Subgroups — C₆₇×#19898 #30699149

Order 17152.

724 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3341	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3325	C₂	1	2	yes	maximal, complemented
3	D₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1706	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1697	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1675	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1667	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1629	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1624	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1589	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₈×C₆₇	—	2	1	no	—
13	#555×C₆₇	C₂²	1	2	yes	complemented
14	D₄²×C₆₇	C₂²	1	2	yes	—
15	#500×C₆₇	C₂²	1	1	yes	characteristic, complemented
16	#493×C₆₇	C₂²	1	1	yes	characteristic
17	#492×C₆₇	C₂²	1	2	yes	complemented
18	#486×C₆₇	C₂²	1	2	yes	complemented
19	#482×C₆₇	—	2	2	no	—
20	#482×C₆₇	—	2	2	no	—
21	#476×C₆₇	C₂²	1	2	yes	—
22	#476×C₆₇	C₂²	1	2	yes	complemented
23	#469×C₆₇	—	2	2	no	—
24	#466×C₆₇	—	2	1	no	—
25	#466×C₆₇	—	2	1	no	—
26	#459×C₆₇	C₂²	1	2	yes	—
27	#459×C₆₇	C₂²	1	2	yes	complemented
28	#447×C₆₇	—	2	2	no	—
29	#447×C₆₇	—	2	2	no	—
30	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
31	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
32	#436×C₆₇	C₂²	1	2	yes	complemented
33	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
34	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
35	#427×C₆₇	—	2	1	no	—
36	#418×C₆₇	—	2	2	no	—
37	#335×C₆₇	C₂²	1	1	yes	characteristic, complemented
38	#335×C₆₇	C₂²	1	1	yes	characteristic, complemented
39	#333×C₆₇	C₂²	1	2	yes	complemented
40	#331×C₆₇	C₂²	1	2	yes	complemented
41	D₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
42	D₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
43	D₄⋊C₈×C₆₇	C₂²	1	2	yes	complemented
44	C₈²×C₆₇	C₂²	1	1	yes	characteristic, complemented
45	C₂²×D₄×C₆₇	—	2	2	no	—
46	C₂²×D₄×C₆₇	—	2	2	no	—
47	C₂×SD₁₆×C₆₇	—	4	2	no	—
48	C₂×D₈×C₆₇	—	2	2	no	—
49	C₂×D₈×C₆₇	—	2	2	no	—
50	C₂×D₈×C₆₇	—	2	1	no	—
51	C₂×D₈×C₆₇	—	2	1	no	—
52	C₂×D₈×C₆₇	—	4	4	no	—
53	C₂×D₈×C₆₇	D₄	1	1	yes	characteristic
54	C₂×D₈×C₆₇	D₄	1	1	yes	characteristic
55	C₂×M₁₆×C₆₇	—	2	2	no	—
56	C₂²×C₈×C₆₇	—	2	1	no	—
57	C₂²×C₈×C₆₇	—	2	1	no	—
58	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
59	#127×C₆₇	C₂³	1	2	yes	—
60	#124×C₆₇	—	4	2	no	—
61	#121×C₆₇	—	2	2	no	—
62	#121×C₆₇	—	2	2	no	—
63	#121×C₆₇	—	2	2	no	—
64	#121×C₆₇	—	2	2	no	—
65	#120×C₆₇	—	4	2	no	—
66	C₄×D₄×C₆₇	C₂³	1	2	yes	—
