Subgroups — C₆₇×#25927 #30705178

Order 17152.

950 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3504	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#3139	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3039	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2642	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2028	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1998	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1995	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1882	C₂	1	4	yes	maximal
9	C₆₇×#1759	C₂	1	1	yes	maximal, characteristic
10	C₂×#137×C₆₇	C₂²	1	2	yes	—
11	#557×C₆₇	—	2	2	no	—
12	#554×C₆₇	—	2	2	no	—
13	#554×C₆₇	—	2	4	no	—
14	#544×C₆₇	—	2	2	no	—
15	#544×C₆₇	C₂²	1	2	yes	—
16	#533×C₆₇	—	2	1	no	—
17	#533×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#125×C₆₇	—	2	1	no	—
20	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
21	#520×C₆₇	C₂²	1	2	yes	—
22	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
23	#491×C₆₇	—	2	1	no	—
24	#491×C₆₇	—	2	1	no	—
25	#487×C₆₇	—	2	2	no	—
26	#462×C₆₇	—	2	2	no	—
27	#461×C₆₇	—	2	2	no	—
28	#423×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×#104×C₆₇	C₂²	1	2	yes	—
30	#419×C₆₇	C₂²	1	2	yes	—
31	#419×C₆₇	C₂²	1	4	yes	—
32	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
33	#400×C₆₇	—	2	4	no	—
34	#391×C₆₇	C₂²	1	4	yes	—
35	#376×C₆₇	C₂²	1	1	yes	characteristic
36	#343×C₆₇	C₂²	1	4	yes	—
37	#339×C₆₇	C₂²	1	4	yes	—
38	#339×C₆₇	C₂²	1	2	yes	—
39	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
40	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
41	#137×C₆₇	—	2	8	no	—
42	C₂×Q₁₆×C₆₇	—	2	2	no	—
43	C₂×SD₁₆×C₆₇	—	2	2	no	—
44	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
45	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
46	C₄⋊Q₈×C₆₇	—	2	4	no	—
47	C₄⋊Q₈×C₆₇	—	2	2	no	—
48	C₄⋊Q₈×C₆₇	—	2	2	no	—
49	C₄⋊Q₈×C₆₇	—	2	2	no	—
50	C₄⋊Q₈×C₆₇	—	2	1	no	—
51	C₄⋊Q₈×C₆₇	—	2	1	no	—
52	C₄⋊Q₈×C₆₇	D₄	1	2	yes	—
53	#125×C₆₇	—	2	4	no	—
54	#125×C₆₇	—	2	4	no	—
55	#125×C₆₇	—	2	2	no	—
56	#125×C₆₇	—	2	2	no	—
57	#125×C₆₇	—	2	1	no	—
58	#125×C₆₇	—	2	1	no	—
59	#125×C₆₇	—	2	2	no	—
60	#125×C₆₇	D₄	1	1	yes	characteristic
61	#125×C₆₇	D₄	1	1	yes	characteristic
62	C₂²⋊Q₈×C₆₇	—	2	4	no	—
63	C₂²⋊Q₈×C₆₇	—	2	4	no	—
64	C₂²⋊Q₈×C₆₇	—	2	4	no	—
65	C₂²⋊Q₈×C₆₇	—	2	2	no	—
66	C₂²⋊Q₈×C₆₇	—	2	4	no	—
67	C₄×Q₈×C₆₇	—	2	4	no	—
68	C₄×Q₈×C₆₇	—	2	2	no	—
69	C₄×Q₈×C₆₇	—	2	2	no	—
70	C₄×Q₈×C₆₇	D₄	1	2	yes	—
71	C₄×D₄×C₆₇	—	2	2	no	—
72	C₄×D₄×C₆₇	—	2	2	no	—
73	C₄×D₄×C₆₇	—	2	4	no	—
74	C₄×D₄×C₆₇	D₄	1	2	yes	—
75	#117×C₆₇	—	2	2	no	—
76	#117×C₆₇	—	2	1	no	—
77	#117×C₆₇	C₂³	1	1	yes	characteristic
78	#117×C₆₇	C₂³	1	2	yes	—
79	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
