Subgroups — C₆₇×#25928 #30705179

Order 17152.

1034 conjugacy classes of subgroups.

The subgroup lattice has 432 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, complemented
2	C₆₇×#3138	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3038	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2641	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2027	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×#1883	C₂	1	4	yes	maximal, complemented
9	C₆₇×#1759	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×AGL(1,ℤ/8)×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×#127×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	#490×C₆₇	—	2	1	no	—
24	#490×C₆₇	—	2	1	no	—
25	#486×C₆₇	—	2	2	no	—
26	#460×C₆₇	—	2	2	no	—
27	#459×C₆₇	—	2	2	no	—
28	#423×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×#104×C₆₇	C₂²	1	2	yes	—
30	#418×C₆₇	C₂²	1	2	yes	—
31	#418×C₆₇	C₂²	1	4	yes	—
32	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
33	#394×C₆₇	—	2	4	no	—
34	#390×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×D₄×C₆₇	—	2	2	no	—
41	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
42	AGL(1,ℤ/8)×C₆₇	—	2	8	no	—
43	C₂×SD₁₆×C₆₇	—	2	2	no	—
44	C₂×D₈×C₆₇	—	2	2	no	—
45	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
46	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
47	C₄⋊Q₈×C₆₇	—	2	4	no	—
48	C₄⋊Q₈×C₆₇	—	2	2	no	—
49	C₄⋊Q₈×C₆₇	—	2	2	no	—
50	C₄⋊Q₈×C₆₇	D₄	1	2	yes	—
51	#127×C₆₇	—	2	2	no	—
52	#127×C₆₇	—	2	1	no	—
53	#127×C₆₇	—	2	1	no	—
54	#125×C₆₇	—	2	4	no	—
55	#125×C₆₇	—	2	4	no	—
56	#125×C₆₇	—	2	2	no	—
57	#125×C₆₇	—	2	2	no	—
58	#125×C₆₇	—	2	1	no	—
59	#125×C₆₇	—	2	1	no	—
60	#125×C₆₇	—	2	2	no	—
61	#125×C₆₇	D₄	1	1	yes	characteristic
62	#125×C₆₇	D₄	1	1	yes	characteristic
63	C₂²⋊Q₈×C₆₇	—	2	4	no	—
64	C₂²⋊Q₈×C₆₇	—	2	4	no	—
65	C₂²⋊Q₈×C₆₇	—	2	4	no	—
66	C₂²⋊Q₈×C₆₇	—	2	2	no	—
67	C₂²⋊Q₈×C₆₇	—	2	4	no	—
68	C₄×Q₈×C₆₇	—	2	4	no	—
69	C₄×Q₈×C₆₇	—	2	2	no	—
70	C₄×Q₈×C₆₇	—	2	2	no	—
71	C₄×Q₈×C₆₇	D₄	1	2	yes	—
72	C₄×D₄×C₆₇	—	2	2	no	—
73	C₄×D₄×C₆₇	—	2	2	no	—
74	C₄×D₄×C₆₇	—	2	4	no	—
75	C₄×D₄×C₆₇	D₄	1	2	yes	—
76	#117×C₆₇	—	2	2	no	—
77	#117×C₆₇	—	2	1	no	—
78	#117×C₆₇	C₂³	1	1	yes	characteristic
79	#117×C₆₇	C₂³	1	2	yes	—
80	C₂×C₄⋊C₄×C₆₇	—	2	4	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×C₂²⋊C₄×C₆₇	—	2	8	no	—
85	C₂×C₄²×C₆₇	—	2	1	no	—
86	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
87	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
88	C₄⋊C₈×C₆₇	—	2	2	no	—
89	#104×C₆₇	—	2	4	no	—
90	#102×C₆₇	—	2	4	no	—
91	#102×C₆₇	—	2	4	no	—
92	#102×C₆₇	—	2	4	no	—
93	#97×C₆₇	—	2	1	no	—
94	C₄×C₈×C₆₇	—	2	1	no	—
95	#95×C₆₇	—	2	2	no	—
96	C₂⁴×C₆₇	—	2	2	no	—
97	#41×C₆₇	—	2	2	no	—
