
90	C₂×Q₈×C₆₇	—	8	2	no	—
91	C₂×D₄×C₆₇	—	2	1	no	—
92	C₂×D₄×C₆₇	—	2	2	no	—
93	C₂×D₄×C₆₇	—	2	1	no	—
94	C₂×D₄×C₆₇	—	2	1	no	—
95	C₂×D₄×C₆₇	—	2	1	no	—
96	C₂×D₄×C₆₇	—	4	1	no	—
97	C₂×D₄×C₆₇	SD₁₆	1	4	yes	—
98	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
99	C₂²×C₄×C₆₇	—	2	1	no	—
100	C₂²×C₄×C₆₇	—	2	2	no	—
101	C₂²×C₄×C₆₇	—	2	2	no	—
102	C₂²×C₄×C₆₇	—	4	1	no	—
103	C₂²×C₄×C₆₇	—	4	1	no	—
104	C₂²×C₄×C₆₇	—	8	2	no	—
105	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
108	SD₁₆×C₆₇	—	4	4	no	—
109	D₈×C₆₇	—	4	4	no	—
110	M₁₆×C₆₇	—	4	2	no	—
111	C₂×C₈×C₆₇	—	2	2	no	—
112	C₂×C₈×C₆₇	—	2	2	no	—
113	C₂×C₈×C₆₇	—	2	2	no	—
114	C₂×C₈×C₆₇	—	4	2	no	—
115	C₂×C₈×C₆₇	—	8	4	no	—
116	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
117	C₄⋊C₄×C₆₇	—	2	1	no	—
118	C₄⋊C₄×C₆₇	—	2	2	no	—
119	C₄⋊C₄×C₆₇	—	2	2	no	—
120	C₄⋊C₄×C₆₇	—	2	1	no	—
121	C₄⋊C₄×C₆₇	—	2	1	no	—
122	C₄⋊C₄×C₆₇	—	4	1	no	—
123	C₄⋊C₄×C₆₇	—	4	1	no	—
124	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
125	C₂²⋊C₄×C₆₇	—	2	2	no	—
126	C₂²⋊C₄×C₆₇	—	2	2	no	—
127	C₂²⋊C₄×C₆₇	—	4	1	no	—
128	C₂²⋊C₄×C₆₇	—	4	1	no	—
129	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
130	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
131	C₄²×C₆₇	#41	1	1	yes	characteristic
132	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
133	C₂³×C₆₇	—	2	2	no	—
134	C₂³×C₆₇	—	2	4	no	—
135	C₂³×C₆₇	—	4	1	no	—
136	C₂³×C₆₇	—	8	1	no	—
137	C₂³×C₆₇	—	8	1	no	—
138	C₂³×C₆₇	#121	1	1	yes	characteristic
139	Q₈×C₆₇	—	4	1	no	—
140	Q₈×C₆₇	—	4	1	no	—
141	Q₈×C₆₇	—	8	2	no	—
142	Q₈×C₆₇	—	8	2	no	—
143	D₄×C₆₇	—	2	2	no	—
144	D₄×C₆₇	—	2	4	no	—
145	D₄×C₆₇	—	4	1	no	—
146	D₄×C₆₇	—	4	1	no	—
147	D₄×C₆₇	—	4	2	no	—
148	D₄×C₆₇	—	8	1	no	—
149	D₄×C₆₇	—	8	2	no	—
150	D₄×C₆₇	C₂×SD₁₆	1	4	yes	—
151	C₂×C₄×C₆₇	—	2	2	no	—
152	C₂×C₄×C₆₇	—	2	2	no	—
153	C₂×C₄×C₆₇	—	4	1	no	—
154	C₂×C₄×C₆₇	—	4	4	no	—
155	C₂×C₄×C₆₇	—	4	1	no	—
156	C₂×C₄×C₆₇	—	4	2	no	—
157	C₂×C₄×C₆₇	—	8	2	no	—
158	C₂×C₄×C₆₇	—	8	1	no	—
159	C₂×C₄×C₆₇	—	8	2	no	—
160	C₂×C₄×C₆₇	—	8	2	no	—
161	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
162	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
163	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
164	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
165	C₂×C₄×C₆₇	#135	1	2	yes	—
166	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
167	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
169	C₈×C₆₇	—	2	4	no	—
170	C₈×C₆₇	—	4	2	no	—
171	C₈×C₆₇	—	4	2	no	—
172	C₈×C₆₇	—	4	2	no	—
173	C₂²×C₆₇	—	2	1	no	—
174	C₂²×C₆₇	—	2	1	no	—
175	C₂²×C₆₇	—	2	4	no	—
176	C₂²×C₆₇	—	4	2	no	—
177	C₂²×C₆₇	—	4	2	no	—
178	C₂²×C₆₇	—	8	4	no	—
179	C₂²×C₆₇	—	8	1	no	—
180	C₂²×C₆₇	—	8	1	no	—
