Subgroups — C₆₇×#25879 #30705130

Order 17152.

1178 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3499	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#3142	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3080	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2642	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2038	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1966	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1963	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1793	C₂	1	4	yes	maximal, complemented
9	C₆₇×#1759	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×2₋¹⁺⁴×C₆₇	C₂²	1	2	yes	—
11	D₄×Q₈×C₆₇	—	2	2	no	—
12	#547×C₆₇	—	2	2	no	—
13	#547×C₆₇	—	2	4	no	—
14	#536×C₆₇	—	2	2	no	—
15	#536×C₆₇	C₂²	1	2	yes	—
16	#532×C₆₇	—	2	1	no	—
17	#532×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
19	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
21	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
22	#491×C₆₇	—	2	1	no	—
23	#490×C₆₇	—	2	1	no	—
24	#488×C₆₇	—	2	2	no	—
25	#479×C₆₇	—	2	2	no	—
26	#478×C₆₇	—	2	2	no	—
27	#428×C₆₇	C₂²	1	2	yes	—
28	#423×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×#104×C₆₇	C₂²	1	2	yes	—
30	C₂×#104×C₆₇	C₂²	1	4	yes	—
31	#417×C₆₇	C₂²	1	4	yes	—
32	#417×C₆₇	C₂²	1	2	yes	—
33	C₂×#101×C₆₇	C₂²	1	4	yes	—
34	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
35	#400×C₆₇	—	2	4	no	—
36	#391×C₆₇	C₂²	1	4	yes	—
37	#376×C₆₇	C₂²	1	1	yes	characteristic
38	#339×C₆₇	C₂²	1	2	yes	—
39	2₋¹⁺⁴×C₆₇	—	2	8	no	—
40	2₋¹⁺⁴×C₆₇	—	4	4	no	—
41	C₂×#41×C₆₇	—	2	2	no	—
42	C₂×#41×C₆₇	—	2	2	no	—
43	C₂×#41×C₆₇	—	2	4	no	—
44	C₂×#41×C₆₇	C₂³	1	2	yes	—
45	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
46	C₂²×Q₈×C₆₇	—	2	2	no	—
47	C₂²×Q₈×C₆₇	—	2	2	no	—
48	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
49	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
50	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
51	C₄⋊Q₈×C₆₇	—	2	4	no	—
52	C₄⋊Q₈×C₆₇	—	2	2	no	—
53	C₄⋊Q₈×C₆₇	—	2	1	no	—
54	C₄⋊Q₈×C₆₇	—	2	1	no	—
55	C₄⋊Q₈×C₆₇	—	2	2	no	—
56	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
57	#127×C₆₇	D₄	1	1	yes	characteristic
58	#125×C₆₇	—	2	2	no	—
59	#125×C₆₇	—	2	1	no	—
60	#125×C₆₇	—	2	1	no	—
61	#124×C₆₇	—	2	2	no	—
62	#124×C₆₇	D₄	1	2	yes	—
63	#123×C₆₇	—	2	8	no	—
64	#123×C₆₇	—	2	4	no	—
65	C₂²⋊Q₈×C₆₇	—	2	8	no	—
66	C₂²⋊Q₈×C₆₇	—	2	4	no	—
67	#121×C₆₇	—	2	4	no	—
68	#121×C₆₇	—	2	2	no	—
69	C₄×Q₈×C₆₇	D₄	1	2	yes	—
70	C₄×D₄×C₆₇	—	2	4	no	—
71	C₄×D₄×C₆₇	—	2	2	no	—
72	C₄×D₄×C₆₇	—	2	2	no	—
73	C₄×D₄×C₆₇	D₄	1	2	yes	—
74	#117×C₆₇	—	2	1	no	—
75	#117×C₆₇	C₂³	1	1	yes	characteristic
76	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
77	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
78	C₂×C₄⋊C₄×C₆₇	D₄	1	4	yes	—
79	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
80	C₂×C₄²×C₆₇	—	2	1	no	—