80	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
81	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
82	C₂×C₄⋊C₄×C₆₇	D₄	1	4	yes	—
83	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
84	C₂×C₄²×C₆₇	—	2	1	no	—
85	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
86	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
87	C₄⋊C₈×C₆₇	—	2	2	no	—
88	#104×C₆₇	—	2	4	no	—
89	#103×C₆₇	—	2	4	no	—
90	#103×C₆₇	—	2	4	no	—
91	#103×C₆₇	—	2	4	no	—
92	#97×C₆₇	—	2	1	no	—
93	C₄×C₈×C₆₇	—	2	1	no	—
94	#95×C₆₇	—	2	2	no	—
95	#95×C₆₇	—	2	8	no	—
96	#41×C₆₇	—	2	2	no	—
97	#41×C₆₇	#41	1	4	yes	—
98	C₂×Q₈×C₆₇	—	2	4	no	—
99	C₂×Q₈×C₆₇	—	2	2	no	—
100	C₂×Q₈×C₆₇	—	2	2	no	—
101	C₂×Q₈×C₆₇	—	2	2	no	—
102	C₂×Q₈×C₆₇	—	2	4	no	—
103	C₂×Q₈×C₆₇	—	2	4	no	—
104	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
105	C₂×D₄×C₆₇	—	2	1	no	—
106	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₆₇	—	2	1	no	—
108	C₂²×C₄×C₆₇	—	2	4	no	—
109	C₂²×C₄×C₆₇	—	2	2	no	—
110	C₂²×C₄×C₆₇	—	2	2	no	—
111	C₂²×C₄×C₆₇	—	2	2	no	—
112	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
113	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
114	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
115	Q₁₆×C₆₇	—	4	4	no	—
116	SD₁₆×C₆₇	—	4	4	no	—
117	M₁₆×C₆₇	—	2	2	no	—
118	M₁₆×C₆₇	—	2	4	no	—
119	M₁₆×C₆₇	—	2	2	no	—
120	C₂×C₈×C₆₇	—	2	2	no	—
121	C₂×C₈×C₆₇	—	2	2	no	—
122	C₄⋊C₄×C₆₇	—	2	4	no	—
123	C₄⋊C₄×C₆₇	—	2	4	no	—
124	C₄⋊C₄×C₆₇	—	2	4	no	—
125	C₄⋊C₄×C₆₇	—	2	4	no	—
126	C₄⋊C₄×C₆₇	—	2	2	no	—
127	C₄⋊C₄×C₆₇	—	2	4	no	—
128	C₄⋊C₄×C₆₇	—	2	4	no	—
129	C₄⋊C₄×C₆₇	—	2	2	no	—
130	C₄⋊C₄×C₆₇	—	2	2	no	—
131	C₄⋊C₄×C₆₇	—	2	2	no	—
132	C₄⋊C₄×C₆₇	—	2	2	no	—
133	C₄⋊C₄×C₆₇	—	2	2	no	—
134	C₄⋊C₄×C₆₇	—	4	2	no	—
135	C₄⋊C₄×C₆₇	—	4	2	no	—
136	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
137	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
138	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
139	C₂²⋊C₄×C₆₇	—	2	4	no	—
140	C₂²⋊C₄×C₆₇	—	2	2	no	—
141	C₂²⋊C₄×C₆₇	—	2	1	no	—
142	C₂²⋊C₄×C₆₇	—	2	2	no	—
143	C₄²×C₆₇	—	2	4	no	—
144	C₄²×C₆₇	—	2	2	no	—
145	C₄²×C₆₇	—	2	1	no	—
146	C₄²×C₆₇	—	2	2	no	—
147	C₄²×C₆₇	—	2	2	no	—
148	C₄²×C₆₇	—	2	2	no	—
149	C₄²×C₆₇	C₂×D₄	1	2	yes	—
150	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
151	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
152	C₂³×C₆₇	—	2	1	no	—
153	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
154	Q₈×C₆₇	—	2	1	no	—
155	Q₈×C₆₇	—	4	2	no	—
156	Q₈×C₆₇	—	4	2	no	—
157	Q₈×C₆₇	—	4	4	no	—
158	Q₈×C₆₇	—	8	2	no	—
159	Q₈×C₆₇	—	8	4	no	—
160	Q₈×C₆₇	#124	1	2	yes	—
161	D₄×C₆₇	—	2	4	no	—
162	D₄×C₆₇	—	2	1	no	—
163	D₄×C₆₇	#124	1	2	yes	—
164	C₂×C₄×C₆₇	—	2	1	no	—
165	C₂×C₄×C₆₇	—	2	4	no	—
166	C₂×C₄×C₆₇	—	2	2	no	—