98	#41×C₆₇	#41	1	4	yes	—
99	C₂×Q₈×C₆₇	—	2	4	no	—
100	C₂×Q₈×C₆₇	—	2	2	no	—
101	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂×D₄×C₆₇	—	2	1	no	—
103	C₂×D₄×C₆₇	—	2	2	no	—
104	C₂×D₄×C₆₇	—	2	2	no	—
105	C₂×D₄×C₆₇	—	2	4	no	—
106	C₂×D₄×C₆₇	—	2	4	no	—
107	C₂×D₄×C₆₇	—	4	2	no	—
108	C₂×D₄×C₆₇	—	4	2	no	—
109	C₂×D₄×C₆₇	—	8	4	no	—
110	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₆₇	—	2	1	no	—
112	C₂²×C₄×C₆₇	—	2	4	no	—
113	C₂²×C₄×C₆₇	—	2	2	no	—
114	C₂²×C₄×C₆₇	—	2	2	no	—
115	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
116	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
117	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
118	SD₁₆×C₆₇	—	4	4	no	—
119	D₈×C₆₇	—	4	4	no	—
120	M₁₆×C₆₇	—	2	2	no	—
121	M₁₆×C₆₇	—	2	4	no	—
122	M₁₆×C₆₇	—	2	2	no	—
123	C₂×C₈×C₆₇	—	2	2	no	—
124	C₂×C₈×C₆₇	—	2	2	no	—
125	C₄⋊C₄×C₆₇	—	2	4	no	—
126	C₄⋊C₄×C₆₇	—	2	4	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	4	no	—
131	C₄⋊C₄×C₆₇	—	2	4	no	—
132	C₄⋊C₄×C₆₇	—	2	2	no	—
133	C₄⋊C₄×C₆₇	—	2	2	no	—
134	C₄⋊C₄×C₆₇	—	2	2	no	—
135	C₄⋊C₄×C₆₇	—	2	2	no	—
136	C₄⋊C₄×C₆₇	—	2	2	no	—
137	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
138	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
139	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
140	C₂²⋊C₄×C₆₇	—	2	4	no	—
141	C₂²⋊C₄×C₆₇	—	2	2	no	—
142	C₂²⋊C₄×C₆₇	—	2	1	no	—
143	C₂²⋊C₄×C₆₇	—	2	2	no	—
144	C₂²⋊C₄×C₆₇	—	8	8	no	—
145	C₄²×C₆₇	—	2	4	no	—
146	C₄²×C₆₇	—	2	2	no	—
147	C₄²×C₆₇	—	2	1	no	—
148	C₄²×C₆₇	—	2	2	no	—
149	C₄²×C₆₇	—	2	2	no	—
150	C₄²×C₆₇	—	2	2	no	—
151	C₄²×C₆₇	C₂×D₄	1	2	yes	—
152	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
153	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
154	C₂³×C₆₇	—	2	1	no	—
155	C₂³×C₆₇	—	4	8	no	—
156	C₂³×C₆₇	—	4	2	no	—
157	C₂³×C₆₇	—	4	4	no	—
158	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
159	Q₈×C₆₇	—	2	1	no	—
160	Q₈×C₆₇	—	8	2	no	—
161	Q₈×C₆₇	—	8	4	no	—
162	Q₈×C₆₇	#124	1	2	yes	—
163	D₄×C₆₇	—	2	4	no	—
164	D₄×C₆₇	—	2	1	no	—
165	D₄×C₆₇	—	4	2	no	—
166	D₄×C₆₇	—	4	2	no	—
167	D₄×C₆₇	—	4	4	no	—
168	D₄×C₆₇	—	8	4	no	—
169	D₄×C₆₇	—	8	4	no	—
170	D₄×C₆₇	#124	1	2	yes	—
171	C₂×C₄×C₆₇	—	2	1	no	—
172	C₂×C₄×C₆₇	—	2	4	no	—
173	C₂×C₄×C₆₇	—	2	2	no	—
174	C₂×C₄×C₆₇	—	2	2	no	—
175	C₂×C₄×C₆₇	—	2	4	no	—
176	C₂×C₄×C₆₇	—	2	2	no	—
177	C₂×C₄×C₆₇	—	4	2	no	—
178	C₂×C₄×C₆₇	—	4	4	no	—
179	C₂×C₄×C₆₇	—	4	4	no	—
180	C₂×C₄×C₆₇	—	4	2	no	—
181	C₂×C₄×C₆₇	—	8	2	no	—
182	C₂×C₄×C₆₇	—	8	4	no	—
183	C₂×C₄×C₆₇	#120	1	2	yes	—
184	C₂×C₄×C₆₇	#120	1	2	yes	—
185	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
186	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
187	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