181	C₂²×C₆₇	—	8	1	no	—
182	C₂²×C₆₇	#465	1	2	yes	—
183	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
184	C₄×C₆₇	—	2	1	no	—
185	C₄×C₆₇	—	2	1	no	—
186	C₄×C₆₇	—	2	1	no	—
187	C₄×C₆₇	—	4	2	no	—
188	C₄×C₆₇	—	8	1	no	—
189	C₄×C₆₇	—	8	2	no	—
190	C₄×C₆₇	#465	1	2	yes	—
191	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
192	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
193	C₄×C₆₇	#577	1	2	yes	—
194	#19391	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
195	C₂×C₆₇	—	2	2	no	—
196	C₂×C₆₇	—	4	4	no	—
197	C₂×C₆₇	—	16	1	no	—
198	C₂×C₆₇	#3042	1	1	yes	characteristic
199	C₂×C₆₇	C₂×#465	1	1	yes	characteristic
200	C₂×C₆₇	#3068	1	1	yes	characteristic
201	#3227	C₂×C₆₇	1	1	yes	characteristic, complemented
202	C₂×#476	C₂×C₆₇	1	1	yes	characteristic, complemented
203	#3036	C₂×C₆₇	1	1	yes	characteristic
204	#2972	C₂×C₆₇	1	1	yes	characteristic, complemented
205	#1677	C₂×C₆₇	1	4	yes	complemented
206	#1671	C₂×C₆₇	1	4	yes	complemented
207	#1570	C₂×C₆₇	1	1	yes	characteristic, complemented
208	#1548	C₂×C₆₇	1	1	yes	characteristic, complemented
209	#1540	C₂×C₆₇	1	1	yes	characteristic, complemented
210	C₆₇	#19391	1	1	yes	characteristic, direct product factor, Sylow
211	C₂×AGL(1,ℤ/8)	—	2	1	no	—
212	#532	C₂²×C₆₇	1	1	yes	characteristic
213	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
214	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
215	#498	C₂²×C₆₇	1	2	yes	complemented
216	#486	C₂²×C₆₇	1	2	yes	complemented
217	#481	—	2	2	no	—
218	#476	—	2	2	no	—
219	#476	—	2	2	no	—
220	#476	C₂²×C₆₇	1	4	yes	—
221	#463	—	2	2	no	—
222	#461	C₂²×C₆₇	1	2	yes	—
223	#459	C₂²×C₆₇	1	2	yes	—
224	C₈×D₄	—	2	2	no	—
225	C₈×D₄	C₂²×C₆₇	1	4	yes	—
226	#431	C₂²×C₆₇	1	1	yes	characteristic
227	#431	C₂²×C₆₇	1	1	yes	characteristic
228	C₂×#106	—	2	1	no	—
229	#423	C₂²×C₆₇	1	1	yes	characteristic
230	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
231	#417	—	2	1	no	—
232	C₂×#102	—	2	1	no	—
233	C₂×C₂²⋊C₈	—	2	2	no	—
234	#334	C₂²×C₆₇	1	4	yes	complemented
235	#333	C₂²×C₆₇	1	4	yes	complemented
236	#332	C₂²×C₆₇	1	2	yes	complemented
237	#331	C₂²×C₆₇	1	2	yes	complemented
238	C₂×#41	—	2	1	no	—
239	C₂²×Q₈	—	2	1	no	—
240	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
241	C₂³×C₄	—	2	1	no	—
242	AGL(1,ℤ/8)	—	4	4	no	—
243	C₂×SD₁₆	—	2	2	no	—
244	C₂×D₈	—	2	2	no	—
245	C₂×M₁₆	—	2	1	no	—
246	C₂²×C₈	—	2	1	no	—
247	C₂²×C₈	—	2	4	no	—
248	C₄⋊Q₈	—	2	1	no	—
249	C₄⋊Q₈	—	2	1	no	—
250	C₄⋊Q₈	—	2	1	no	—
251	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
252	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
253	#127	C₂³×C₆₇	1	1	yes	characteristic
254	#124	—	2	1	no	—
255	#121	—	2	1	no	—