167	C₂×C₄×C₆₇	—	2	2	no	—
168	C₂×C₄×C₆₇	—	2	4	no	—
169	C₂×C₄×C₆₇	—	2	2	no	—
170	C₂×C₄×C₆₇	—	4	2	no	—
171	C₂×C₄×C₆₇	—	4	4	no	—
172	C₂×C₄×C₆₇	—	4	4	no	—
173	C₂×C₄×C₆₇	—	4	2	no	—
174	C₂×C₄×C₆₇	—	4	2	no	—
175	C₂×C₄×C₆₇	—	4	4	no	—
176	C₂×C₄×C₆₇	—	8	2	no	—
177	C₂×C₄×C₆₇	—	8	4	no	—
178	C₂×C₄×C₆₇	#120	1	2	yes	—
179	C₂×C₄×C₆₇	#120	1	2	yes	—
180	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
181	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
182	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
183	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
184	C₈×C₆₇	—	4	2	no	—
185	C₈×C₆₇	—	4	2	no	—
186	C₂²×C₆₇	—	2	2	no	—
187	C₂²×C₆₇	—	2	2	no	—
188	C₂²×C₆₇	—	4	2	no	—
189	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
190	C₂²×C₆₇	#548	1	2	yes	—
191	C₄×C₆₇	—	2	2	no	—
192	C₄×C₆₇	—	2	2	no	—
193	C₄×C₆₇	—	2	2	no	—
194	C₄×C₆₇	—	4	4	no	—
195	C₄×C₆₇	—	4	2	no	—
196	C₄×C₆₇	—	4	2	no	—
197	C₄×C₆₇	—	4	2	no	—
198	C₄×C₆₇	—	8	2	no	—
199	C₄×C₆₇	—	8	2	no	—
200	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
201	C₄×C₆₇	#536	1	1	yes	characteristic
202	C₄×C₆₇	#548	1	2	yes	—
203	#25927	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
204	C₂×C₆₇	—	2	2	no	—
205	C₂×C₆₇	—	4	2	no	—
206	C₂×C₆₇	#2436	1	1	yes	characteristic
207	C₂×C₆₇	#3022	1	2	yes	—
208	#3504	C₂×C₆₇	1	1	yes	characteristic
209	#3139	C₂×C₆₇	1	1	yes	characteristic, complemented
210	#3039	C₂×C₆₇	1	1	yes	characteristic
211	#2642	C₂×C₆₇	1	1	yes	characteristic, complemented
212	#2028	C₂×C₆₇	1	4	yes	complemented
213	#1998	C₂×C₆₇	1	1	yes	characteristic, complemented
214	#1995	C₂×C₆₇	1	1	yes	characteristic, complemented
215	#1882	C₂×C₆₇	1	4	yes	—
216	#1759	C₂×C₆₇	1	1	yes	characteristic
217	C₆₇	#25927	1	1	yes	characteristic, direct product factor, Sylow
218	C₂×#137	C₂²×C₆₇	1	2	yes	—
219	#557	—	2	2	no	—
220	#554	—	2	2	no	—
221	#554	—	2	4	no	—
222	#544	—	2	2	no	—
223	#544	C₂²×C₆₇	1	2	yes	—
224	#533	—	2	1	no	—
225	#533	C₂²×C₆₇	1	1	yes	characteristic
226	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
227	C₂×#125	—	2	1	no	—
228	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
229	#520	C₂²×C₆₇	1	2	yes	—
230	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
231	#491	—	2	1	no	—
232	#491	—	2	1	no	—
233	#487	—	2	2	no	—
234	#462	—	2	2	no	—
235	#461	—	2	2	no	—
236	#423	C₂²×C₆₇	1	1	yes	characteristic
237	C₂×#104	C₂²×C₆₇	1	2	yes	—
238	#419	C₂²×C₆₇	1	4	yes	—
239	#419	C₂²×C₆₇	1	2	yes	—
240	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
241	#400	—	2	4	no	—
242	#391	C₂²×C₆₇	1	4	yes	—
243	#376	C₂²×C₆₇	1	1	yes	characteristic
244	#343	C₂²×C₆₇	1	4	yes	—
245	#339	C₂²×C₆₇	1	4	yes	—
246	#339	C₂²×C₆₇	1	2	yes	—
247	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
