Subgroups — C₆₇×#21210 #30700461

Order 17152.

1570 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×#3449	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2905	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#421×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₂²×#104×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₄×#117×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1963	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1962	C₂	1	4	yes	maximal, complemented
8	C₂²×#41×C₆₇	C₂²	1	2	yes	—
9	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
10	#537×C₆₇	—	2	2	no	—
11	#536×C₆₇	C₄	1	4	yes	complemented
12	#535×C₆₇	C₄	1	4	yes	complemented
13	#534×C₆₇	—	2	2	no	—
14	#532×C₆₇	C₂²	1	4	yes	—
15	#525×C₆₇	—	2	4	no	—
16	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
19	#424×C₆₇	—	2	4	no	—
20	#423×C₆₇	C₂²	1	4	yes	complemented
21	#421×C₆₇	—	2	8	no	—
22	C₂×#104×C₆₇	—	2	8	no	—
23	C₂×#104×C₆₇	C₂²	1	4	yes	complemented
24	C₂×#104×C₆₇	C₂²	1	4	yes	complemented
25	C₂×#104×C₆₇	C₂²	1	8	yes	complemented
26	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
27	C₄×C₂²⋊C₄×C₆₇	—	2	2	no	—
28	#376×C₆₇	C₂²	1	2	yes	complemented
29	C₄³×C₆₇	C₂²	1	2	yes	complemented
30	C₂×#41×C₆₇	—	2	2	no	—
31	C₂×#41×C₆₇	—	2	2	no	—
32	C₂×#41×C₆₇	—	2	4	no	—
33	C₂×#41×C₆₇	—	2	4	no	—
34	C₂×#41×C₆₇	—	2	4	no	—
35	C₂×#41×C₆₇	—	2	4	no	—
36	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
37	C₂×#41×C₆₇	C₂³	1	4	yes	—
38	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
39	C₂²×D₄×C₆₇	—	2	2	no	—
40	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
41	C₂³×C₄×C₆₇	—	2	1	no	—
42	C₂³×C₄×C₆₇	—	2	2	no	—
43	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
44	C₂×M₁₆×C₆₇	—	2	2	no	—
45	C₂×M₁₆×C₆₇	—	2	4	no	—
46	C₂×M₁₆×C₆₇	—	2	4	no	—
47	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
48	C₂²×C₈×C₆₇	—	2	2	no	—
49	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	complemented
50	#127×C₆₇	C₂×C₄	1	4	yes	complemented
51	#124×C₆₇	C₂×C₄	1	8	yes	complemented
52	#123×C₆₇	—	2	8	no	—
53	C₂²⋊Q₈×C₆₇	—	2	8	no	—
54	#121×C₆₇	—	2	8	no	—
55	#121×C₆₇	—	2	8	no	—
56	#120×C₆₇	—	2	8	no	—
57	C₄×D₄×C₆₇	—	2	8	no	—
58	#117×C₆₇	—	2	2	no	—
59	#117×C₆₇	C₂×C₄	1	4	yes	—
60	#117×C₆₇	D₄	1	4	yes	—
61	#117×C₆₇	D₄	1	4	yes	complemented
62	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
63	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
64	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
65	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
66	C₂×C₄²×C₆₇	—	2	1	no	—
67	C₂×C₄²×C₆₇	—	2	2	no	—
68	C₂×C₄²×C₆₇	—	2	4	no	—
69	C₂×C₄²×C₆₇	D₄	1	4	yes	complemented
70	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
73	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
74	C₄⋊C₈×C₆₇	—	2	8	no	—
75	#104×C₆₇	—	2	8	no	—
76	#104×C₆₇	—	2	8	no	—
77	#104×C₆₇	—	2	16	no	—
78	#95×C₆₇	—	2	4	no	—
79	C₂⁴×C₆₇	—	2	2	no	—
80	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
81	#41×C₆₇	—	2	8	no	—
82	#41×C₆₇	—	2	8	no	—
83	#41×C₆₇	—	2	4	no	—
84	#41×C₆₇	—	2	4	no	—
85	#41×C₆₇	—	2	4	no	—
86	#41×C₆₇	—	2	4	no	—
87	#41×C₆₇	—	4	4	no	—
88	#41×C₆₇	—	4	4	no	—
89	#41×C₆₇	—	4	4	no	—
90	#41×C₆₇	—	4	4	no	—