67	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
68	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
69	#117×C₆₇	—	2	2	no	—
70	#117×C₆₇	—	2	2	no	—
71	#107×C₆₇	—	2	1	no	—
72	#107×C₆₇	—	2	1	no	—
73	#107×C₆₇	D₄	1	1	yes	characteristic
74	#107×C₆₇	D₄	1	1	yes	characteristic
75	#106×C₆₇	—	2	2	no	—
76	#106×C₆₇	—	2	2	no	—
77	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
78	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
79	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
80	#102×C₆₇	—	2	2	no	—
81	#102×C₆₇	—	2	2	no	—
82	#102×C₆₇	—	2	2	no	—
83	#102×C₆₇	—	2	2	no	—
84	#102×C₆₇	—	2	2	no	—
85	#102×C₆₇	—	2	2	no	—
86	#102×C₆₇	D₄	1	1	yes	characteristic
87	#102×C₆₇	D₄	1	2	yes	—
88	#102×C₆₇	D₄	1	1	yes	characteristic
89	C₂²⋊C₈×C₆₇	—	2	1	no	—
90	C₂²⋊C₈×C₆₇	—	2	2	no	—
91	C₂²⋊C₈×C₆₇	—	2	1	no	—
92	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
95	C₂⁴×C₆₇	—	4	2	no	—
96	C₂×Q₈×C₆₇	—	2	2	no	—
97	C₂×D₄×C₆₇	—	2	2	no	—
98	C₂×D₄×C₆₇	—	2	2	no	—
99	C₂×D₄×C₆₇	—	2	2	no	—
100	C₂×D₄×C₆₇	—	4	2	no	—
101	C₂×D₄×C₆₇	—	4	2	no	—
102	C₂×D₄×C₆₇	—	4	4	no	—
103	C₂×D₄×C₆₇	—	4	4	no	—
104	C₂×D₄×C₆₇	—	4	2	no	—
105	C₂×D₄×C₆₇	—	8	2	no	—
106	C₂×D₄×C₆₇	—	8	2	no	—
107	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
108	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
109	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₆₇	—	2	1	no	—
111	C₂²×C₄×C₆₇	—	2	2	no	—
112	C₂²×C₄×C₆₇	—	2	1	no	—
113	SD₁₆×C₆₇	—	8	4	no	—
114	D₈×C₆₇	—	2	2	no	—
115	D₈×C₆₇	—	2	2	no	—
116	D₈×C₆₇	—	4	4	no	—
117	D₈×C₆₇	—	4	2	no	—
118	D₈×C₆₇	—	8	1	no	—
119	D₈×C₆₇	—	8	2	no	—
120	M₁₆×C₆₇	—	8	2	no	—
121	C₂×C₈×C₆₇	—	2	1	no	—
122	C₂×C₈×C₆₇	—	2	1	no	—
123	C₂×C₈×C₆₇	—	2	2	no	—
124	C₂×C₈×C₆₇	—	2	1	no	—
125	C₂×C₈×C₆₇	—	4	2	no	—
126	C₂×C₈×C₆₇	—	4	2	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	2	no	—
132	C₄⋊C₄×C₆₇	—	2	2	no	—
133	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
134	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
135	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
136	C₂²⋊C₄×C₆₇	—	2	1	no	—
137	C₂²⋊C₄×C₆₇	—	2	2	no	—
138	C₂²⋊C₄×C₆₇	—	2	1	no	—
139	C₂²⋊C₄×C₆₇	—	4	2	no	—
140	C₂²⋊C₄×C₆₇	—	4	2	no	—
141	C₂²⋊C₄×C₆₇	—	4	2	no	—
142	C₄²×C₆₇	—	4	2	no	—
143	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
144	C₂³×C₆₇	—	2	1	no	—
145	C₂³×C₆₇	—	2	1	no	—
146	C₂³×C₆₇	—	2	2	no	—
147	C₂³×C₆₇	—	4	2	no	—
148	C₂³×C₆₇	—	8	4	no	—
149	C₂³×C₆₇	—	8	4	no	—
150	C₂³×C₆₇	—	16	2	no	—
151	Q₈×C₆₇	—	8	2	no	—
152	D₄×C₆₇	—	2	1	no	—
153	D₄×C₆₇	—	2	2	no	—
154	D₄×C₆₇	—	2	1	no	—
155	D₄×C₆₇	—	2	1	no	—
156	D₄×C₆₇	—	2	2	no	—