188	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
189	C₈×C₆₇	—	4	2	no	—
190	C₈×C₆₇	—	4	2	no	—
191	C₂²×C₆₇	—	2	2	no	—
192	C₂²×C₆₇	—	2	2	no	—
193	C₂²×C₆₇	—	4	2	no	—
194	C₂²×C₆₇	—	8	8	no	—
195	C₂²×C₆₇	—	8	2	no	—
196	C₂²×C₆₇	—	8	4	no	—
197	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
198	C₂²×C₆₇	#548	1	2	yes	—
199	C₄×C₆₇	—	2	2	no	—
200	C₄×C₆₇	—	2	2	no	—
201	C₄×C₆₇	—	2	2	no	—
202	C₄×C₆₇	—	4	4	no	—
203	C₄×C₆₇	—	4	2	no	—
204	C₄×C₆₇	—	4	2	no	—
205	C₄×C₆₇	—	4	2	no	—
206	C₄×C₆₇	—	8	2	no	—
207	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
208	C₄×C₆₇	#536	1	1	yes	characteristic
209	C₄×C₆₇	#548	1	2	yes	—
210	#25928	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
211	C₂×C₆₇	—	2	2	no	—
212	C₂×C₆₇	—	4	2	no	—
213	C₂×C₆₇	—	16	2	no	—
214	C₂×C₆₇	#2436	1	1	yes	characteristic
215	C₂×C₆₇	#3021	1	2	yes	—
216	#3504	C₂×C₆₇	1	1	yes	characteristic, complemented
217	#3138	C₂×C₆₇	1	1	yes	characteristic, complemented
218	#3038	C₂×C₆₇	1	1	yes	characteristic
219	#2641	C₂×C₆₇	1	1	yes	characteristic, complemented
220	#2027	C₂×C₆₇	1	4	yes	complemented
221	#1998	C₂×C₆₇	1	1	yes	characteristic, complemented
222	#1995	C₂×C₆₇	1	1	yes	characteristic, complemented
223	#1883	C₂×C₆₇	1	4	yes	complemented
224	#1759	C₂×C₆₇	1	1	yes	characteristic, complemented
225	C₆₇	#25928	1	1	yes	characteristic, direct product factor, Sylow
226	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	2	yes	—
227	#557	—	2	2	no	—
228	#554	—	2	2	no	—
229	#554	—	2	4	no	—
230	#544	—	2	2	no	—
231	#544	C₂²×C₆₇	1	2	yes	—
232	#533	—	2	1	no	—
233	#533	C₂²×C₆₇	1	1	yes	characteristic
234	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
235	C₂×#125	—	2	1	no	—
236	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
237	#520	C₂²×C₆₇	1	2	yes	—
238	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
239	#490	—	2	1	no	—
240	#490	—	2	1	no	—
241	#486	—	2	2	no	—
242	#460	—	2	2	no	—
243	#459	—	2	2	no	—
244	#423	C₂²×C₆₇	1	1	yes	characteristic
245	C₂×#104	C₂²×C₆₇	1	2	yes	—
246	#418	C₂²×C₆₇	1	4	yes	—
247	#418	C₂²×C₆₇	1	2	yes	—
248	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
249	#394	—	2	4	no	—
250	#390	C₂²×C₆₇	1	4	yes	—
251	#376	C₂²×C₆₇	1	1	yes	characteristic
252	#343	C₂²×C₆₇	1	4	yes	—
253	#339	C₂²×C₆₇	1	4	yes	—
254	#339	C₂²×C₆₇	1	2	yes	—
255	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
256	C₂²×D₄	—	2	2	no	—
257	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
258	AGL(1,ℤ/8)	—	2	8	no	—
259	C₂×SD₁₆	—	2	2	no	—
260	C₂×D₈	—	2	2	no	—
261	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
262	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
263	C₄⋊Q₈	—	2	4	no	—