91	C₂×Q₈×C₆₇	—	2	2	no	—
92	C₂×Q₈×C₆₇	—	2	4	no	—
93	C₂×Q₈×C₆₇	—	2	4	no	—
94	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
95	C₂×D₄×C₆₇	—	2	4	no	—
96	C₂×D₄×C₆₇	—	2	4	no	—
97	C₂×D₄×C₆₇	—	2	2	no	—
98	C₂×D₄×C₆₇	—	2	8	no	—
99	C₂×D₄×C₆₇	—	2	4	no	—
100	C₂×D₄×C₆₇	—	2	4	no	—
101	C₂×D₄×C₆₇	—	4	4	no	—
102	C₂×D₄×C₆₇	—	4	4	no	—
103	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
104	C₂²×C₄×C₆₇	—	2	1	no	—
105	C₂²×C₄×C₆₇	—	2	2	no	—
106	C₂²×C₄×C₆₇	—	2	4	no	—
107	C₂²×C₄×C₆₇	—	2	2	no	—
108	C₂²×C₄×C₆₇	—	2	1	no	—
109	C₂²×C₄×C₆₇	—	2	4	no	—
110	C₂²×C₄×C₆₇	—	2	4	no	—
111	C₂²×C₄×C₆₇	—	4	4	no	—
112	C₂²×C₄×C₆₇	—	4	4	no	—
113	C₂²×C₄×C₆₇	—	4	4	no	—
114	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
115	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
116	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
118	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
119	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
120	C₂²×C₄×C₆₇	#41	1	2	yes	—
121	C₂²×C₄×C₆₇	#41	1	2	yes	—
122	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
123	M₁₆×C₆₇	—	2	4	no	—
124	M₁₆×C₆₇	—	2	4	no	—
125	M₁₆×C₆₇	—	4	4	no	—
126	C₂×C₈×C₆₇	—	2	4	no	—
127	C₂×C₈×C₆₇	—	4	4	no	—
128	C₄⋊C₄×C₆₇	—	2	8	no	—
129	C₄⋊C₄×C₆₇	—	2	4	no	—
130	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
131	C₂²⋊C₄×C₆₇	—	2	4	no	—
132	C₂²⋊C₄×C₆₇	—	2	16	no	—
133	C₂²⋊C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
134	C₄²×C₆₇	—	2	4	no	—
135	C₄²×C₆₇	—	2	1	no	—
136	C₄²×C₆₇	—	2	1	no	—
137	C₄²×C₆₇	—	2	2	no	—
138	C₄²×C₆₇	—	4	8	no	—
139	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
140	C₄²×C₆₇	C₂×D₄	1	2	yes	—
141	C₄²×C₆₇	C₂×D₄	1	2	yes	—
142	C₄²×C₆₇	C₂×D₄	1	4	yes	—
143	C₄²×C₆₇	C₂×D₄	1	4	yes	—
144	C₂³×C₆₇	—	2	4	no	—
145	C₂³×C₆₇	—	4	2	no	—
146	C₂³×C₆₇	—	4	4	no	—
147	C₂³×C₆₇	—	8	4	no	—
148	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
149	C₂³×C₆₇	#121	1	2	yes	—
150	C₂³×C₆₇	#121	1	2	yes	—
151	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
152	Q₈×C₆₇	—	2	4	no	—
153	Q₈×C₆₇	—	2	4	no	—
154	Q₈×C₆₇	—	4	4	no	—
155	D₄×C₆₇	—	2	4	no	—
156	D₄×C₆₇	—	2	4	no	—
157	D₄×C₆₇	—	4	4	no	—
158	D₄×C₆₇	—	4	4	no	—
159	D₄×C₆₇	—	4	8	no	—
160	D₄×C₆₇	—	4	4	no	—
161	D₄×C₆₇	—	8	8	no	—
162	C₂×C₄×C₆₇	—	2	4	no	—
163	C₂×C₄×C₆₇	—	2	2	no	—
164	C₂×C₄×C₆₇	—	2	4	no	—
165	C₂×C₄×C₆₇	—	2	2	no	—
166	C₂×C₄×C₆₇	—	2	4	no	—
167	C₂×C₄×C₆₇	—	4	4	no	—
168	C₂×C₄×C₆₇	—	4	8	no	—
169	C₂×C₄×C₆₇	—	4	4	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	4	no	—
174	C₂×C₄×C₆₇	—	4	8	no	—
175	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	8	yes	—
176	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
177	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
178	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
179	C₂×C₄×C₆₇	#121	1	2	yes	—
180	C₂×C₄×C₆₇	#121	1	2	yes	—
181	C₂×C₄×C₆₇	#124	1	4	yes	—
182	C₂×C₄×C₆₇	#127	1	4	yes	—
183	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
184	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
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	1	yes	characteristic
188	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
189	C₈×C₆₇	—	4	4	no	—
190	C₂²×C₆₇	—	2	2	no	—
