Subgroups — C₆₇×#19196 #30698447

Order 17152.

714 conjugacy classes of subgroups.

The subgroup lattice has 424 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	C₆₇×#3205	C₂	1	1	yes	maximal, characteristic
3	C₂×#486×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2931	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1697	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1677	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1564	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1548	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1545	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#131×C₆₇	—	2	1	no	—
11	#532×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
14	#498×C₆₇	—	2	1	no	—
15	#493×C₆₇	—	2	1	no	—
16	#486×C₆₇	—	2	1	no	—
17	#486×C₆₇	—	2	1	no	—
18	#486×C₆₇	—	2	1	no	—
19	#486×C₆₇	—	2	1	no	—
20	#486×C₆₇	C₂²	1	2	yes	—
21	#486×C₆₇	C₂²	1	2	yes	—
22	#481×C₆₇	—	2	2	no	—
23	#476×C₆₇	C₂²	1	4	yes	complemented
24	#459×C₆₇	C₂²	1	4	yes	—
25	#447×C₆₇	—	2	2	no	—
26	#436×C₆₇	—	2	1	no	—
27	#436×C₆₇	C₂²	1	2	yes	complemented
28	C₈×D₄×C₆₇	—	2	1	no	—
29	C₈×D₄×C₆₇	C₂²	1	2	yes	complemented
30	#431×C₆₇	C₂²	1	1	yes	characteristic
31	#431×C₆₇	C₂²	1	1	yes	characteristic
32	#423×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×#102×C₆₇	—	2	1	no	—
34	C₂×#102×C₆₇	—	2	1	no	—
35	#408×C₆₇	—	2	1	no	—
36	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
37	#335×C₆₇	C₂²	1	2	yes	complemented
38	#334×C₆₇	C₂²	1	2	yes	complemented
39	#333×C₆₇	C₂²	1	4	yes	complemented
40	#331×C₆₇	C₂²	1	4	yes	—
41	C₂×#41×C₆₇	—	2	1	no	—
42	C₂²×Q₈×C₆₇	—	2	1	no	—
43	C₂²×D₄×C₆₇	—	2	1	no	—
44	C₂²×D₄×C₆₇	—	2	1	no	—
45	C₂²×D₄×C₆₇	—	2	1	no	—
46	C₂×D₈×C₆₇	—	2	4	no	—
47	#131×C₆₇	—	4	4	no	—
48	C₂×M₁₆×C₆₇	—	2	1	no	—
49	C₂×M₁₆×C₆₇	—	2	2	no	—
50	C₂²×C₈×C₆₇	—	2	2	no	—
51	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
53	C₄⋊Q₈×C₆₇	—	2	1	no	—
54	C₄⋊Q₈×C₆₇	—	2	1	no	—
55	C₄⋊Q₈×C₆₇	—	2	1	no	—
56	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
57	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
58	#127×C₆₇	—	2	1	no	—
59	#127×C₆₇	—	2	1	no	—
60	#127×C₆₇	—	2	1	no	—
61	#127×C₆₇	D₄	1	1	yes	characteristic
62	#127×C₆₇	C₂³	1	2	yes	—
63	#124×C₆₇	—	2	1	no	—
64	#121×C₆₇	—	2	1	no	—
65	#121×C₆₇	D₄	1	2	yes	—
66	C₄×D₄×C₆₇	—	2	1	no	—
67	C₄×D₄×C₆₇	C₂³	1	2	yes	—
68	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
69	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
70	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
71	#106×C₆₇	—	2	4	no	—
72	C₄⋊C₈×C₆₇	—	2	1	no	—
73	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
74	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
75	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
76	#102×C₆₇	—	2	4	no	—
77	#102×C₆₇	—	2	2	no	—
78	#102×C₆₇	—	2	2	no	—
79	#102×C₆₇	—	2	2	no	—
80	#102×C₆₇	—	2	2	no	—
81	#102×C₆₇	—	2	4	no	—
82	C₂²⋊C₈×C₆₇	—	2	1	no	—