157	D₄×C₆₇	—	4	2	no	—
158	D₄×C₆₇	—	4	2	no	—
159	D₄×C₆₇	—	4	2	no	—
160	D₄×C₆₇	—	4	2	no	—
161	D₄×C₆₇	—	8	2	no	—
162	D₄×C₆₇	—	8	4	no	—
163	D₄×C₆₇	—	8	4	no	—
164	D₄×C₆₇	—	16	2	no	—
165	D₄×C₆₇	#135	1	2	yes	—
166	C₂×C₄×C₆₇	—	2	1	no	—
167	C₂×C₄×C₆₇	—	2	2	no	—
168	C₂×C₄×C₆₇	—	2	1	no	—
169	C₂×C₄×C₆₇	—	4	2	no	—
170	C₂×C₄×C₆₇	—	4	1	no	—
171	C₂×C₄×C₆₇	—	4	1	no	—
172	C₂×C₄×C₆₇	—	4	2	no	—
173	C₂×C₄×C₆₇	—	4	1	no	—
174	C₂×C₄×C₆₇	—	4	1	no	—
175	C₂×C₄×C₆₇	—	4	2	no	—
176	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
179	C₈×C₆₇	—	2	2	no	—
180	C₈×C₆₇	—	2	1	no	—
181	C₈×C₆₇	—	2	1	no	—
182	C₈×C₆₇	—	2	1	no	—
183	C₈×C₆₇	—	4	1	no	—
184	C₈×C₆₇	—	4	1	no	—
185	C₈×C₆₇	—	4	2	no	—
186	C₈×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
187	C₂²×C₆₇	—	2	2	no	—
188	C₂²×C₆₇	—	4	1	no	—
189	C₂²×C₆₇	—	4	1	no	—
190	C₂²×C₆₇	—	4	1	no	—
191	C₂²×C₆₇	—	4	2	no	—
192	C₂²×C₆₇	—	4	2	no	—
193	C₂²×C₆₇	—	4	1	no	—
194	C₂²×C₆₇	—	4	1	no	—
195	C₂²×C₆₇	—	4	2	no	—
196	C₂²×C₆₇	—	8	2	no	—
197	C₂²×C₆₇	—	8	2	no	—
198	C₂²×C₆₇	—	8	2	no	—
199	C₂²×C₆₇	—	16	8	no	—
200	C₂²×C₆₇	D₄²	1	1	yes	characteristic
201	C₄×C₆₇	—	2	1	no	—
202	C₄×C₆₇	—	4	1	no	—
203	C₄×C₆₇	—	4	1	no	—
204	C₄×C₆₇	—	4	2	no	—
205	C₄×C₆₇	—	8	2	no	—
206	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
207	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
208	C₄×C₆₇	#576	1	1	yes	characteristic
209	C₄×C₆₇	#577	1	1	yes	characteristic
210	#19898	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
211	C₂×C₆₇	—	4	2	no	—
212	C₂×C₆₇	—	8	1	no	—
213	C₂×C₆₇	—	8	2	no	—
214	C₂×C₆₇	—	16	2	no	—
215	C₂×C₆₇	#3272	1	1	yes	characteristic
216	C₂×C₆₇	#3280	1	1	yes	characteristic
217	C₂×C₆₇	#3292	1	1	yes	characteristic
218	#3341	C₂×C₆₇	1	1	yes	characteristic, complemented
219	#3325	C₂×C₆₇	1	2	yes	complemented
220	D₄×D₈	C₂×C₆₇	1	1	yes	characteristic, complemented
221	#1706	C₂×C₆₇	1	1	yes	characteristic, complemented
222	#1697	C₂×C₆₇	1	2	yes	complemented
223	#1675	C₂×C₆₇	1	1	yes	characteristic, complemented
224	#1667	C₂×C₆₇	1	1	yes	characteristic, complemented
225	#1629	C₂×C₆₇	1	2	yes	complemented
226	#1624	C₂×C₆₇	1	2	yes	complemented
227	#1589	C₂×C₆₇	1	1	yes	characteristic, complemented
228	C₈×D₈	C₂×C₆₇	1	1	yes	characteristic, complemented
229	C₆₇	#19898	1	1	yes	characteristic, direct product factor, Sylow
230	C₂²×D₈	—	2	1	no	—
231	#555	C₂²×C₆₇	1	2	yes	complemented
232	D₄²	C₂²×C₆₇	1	2	yes	—
233	#500	C₂²×C₆₇	1	1	yes	characteristic, complemented
234	#493	C₂²×C₆₇	1	1	yes	characteristic
235	#492	C₂²×C₆₇	1	2	yes	complemented
236	#486	C₂²×C₆₇	1	2	yes	complemented
237	#482	—	2	2	no	—