191	C₂²×C₆₇	—	2	4	no	—
192	C₂²×C₆₇	—	4	2	no	—
193	C₂²×C₆₇	—	4	2	no	—
194	C₂²×C₆₇	—	4	8	no	—
195	C₂²×C₆₇	—	4	4	no	—
196	C₂²×C₆₇	—	8	8	no	—
197	C₂²×C₆₇	#390	1	4	yes	—
198	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
199	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
200	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
201	C₄×C₆₇	—	2	4	no	—
202	C₄×C₆₇	—	2	8	no	—
203	C₄×C₆₇	—	4	4	no	—
204	C₄×C₆₇	—	4	2	no	—
205	C₄×C₆₇	—	4	4	no	—
206	C₄×C₆₇	#390	1	4	yes	—
207	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
208	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
209	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
210	C₄×C₆₇	C₂×#127	1	1	yes	characteristic
211	#21210	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
212	C₂×C₆₇	—	2	1	no	—
213	C₂×C₆₇	—	2	1	no	—
214	C₂×C₆₇	—	2	4	no	—
215	C₂×C₆₇	—	8	4	no	—
216	C₂×C₆₇	C₂×#390	1	1	yes	characteristic
217	C₂×C₆₇	#2901	1	2	yes	—
218	#3449	C₂×C₆₇	1	1	yes	characteristic
219	#2905	C₂×C₆₇	1	1	yes	characteristic, complemented
220	C₂×#421	C₂×C₆₇	1	2	yes	complemented
221	C₂²×#104	C₂×C₆₇	1	2	yes	complemented
222	C₄×#117	C₂×C₆₇	1	1	yes	characteristic, complemented
223	#1963	C₂×C₆₇	1	4	yes	complemented
224	#1962	C₂×C₆₇	1	4	yes	complemented
225	C₆₇	#21210	1	1	yes	characteristic, direct product factor, Sylow
226	C₂²×#41	C₂²×C₆₇	1	2	yes	—
227	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
228	#537	—	2	2	no	—
229	#536	C₄×C₆₇	1	4	yes	complemented
230	#535	C₄×C₆₇	1	4	yes	complemented
231	#534	—	2	2	no	—
232	#532	C₂²×C₆₇	1	4	yes	—
233	#525	—	2	4	no	—
234	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
235	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
236	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
237	#424	—	2	4	no	—
238	#423	C₂²×C₆₇	1	4	yes	complemented
239	#421	—	2	8	no	—
240	C₂×#104	—	2	8	no	—
241	C₂×#104	C₂²×C₆₇	1	4	yes	complemented
242	C₂×#104	C₂²×C₆₇	1	4	yes	complemented
243	C₂×#104	C₂²×C₆₇	1	8	yes	complemented
244	C₄×C₄⋊C₄	—	2	2	no	—
245	C₄×C₂²⋊C₄	—	2	2	no	—
246	#376	C₂²×C₆₇	1	2	yes	complemented
247	C₄³	C₂²×C₆₇	1	2	yes	complemented
248	C₂×#41	—	2	2	no	—
249	C₂×#41	—	2	2	no	—
250	C₂×#41	—	2	4	no	—
251	C₂×#41	—	2	4	no	—
252	C₂×#41	—	2	4	no	—
253	C₂×#41	—	2	4	no	—
254	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
255	C₂×#41	C₂³×C₆₇	1	4	yes	—
256	C₂²×Q₈	C₂×C₄×C₆₇	1	2	yes	—
257	C₂²×D₄	—	2	2	no	—
258	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
259	C₂³×C₄	—	2	1	no	—
260	C₂³×C₄	—	2	2	no	—
261	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
262	C₂×M₁₆	—	2	2	no	—
263	C₂×M₁₆	—	2	4	no	—
264	C₂×M₁₆	—	2	4	no	—
265	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
266	C₂²×C₈	—	2	2	no	—
267	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	complemented
268	#127	C₂×C₄×C₆₇	1	4	yes	complemented
269	#124	C₂×C₄×C₆₇	1	8	yes	complemented
270	#123	—	2	8	no	—
271	C₂²⋊Q₈	—	2	8	no	—
272	#121	—	2	8	no	—
273	#121	—	2	8	no	—
274	#120	—	2	8	no	—
275	C₄×D₄	—	2	8	no	—
276	#117	—	2	2	no	—
277	#117	C₂×C₄×C₆₇	1	4	yes	—
278	#117	D₄×C₆₇	1	4	yes	—
279	#117	D₄×C₆₇	1	4	yes	complemented
280	C₂×C₄⋊C₄	—	2	1	no	—
281	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