264	C₄⋊Q₈	—	2	2	no	—
265	C₄⋊Q₈	—	2	2	no	—
266	C₄⋊Q₈	D₄×C₆₇	1	2	yes	—
267	#127	—	2	2	no	—
268	#127	—	2	1	no	—
269	#127	—	2	1	no	—
270	#125	—	2	4	no	—
271	#125	—	2	4	no	—
272	#125	—	2	2	no	—
273	#125	—	2	2	no	—
274	#125	—	2	1	no	—
275	#125	—	2	1	no	—
276	#125	—	2	2	no	—
277	#125	D₄×C₆₇	1	1	yes	characteristic
278	#125	D₄×C₆₇	1	1	yes	characteristic
279	C₂²⋊Q₈	—	2	4	no	—
280	C₂²⋊Q₈	—	2	4	no	—
281	C₂²⋊Q₈	—	2	4	no	—
282	C₂²⋊Q₈	—	2	2	no	—
283	C₂²⋊Q₈	—	2	4	no	—
284	C₄×Q₈	—	2	4	no	—
285	C₄×Q₈	—	2	2	no	—
286	C₄×Q₈	—	2	2	no	—
287	C₄×Q₈	D₄×C₆₇	1	2	yes	—
288	C₄×D₄	—	2	2	no	—
289	C₄×D₄	—	2	2	no	—
290	C₄×D₄	—	2	4	no	—
291	C₄×D₄	D₄×C₆₇	1	2	yes	—
292	#117	—	2	2	no	—
293	#117	—	2	1	no	—
294	#117	C₂³×C₆₇	1	1	yes	characteristic
295	#117	C₂³×C₆₇	1	2	yes	—
296	C₂×C₄⋊C₄	—	2	4	no	—
297	C₂×C₄⋊C₄	—	2	2	no	—
298	C₂×C₄⋊C₄	D₄×C₆₇	1	4	yes	—
299	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
300	C₂×C₂²⋊C₄	—	2	8	no	—
301	C₂×C₄²	—	2	1	no	—
302	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
303	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
304	C₄⋊C₈	—	2	2	no	—
305	#104	—	2	4	no	—
306	#102	—	2	4	no	—
307	#102	—	2	4	no	—
308	#102	—	2	4	no	—
309	#97	—	2	1	no	—
310	C₄×C₈	—	2	1	no	—
311	#95	—	2	2	no	—
312	C₂⁴	—	2	2	no	—
313	#41	—	2	2	no	—
314	#41	#41×C₆₇	1	4	yes	—
315	C₂×Q₈	—	2	4	no	—
316	C₂×Q₈	—	2	2	no	—
317	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
318	C₂×D₄	—	2	1	no	—
319	C₂×D₄	—	2	2	no	—
320	C₂×D₄	—	2	2	no	—
321	C₂×D₄	—	2	4	no	—
322	C₂×D₄	—	2	4	no	—
323	C₂×D₄	—	4	2	no	—
324	C₂×D₄	—	4	2	no	—
325	C₂×D₄	—	8	4	no	—
326	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
327	C₂²×C₄	—	2	1	no	—
328	C₂²×C₄	—	2	4	no	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	—	2	2	no	—
331	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
332	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
333	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
334	SD₁₆	—	4	4	no	—
335	D₈	—	4	4	no	—
336	M₁₆	—	2	2	no	—
337	M₁₆	—	2	4	no	—
338	M₁₆	—	2	2	no	—
339	C₂×C₈	—	2	2	no	—
340	C₂×C₈	—	2	2	no	—
341	C₄⋊C₄	—	2	4	no	—
342	C₄⋊C₄	—	2	4	no	—
343	C₄⋊C₄	—	2	4	no	—
344	C₄⋊C₄	—	2	4	no	—
345	C₄⋊C₄	—	2	2	no	—
346	C₄⋊C₄	—	2	4	no	—
347	C₄⋊C₄	—	2	4	no	—
348	C₄⋊C₄	—	2	2	no	—
349	C₄⋊C₄	—	2	2	no	—
350	C₄⋊C₄	—	2	2	no	—
351	C₄⋊C₄	—	2	2	no	—
352	C₄⋊C₄	—	2	2	no	—
353	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
354	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
355	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
356	C₂²⋊C₄	—	2	4	no	—
357	C₂²⋊C₄	—	2	2	no	—
358	C₂²⋊C₄	—	2	1	no	—
359	C₂²⋊C₄	—	2	2	no	—
360	C₂²⋊C₄	—	8	8	no	—
361	C₄²	—	2	4	no	—
362	C₄²	—	2	2	no	—
363	C₄²	—	2	1	no	—