81	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
82	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
83	#107×C₆₇	—	2	2	no	—
84	#106×C₆₇	—	2	2	no	—
85	C₄⋊C₈×C₆₇	—	2	2	no	—
86	#104×C₆₇	—	2	4	no	—
87	#104×C₆₇	—	4	4	no	—
88	#103×C₆₇	—	2	2	no	—
89	#103×C₆₇	—	2	4	no	—
90	#102×C₆₇	—	2	2	no	—
91	#102×C₆₇	—	2	4	no	—
92	#101×C₆₇	—	4	4	no	—
93	#97×C₆₇	—	2	1	no	—
94	C₄×C₈×C₆₇	—	2	1	no	—
95	#95×C₆₇	—	2	2	no	—
96	#95×C₆₇	—	2	8	no	—
97	#41×C₆₇	—	2	2	no	—
98	#41×C₆₇	—	2	4	no	—
99	#41×C₆₇	—	2	4	no	—
100	#41×C₆₇	—	4	8	no	—
101	#41×C₆₇	—	4	16	no	—
102	#41×C₆₇	—	4	4	no	—
103	#41×C₆₇	—	4	4	no	—
104	#41×C₆₇	#41	1	4	yes	—
105	C₂×Q₈×C₆₇	—	2	4	no	—
106	C₂×Q₈×C₆₇	—	2	2	no	—
107	C₂×Q₈×C₆₇	—	2	2	no	—
108	C₂×Q₈×C₆₇	—	2	4	no	—
109	C₂×Q₈×C₆₇	—	2	4	no	—
110	C₂×Q₈×C₆₇	—	4	8	no	—
111	C₂×Q₈×C₆₇	—	4	4	no	—
112	C₂×Q₈×C₆₇	—	4	4	no	—
113	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
114	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₂×D₄×C₆₇	—	2	1	no	—
116	C₂×D₄×C₆₇	—	2	4	no	—
117	C₂×D₄×C₆₇	—	2	2	no	—
118	C₂×D₄×C₆₇	—	2	2	no	—
119	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
120	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₆₇	—	2	1	no	—
122	C₂²×C₄×C₆₇	—	2	4	no	—
123	C₂²×C₄×C₆₇	—	2	4	no	—
124	C₂²×C₄×C₆₇	—	2	2	no	—
125	C₂²×C₄×C₆₇	—	2	2	no	—
126	C₂²×C₄×C₆₇	—	2	2	no	—
127	C₂²×C₄×C₆₇	—	2	2	no	—
128	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
129	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
130	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
131	M₁₆×C₆₇	—	2	2	no	—
132	M₁₆×C₆₇	—	2	2	no	—
133	M₁₆×C₆₇	—	2	4	no	—
134	C₂×C₈×C₆₇	—	2	2	no	—
135	C₂×C₈×C₆₇	—	2	2	no	—
136	C₄⋊C₄×C₆₇	—	2	8	no	—
137	C₄⋊C₄×C₆₇	—	2	2	no	—
138	C₄⋊C₄×C₆₇	—	2	4	no	—
139	C₄⋊C₄×C₆₇	—	2	4	no	—
140	C₄⋊C₄×C₆₇	—	2	2	no	—
141	C₄⋊C₄×C₆₇	—	2	2	no	—
142	C₄⋊C₄×C₆₇	—	2	2	no	—
143	C₄⋊C₄×C₆₇	—	4	2	no	—
144	C₄⋊C₄×C₆₇	—	4	2	no	—
145	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
146	C₂²⋊C₄×C₆₇	—	2	8	no	—
147	C₂²⋊C₄×C₆₇	—	2	1	no	—
148	C₂²⋊C₄×C₆₇	—	2	2	no	—
149	C₂²⋊C₄×C₆₇	—	2	4	no	—
150	C₄²×C₆₇	—	2	2	no	—
151	C₄²×C₆₇	—	2	1	no	—
152	C₄²×C₆₇	—	2	2	no	—
153	C₄²×C₆₇	—	2	2	no	—
154	C₄²×C₆₇	C₂×D₄	1	2	yes	—
155	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
156	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
157	C₂³×C₆₇	—	2	1	no	—
158	C₂³×C₆₇	—	2	2	no	—
159	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
160	Q₈×C₆₇	—	2	1	no	—
161	Q₈×C₆₇	—	2	4	no	—
162	Q₈×C₆₇	—	4	8	no	—
163	Q₈×C₆₇	—	4	2	no	—
164	Q₈×C₆₇	—	4	2	no	—
165	Q₈×C₆₇	—	4	4	no	—
166	Q₈×C₆₇	#124	1	2	yes	—
167	D₄×C₆₇	—	2	4	no	—
168	D₄×C₆₇	—	2	1	no	—
169	D₄×C₆₇	—	2	4	no	—
170	D₄×C₆₇	—	4	4	no	—
171	D₄×C₆₇	—	4	8	no	—
172	D₄×C₆₇	—	4	4	no	—
173	D₄×C₆₇	#124	1	2	yes	—
174	C₂×C₄×C₆₇	—	2	1	no	—
175	C₂×C₄×C₆₇	—	2	4	no	—
176	C₂×C₄×C₆₇	—	2	2	no	—
177	C₂×C₄×C₆₇	—	2	2	no	—
178	C₂×C₄×C₆₇	—	2	4	no	—