282	C₂×C₂²⋊C₄	—	2	1	no	—
283	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
284	C₂×C₄²	—	2	1	no	—
285	C₂×C₄²	—	2	2	no	—
286	C₂×C₄²	—	2	4	no	—
287	C₂×C₄²	D₄×C₆₇	1	4	yes	complemented
288	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
289	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
290	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
291	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
292	C₄⋊C₈	—	2	8	no	—
293	#104	—	2	8	no	—
294	#104	—	2	8	no	—
295	#104	—	2	16	no	—
296	#95	—	2	4	no	—
297	C₂⁴	—	2	2	no	—
298	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
299	#41	—	2	8	no	—
300	#41	—	2	8	no	—
301	#41	—	2	4	no	—
302	#41	—	2	4	no	—
303	#41	—	2	4	no	—
304	#41	—	2	4	no	—
305	#41	—	4	4	no	—
306	#41	—	4	4	no	—
307	#41	—	4	4	no	—
308	#41	—	4	4	no	—
309	C₂×Q₈	—	2	2	no	—
310	C₂×Q₈	—	2	4	no	—
311	C₂×Q₈	—	2	4	no	—
312	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
313	C₂×D₄	—	2	4	no	—
314	C₂×D₄	—	2	4	no	—
315	C₂×D₄	—	2	2	no	—
316	C₂×D₄	—	2	8	no	—
317	C₂×D₄	—	2	4	no	—
318	C₂×D₄	—	2	4	no	—
319	C₂×D₄	—	4	4	no	—
320	C₂×D₄	—	4	4	no	—
321	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
322	C₂²×C₄	—	2	1	no	—
323	C₂²×C₄	—	2	2	no	—
324	C₂²×C₄	—	2	4	no	—
325	C₂²×C₄	—	2	1	no	—
326	C₂²×C₄	—	2	2	no	—
327	C₂²×C₄	—	2	4	no	—
328	C₂²×C₄	—	2	4	no	—
329	C₂²×C₄	—	4	4	no	—
330	C₂²×C₄	—	4	4	no	—
331	C₂²×C₄	—	4	4	no	—
332	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
333	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
334	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
335	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
336	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
337	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
338	C₂²×C₄	#41×C₆₇	1	2	yes	—
339	C₂²×C₄	#41×C₆₇	1	2	yes	—
340	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
341	M₁₆	—	2	4	no	—
342	M₁₆	—	2	4	no	—
343	M₁₆	—	4	4	no	—
344	C₂×C₈	—	2	4	no	—
345	C₂×C₈	—	4	4	no	—
346	C₄⋊C₄	—	2	8	no	—
347	C₄⋊C₄	—	2	4	no	—
348	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
349	C₂²⋊C₄	—	2	4	no	—
350	C₂²⋊C₄	—	2	16	no	—
351	C₂²⋊C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
352	C₄²	—	2	4	no	—
353	C₄²	—	2	1	no	—
354	C₄²	—	2	1	no	—
355	C₄²	—	2	2	no	—
356	C₄²	—	4	8	no	—
357	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
358	C₄²	C₂×D₄×C₆₇	1	2	yes	—
359	C₄²	C₂×D₄×C₆₇	1	2	yes	—
360	C₄²	C₂×D₄×C₆₇	1	4	yes	—
361	C₄²	C₂×D₄×C₆₇	1	4	yes	—
362	C₂³	—	2	4	no	—
363	C₂³	—	4	2	no	—
364	C₂³	—	4	4	no	—
365	C₂³	—	8	4	no	—
366	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
367	C₂³	#121×C₆₇	1	2	yes	—
368	C₂³	#121×C₆₇	1	2	yes	—
369	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
370	Q₈	—	2	4	no	—
371	Q₈	—	2	4	no	—
372	Q₈	—	4	4	no	—
373	D₄	—	2	4	no	—
374	D₄	—	2	4	no	—
375	D₄	—	4	4	no	—
376	D₄	—	4	4	no	—
377	D₄	—	4	8	no	—
378	D₄	—	4	4	no	—
379	D₄	—	8	8	no	—
380	C₂×C₄	—	2	4	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	2	4	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	4	no	—
385	C₂×C₄	—	4	4	no	—
386	C₂×C₄	—	4	8	no	—
387	C₂×C₄	—	4	4	no	—
388	C₂×C₄	—	4	2	no	—
389	C₂×C₄	—	4	4	no	—
390	C₂×C₄	—	4	4	no	—
391	C₂×C₄	—	4	4	no	—