83	C₂²⋊C₈×C₆₇	—	2	1	no	—
84	C₂²⋊C₈×C₆₇	—	2	2	no	—
85	C₂²⋊C₈×C₆₇	—	2	2	no	—
86	C₄×C₈×C₆₇	—	2	1	no	—
87	C₄×C₈×C₆₇	—	2	1	no	—
88	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
89	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
90	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
92	C₂⁴×C₆₇	—	4	1	no	—
93	#41×C₆₇	—	4	2	no	—
94	#41×C₆₇	—	4	2	no	—
95	C₂×Q₈×C₆₇	—	2	1	no	—
96	C₂×Q₈×C₆₇	—	2	2	no	—
97	C₂×Q₈×C₆₇	—	4	1	no	—
98	C₂×Q₈×C₆₇	—	8	2	no	—
99	C₂×D₄×C₆₇	—	2	2	no	—
100	C₂×D₄×C₆₇	—	2	1	no	—
101	C₂×D₄×C₆₇	—	2	1	no	—
102	C₂×D₄×C₆₇	—	2	2	no	—
103	C₂×D₄×C₆₇	—	2	2	no	—
104	C₂×D₄×C₆₇	—	2	2	no	—
105	C₂×D₄×C₆₇	—	4	1	no	—
106	C₂×D₄×C₆₇	—	4	1	no	—
107	C₂×D₄×C₆₇	—	4	1	no	—
108	C₂×D₄×C₆₇	—	4	2	no	—
109	C₂×D₄×C₆₇	—	4	2	no	—
110	C₂×D₄×C₆₇	—	8	2	no	—
111	C₂×D₄×C₆₇	—	8	2	no	—
112	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
113	C₂²×C₄×C₆₇	—	2	2	no	—
114	C₂²×C₄×C₆₇	—	4	1	no	—
115	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
118	D₈×C₆₇	—	8	4	no	—
119	M₁₆×C₆₇	—	4	4	no	—
120	M₁₆×C₆₇	—	4	2	no	—
121	C₂×C₈×C₆₇	—	2	2	no	—
122	C₂×C₈×C₆₇	—	2	2	no	—
123	C₂×C₈×C₆₇	—	2	2	no	—
124	C₂×C₈×C₆₇	—	2	2	no	—
125	C₂×C₈×C₆₇	—	2	2	no	—
126	C₂×C₈×C₆₇	—	4	4	no	—
127	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
128	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
130	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
131	C₄⋊C₄×C₆₇	—	2	1	no	—
132	C₄⋊C₄×C₆₇	—	2	2	no	—
133	C₄⋊C₄×C₆₇	—	2	2	no	—
134	C₄⋊C₄×C₆₇	—	4	1	no	—
135	C₄⋊C₄×C₆₇	—	4	1	no	—
136	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
137	C₂²⋊C₄×C₆₇	—	2	2	no	—
138	C₂²⋊C₄×C₆₇	—	2	2	no	—
139	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
140	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
141	C₄²×C₆₇	#41	1	1	yes	characteristic
142	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
143	C₂³×C₆₇	—	2	2	no	—
144	C₂³×C₆₇	—	4	2	no	—
145	C₂³×C₆₇	—	8	2	no	—
146	C₂³×C₆₇	—	8	2	no	—
147	C₂³×C₆₇	—	8	2	no	—
148	C₂³×C₆₇	#121	1	1	yes	characteristic
149	Q₈×C₆₇	—	4	2	no	—
150	Q₈×C₆₇	—	8	2	no	—
151	Q₈×C₆₇	—	8	2	no	—
152	D₄×C₆₇	—	2	4	no	—
153	D₄×C₆₇	—	4	4	no	—
154	D₄×C₆₇	—	4	2	no	—
155	D₄×C₆₇	—	4	2	no	—
156	D₄×C₆₇	—	4	4	no	—
157	D₄×C₆₇	—	4	2	no	—
158	D₄×C₆₇	—	4	2	no	—
159	D₄×C₆₇	—	8	2	no	—
160	D₄×C₆₇	—	8	2	no	—
161	D₄×C₆₇	—	8	2	no	—
162	D₄×C₆₇	—	8	2	no	—
163	C₂×C₄×C₆₇	—	2	2	no	—
164	C₂×C₄×C₆₇	—	2	2	no	—
165	C₂×C₄×C₆₇	—	2	2	no	—
166	C₂×C₄×C₆₇	—	4	2	no	—
167	C₂×C₄×C₆₇	—	4	2	no	—
168	C₂×C₄×C₆₇	—	4	2	no	—
169	C₂×C₄×C₆₇	—	8	2	no	—
170	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
171	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
172	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
173	C₂×C₄×C₆₇	#135	1	2	yes	—
174	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
175	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
176	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
178	C₈×C₆₇	—	2	1	no	—