179	C₂×C₄×C₆₇	—	2	8	no	—
180	C₂×C₄×C₆₇	—	4	2	no	—
181	C₂×C₄×C₆₇	—	4	4	no	—
182	C₂×C₄×C₆₇	—	4	8	no	—
183	C₂×C₄×C₆₇	—	4	8	no	—
184	C₂×C₄×C₆₇	—	4	2	no	—
185	C₂×C₄×C₆₇	—	4	2	no	—
186	C₂×C₄×C₆₇	—	4	4	no	—
187	C₂×C₄×C₆₇	—	4	4	no	—
188	C₂×C₄×C₆₇	—	4	2	no	—
189	C₂×C₄×C₆₇	—	4	2	no	—
190	C₂×C₄×C₆₇	#120	1	2	yes	—
191	C₂×C₄×C₆₇	#120	1	2	yes	—
192	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
193	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
195	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
196	C₈×C₆₇	—	4	2	no	—
197	C₈×C₆₇	—	4	2	no	—
198	C₂²×C₆₇	—	2	2	no	—
199	C₂²×C₆₇	—	2	2	no	—
200	C₂²×C₆₇	—	4	2	no	—
201	C₂²×C₆₇	—	4	2	no	—
202	C₂²×C₆₇	—	4	4	no	—
203	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
204	C₂²×C₆₇	#546	1	2	yes	—
205	C₄×C₆₇	—	2	2	no	—
206	C₄×C₆₇	—	2	2	no	—
207	C₄×C₆₇	—	2	2	no	—
208	C₄×C₆₇	—	4	4	no	—
209	C₄×C₆₇	—	4	2	no	—
210	C₄×C₆₇	—	4	2	no	—
211	C₄×C₆₇	—	8	2	no	—
212	C₄×C₆₇	—	8	2	no	—
213	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
214	C₄×C₆₇	#535	1	1	yes	characteristic
215	C₄×C₆₇	#546	1	2	yes	—
216	#25879	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
217	C₂×C₆₇	—	2	2	no	—
218	C₂×C₆₇	—	4	2	no	—
219	C₂×C₆₇	—	8	2	no	—
220	C₂×C₆₇	#2435	1	1	yes	characteristic
221	C₂×C₆₇	#3020	1	2	yes	—
222	#3499	C₂×C₆₇	1	1	yes	characteristic
223	#3142	C₂×C₆₇	1	1	yes	characteristic, complemented
224	#3080	C₂×C₆₇	1	1	yes	characteristic, complemented
225	#2642	C₂×C₆₇	1	1	yes	characteristic, complemented
226	#2038	C₂×C₆₇	1	4	yes	complemented
227	#1966	C₂×C₆₇	1	1	yes	characteristic, complemented
228	#1963	C₂×C₆₇	1	1	yes	characteristic, complemented
229	#1793	C₂×C₆₇	1	4	yes	complemented
230	#1759	C₂×C₆₇	1	1	yes	characteristic, complemented
231	C₆₇	#25879	1	1	yes	characteristic, direct product factor, Sylow
232	C₂×2₋¹⁺⁴	C₂²×C₆₇	1	2	yes	—
233	D₄×Q₈	—	2	2	no	—
234	#547	—	2	2	no	—
235	#547	—	2	4	no	—
236	#536	—	2	2	no	—
237	#536	C₂²×C₆₇	1	2	yes	—
238	#532	—	2	1	no	—
239	#532	C₂²×C₆₇	1	1	yes	characteristic
240	C₂×C₄⋊Q₈	—	2	1	no	—
241	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
242	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
243	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
244	#491	—	2	1	no	—
245	#490	—	2	1	no	—
246	#488	—	2	2	no	—
247	#479	—	2	2	no	—
248	#478	—	2	2	no	—
249	#428	C₂²×C₆₇	1	2	yes	—
250	#423	C₂²×C₆₇	1	1	yes	characteristic
251	C₂×#104	C₂²×C₆₇	1	2	yes	—
252	C₂×#104	C₂²×C₆₇	1	4	yes	—
253	#417	C₂²×C₆₇	1	2	yes	—
254	#417	C₂²×C₆₇	1	4	yes	—
255	C₂×#101	C₂²×C₆₇	1	4	yes	—
256	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
257	#400	—	2	4	no	—
258	#391	C₂²×C₆₇	1	4	yes	—
259	#376	C₂²×C₆₇	1	1	yes	characteristic
260	#339	C₂²×C₆₇	1	2	yes	—
261	2₋¹⁺⁴	—	2	8	no	—
262	2₋¹⁺⁴	—	4	4	no	—
263	C₂×#41	—	2	2	no	—
264	C₂×#41	—	2	2	no	—
265	C₂×#41	—	2	4	no	—
266	C₂×#41	C₂³×C₆₇	1	2	yes	—