238	#482	—	2	2	no	—
239	#476	C₂²×C₆₇	1	2	yes	—
240	#476	C₂²×C₆₇	1	2	yes	complemented
241	#469	—	2	2	no	—
242	#466	—	2	1	no	—
243	#466	—	2	1	no	—
244	#459	C₂²×C₆₇	1	2	yes	—
245	#459	C₂²×C₆₇	1	2	yes	complemented
246	#447	—	2	2	no	—
247	#447	—	2	2	no	—
248	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
249	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
250	#436	C₂²×C₆₇	1	2	yes	complemented
251	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic
252	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic, complemented
253	#427	—	2	1	no	—
254	#418	—	2	2	no	—
255	#335	C₂²×C₆₇	1	1	yes	characteristic, complemented
256	#335	C₂²×C₆₇	1	1	yes	characteristic, complemented
257	#333	C₂²×C₆₇	1	2	yes	complemented
258	#331	C₂²×C₆₇	1	2	yes	complemented
259	D₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
260	D₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
261	D₄⋊C₈	C₂²×C₆₇	1	2	yes	complemented
262	C₈²	C₂²×C₆₇	1	1	yes	characteristic, complemented
263	C₂²×D₄	—	2	2	no	—
264	C₂²×D₄	—	2	2	no	—
265	C₂×SD₁₆	—	4	2	no	—
266	C₂×D₈	—	2	2	no	—
267	C₂×D₈	—	2	2	no	—
268	C₂×D₈	—	2	1	no	—
269	C₂×D₈	—	2	1	no	—
270	C₂×D₈	—	4	4	no	—
271	C₂×D₈	D₄×C₆₇	1	1	yes	characteristic
272	C₂×D₈	D₄×C₆₇	1	1	yes	characteristic
273	C₂×M₁₆	—	2	2	no	—
274	C₂²×C₈	—	2	1	no	—
275	C₂²×C₈	—	2	1	no	—
276	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
277	#127	C₂³×C₆₇	1	2	yes	—
278	#124	—	4	2	no	—
279	#121	—	2	2	no	—
280	#121	—	2	2	no	—
281	#121	—	2	2	no	—
282	#121	—	2	2	no	—
283	#120	—	4	2	no	—
284	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
285	C₄×D₄	C₂³×C₆₇	1	2	yes	—
286	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
287	#117	—	2	2	no	—
288	#117	—	2	2	no	—
289	#107	—	2	1	no	—
290	#107	—	2	1	no	—
291	#107	D₄×C₆₇	1	1	yes	characteristic
292	#107	D₄×C₆₇	1	1	yes	characteristic
293	#106	—	2	2	no	—
294	#106	—	2	2	no	—
295	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
296	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
297	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
298	#102	—	2	2	no	—
299	#102	—	2	2	no	—
300	#102	—	2	2	no	—
301	#102	—	2	2	no	—
302	#102	—	2	2	no	—
303	#102	—	2	2	no	—
304	#102	D₄×C₆₇	1	1	yes	characteristic
305	#102	D₄×C₆₇	1	2	yes	—
306	#102	D₄×C₆₇	1	1	yes	characteristic
307	C₂²⋊C₈	—	2	1	no	—
308	C₂²⋊C₈	—	2	1	no	—
309	C₂²⋊C₈	—	2	2	no	—
310	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
311	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
312	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
313	C₂⁴	—	4	2	no	—