392	C₂×C₄	—	4	8	no	—
393	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	8	yes	—
394	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
395	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
396	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
397	C₂×C₄	#121×C₆₇	1	2	yes	—
398	C₂×C₄	#121×C₆₇	1	2	yes	—
399	C₂×C₄	#124×C₆₇	1	4	yes	—
400	C₂×C₄	#127×C₆₇	1	4	yes	—
401	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
402	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
403	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
404	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
405	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
406	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
407	C₈	—	4	4	no	—
408	C₂²	—	2	2	no	—
409	C₂²	—	2	4	no	—
410	C₂²	—	4	2	no	—
411	C₂²	—	4	2	no	—
412	C₂²	—	4	4	no	—
413	C₂²	—	4	8	no	—
414	C₂²	—	8	8	no	—
415	C₂²	#390×C₆₇	1	4	yes	—
416	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
417	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
418	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
419	C₄	—	2	4	no	—
420	C₄	—	2	8	no	—
421	C₄	—	4	2	no	—
422	C₄	—	4	4	no	—
423	C₄	—	4	4	no	—
424	C₄	#390×C₆₇	1	4	yes	—
425	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
426	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
427	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
428	C₄	C₂×#127×C₆₇	1	1	yes	characteristic
429	C₂	—	2	1	no	—
430	C₂	—	2	1	no	—
431	C₂	—	2	4	no	—
432	C₂	—	8	4	no	—
433	C₂	C₂×#390×C₆₇	1	1	yes	characteristic
434	C₂	C₆₇×#2901	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂²×C₂²⋊C₄
prime radical	C₆₇×#3449	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2905	C₂	semidirect
3. C₂×#421×C₆₇	C₂	semidirect
4. C₂²×#104×C₆₇	C₂	semidirect
5. C₄×#117×C₆₇	C₂	semidirect
6. C₆₇×#1963	C₂	semidirect
7. C₆₇×#1962	C₂	semidirect
11. #536×C₆₇	C₄	semidirect
12. #535×C₆₇	C₄	semidirect
20. #423×C₆₇	C₂²	semidirect
23. C₂×#104×C₆₇	C₂²	semidirect
24. C₂×#104×C₆₇	C₂²	semidirect
25. C₂×#104×C₆₇	C₂²	semidirect
28. #376×C₆₇	C₂²	semidirect
29. C₄³×C₆₇	C₂²	semidirect
49. C₄⋊Q₈×C₆₇	C₂×C₄	semidirect
50. #127×C₆₇	C₂×C₄	semidirect
51. #124×C₆₇	C₂×C₄	semidirect
61. #117×C₆₇	D₄	semidirect
69. C₂×C₄²×C₆₇	D₄	semidirect
211. #21210	C₆₇	direct (N × H)
219. #2905	C₂×C₆₇	semidirect
220. C₂×#421	C₂×C₆₇	semidirect
221. C₂²×#104	C₂×C₆₇	semidirect
222. C₄×#117	C₂×C₆₇	semidirect
223. #1963	C₂×C₆₇	semidirect
224. #1962	C₂×C₆₇	semidirect
225. C₆₇	#21210	direct (N × H)
229. #536	C₄×C₆₇	semidirect
230. #535	C₄×C₆₇	semidirect
238. #423	C₂²×C₆₇	semidirect
241. C₂×#104	C₂²×C₆₇	semidirect
242. C₂×#104	C₂²×C₆₇	semidirect
243. C₂×#104	C₂²×C₆₇	semidirect
246. #376	C₂²×C₆₇	semidirect
247. C₄³	C₂²×C₆₇	semidirect
267. C₄⋊Q₈	C₂×C₄×C₆₇	semidirect
268. #127	C₂×C₄×C₆₇	semidirect
269. #124	C₂×C₄×C₆₇	semidirect
279. #117	D₄×C₆₇	semidirect
287. C₂×C₄²	D₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3449	2
C₆₇×#2905	2
C₂×#421×C₆₇	2
C₂²×#104×C₆₇	2
C₄×#117×C₆₇	2
C₆₇×#1963	2
C₆₇×#1962	2
#21210	67

Subgroups — C67×#21210 #30700461

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21210 #30700461