179	C₈×C₆₇	—	2	1	no	—
180	C₈×C₆₇	—	2	1	no	—
181	C₈×C₆₇	—	2	1	no	—
182	C₈×C₆₇	—	4	2	no	—
183	C₈×C₆₇	—	4	2	no	—
184	C₈×C₆₇	—	4	2	no	—
185	C₂²×C₆₇	—	2	1	no	—
186	C₂²×C₆₇	—	2	1	no	—
187	C₂²×C₆₇	—	4	2	no	—
188	C₂²×C₆₇	—	4	2	no	—
189	C₂²×C₆₇	—	4	2	no	—
190	C₂²×C₆₇	—	8	2	no	—
191	C₂²×C₆₇	—	8	2	no	—
192	C₂²×C₆₇	—	8	2	no	—
193	C₂²×C₆₇	—	16	4	no	—
194	C₂²×C₆₇	#463	1	2	yes	—
195	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
196	C₄×C₆₇	—	2	1	no	—
197	C₄×C₆₇	—	2	1	no	—
198	C₄×C₆₇	—	4	2	no	—
199	C₄×C₆₇	—	8	2	no	—
200	C₄×C₆₇	#463	1	2	yes	—
201	C₄×C₆₇	#472	1	2	yes	—
202	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
203	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
204	C₄×C₆₇	#577	1	2	yes	—
205	#19196	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
206	C₂×C₆₇	—	2	2	no	—
207	C₂×C₆₇	—	8	2	no	—
208	C₂×C₆₇	—	16	2	no	—
209	C₂×C₆₇	C₂×#463	1	1	yes	characteristic
210	C₂×C₆₇	#3057	1	1	yes	characteristic
211	C₂×C₆₇	C₂×#472	1	1	yes	characteristic
212	#3227	C₂×C₆₇	1	1	yes	characteristic, complemented
213	#3205	C₂×C₆₇	1	1	yes	characteristic
214	C₂×#486	C₂×C₆₇	1	1	yes	characteristic, complemented
215	#2931	C₂×C₆₇	1	1	yes	characteristic, complemented
216	#1697	C₂×C₆₇	1	4	yes	complemented
217	#1677	C₂×C₆₇	1	4	yes	complemented
218	#1564	C₂×C₆₇	1	1	yes	characteristic, complemented
219	#1548	C₂×C₆₇	1	1	yes	characteristic, complemented
220	#1545	C₂×C₆₇	1	1	yes	characteristic, complemented
221	C₆₇	#19196	1	1	yes	characteristic, direct product factor, Sylow
222	C₂×#131	—	2	1	no	—
223	#532	C₂²×C₆₇	1	1	yes	characteristic
224	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
225	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
226	#498	—	2	1	no	—
227	#493	—	2	1	no	—
228	#486	—	2	1	no	—
229	#486	—	2	1	no	—
230	#486	—	2	1	no	—
231	#486	—	2	1	no	—
232	#486	C₂²×C₆₇	1	2	yes	—
233	#486	C₂²×C₆₇	1	2	yes	—
234	#481	—	2	2	no	—
235	#476	C₂²×C₆₇	1	4	yes	complemented
236	#459	C₂²×C₆₇	1	4	yes	—
237	#447	—	2	2	no	—
238	#436	—	2	1	no	—
239	#436	C₂²×C₆₇	1	2	yes	complemented
240	C₈×D₄	—	2	1	no	—
241	C₈×D₄	C₂²×C₆₇	1	2	yes	complemented
242	#431	C₂²×C₆₇	1	1	yes	characteristic
243	#431	C₂²×C₆₇	1	1	yes	characteristic
244	#423	C₂²×C₆₇	1	1	yes	characteristic
245	C₂×#102	—	2	1	no	—
246	C₂×#102	—	2	1	no	—
247	#408	—	2	1	no	—
248	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
249	#335	C₂²×C₆₇	1	2	yes	complemented
250	#334	C₂²×C₆₇	1	2	yes	complemented
251	#333	C₂²×C₆₇	1	4	yes	complemented
252	#331	C₂²×C₆₇	1	4	yes	—
253	C₂×#41	—	2	1	no	—
254	C₂²×Q₈	—	2	1	no	—
255	C₂²×D₄	—	2	1	no	—
256	C₂²×D₄	—	2	1	no	—
257	C₂²×D₄	—	2	1	no	—
258	C₂×D₈	—	2	4	no	—
259	#131	—	4	4	no	—
260	C₂×M₁₆	—	2	1	no	—
261	C₂×M₁₆	—	2	2	no	—
262	C₂²×C₈	—	2	2	no	—
263	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
264	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
265	C₄⋊Q₈	—	2	1	no	—
266	C₄⋊Q₈	—	2	1	no	—
267	C₄⋊Q₈	—	2	1	no	—