267	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
268	C₂²×Q₈	—	2	2	no	—
269	C₂²×Q₈	—	2	2	no	—
270	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
271	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
272	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
273	C₄⋊Q₈	—	2	4	no	—
274	C₄⋊Q₈	—	2	2	no	—
275	C₄⋊Q₈	—	2	1	no	—
276	C₄⋊Q₈	—	2	1	no	—
277	C₄⋊Q₈	—	2	2	no	—
278	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
279	#127	D₄×C₆₇	1	1	yes	characteristic
280	#125	—	2	2	no	—
281	#125	—	2	1	no	—
282	#125	—	2	1	no	—
283	#124	—	2	2	no	—
284	#124	D₄×C₆₇	1	2	yes	—
285	#123	—	2	8	no	—
286	#123	—	2	4	no	—
287	C₂²⋊Q₈	—	2	8	no	—
288	C₂²⋊Q₈	—	2	4	no	—
289	#121	—	2	4	no	—
290	#121	—	2	2	no	—
291	C₄×Q₈	D₄×C₆₇	1	2	yes	—
292	C₄×D₄	—	2	4	no	—
293	C₄×D₄	—	2	2	no	—
294	C₄×D₄	—	2	2	no	—
295	C₄×D₄	D₄×C₆₇	1	2	yes	—
296	#117	—	2	1	no	—
297	#117	C₂³×C₆₇	1	1	yes	characteristic
298	C₂×C₄⋊C₄	—	2	4	no	—
299	C₂×C₄⋊C₄	—	2	2	no	—
300	C₂×C₄⋊C₄	D₄×C₆₇	1	4	yes	—
301	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
302	C₂×C₄²	—	2	1	no	—
303	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
304	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
305	#107	—	2	2	no	—
306	#106	—	2	2	no	—
307	C₄⋊C₈	—	2	2	no	—
308	#104	—	2	4	no	—
309	#104	—	4	4	no	—
310	#103	—	2	2	no	—
311	#103	—	2	4	no	—
312	#102	—	2	2	no	—
313	#102	—	2	4	no	—
314	#101	—	4	4	no	—
315	#97	—	2	1	no	—
316	C₄×C₈	—	2	1	no	—
317	#95	—	2	2	no	—
318	#95	—	2	8	no	—
319	#41	—	2	2	no	—
320	#41	—	2	4	no	—
321	#41	—	2	4	no	—
322	#41	—	4	8	no	—
323	#41	—	4	16	no	—
324	#41	—	4	4	no	—
325	#41	—	4	4	no	—
326	#41	#41×C₆₇	1	4	yes	—
327	C₂×Q₈	—	2	4	no	—
328	C₂×Q₈	—	2	2	no	—
329	C₂×Q₈	—	2	2	no	—
330	C₂×Q₈	—	2	4	no	—
331	C₂×Q₈	—	2	4	no	—
332	C₂×Q₈	—	4	8	no	—
333	C₂×Q₈	—	4	4	no	—
334	C₂×Q₈	—	4	4	no	—
335	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
336	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
337	C₂×D₄	—	2	1	no	—
338	C₂×D₄	—	2	4	no	—
339	C₂×D₄	—	2	2	no	—
340	C₂×D₄	—	2	2	no	—
341	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
342	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
343	C₂²×C₄	—	2	1	no	—
344	C₂²×C₄	—	2	4	no	—
345	C₂²×C₄	—	2	4	no	—
346	C₂²×C₄	—	2	2	no	—
347	C₂²×C₄	—	2	2	no	—
348	C₂²×C₄	—	2	2	no	—
349	C₂²×C₄	—	2	2	no	—
350	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
351	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
352	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
353	M₁₆	—	2	2	no	—
354	M₁₆	—	2	2	no	—
355	M₁₆	—	2	4	no	—
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	—	2	2	no	—
358	C₄⋊C₄	—	2	8	no	—
359	C₄⋊C₄	—	2	2	no	—
360	C₄⋊C₄	—	2	4	no	—
361	C₄⋊C₄	—	2	4	no	—
362	C₄⋊C₄	—	2	2	no	—
363	C₄⋊C₄	—	2	2	no	—
364	C₄⋊C₄	—	2	2	no	—
365	C₄⋊C₄	—	4	2	no	—
366	C₄⋊C₄	—	4	2	no	—
367	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