364	C₄²	—	2	2	no	—
365	C₄²	—	2	2	no	—
366	C₄²	—	2	2	no	—
367	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
368	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
369	C₄²	C₂×D₄×C₆₇	1	2	yes	—
370	C₂³	—	2	1	no	—
371	C₂³	—	4	8	no	—
372	C₂³	—	4	2	no	—
373	C₂³	—	4	4	no	—
374	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
375	Q₈	—	2	1	no	—
376	Q₈	—	8	2	no	—
377	Q₈	—	8	4	no	—
378	Q₈	#124×C₆₇	1	2	yes	—
379	D₄	—	2	4	no	—
380	D₄	—	2	1	no	—
381	D₄	—	4	2	no	—
382	D₄	—	4	2	no	—
383	D₄	—	4	4	no	—
384	D₄	—	8	4	no	—
385	D₄	—	8	4	no	—
386	D₄	#124×C₆₇	1	2	yes	—
387	C₂×C₄	—	2	1	no	—
388	C₂×C₄	—	2	4	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	4	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	4	2	no	—
394	C₂×C₄	—	4	4	no	—
395	C₂×C₄	—	4	4	no	—
396	C₂×C₄	—	4	2	no	—
397	C₂×C₄	—	8	2	no	—
398	C₂×C₄	—	8	4	no	—
399	C₂×C₄	#120×C₆₇	1	2	yes	—
400	C₂×C₄	#120×C₆₇	1	2	yes	—
401	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
402	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
403	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
404	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
405	C₈	—	4	2	no	—
406	C₈	—	4	2	no	—
407	C₂²	—	2	2	no	—
408	C₂²	—	2	2	no	—
409	C₂²	—	4	2	no	—
410	C₂²	—	8	8	no	—
411	C₂²	—	8	4	no	—
412	C₂²	—	8	2	no	—
413	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
414	C₂²	#548×C₆₇	1	2	yes	—
415	C₄	—	2	2	no	—
416	C₄	—	2	2	no	—
417	C₄	—	2	2	no	—
418	C₄	—	4	4	no	—
419	C₄	—	4	2	no	—
420	C₄	—	4	2	no	—
421	C₄	—	4	2	no	—
422	C₄	—	8	2	no	—
423	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
424	C₄	#536×C₆₇	1	1	yes	characteristic
425	C₄	#548×C₆₇	1	2	yes	—
426	C₂	—	2	2	no	—
427	C₂	—	4	2	no	—
428	C₂	—	16	2	no	—
429	C₂	C₆₇×#2436	1	1	yes	characteristic
430	C₂	C₆₇×#3021	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₆₇×#3038	C₂
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
1. C₆₇×#3504	C₂	semidirect
2. C₆₇×#3138	C₂	semidirect
4. C₆₇×#2641	C₂	semidirect
5. C₆₇×#2027	C₂	semidirect
6. C₆₇×#1998	C₂	semidirect
7. C₆₇×#1995	C₂	semidirect
8. C₆₇×#1883	C₂	semidirect
9. C₆₇×#1759	C₂	semidirect
210. #25928	C₆₇	direct (N × H)
216. #3504	C₂×C₆₇	semidirect
217. #3138	C₂×C₆₇	semidirect
219. #2641	C₂×C₆₇	semidirect
220. #2027	C₂×C₆₇	semidirect
221. #1998	C₂×C₆₇	semidirect
222. #1995	C₂×C₆₇	semidirect
223. #1883	C₂×C₆₇	semidirect
224. #1759	C₂×C₆₇	semidirect
225. C₆₇	#25928	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×#3138	2
C₆₇×#3038	2
C₆₇×#2641	2
C₆₇×#2027	2
C₆₇×#1998	2
C₆₇×#1995	2
C₆₇×#1883	2
C₆₇×#1759	2
#25928	67

Subgroups — C67×#25928 #30705179

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#25928 #30705179