256	#121	—	2	1	no	—
257	C₄×D₄	—	2	1	no	—
258	C₄×D₄	—	2	1	no	—
259	C₄×D₄	—	2	1	no	—
260	C₄×D₄	—	2	1	no	—
261	C₄×D₄	D₄×C₆₇	1	2	yes	—
262	C₄×D₄	C₂³×C₆₇	1	2	yes	—
263	C₂×C₄⋊C₄	—	2	1	no	—
264	C₂×C₄⋊C₄	—	2	1	no	—
265	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
266	C₂×C₂²⋊C₄	—	2	1	no	—
267	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
268	#106	—	2	4	no	—
269	#106	—	2	4	no	—
270	C₄⋊C₈	—	2	1	no	—
271	C₄⋊C₈	—	2	1	no	—
272	C₄⋊C₈	D₄×C₆₇	1	2	yes	—
273	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
274	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
275	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
276	#103	—	2	2	no	—
277	#102	—	2	4	no	—
278	#102	—	2	4	no	—
279	#102	—	2	2	no	—
280	C₂²⋊C₈	—	2	1	no	—
281	C₂²⋊C₈	—	2	4	no	—
282	C₂²⋊C₈	D₄×C₆₇	1	2	yes	—
283	C₄×C₈	—	2	1	no	—
284	C₄×C₈	C₂³×C₆₇	1	2	yes	—
285	C₂⁴	—	2	1	no	—
286	#41	—	4	2	no	—
287	#41	—	4	2	no	—
288	C₂×Q₈	—	2	1	no	—
289	C₂×Q₈	—	2	2	no	—
290	C₂×Q₈	—	4	1	no	—
291	C₂×Q₈	—	8	2	no	—
292	C₂×D₄	—	2	1	no	—
293	C₂×D₄	—	2	2	no	—
294	C₂×D₄	—	2	1	no	—
295	C₂×D₄	—	2	1	no	—
296	C₂×D₄	—	2	1	no	—
297	C₂×D₄	—	4	1	no	—
298	C₂×D₄	SD₁₆×C₆₇	1	4	yes	—
299	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
300	C₂²×C₄	—	2	1	no	—
301	C₂²×C₄	—	2	2	no	—
302	C₂²×C₄	—	2	2	no	—
303	C₂²×C₄	—	4	1	no	—
304	C₂²×C₄	—	4	1	no	—
305	C₂²×C₄	—	8	2	no	—
306	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
307	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
308	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
309	SD₁₆	—	4	4	no	—
310	D₈	—	4	4	no	—
311	M₁₆	—	4	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	4	2	no	—
316	C₂×C₈	—	8	4	no	—
317	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
318	C₄⋊C₄	—	2	1	no	—
319	C₄⋊C₄	—	2	2	no	—
320	C₄⋊C₄	—	2	2	no	—
321	C₄⋊C₄	—	2	1	no	—
322	C₄⋊C₄	—	2	1	no	—
323	C₄⋊C₄	—	4	1	no	—
324	C₄⋊C₄	—	4	1	no	—
325	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
326	C₂²⋊C₄	—	2	2	no	—
327	C₂²⋊C₄	—	2	2	no	—
328	C₂²⋊C₄	—	4	1	no	—
329	C₂²⋊C₄	—	4	1	no	—
330	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
331	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
332	C₄²	#41×C₆₇	1	1	yes	characteristic
333	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
334	C₂³	—	2	2	no	—
335	C₂³	—	2	4	no	—
336	C₂³	—	4	1	no	—
337	C₂³	—	8	1	no	—
338	C₂³	—	8	1	no	—
339	C₂³	#121×C₆₇	1	1	yes	characteristic
340	Q₈	—	4	1	no	—
341	Q₈	—	4	1	no	—
342	Q₈	—	8	2	no	—
343	Q₈	—	8	2	no	—
344	D₄	—	2	4	no	—
345	D₄	—	2	2	no	—
346	D₄	—	4	1	no	—
347	D₄	—	4	1	no	—
348	D₄	—	4	2	no	—
349	D₄	—	8	1	no	—
350	D₄	—	8	2	no	—
351	D₄	C₂×SD₁₆×C₆₇	1	4	yes	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	4	1	no	—