314	C₂×Q₈	—	2	2	no	—
315	C₂×D₄	—	2	2	no	—
316	C₂×D₄	—	2	2	no	—
317	C₂×D₄	—	2	2	no	—
318	C₂×D₄	—	4	2	no	—
319	C₂×D₄	—	4	2	no	—
320	C₂×D₄	—	4	4	no	—
321	C₂×D₄	—	4	4	no	—
322	C₂×D₄	—	4	2	no	—
323	C₂×D₄	—	8	2	no	—
324	C₂×D₄	—	8	2	no	—
325	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
326	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
327	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
328	C₂²×C₄	—	2	1	no	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	—	2	1	no	—
331	SD₁₆	—	8	4	no	—
332	D₈	—	2	2	no	—
333	D₈	—	2	2	no	—
334	D₈	—	4	4	no	—
335	D₈	—	4	2	no	—
336	D₈	—	8	1	no	—
337	D₈	—	8	2	no	—
338	M₁₆	—	8	2	no	—
339	C₂×C₈	—	2	1	no	—
340	C₂×C₈	—	2	1	no	—
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	—	2	1	no	—
343	C₂×C₈	—	4	2	no	—
344	C₂×C₈	—	4	2	no	—
345	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
346	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
347	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
348	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
349	C₄⋊C₄	—	2	2	no	—
350	C₄⋊C₄	—	2	2	no	—
351	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
352	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
353	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
354	C₂²⋊C₄	—	2	1	no	—
355	C₂²⋊C₄	—	2	2	no	—
356	C₂²⋊C₄	—	2	1	no	—
357	C₂²⋊C₄	—	4	2	no	—
358	C₂²⋊C₄	—	4	2	no	—
359	C₂²⋊C₄	—	4	2	no	—
360	C₄²	—	4	2	no	—
361	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
362	C₂³	—	2	1	no	—
363	C₂³	—	2	1	no	—
364	C₂³	—	2	2	no	—
365	C₂³	—	4	2	no	—
366	C₂³	—	8	4	no	—
367	C₂³	—	8	4	no	—
368	C₂³	—	16	2	no	—
369	Q₈	—	8	2	no	—
370	D₄	—	2	1	no	—
371	D₄	—	2	2	no	—
372	D₄	—	2	1	no	—
373	D₄	—	2	1	no	—
374	D₄	—	2	2	no	—
375	D₄	—	4	2	no	—
376	D₄	—	4	2	no	—
377	D₄	—	4	2	no	—
378	D₄	—	4	2	no	—
379	D₄	—	8	2	no	—
380	D₄	—	8	4	no	—
381	D₄	—	8	4	no	—
382	D₄	—	16	2	no	—
383	D₄	#135×C₆₇	1	2	yes	—
384	C₂×C₄	—	2	1	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	4	2	no	—
388	C₂×C₄	—	4	1	no	—
389	C₂×C₄	—	4	1	no	—
390	C₂×C₄	—	4	2	no	—
391	C₂×C₄	—	4	1	no	—
392	C₂×C₄	—	4	1	no	—
393	C₂×C₄	—	4	2	no	—
394	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
395	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
396	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
397	C₈	—	2	2	no	—
398	C₈	—	2	1	no	—
399	C₈	—	2	1	no	—
400	C₈	—	2	1	no	—
401	C₈	—	4	1	no	—
402	C₈	—	4	1	no	—
403	C₈	—	4	2	no	—
404	C₈	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
405	C₂²	—	2	2	no	—
406	C₂²	—	4	1	no	—
407	C₂²	—	4	1	no	—
408	C₂²	—	4	1	no	—
409	C₂²	—	4	2	no	—