268	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
269	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
270	#127	—	2	1	no	—
271	#127	—	2	1	no	—
272	#127	—	2	1	no	—
273	#127	D₄×C₆₇	1	1	yes	characteristic
274	#127	C₂³×C₆₇	1	2	yes	—
275	#124	—	2	1	no	—
276	#121	—	2	1	no	—
277	#121	D₄×C₆₇	1	2	yes	—
278	C₄×D₄	—	2	1	no	—
279	C₄×D₄	C₂³×C₆₇	1	2	yes	—
280	C₂×C₄⋊C₄	—	2	1	no	—
281	C₂×C₄⋊C₄	—	2	1	no	—
282	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
283	#106	—	2	4	no	—
284	C₄⋊C₈	—	2	1	no	—
285	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
286	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
287	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
288	#102	—	2	4	no	—
289	#102	—	2	2	no	—
290	#102	—	2	2	no	—
291	#102	—	2	2	no	—
292	#102	—	2	2	no	—
293	#102	—	2	4	no	—
294	C₂²⋊C₈	—	2	1	no	—
295	C₂²⋊C₈	—	2	1	no	—
296	C₂²⋊C₈	—	2	2	no	—
297	C₂²⋊C₈	—	2	2	no	—
298	C₄×C₈	—	2	1	no	—
299	C₄×C₈	—	2	1	no	—
300	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
301	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
302	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
303	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
304	C₂⁴	—	4	1	no	—
305	#41	—	4	2	no	—
306	#41	—	4	2	no	—
307	C₂×Q₈	—	2	1	no	—
308	C₂×Q₈	—	2	2	no	—
309	C₂×Q₈	—	4	1	no	—
310	C₂×Q₈	—	8	2	no	—
311	C₂×D₄	—	2	2	no	—
312	C₂×D₄	—	2	1	no	—
313	C₂×D₄	—	2	1	no	—
314	C₂×D₄	—	2	2	no	—
315	C₂×D₄	—	2	2	no	—
316	C₂×D₄	—	2	2	no	—
317	C₂×D₄	—	4	1	no	—
318	C₂×D₄	—	4	1	no	—
319	C₂×D₄	—	4	1	no	—
320	C₂×D₄	—	4	2	no	—
321	C₂×D₄	—	4	2	no	—
322	C₂×D₄	—	8	2	no	—
323	C₂×D₄	—	8	2	no	—
324	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
325	C₂²×C₄	—	2	2	no	—
326	C₂²×C₄	—	4	1	no	—
327	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
328	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
329	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
330	D₈	—	8	4	no	—
331	M₁₆	—	4	4	no	—
332	M₁₆	—	4	2	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	2	2	no	—
335	C₂×C₈	—	2	2	no	—
336	C₂×C₈	—	2	2	no	—
337	C₂×C₈	—	2	2	no	—
338	C₂×C₈	—	4	4	no	—
339	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
340	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
341	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
342	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
343	C₄⋊C₄	—	2	1	no	—
344	C₄⋊C₄	—	2	2	no	—
345	C₄⋊C₄	—	2	2	no	—
346	C₄⋊C₄	—	4	1	no	—
347	C₄⋊C₄	—	4	1	no	—
348	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
349	C₂²⋊C₄	—	2	2	no	—
350	C₂²⋊C₄	—	2	2	no	—
351	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
352	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
353	C₄²	#41×C₆₇	1	1	yes	characteristic
354	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
355	C₂³	—	2	2	no	—
356	C₂³	—	4	2	no	—
357	C₂³	—	8	2	no	—
358	C₂³	—	8	2	no	—
359	C₂³	—	8	2	no	—
360	C₂³	#121×C₆₇	1	1	yes	characteristic