248	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
249	#137	—	2	8	no	—
250	C₂×Q₁₆	—	2	2	no	—
251	C₂×SD₁₆	—	2	2	no	—
252	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
253	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
254	C₄⋊Q₈	—	2	4	no	—
255	C₄⋊Q₈	—	2	2	no	—
256	C₄⋊Q₈	—	2	2	no	—
257	C₄⋊Q₈	—	2	2	no	—
258	C₄⋊Q₈	—	2	1	no	—
259	C₄⋊Q₈	—	2	1	no	—
260	C₄⋊Q₈	D₄×C₆₇	1	2	yes	—
261	#125	—	2	4	no	—
262	#125	—	2	4	no	—
263	#125	—	2	2	no	—
264	#125	—	2	2	no	—
265	#125	—	2	1	no	—
266	#125	—	2	1	no	—
267	#125	—	2	2	no	—
268	#125	D₄×C₆₇	1	1	yes	characteristic
269	#125	D₄×C₆₇	1	1	yes	characteristic
270	C₂²⋊Q₈	—	2	4	no	—
271	C₂²⋊Q₈	—	2	4	no	—
272	C₂²⋊Q₈	—	2	4	no	—
273	C₂²⋊Q₈	—	2	2	no	—
274	C₂²⋊Q₈	—	2	4	no	—
275	C₄×Q₈	—	2	4	no	—
276	C₄×Q₈	—	2	2	no	—
277	C₄×Q₈	—	2	2	no	—
278	C₄×Q₈	D₄×C₆₇	1	2	yes	—
279	C₄×D₄	—	2	2	no	—
280	C₄×D₄	—	2	2	no	—
281	C₄×D₄	—	2	4	no	—
282	C₄×D₄	D₄×C₆₇	1	2	yes	—
283	#117	—	2	2	no	—
284	#117	—	2	1	no	—
285	#117	C₂³×C₆₇	1	1	yes	characteristic
286	#117	C₂³×C₆₇	1	2	yes	—
287	C₂×C₄⋊C₄	—	2	4	no	—
288	C₂×C₄⋊C₄	—	2	2	no	—
289	C₂×C₄⋊C₄	—	2	2	no	—
290	C₂×C₄⋊C₄	D₄×C₆₇	1	4	yes	—
291	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
292	C₂×C₄²	—	2	1	no	—
293	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
294	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
295	C₄⋊C₈	—	2	2	no	—
296	#104	—	2	4	no	—
297	#103	—	2	4	no	—
298	#103	—	2	4	no	—
299	#103	—	2	4	no	—
300	#97	—	2	1	no	—
301	C₄×C₈	—	2	1	no	—
302	#95	—	2	2	no	—
303	#95	—	2	8	no	—
304	#41	—	2	2	no	—
305	#41	#41×C₆₇	1	4	yes	—
306	C₂×Q₈	—	2	4	no	—
307	C₂×Q₈	—	2	2	no	—
308	C₂×Q₈	—	2	2	no	—
309	C₂×Q₈	—	2	2	no	—
310	C₂×Q₈	—	2	4	no	—
311	C₂×Q₈	—	2	4	no	—
312	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
313	C₂×D₄	—	2	1	no	—
314	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	—	2	4	no	—
317	C₂²×C₄	—	2	2	no	—
318	C₂²×C₄	—	2	2	no	—
319	C₂²×C₄	—	2	2	no	—
320	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
321	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
322	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
323	Q₁₆	—	4	4	no	—
324	SD₁₆	—	4	4	no	—
325	M₁₆	—	2	2	no	—
326	M₁₆	—	2	4	no	—
327	M₁₆	—	2	2	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	2	2	no	—
330	C₄⋊C₄	—	2	4	no	—
331	C₄⋊C₄	—	2	4	no	—
332	C₄⋊C₄	—	2	4	no	—
333	C₄⋊C₄	—	2	4	no	—
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	—	2	4	no	—
336	C₄⋊C₄	—	2	4	no	—
337	C₄⋊C₄	—	2	2	no	—
338	C₄⋊C₄	—	2	2	no	—
339	C₄⋊C₄	—	2	2	no	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	—	2	2	no	—
342	C₄⋊C₄	—	4	2	no	—
343	C₄⋊C₄	—	4	2	no	—
344	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