410	C₂²	—	4	2	no	—
411	C₂²	—	4	1	no	—
412	C₂²	—	4	1	no	—
413	C₂²	—	4	2	no	—
414	C₂²	—	8	2	no	—
415	C₂²	—	8	2	no	—
416	C₂²	—	8	2	no	—
417	C₂²	—	16	8	no	—
418	C₂²	D₄²×C₆₇	1	1	yes	characteristic
419	C₄	—	2	1	no	—
420	C₄	—	4	1	no	—
421	C₄	—	4	1	no	—
422	C₄	—	4	2	no	—
423	C₄	—	8	2	no	—
424	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
425	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
426	C₄	#576×C₆₇	1	1	yes	characteristic
427	C₄	#577×C₆₇	1	1	yes	characteristic
428	C₂	—	4	2	no	—
429	C₂	—	8	2	no	—
430	C₂	—	8	1	no	—
431	C₂	—	16	2	no	—
432	C₂	C₆₇×#3272	1	1	yes	characteristic
433	C₂	C₆₇×#3280	1	1	yes	characteristic
434	C₂	C₆₇×#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3341	C₂	semidirect
2. C₆₇×#3325	C₂	semidirect
3. D₄×D₈×C₆₇	C₂	semidirect
4. C₆₇×#1706	C₂	semidirect
5. C₆₇×#1697	C₂	semidirect
6. C₆₇×#1675	C₂	semidirect
7. C₆₇×#1667	C₂	semidirect
8. C₆₇×#1629	C₂	semidirect
9. C₆₇×#1624	C₂	semidirect
10. C₆₇×#1589	C₂	semidirect
11. C₈×D₈×C₆₇	C₂	semidirect
13. #555×C₆₇	C₂²	semidirect
15. #500×C₆₇	C₂²	semidirect
17. #492×C₆₇	C₂²	semidirect
18. #486×C₆₇	C₂²	semidirect
22. #476×C₆₇	C₂²	semidirect
27. #459×C₆₇	C₂²	semidirect
32. #436×C₆₇	C₂²	semidirect
34. C₈×D₄×C₆₇	C₂²	semidirect
37. #335×C₆₇	C₂²	semidirect
38. #335×C₆₇	C₂²	semidirect
39. #333×C₆₇	C₂²	semidirect
40. #331×C₆₇	C₂²	semidirect
42. D₄⋊C₈×C₆₇	C₂²	semidirect
43. D₄⋊C₈×C₆₇	C₂²	semidirect
44. C₈²×C₆₇	C₂²	semidirect
210. #19898	C₆₇	direct (N × H)
218. #3341	C₂×C₆₇	semidirect
219. #3325	C₂×C₆₇	semidirect
220. D₄×D₈	C₂×C₆₇	semidirect
221. #1706	C₂×C₆₇	semidirect
222. #1697	C₂×C₆₇	semidirect
223. #1675	C₂×C₆₇	semidirect
224. #1667	C₂×C₆₇	semidirect
225. #1629	C₂×C₆₇	semidirect
226. #1624	C₂×C₆₇	semidirect
227. #1589	C₂×C₆₇	semidirect
228. C₈×D₈	C₂×C₆₇	semidirect
229. C₆₇	#19898	direct (N × H)
231. #555	C₂²×C₆₇	semidirect
233. #500	C₂²×C₆₇	semidirect
235. #492	C₂²×C₆₇	semidirect
236. #486	C₂²×C₆₇	semidirect
240. #476	C₂²×C₆₇	semidirect
245. #459	C₂²×C₆₇	semidirect
250. #436	C₂²×C₆₇	semidirect
252. C₈×D₄	C₂²×C₆₇	semidirect
255. #335	C₂²×C₆₇	semidirect
256. #335	C₂²×C₆₇	semidirect
257. #333	C₂²×C₆₇	semidirect
258. #331	C₂²×C₆₇	semidirect
260. D₄⋊C₈	C₂²×C₆₇	semidirect
261. D₄⋊C₈	C₂²×C₆₇	semidirect
262. C₈²	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×#3341	2
C₆₇×#3325	2
D₄×D₈×C₆₇	2
C₆₇×#1706	2
C₆₇×#1697	2
C₆₇×#1675	2
C₆₇×#1667	2
C₆₇×#1629	2
C₆₇×#1624	2
C₆₇×#1589	2
C₈×D₈×C₆₇	2
#19898	67

Subgroups — C67×#19898 #30699149

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#19898 #30699149