361	Q₈	—	4	2	no	—
362	Q₈	—	8	2	no	—
363	Q₈	—	8	2	no	—
364	D₄	—	2	4	no	—
365	D₄	—	4	4	no	—
366	D₄	—	4	2	no	—
367	D₄	—	4	2	no	—
368	D₄	—	4	2	no	—
369	D₄	—	4	2	no	—
370	D₄	—	4	4	no	—
371	D₄	—	8	2	no	—
372	D₄	—	8	2	no	—
373	D₄	—	8	2	no	—
374	D₄	—	8	2	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	4	2	no	—
379	C₂×C₄	—	4	2	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	8	2	no	—
382	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
383	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
384	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
385	C₂×C₄	#135×C₆₇	1	2	yes	—
386	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
387	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
388	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
389	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
390	C₈	—	2	1	no	—
391	C₈	—	2	1	no	—
392	C₈	—	2	1	no	—
393	C₈	—	2	1	no	—
394	C₈	—	4	2	no	—
395	C₈	—	4	2	no	—
396	C₈	—	4	2	no	—
397	C₂²	—	2	1	no	—
398	C₂²	—	2	1	no	—
399	C₂²	—	4	2	no	—
400	C₂²	—	4	2	no	—
401	C₂²	—	4	2	no	—
402	C₂²	—	8	2	no	—
403	C₂²	—	8	2	no	—
404	C₂²	—	8	2	no	—
405	C₂²	—	16	4	no	—
406	C₂²	#463×C₆₇	1	2	yes	—
407	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
408	C₄	—	2	1	no	—
409	C₄	—	2	1	no	—
410	C₄	—	4	2	no	—
411	C₄	—	8	2	no	—
412	C₄	#463×C₆₇	1	2	yes	—
413	C₄	#472×C₆₇	1	2	yes	—
414	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
415	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
416	C₄	#577×C₆₇	1	2	yes	—
417	C₂	—	2	2	no	—
418	C₂	—	8	2	no	—
419	C₂	—	16	2	no	—
420	C₂	C₂×#463×C₆₇	1	1	yes	characteristic
421	C₂	C₆₇×#3057	1	1	yes	characteristic
422	C₂	C₂×#472×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	C₆₇×#3205	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
3. C₂×#486×C₆₇	C₂	semidirect
4. C₆₇×#2931	C₂	semidirect
5. C₆₇×#1697	C₂	semidirect
6. C₆₇×#1677	C₂	semidirect
7. C₆₇×#1564	C₂	semidirect
8. C₆₇×#1548	C₂	semidirect
9. C₆₇×#1545	C₂	semidirect
23. #476×C₆₇	C₂²	semidirect
27. #436×C₆₇	C₂²	semidirect
29. C₈×D₄×C₆₇	C₂²	semidirect
37. #335×C₆₇	C₂²	semidirect
38. #334×C₆₇	C₂²	semidirect
39. #333×C₆₇	C₂²	semidirect
205. #19196	C₆₇	direct (N × H)
212. #3227	C₂×C₆₇	semidirect
214. C₂×#486	C₂×C₆₇	semidirect
215. #2931	C₂×C₆₇	semidirect
216. #1697	C₂×C₆₇	semidirect
217. #1677	C₂×C₆₇	semidirect
218. #1564	C₂×C₆₇	semidirect
219. #1548	C₂×C₆₇	semidirect
220. #1545	C₂×C₆₇	semidirect
221. C₆₇	#19196	direct (N × H)
235. #476	C₂²×C₆₇	semidirect
239. #436	C₂²×C₆₇	semidirect
241. C₈×D₄	C₂²×C₆₇	semidirect
249. #335	C₂²×C₆₇	semidirect
250. #334	C₂²×C₆₇	semidirect
251. #333	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
C₆₇×#3205	2
C₂×#486×C₆₇	2
C₆₇×#2931	2
C₆₇×#1697	2
C₆₇×#1677	2
C₆₇×#1564	2
C₆₇×#1548	2
C₆₇×#1545	2
#19196	67

Subgroups — C67×#19196 #30698447

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#19196 #30698447