355	C₂×C₄	—	4	4	no	—
356	C₂×C₄	—	4	1	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	8	2	no	—
359	C₂×C₄	—	8	1	no	—
360	C₂×C₄	—	8	2	no	—
361	C₂×C₄	—	8	2	no	—
362	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
363	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
364	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
365	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
366	C₂×C₄	#135×C₆₇	1	2	yes	—
367	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
368	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
369	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
370	C₈	—	2	4	no	—
371	C₈	—	4	2	no	—
372	C₈	—	4	2	no	—
373	C₈	—	4	2	no	—
374	C₂²	—	2	1	no	—
375	C₂²	—	2	1	no	—
376	C₂²	—	2	4	no	—
377	C₂²	—	4	2	no	—
378	C₂²	—	4	2	no	—
379	C₂²	—	8	4	no	—
380	C₂²	—	8	1	no	—
381	C₂²	—	8	1	no	—
382	C₂²	—	8	1	no	—
383	C₂²	#465×C₆₇	1	2	yes	—
384	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
385	C₄	—	2	1	no	—
386	C₄	—	2	1	no	—
387	C₄	—	2	1	no	—
388	C₄	—	4	2	no	—
389	C₄	—	8	1	no	—
390	C₄	—	8	2	no	—
391	C₄	#465×C₆₇	1	2	yes	—
392	C₄	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
393	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
394	C₄	#577×C₆₇	1	2	yes	—
395	C₂	—	2	2	no	—
396	C₂	—	4	4	no	—
397	C₂	—	16	1	no	—
398	C₂	C₆₇×#3042	1	1	yes	characteristic
399	C₂	C₂×#465×C₆₇	1	1	yes	characteristic
400	C₂	C₆₇×#3068	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	C₆₇×#3036	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3227	C₂	semidirect
2. C₂×#476×C₆₇	C₂	semidirect
4. C₆₇×#2972	C₂	semidirect
5. C₆₇×#1677	C₂	semidirect
6. C₆₇×#1671	C₂	semidirect
7. C₆₇×#1570	C₂	semidirect
8. C₆₇×#1548	C₂	semidirect
9. C₆₇×#1540	C₂	semidirect
14. #498×C₆₇	C₂²	semidirect
15. #486×C₆₇	C₂²	semidirect
33. #334×C₆₇	C₂²	semidirect
34. #333×C₆₇	C₂²	semidirect
35. #332×C₆₇	C₂²	semidirect
36. #331×C₆₇	C₂²	semidirect
194. #19391	C₆₇	direct (N × H)
201. #3227	C₂×C₆₇	semidirect
202. C₂×#476	C₂×C₆₇	semidirect
204. #2972	C₂×C₆₇	semidirect
205. #1677	C₂×C₆₇	semidirect
206. #1671	C₂×C₆₇	semidirect
207. #1570	C₂×C₆₇	semidirect
208. #1548	C₂×C₆₇	semidirect
209. #1540	C₂×C₆₇	semidirect
210. C₆₇	#19391	direct (N × H)
215. #498	C₂²×C₆₇	semidirect
216. #486	C₂²×C₆₇	semidirect
234. #334	C₂²×C₆₇	semidirect
235. #333	C₂²×C₆₇	semidirect
236. #332	C₂²×C₆₇	semidirect
237. #331	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3227	2
C₂×#476×C₆₇	2
C₆₇×#3036	2
C₆₇×#2972	2
C₆₇×#1677	2
C₆₇×#1671	2
C₆₇×#1570	2
C₆₇×#1548	2
C₆₇×#1540	2
#19391	67

Subgroups — C67×#19391 #30698642

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#19391 #30698642