368	C₂²⋊C₄	—	2	8	no	—
369	C₂²⋊C₄	—	2	1	no	—
370	C₂²⋊C₄	—	2	2	no	—
371	C₂²⋊C₄	—	2	4	no	—
372	C₄²	—	2	2	no	—
373	C₄²	—	2	1	no	—
374	C₄²	—	2	2	no	—
375	C₄²	—	2	2	no	—
376	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
377	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
378	C₄²	C₂×D₄×C₆₇	1	2	yes	—
379	C₂³	—	2	1	no	—
380	C₂³	—	2	2	no	—
381	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
382	Q₈	—	2	1	no	—
383	Q₈	—	2	4	no	—
384	Q₈	—	4	8	no	—
385	Q₈	—	4	2	no	—
386	Q₈	—	4	2	no	—
387	Q₈	—	4	4	no	—
388	Q₈	#124×C₆₇	1	2	yes	—
389	D₄	—	2	4	no	—
390	D₄	—	2	1	no	—
391	D₄	—	2	4	no	—
392	D₄	—	4	4	no	—
393	D₄	—	4	8	no	—
394	D₄	—	4	4	no	—
395	D₄	#124×C₆₇	1	2	yes	—
396	C₂×C₄	—	2	1	no	—
397	C₂×C₄	—	2	4	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	4	no	—
401	C₂×C₄	—	2	8	no	—
402	C₂×C₄	—	4	2	no	—
403	C₂×C₄	—	4	8	no	—
404	C₂×C₄	—	4	4	no	—
405	C₂×C₄	—	4	8	no	—
406	C₂×C₄	—	4	2	no	—
407	C₂×C₄	—	4	2	no	—
408	C₂×C₄	—	4	4	no	—
409	C₂×C₄	—	4	4	no	—
410	C₂×C₄	—	4	2	no	—
411	C₂×C₄	—	4	2	no	—
412	C₂×C₄	#120×C₆₇	1	2	yes	—
413	C₂×C₄	#120×C₆₇	1	2	yes	—
414	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
415	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
416	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
417	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
418	C₈	—	4	2	no	—
419	C₈	—	4	2	no	—
420	C₂²	—	2	2	no	—
421	C₂²	—	2	2	no	—
422	C₂²	—	4	2	no	—
423	C₂²	—	4	4	no	—
424	C₂²	—	4	2	no	—
425	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
426	C₂²	#546×C₆₇	1	2	yes	—
427	C₄	—	2	2	no	—
428	C₄	—	2	2	no	—
429	C₄	—	2	2	no	—
430	C₄	—	4	2	no	—
431	C₄	—	4	4	no	—
432	C₄	—	4	2	no	—
433	C₄	—	8	2	no	—
434	C₄	—	8	2	no	—
435	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
436	C₄	#535×C₆₇	1	1	yes	characteristic
437	C₄	#546×C₆₇	1	2	yes	—
438	C₂	—	2	2	no	—
439	C₂	—	4	2	no	—
440	C₂	—	8	2	no	—
441	C₂	C₆₇×#2435	1	1	yes	characteristic
442	C₂	C₆₇×#3020	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₆₇×#3499	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
2. C₆₇×#3142	C₂	semidirect
3. C₆₇×#3080	C₂	semidirect
4. C₆₇×#2642	C₂	semidirect
5. C₆₇×#2038	C₂	semidirect
6. C₆₇×#1966	C₂	semidirect
7. C₆₇×#1963	C₂	semidirect
8. C₆₇×#1793	C₂	semidirect
9. C₆₇×#1759	C₂	semidirect
216. #25879	C₆₇	direct (N × H)
223. #3142	C₂×C₆₇	semidirect
224. #3080	C₂×C₆₇	semidirect
225. #2642	C₂×C₆₇	semidirect
226. #2038	C₂×C₆₇	semidirect
227. #1966	C₂×C₆₇	semidirect
228. #1963	C₂×C₆₇	semidirect
229. #1793	C₂×C₆₇	semidirect
230. #1759	C₂×C₆₇	semidirect
231. C₆₇	#25879	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×#3499	2
C₆₇×#3142	2
C₆₇×#3080	2
C₆₇×#2642	2
C₆₇×#2038	2
C₆₇×#1966	2
C₆₇×#1963	2
C₆₇×#1793	2
C₆₇×#1759	2
#25879	67

Subgroups — C67×#25879 #30705130

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#25879 #30705130