345	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
346	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
347	C₂²⋊C₄	—	2	4	no	—
348	C₂²⋊C₄	—	2	2	no	—
349	C₂²⋊C₄	—	2	1	no	—
350	C₂²⋊C₄	—	2	2	no	—
351	C₄²	—	2	4	no	—
352	C₄²	—	2	2	no	—
353	C₄²	—	2	1	no	—
354	C₄²	—	2	2	no	—
355	C₄²	—	2	2	no	—
356	C₄²	—	2	2	no	—
357	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
358	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
359	C₄²	C₂×D₄×C₆₇	1	2	yes	—
360	C₂³	—	2	1	no	—
361	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
362	Q₈	—	2	1	no	—
363	Q₈	—	4	2	no	—
364	Q₈	—	4	2	no	—
365	Q₈	—	4	4	no	—
366	Q₈	—	8	2	no	—
367	Q₈	—	8	4	no	—
368	Q₈	#124×C₆₇	1	2	yes	—
369	D₄	—	2	4	no	—
370	D₄	—	2	1	no	—
371	D₄	#124×C₆₇	1	2	yes	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	2	4	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	4	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	4	2	no	—
379	C₂×C₄	—	4	4	no	—
380	C₂×C₄	—	4	4	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	4	no	—
384	C₂×C₄	—	8	2	no	—
385	C₂×C₄	—	8	4	no	—
386	C₂×C₄	#120×C₆₇	1	2	yes	—
387	C₂×C₄	#120×C₆₇	1	2	yes	—
388	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
389	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
390	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
391	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
392	C₈	—	4	2	no	—
393	C₈	—	4	2	no	—
394	C₂²	—	2	2	no	—
395	C₂²	—	2	2	no	—
396	C₂²	—	4	2	no	—
397	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
398	C₂²	#548×C₆₇	1	2	yes	—
399	C₄	—	2	2	no	—
400	C₄	—	2	2	no	—
401	C₄	—	2	2	no	—
402	C₄	—	4	4	no	—
403	C₄	—	4	2	no	—
404	C₄	—	4	2	no	—
405	C₄	—	4	2	no	—
406	C₄	—	8	2	no	—
407	C₄	—	8	2	no	—
408	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
409	C₄	#536×C₆₇	1	1	yes	characteristic
410	C₄	#548×C₆₇	1	2	yes	—
411	C₂	—	2	2	no	—
412	C₂	—	4	2	no	—
413	C₂	C₆₇×#2436	1	1	yes	characteristic
414	C₂	C₆₇×#3022	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#41×C₆₇	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ 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
2. C₆₇×#3139	C₂	semidirect
4. C₆₇×#2642	C₂	semidirect
5. C₆₇×#2028	C₂	semidirect
6. C₆₇×#1998	C₂	semidirect
7. C₆₇×#1995	C₂	semidirect
203. #25927	C₆₇	direct (N × H)
209. #3139	C₂×C₆₇	semidirect
211. #2642	C₂×C₆₇	semidirect
212. #2028	C₂×C₆₇	semidirect
213. #1998	C₂×C₆₇	semidirect
214. #1995	C₂×C₆₇	semidirect
217. C₆₇	#25927	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#3504	2
C₆₇×#3139	2
C₆₇×#3039	2
C₆₇×#2642	2
C₆₇×#2028	2
C₆₇×#1998	2
C₆₇×#1995	2
C₆₇×#1882	2
C₆₇×#1759	2
#25927	67

Subgroups — C67×#25927 #30705178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#25927 #30705178